**UPDATE on 10/8/2015: Explained how to model a different objective function (Requirement 5).**

Yesterday, I wrote a post describing an optimization model for picking a set of players for a fantasy football team that maximizes the teams’ point projection, while respecting a given budget and team composition constraints. In this post I’ll assume you’re familiar with that model. (If you are not, please spend a few minutes reading this first.)

Fellow O.R. blogger and Analytics expert Matthew Galati pointed out that my model did not include all of the team-building constraints that appear on popular fantasy football web sites. Therefore, I’m writing this follow-up post to address this issue. (Thanks, Matthew!) My MBA student Kevin Bustillo was kind enough to compile a list of rules from three sites for me. (Thanks, Kevin!) After looking at them, it seems my previous model fails to deal with three kinds of requirements:

- Rosters must include players from at least different NFL teams ( for Draft Kings and for both Fan Duel and Yahoo!).
- Rosters cannot have more than players from the same team ( for Fan Duel and for Yahoo! Draft Kings does not seem to have this requirement).
- Players in the roster must represent at least different football games (Only Draft Kings seems to have this requirement, with ).

Let’s see what the math would look like for each of the three requirements above. (Converting this math into Excel formulas shouldn’t be a problem if you follow the methodology I used in my previous post.) I’ll be using the same variables I had before (recall that binary variable indicates whether or not player is on the team).

**Requirement 1**

Last time I checked, the NFL had 32 teams, so let’s index them with the letter and create 32 new binary variables called , each of which is equal to 1 when at least one player from team is on our team, and equal to zero otherwise. The requirement that our team must include players from at least teams can be written as this constraint:

The above constraint alone, however, won’t do anything unless the variables are connected with the variables via additional constraints. The behavior that we want to enforce is that a given can only be allowed to equal 1, if at least one of the players from team has its corresponding variable equal to 1. To make this happen, we add the constraint below for each team :

For example, if the Miami Dolphins are team number 1 and their players are numbered from 1 to 20, this constraint would look like this:

**Requirement 2**

Repeat the following constraint for every team :

Assuming again that the first 2o players represent all the players from the Miami Dolphins, this constraint on Fan Duel would look like this:

**Requirement 3**

My understanding of this requirement is that it applies to short-term leagues that get decided after a given collection of games takes place (it could even be a single-day league). This could be implemented in a way that’s very similar to what I did for requirement 1. Create one binary variable for each game . It will be equal to 1 if your team includes at least one player who’s participating in game , and equal to zero otherwise. Then, you need this constraint

as well as the constraint below repeated for each game :

**Additional Requirements Submitted by Readers**

I earlier claimed that this model can be adapted to fit fantasy leagues other than football. So here’s a question I received from one of my readers:

For fantasy baseball, some players can play multiple positions. E.g. Miguel Cabrera can play 1B or 3B. I currently use OpenSolver for DFS and haven’t found a good way to incorporate this into my model. Any ideas?

Let’s call this…

**Requirement 4: What if some players can be added to the team at one of several positions?**

Here’s how to take care of this. Given a player , let the index represent the different positions he/she can play. Instead of having a binary variable representing whether or not is on the team, we have binary variables (as many as there are possible values for ) representing whether or not player is on the team at position . Because a player can either not be picked or picked to play one position, we need the following constraint for each of these multi-position players:

Because we have replaced with a collection of ‘s, we need to replace all occurrences of in our model with .

In the Miguel Cabrera example above, let’s say Cabrera’s player ID (the index ) is 3, and that represents the first-base position, and represents the third-base position. The constraint above would become

And we would replace all occurrences of in our model with .

That’s it!

Reader rs181602 asked me the following question:

I was wondering, is there a way to add an additional constraint that maximizes the minimum rating of the chosen players, if each player has some rating score. I tried to think that out, but can’t seem to get it to be linear.

Let’s call this…

**Requirement 5: What if I want to maximize the point projection of the worst player on the team? (In other words, how do I make my worst player as good as possible?)**

It’s possible to write a linear model to accomplish this. Technically speaking, we would be changing the objective function from maximizing the total point projection of all players on the team to maximizing the point projection of the worst player on the team. (There’s a way to do both together (sort of). I’ll say a few words about that later on.)

Here we go. Because we don’t know what the projection of the worst player is, let’s create a variable to represent it and call it . The objective then becomes:

You might have imagined, however, that this isn’t enough. We defined in words what we want to be, but we still need formulas to make behave the way we want. Let be the largest point projection among all players that could potentially be on our team. It should be clear to you that the constraint is a valid ceiling on the value of . In fact, the value of will be limited above by 9 values/ceilings: the 9 point projections of the players on the team. We want the lowest of these ceilings to be as high as possible.

When a player is **not** on the team (), his point projection should not interfere with the value of . When player **is** on the team (), we would like to become a ceiling for , by enforcing . The way to make this happen is to write a constraint that changes its behavior depending on the value of , as follows:

We need one of these for each player. To see why the constraint above works, consider the two possibilities for . When (player not on the team), the constraint reduces to (the obvious ceiling), and when (player on the team), the constraint reduces to (the ceiling we want to push up).

**BONUS:** What if I want, among all possible teams that have the maximum total point projection, the one team whose worst player is as good as possible? To do this, you solve two optimization problems. First solve the original model maximizing the total point projection. Then switch to this model and include a constraint saying that the total point projection of your team (the objective formula of the first model) should equal the total maximum value you found earlier.

That’s it!

And that does it, folks!

Does your league have other requirements I have not addressed here? If so, let me know in the comments. I’m sure most (if not all) of them can be incorporated.

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*Note 2: Although the title says “Fantasy Football”, the model I describe below can, in principle, be modified to fit any fantasy league for any sport.*

I’ve been recently approached by several people (some students, some friends) regarding the creation of optimal teams for fantasy football leagues. With the recent surge of betting sites like Fan Duel and Draft Kings, this has become a multi-million (or should I say, billion?) dollar industry. So I figured I’d write down a simple recipe to help everybody out. We’re about to use **Prescriptive Analytics** to bet on sports. Are you ready? Let’s do this! I’ll start with the math model and then show you how to make it all work using a spreadsheet.

**The Rules**

The fantasy football team rules state that a team must consist of:

- 1 quarterback (QB)
- 2 running backs (RB)
- 3 wide receivers (WR)
- 1 tight end (TE)
- 1 kicker
- 1 defense

Some leagues also have what’s called a “flex player”, which could be either a RB, WR, or TE. I’ll explain how to handle the flex player below. In addition, players have a cost and the person creating the team has a budget, call it , to abide by (usually is $50,000 or $60,000).

**The Data**

For each player , we are given the cost mentioned above, call it , and a point projection . The latter is an estimate of how many points we expect that player to score in a given week or game. When it comes to the defense, although it doesn’t always score, there’s also a way to calculate points for it (e.g. points prevented). How do these point projections get calculated, you may ask? This is where **Predictive Analytics** come into play. It’s essentially forecasting. You look at past/recent performance, you look at the upcoming opponent, you look at players’ health, etc. There are web sites that provide you with these projections, or you can calculate your own. The more accurate you are at these predictions, the more likely you are to cash in on the bets. Here, we’ll take these numbers as given.

**The Optimization Model**

The main decisions to be made are simple: which players should be on our team? This can be modeled as a yes/no decision variable for each player. So let’s create a binary variable called which can only take two values: it’s equal to the value 1 when player is on our team, and it’s equal to the value zero when player is not on our team. The value of (the player ID) ranges from 1 to the total number of players available to us.

Our objective is to create a team with the largest possible aggregate value of projected points. That is, we want to maximize the sum of point projections of all players we include on the team. This formula looks like this:

The formula above works because when a player is on the team (), its gets multiplied by one and is added to the sum, and when a player isn’t on the team () its gets multiplied by zero and doesn’t get added to the final sum. The mechanism I just described is the main idea behind what makes all formulas in this model work. For example, if the point predictions for the first 3 players are 12, 20, and 10, the maximization function start as:

The budget constraint can be written by saying that the sum of the costs of all players on our team has to be less than or equal to our budget , like this:

For example, if the first 3 players cost 9000, 8500, and 11000, and our budget is 60,000, the above formula would look like this: .

To enforce that the team has the right number of players in each position, we do it position by position. For example, to require that the team have one quarterback, we write:

To require that the team have two running backs and three wide receivers, we write:

The constraints for the remaining positions would be:

**The Curious Case of the Flex Player**

The flex player adds an interesting twist to this model. It’s a player that, if I understand correctly, takes the place of the kicker (meaning we would not have the kicker constraint above) and can be either a RB, WR, or TE. Therefore, right away, we have a new decision to make: what kind of player should the flex be? Let’s create three new yes/no variables to represent this decision: , , and . These variables mean, respectively: is the flex RB?, is the flex WR?, and is the flex TE? To indicate that only one of these things can be true, we write the constraint below:

In addition, having a flex player is equivalent to increasing the right-hand side of the constraints that count the number of RB, WR, and TE by one, but only for a single one of those constraints. We achieve this by changing these constraints from the format they had above to the following:

Note that because only one of the variables can be equal to 1, only one of the three constraints above will have its right-hand side increased from its original value of 2, 3, or 1.

**Other Potential Requirements**

Due to personal preference, inside information, or other esoteric considerations, one might want to include other requirements in this model. For example, if I want the best team that includes player number 8 and excludes player number 22, I simply have to force the x variable of player 8 to be 1, and the x variable of player 22 to be zero. Another constraint that may come in handy is to say that if player 9 is on the team, then player 10 also has to be on the team. This is achieved by:

If you wanted the opposite, that is if player 9 is on the team then player 10 is NOT on the team, you’d write:

Other conditions along these lines are also possible.

**Putting It All Together**

If you were patient enough to stick with me all the way through here, you’re eager to put this math to work. Let’s do it using Microsoft Excel. Start by downloading this spreadsheet and opening it on your computer. Here’s what it contains:

- Column A: list of player names.
- Column B: yes/no decisions for whether a player is on the team (these are the x variables that Excel Solver will compute for us).
- Columns C through H: flags indicating whether or not a player is of a given type (0 = no, 1 = yes).
- Columns I and J: the cost and point projections for each player.

Now scroll down so that you can see rows 144 through 150. The cells in column B are currently empty because we haven’t chosen which players to add to the team yet. But if those choices had been made (that is, if we had filled column B with 0’s and 1’s), multiplying column B with column C in a cell-wise fashion and adding it all up would tell you how many quarterbacks you have. I have included this multiplication in cell C144 using the SUMPRODUCT formula. In a similar fashion, cells D144:H144 calculate how many players of each kind we’d have once the cells in column B receive values. The calculations of total team cost and total projected points for the team are analogous to the previous calculations and also use the SUMPRODUCT formula (see cells I144 and J144). You can try picking some players by hand (putting 1’s in some cells of column B) to see how the values of the cells in row 144 will change.

If you now open the Excel Solver window (under the Data tab, if your Solver add-in is active), you’ll see that I already have the entire model set up for you. If you’ve never used Excel Solver before, the following two-part video will get you started with it: part 1 and part 2.

The objective cell is J144, and that’s what we want to maximize. The variables (a.k.a. changing cells) are the player selections in column B, plus the flex-player type decisions (cells D147:F147). The constraints say that: (1) the actual number of players of each type (C144:H144) are equal to the desired number of each type (C146:H146), (2) the total cost of the team (I144) doesn’t exceed the budget (I146), (3) the three flex-player binary variables add up to 1 (D150 = F150), and, (4) all variables in the problem are binary. (I set the required number of kickers in cell G146 to zero because we are using the flex-player option. If you can have both a flex player and a kicker, just type a 1 in cell G146.) If you click on the “Solve” button, you’ll see that the best answer is a team that costs exactly $50,000 and has a total projected point value of 78.3. Its flex player ended up being an RB.

This model is small enough that I can solve it with the free student version of Excel Solver (which comes by default with any Office installation). If you happen to have more players and your total variable count exceeds 200, the free solver won’t work. But don’t despair! There exists a great Solver add-in for Excel that is also free and has no size limit. It’s called OpenSolver, and it will work with the exact same setup I have here.

That’s it! If you have any questions or remarks, feel free to leave me a note in the comments below.

**UPDATE:** In a follow-up post, I explain how to model a few additional fantasy-league requirements that are not included in the model above.

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**Two Tenure-Track Faculty Positions in Management Science**

**School of Business Administration**

**University of Miami**

**Coral Gables, Florida, USA**

The Management Science Department at the University of Miami’s School of Business Administration invites applications for two tenure-track faculty positions at the Assistant Professor level to begin in the Fall of 2016 subject to budgetary approval. Salaries are extremely competitive and commensurate with background and experience. Generous summer research support is anticipated from the School of Business.

Applicants with research interests in all areas of analytics will be considered. The Management Science Department consists of a diverse group of faculty with expertise in statistics and operations research. Duties will include research and teaching at both the graduate and undergraduate levels.

Applicants should possess, or be close to completing, a Ph.D. in statistics, operations research, or a related discipline by the start date of employment. Applications should be submitted by e-mail to MASrecruiting@bus.miami.edu, and should include the following: a curriculum vitae, up to three representative publications, brief research and teaching statements, an official graduate transcript, information about teaching experience and performance evaluations (if available), and three letters of recommendation. All applications completed by December 1, 2015 will receive full consideration, but candidates are urged to submit all required material as soon as possible. Applications will be accepted until the positions are filled.

The University of Miami offers a comprehensive benefits package including medical and dental benefits, tuition remission, vacation, paid holidays, and much more. The University of Miami is an Equal Opportunity/Affirmative Action Employer.

**Clinical Faculty Position Opening For 2015-2016**

**Management Science Department**

**School of Business Administration**

**University of Miami**

**Coral Gables, Florida, USA**

The School of Business Administration at the University of Miami is currently seeking applications for a non-tenure track Clinical faculty position in the Management Science Department to begin in the Fall of 2016 subject to budgetary approval. Salaries are extremely competitive and commensurate with background and experience.

Applicants with research interests in all areas of analytics will be considered. The Management Science Department consists of a diverse group of faculty with expertise in statistics and operations research. The Department offers a major/minor and has a Specialty Master Program in Business Analytics. The selected candidates will be expected to teach business analytics classes, supervise students’ projects, and contribute to program outreach efforts to establish/strengthen relationships with industry leaders in Business Analytics. They are also expected to be intellectually active and committed to career-long professional development. Writing and publishing are valued activities as means of disseminating knowledge.

Applicants should possess, or be close to completing, a Ph.D. in statistics, operations research, or a related discipline by the start date of employment. We are particularly interested in individuals who have extensive experience in areas related to business analytics such as data visualization, data mining, and machine learning. Candidates with a master’s degree and exceptional industry experience equivalent to a doctorate will be considered. Applications should be submitted by e-mail to MASrecruiting@bus.miami.edu, and should include the following: a curriculum vitae, brief research and teaching statements (for candidates from the academia) or a statement of professional achievement (for candidates from the industry), information about teaching experience and performance evaluations (if available), and three letters of recommendation. All applications completed by December 1, 2015 will receive full consideration, but candidates are urged to submit all required material as soon as possible. Applications will be accepted until the position is filled.

The University of Miami offers a comprehensive benefits package including medical and dental benefits, tuition remission, vacation, paid holidays, and much more. The University of Miami is an Equal Opportunity/Affirmative Action Employer.

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The Management Science Department at the University of Miami’s School of Business Administration invites applications for a non-tenure-track Research Assistant Professor position to begin in the Fall of 2015. The Management Science Department is a diverse group of faculty with expertise in several areas within Operations Research and Analytics, including statistics and machine learning, optimization, simulation, and quality management. Duties will include research, teaching at both the graduate and undergraduate levels, and advising undergraduate students seeking majors/minors in Management Science or Business Analytics.

Applicants should possess a PhD in operations research or a related discipline by the start date of employment. Applications should be submitted by e-mail to facultyaffairs@bus.miami.edu, and should include the following: a curriculum vitae, up to three representative publications, brief research and teaching statements, an official graduate transcript, information about teaching experience and performance evaluations, and three letters of recommendation. Applications will be reviewed as they arrive. The position will remain open until filled.

The University of Miami offers a comprehensive benefits package including medical and dental benefits, tuition remission, vacation, paid holidays, and much more. The University of Miami is an Equal Opportunity/Affirmative Action Employer.

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**Tenure-Track Faculty Position in Management Science (Big Data Analytics)**

The Management Science Department at the University of Miami’s School of Business Administration invites applications for a tenure-track faculty position at the junior or advanced Assistant Professor level to begin in the Fall of 2015. Exceptional candidates at higher ranks will be considered subject to additional approval from the administration. Salaries are extremely competitive and commensurate with background and experience. This is a nine-month appointment but generous summer research support is anticipated from the School of Business.

Applicants with research interests in all areas of Analytics will be considered, although primary consideration will be given to those with expertise in Big Data Analytics and the computational challenges of dealing with large data sets. Expertise in, or experience with, one or more of the following is particularly welcome: MapReduce/Hadoop, Mahout, Cassandra, cloud computing, mobile/wearable technologies, social media analytics, recommendation systems, data mining and machine learning, and text mining. The Management Science Department is a diverse group of faculty with expertise in several areas within Operations Research and Analytics, including statistics and machine learning, optimization, simulation, and quality management. Duties will include research and teaching at the graduate and undergraduate levels.

Applicants should possess, or be close to completing, a PhD in computer science, operations research, statistics, or a related discipline by the start date of employment. Applications should be submitted by e-mail to facultyaffairs@bus.miami.edu, and should include the following: a curriculum vitae, up to three representative publications, brief research and teaching statements, an official graduate transcript (for the junior Assistant Professor level), information about teaching experience and performance evaluations, and three letters of recommendation. All applications completed by December 1, 2014 will receive full consideration, but candidates are urged to submit all required material as soon as possible. Applications will be accepted until the position is filled.

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htlatex file.tex “xhtml,charset=utf-8″ ” -cunihtf -utf8″

Overall, I’m very happy with the results produced by htlatex. Nevertheless, as I loaded file.html on my iPhone, I noticed that mobile Safari does not render all ligatures properly. For example, it has no problem with the ‘fi’ ligature, but it displays a hollow square in place of the characters for ‘ff’ and ‘ffi’ ligatures. I have not tested other mobile browsers, so I’m not sure if this is only an issue with mobile Safari. Safari on my desktop computer does not exhibit this problem.

To be safe, I thought I’d be better off removing all ligatures from the HTML file, which led me to search around for their UTF-8 codes and to write a little command-shell script that uses Perl to perform the task. Since this might turn out to be useful to someone else out there, I decided to post my shell script here. Use it at your own risk and enjoy!

perl -pi -e ‘s/\xef\xac\x80/ff/g’ file.html

perl -pi -e ‘s/\xef\xac\x81/fi/g’ file.html

perl -pi -e ‘s/\xef\xac\x82/fl/g’ file.html

perl -pi -e ‘s/\xef\xac\x83/ffi/g’ file.html

perl -pi -e ‘s/\xef\xac\x84/ffl/g’ file.html

perl -pi -e ‘s/\xc5\x92/OE/g’ file.html

perl -pi -e ‘s/\xc5\x93/oe/g’ file.html

perl -pi -e ‘s/\xc3\x86/AE/g’ file.html

perl -pi -e ‘s/\xc3\xa6/ae/g’ file.html

perl -pi -e ‘s/\xef\xac\x86/st/g’ file.html

perl -pi -e ‘s/\xc4\xb2/IJ/g’ file.html

perl -pi -e ‘s/\xc4\xb3/ij/g’ file.html

By the way, I’m only concerned with Latin ligatures, but you can find UTF-8 codes for other ligatures on this page. Bonus: here’s another useful article related to this topic: The Absolute Minimum Every Software Developer Absolutely, Positively Must Know About Unicode and Character Sets (No Excuses!).

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Today, my wife brought to my attention The Bulwer-Lytton Fiction Contest, which, according to their web page, consists of the following:

Since 1982 the English Department at San Jose State University has sponsored the Bulwer-Lytton Fiction Contest, a whimsical literary competition that challenges entrants to compose the opening sentence to the worst of all possible novels. The contest (hereafter referred to as the BLFC) was the brainchild (or Rosemary’s baby) of Professor Scott Rice, whose graduate school excavations unearthed the source of the line “It was a dark and stormy night.” Sentenced to write a seminar paper on a minor Victorian novelist, he chose the man with the funny hyphenated name, Edward George Bulwer-Lytton, who was best known for perpetrating

The Last Days of Pompeii,Eugene Aram,Rienzi,The Caxtons,The Coming Race, and – not least –Paul Clifford, whose famous opener has been plagiarized repeatedly by the cartoon beagle Snoopy. No less impressively, Lytton coined phrases that have become common parlance in our language: “the pen is mightier than the sword,” “the great unwashed,” and “the almighty dollar” (the latter fromThe Coming Race, now available from Broadview Press).

Just like an awful first sentence can be a good indicator of a terrible book, the converse can also be true. Take, for example, the first sentence of Stephen King’s The Dark Tower series, which I happen to be reading (and loving) as we speak:

The man in black fled across the desert, and the gunslinger followed.

It’s such a strong, mysterious, and captivating sentence…

…which brings me to the point of this post. If it’s going to be difficult to write *The Great Analytics Novel*, what if we start by thinking about what would be the perfect, most compelling sentence to start such a novel? Yes, I propose a contest. Let’s use our artistic abilities and suggest starting sentences. Feel free to add them as comments to this post. Who knows? Maybe someone will get inspired and start writing the novel.

Here’s mine:

Upon using the word “mathematical” he knew he had lost the battle for, despite the dramatic cost savings, their logical reasoning was instantly halted, like a snowshoe hare frozen in fear of its chief predator: the Canada lynx.

I can’t wait to read your submissions!

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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 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 to represent the machine assigned to job . 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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After playing in the Miami Heat’s first five preseason games, LeBron James sat out Saturday night’s 121-96 victory over the San Antonio Spurs to rest…James said the decision to sit was part of the team’s “maintenance” process. Heat teammate Dwyane Wade played Saturday and scored 25 points in 26 minutes, but previously skipped three preseason games…”No, no injuries — just not suiting up,” James said. “It’s OK for LeBron to take one off.”

The key term here is *maintenance process*. You may also recall that, back in November 2012, the Spurs were fined $250,000 by the league after coach Popovich sent Duncan, Parker, Ginobili, and Green home right before a game against the Miami Heat.

So we want to rest our players to keep them healthy, but this cannot come at the expense of losing games. There are many factors to be taken into account here, such as players’ current physical condition, strength and tightness of schedule, and match-ups (how well a team stacks up against another team), to name a few. This is definitely not an easy problem. However, some insight is better than no insight at all. Therefore, let’s see what we can do with a simple O.R. model, and then we can talk about the strengths and weaknesses of our initial approach. (Here’s where *you*, dear reader, are supposed to chime in!)

Let’s begin with two simple assumptions: (i) when it comes to resting, we have to take players’ individual needs into account, i.e., we’ll use player-specific data; and (ii) when it comes to the likelihood of beating an opposing team, it’s better to think in terms of full lineups, rather than in terms of individual players, i.e., we’ll use lineup-specific data. The data in assumption (i) comes from doctors, players’ medical records, and coaches’ strategies. In essence, it boils down to one number: how many minutes, at most, should each player play in each game, under ideal circumstances. A useful measure of the strength of a lineup is its adjusted plus-minus score (see, for example, the work of Wayne Winston and his book Mathletics). In summary, it’s a number that tells you how many points a given lineup plays above (or below) an average lineup in the league over 48 minutes (or over 100 possessions, or another metric of reference).

For the sake of explanation, I’ll pretend to be in charge of resting Miami Heat players (surprise!). I’ll refer to a generic lineup by the letter (), to a generic player by the letter ( LeBron, D-Wade, …, Andersen (Bird Man)), and to a generic game by the letter .

We’re now ready to begin. Fasten your seat belts!

**What are the decisions to be made?** Let’s consider a planning horizon that consists of the next 7 games (or pick your favorite number). So . For the Heat, the first 7 games of the 2013-2014 season are against the following teams: Bulls, 76ers, Nets, Wizards, Raptors, Clippers, and Celtics. For each one of my potential lineups and each game , I want to figure out the number of minutes I should use lineup during game . Because this is an unknown number right now, it’s a variable in the model. Let’s call it . Note it’s also OK to think of as a percentage, rather than minutes. I’ll adopt the latter interpretation.

**What are the constraints in this problem?** There are three main constraints to worry about: (a) make sure to pick enough lineups to play each game in its entirety; (b) make sure your lineups are good enough to hopefully beat your opponents in each game; (c) keep track of players’ minutes, and don’t let them get out of hand. The next step is to represent each constraint mathematically.

**Constraint (a):** Pick enough lineups to completely cover each game. For every game , we want to impose the following constraint:

This means that if we sum the percentage of time each lineup is used during game , we reach 100%.

**Constraint (b):** Choose your lineups so that you expect to score enough points in every game to beat your opponents. In this example, I’ll focus on plus-minus scores, but as a coach you could focus on any metric that matters to you. Given a lineup , let be its adjusted plus-minus score. For example, the lineup of LeBron, Wade, Bosh, Chalmers, and Allen in the 2012-2013 season had the amazing score of +36.9 (you can obtain these numbers, and many other neat statistics, from the web site stats.nba.com). Now let’s say you have the plus-minus score of your opponent in game , which we’ll call . One way to increase your chances of victory is by requiring that the expected plus-minus score of your lineup combination in game exceed by a certain amount. Therefore, for every game , we write the following constraint:

I want to emphasize two things. First, can be *any* measure of goodness of your lineup, and it can take into account the specific opponent in game . Likewise, can be any measure of goodness of team , as long as it’s consistent with . Second, you’re not restricted to having only one of these constraints. If many measures of goodness matter to you, add them all in. For example, if you’re playing a team that’s particularly good at rebounding and you believe that rebounding is the key to beating them (e.g. Heat vs. Pacers), then either replace the constraint above with the analogous rebounding version, or include the rebounding version in addition to the constraint above. Finally, note that I picked 0.5 as a fixed amount by which to exceed , but it could be any number you wish, of course. It can even be a number that varies depending on the opponent.

**Constraint (c):** Keep track of how many minutes your players are playing above and beyond what you’d like them to play. For any given player and any given game , let be ‘s ideal number of playing minutes in game (make it zero if you want the player to sit out). When it’s not possible to match exactly, we need to know how many minutes player played under or over . Let’s call these two unknown numbers (variables) and , respectively. So, for every player and game , we write the following constraint:

The expression “ that includes ” under the summation means that we’re summing variables for all lineups of which is a member. We’re multiplying the summation by 48 minutes because is in percentage points and is in minutes.

**What is our goal? (a.k.a. objective function)** It’s simple: we don’t want players to play too many minutes above . Because this overage amount is captured by variable , we can write our goal as:

This minimizes the total overage in playing minutes. For a more balanced solution, it’s also possible to minimize the maximum overage over all players, or add weights in front of the variables to give preference to some players.

**Now what?** Well, the next step would be to solve this model and see what happens. I created a Microsoft Excel spreadsheet that can be solved with Excel Solver or OpenSolver. You can download it from here. Feel free to adapt it to your own needs and play around with it (this is the fun part!). Because my model was limited in size (I can’t use OpenSolver on my Mac at home), the solution isn’t very good (too many overage minutes). However, by adding more players and more lineups, the quality will certainly improve (use OpenSolver to break free from limits on model size). Here are some notes to help you understand the spreadsheet:

- Variables are in the range B18:H25.
- Variables and are in ranges B56:J62 and B65:J71, respectively.
- Constraints (a) are implemented in rows 27, 28, 29.
- Constraints (b) are implemented in rows 33, 34, 35.
- The left-hand side of constraints (c) are in the range B74:J80. This range is required to be equal to the range B47:J53 (where the are) inside the Solver window.
- The objective function whose formula appears above is in cell J21.

**What are the pros and cons of this model?** Can you make it better? No model is perfect. There are always real-life details that get omitted. The art of modeling is creating a model that is detailed enough to provide useful answers, but not too detailed to the point of requiring an unreasonable amount of time to solve. The definitions of “detailed enough” and “unreasonable amount of time” are mostly client-specific. (What would please Erik Spoelstra and his coaching staff?) What do you think are the main strengths and weaknesses in the model I describe above? What would you change? Good data is a big issue in this particular case. If you don’t like my data, can you propose alternative sources that are practical? I believe there’s plenty to talk about in this context, and I’m looking forward to receiving your feedback. Maybe we can converge to a model that is good enough for me to go knocking on the Miami Heat’s door! (Don’t worry. In the unlikely event they open the door, I’ll share the consulting fees.)

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