1. MgtOp 470—Business Modeling with Spreadsheets Professor Munson
Topic 11
Sports Applications of Management Science

2. “Assigning Season Tickets Fairly”
Problem: Friends purchase a block of Seattle Mariners season tickets for all 81 games and must divide them amongst themselves
Solution: Mixed-integer programming (modification of the assignment problem)

3. Difficulties with Simple Rankings
If want one game in a series, member might rank all three highly and then get all three!
Games might not be spread out over the season
Persons traveling from out of town might want the whole series
Generally harder for the computer to solve than a binary program with many constraints

4. Some Extra Constraints
No more than one game in any three-week stretch
No more than one game per month
Have the games spread out over the season
No more than one game in each series
No more than one game in each weekday series
No more than two games on any weekend
No more than one pair of tickets per home stand
No more than one game against any given opponent
No games in April
A game with all four tickets in August
At least two days between any two games
Games with all four tickets should occur on Thursday, Friday, Saturday, or Sunday
No games while on vacation
No games with all four tickets on school nights
Games for both Friday and Saturday or for neither

5. Notation for the Model Let n = number of participating buyers
Let pi = number of games for which person i wanted 2 tickets
Let qi = number of games for which person i wanted 4 tickets
Let pij = 1 if person i is assigned 2 tickets to game j
Let qij = 1 if person i is assigned 4 tickets to game j
Let αij = weight that person i assigns to game j
Let e = average satisfaction of the most dissatisfied person
Let c = a large number (10,000)

6. The Formulation

7. Conclusions This approach effectively incorporated multiple objectives
Solution time was unwieldy, but near-optimal solutions worked well
Most tickets were assigned to people who really wanted to go
Should there be limits/rules regarding personal constraints?

8. “Using Bivalent Integer Programming to Select Teams for Intercollegiate Women’s Gymnastics Competition”
College gymnasts must be assigned to events according to the following rules:
Four events: vault, uneven bars, balance beam, and floor exercise
Each team can enter up to six gymnasts per event
Top five scores included in the team score
Must be at least four all-around participants

9. The Problem Coach must compute expected scores for different lineups
This is an onerous combinatorial problem of nontrivial magnitude
Competitions are often extremely close, so heuristics may not work well enough (1% from optimal may reduce score by 1.8 points)
Potential heuristics?

10. Notation for the Model Let N be the number of team members
Let Sij = expected score of gymnast i in event j
Xij = 1 if gymnast i is a specialist in event j
Yi = 1 if gymnast i is an all-arounder

11. The Formulation

12. Issues
Program can be written as an LP by adding to the objective function coefficients of the Y variables
What should Sij represent?

13. Other Issues Is there a better objective function?
Program can easily be re-run to account for injuries
Results are useful to explain to the participants themselves why the selections were made (e.g., must have 4 all-arounders)
Model may provide alternate optima that could allow for rotations

14. “Baseball, Optimization, and the World Wide Web,”
Playoff spots in major league baseball
“Has my team clinched a playoff spot?”
“Has my team been eliminated?”
“How many wins does my team need to clinch a playoff spot?”
“My team must win at least how many games to avoid elimination?”
Problem: “Games back” and “magic number” are overly conservative and do not account for the remaining schedule
Solution: Optimization

15. MLB Playoff System Each team plays 162 regular season games
Each of two leagues (American and National) invites 4 teams: 3 Division winners plus the best second-place team (by record)
Ties are broken by a one-game playoff
Each league has a tournament to determine the pennant winner, and those two teams meet in the World Series of Major League Baseball

16. Elimination Questions
When has a team been eliminated from the playoffs?
Traditional approach: If a team trails the first-place team in wins by more games than it has remaining, it is eliminated
Problem: Doesn’t consider the schedules of all other teams

17. Example: Detroit Tigers, 8/30/96
Team
Wins
Losses
Games Back
Games Left
New York 75 59 — 28
Baltimore 71 63 4
Boston 69 66
6.5 27
Toronto 72
12.5
Detroit 49 86
26.5
Opponents
Games Remaining
Baltimore vs. Boston 2
Baltimore vs. New York 3
Baltimore vs. Toronto 7
Boston vs. New York 8
Boston vs. Toronto
New York vs. Toronto

18. Hypothetical: Detroit wins final 27 games; N. Y
Hypothetical: Detroit wins final 27 games; N.Y. beats Boston once and loses other 27 games; Boston beats N.Y. 7 times and loses other 20 games
Team
Wins
Losses
Games Back
Detroit 76 86 —
Boston
New York
Baltimore ?
Toronto

19. Elimination “Numbers”
(1) First-Place Elimination Number: Minimum number of remaining games that the team must win to have any chance of finishing in first place in the division
(2) Play-Off Elimination Number: Minimum number of remaining games that the team must win to have any chance of earning a play-off spot (whether division or wild-card)

20. Clinch Questions When has a team clinched a play-off spot?
Traditional approach for the first-place team: “Magic Number” =
games remaining − (losses of 2nd place team − losses of 1st place team)
Any combination of wins by the first-place team and losses by the second-place team totaling the magic number guarantees at least a tie for the top spot
When the magic number drops to 0, the first-place team has won
Problems: Only works for 1st place team and is not independent of other teams

21. Clinch “Numbers”
(1) 1st Place Clinch Number: Minimum number of games which, if won, guarantees that the team finishes in at least a tie for first place
(2) Play-Off Clinch Number: Minimum number of games which, if won, guarantees that the team has a position in the play-offs (either as a division winner or as the wild-card team)

22. Solution
The clinch and elimination numbers are reported for each team every day during the baseball season
A software system updates the standings nightly, runs the solution algorithms, and reports the results—all automatically
Uses a free Internet news service, Infobeat, to automatically send an message each night containing final scores of MLB games

23. As of Start of Play, Tuesday, April 20, 2010
As of Start of Play, Tuesday, April 20, 2010

24. Notation xij = number of future games team i wins against team j
gij = number of games remaining between teams i and j
Dk = set of teams in Division k
vk = Division k’s first-place-elimination threshhold
wi = team i’s current number of wins

25. Formulation for 1st Place Elimination
For any team i, vk − wi is its 1st place elimination #

26. Conclusions
The article also provides optimization programs for two of the other “numbers.” The first-place clinch number can be found via the provided arithmetic calculations
Developed and housed at Cal-Berkeley
The home page has other educational links describing OR applications

27. “Scheduling the Chilean Soccer League by Integer Programming”
Soccer is Chile’s most popular sport
But in the early 2000s, fan support began to waiver
Also, the current manual schedule lacked fairness, created poor travel conditions for teams and TV, and generated poor end-of-season matchups
An integer programming model came to the rescue

28. Map of Chile

29. League Conditions in Chile
Two Divisions: First (20 teams) and Second (12 teams)
Two Seasons: Opening and Closing
First Division is divided into 4 groups
Three dominant teams: All of their games are shown on TV; plus 1-2 other TV games per week
19-week regular season
2 teams from each group make the playoffs

30. Objective Function
Objective: Push “decisive” games towards the end of the season (feasibility more important than optimality)
Let xijk = 1 if team i plays at home against team j in round k
Let t(e) = set of teams in group e

31. Constraints Basic Scheduling Home and Away Sequence
Each team must play each other team once
Either 9 or 10 games must be at home
Home and Away Sequence
No more than two consecutive home games
No more than two consecutive away games
1st 2 games must alternate home-away and last 4 games must alternate home-away

32. More Constraints Home Game Balance Against Group Rivals
Two group opponents played at home and 2 away
Geographic Constraints for Double Away Game Sequences
No 2-game trips from North (South) to South (North)
At least one game in own region

33. Even More Constraints Constraints on the Three Highly Popular Teams
Must play either COLO at home and UCH away or vice-versa
The 3 popular teams must play between weeks 10 and 16
Each of the three plays exactly one home game against the other two
TV Constraints
The 3 teams must play in Central or either North/South (but not North and South in same wk.)
None plays away from home during first 5 rounds (summertime when mobile TV is less available)

34. Still More Constraints
Constraints on Strong Teams
No team may play 2 straight games against the set of four strongest teams
Games between the 4th best team and the three best must be played between rounds 4 and 18
Crossed Teams
Teams that share the same stadium cannot both be home during the same week
Regional Classic Matchups
Regional rivals must play between rounds 8 and 18

35. Can You Believe More Constraints?
Santiago Games
Santiago must host between 2 and 4 home games each week (there are 7 teams from Santiago)
The four least popular Santiago teams should not play each other during the first five rounds (summertime) because attendance would be low
Tourism-Related Constraints
Each team in a tourist area (e.g., the beach) plays at home against at least one of the popular teams during the first five rounds (summertime)

36. Impacts 35% rise in attendance during the first year
74% increase in rival game attendance
More than 100% revenue increase in certain regions
TV stations saved money
NO COMPLAINTS!!!