1. Optimization Problems in Sports
Celso C. Ribeiro
Joint work with S. Urrutia,
Duarte, T. Noronha,
E.H. Haeusler, R. Melo,
Guedes, F. Costa,
S. Martins, R. Capua et al.

2. Motivation Optimization in sports is a field of increasing interest:
Traveling tournament problem
Playoff elimination
Tournament scheduling
Referee assignment
Regional amateur leagues in the US (baseball, basketball, soccer): hundreds of games every weekend in different divisions
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3. Motivation
Sports competitions involve many economic and logistic issues
Multiple decision makers: federations, TV, teams, security authorities, ...
Conflicting objectives:
Maximize revenue (attractive games in specific days)
Minimize costs (traveled distance)
Maximize athlete performance (time to rest)
Fairness (avoid playing all strong teams in a row)
Avoid conflicts (teams with a history of conflicts playing at the same place)

4. Motivation Professional sports: Millions of fans
Multiple agents: organizers, media, fans, players, security forces, ...
Big investments:
Belgacom TV: €12 million per year for soccer broadcasting rights
Baseball US: > US$ 500 millions
Basketball US: > US$ 600 millions
Main problems: maximize revenues, optimize logistic, maximize fairness, minimize conflicts, etc.

5. If San Lorenzo would have won the first two games, the tournament would have been decided and the third game would have no importance!
Taxi driver the night before: “the only fair solution is that San Lorenzo and Boca play at Tigre’s, Boca and Tigre at San Lorenzo's, and Tigre and San Lorenzo at Boca’s, but these guys never do the right thing!”
La Havana, March 2009

6. Fairness issues: “The International Rugby Board (IRB) has admitted the World Cup draw was unfairly stacked against poorer countries so tournament organisers could maximise their profits.”
(2003)

7. Motivation Amateur sports: Different problems and applications
Thousands of athletes
Athletes pay for playing
Large number of simultaneous events
Amateur leagues do not involve as much money as professional leagues but, on the other hand, amateur competitions abound

8. Motivation
In a single league in California there might be up to 500 soccer games in a weekend, to be refereed by hundreds of certified referees
MOSA (Monmouth & Ocean Counties Soccer Association) League (NJ): boys & girls, ages 8-18, six divisions per age/gender group, six teams per division: 396 games every Sunday (US$ 40 per ref; U$ 20 per linesman, two linesmen)
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9. Optimization problems in sports
Examples:
Qualification/elimination problems
Tournament scheduling
Referee assignment
Tournament planning (teams? dates? rules?)
League assignment (which teams in each league?)
Player selection
Carry-over minimization
Practice assignment
...
Optimal strategies for curling!

10. Qualification/elimination problems
How all this work started in
Team managers, players, fans and the press are often eager to know the chances of a team to be qualified for the playoffs of a given competition
Press often makes false announcements based on unclear forecasts that are often biased and wrong (“any team with 54 points will qualify”)

11. Qualification/elimination problems
Two basic approaches:
Probabilistic model + simulation (abound in the sports press, journalists love but do not understand: “Probability that Estudiantes wins is 14,87%”; “Probability that Fluminense will be downgraded next year is 1%”)
Number of points to qualify: ìnteger programming application, doctorate thesis of Sebastián Urrutia (“easy” only in the last round!)

12. Qualification/elimination problems
How many points a team should make to:
… be sure of finishing among the p teams in the first positions? (sufficient condition for play-offs qualification)
… have a chance of finishing among the p teams in the first positions? (necessary condition for play-offs qualification):
Integer programming model determines the maximum number K of points a team can make such as that p other teams can still make more than K points.
Team must win K+1 points to qualify.

13. Tournament scheduling
Timetabling is the major area of applications: game scheduling is a difficult task, involving different types of constraints, logistic issues, multiple objectives, and several decision makers
Round robin schedules:
Every team plays each other a fixed number of times
Every team plays once in each round
Single (SRR) or double (DRR) round robin
Mirrored DRR: two phases with games in same order

14. Tournament scheduling
Problems:
Minimize distance (costs)
Minimize breaks (fairness and equilibrium, every two rounds there is a game in the city)
Balanced tournaments (even distribution of fields used by the teams: n teams, n/2 fields, SRR with n-1 games/team, 2 games/team in n/2-1 fields and 1 in the other)
Minimize carry over effect (maximize fairness, polygon method)

15. Polygon method 6
Example:
“polygon method” for n=6 1 5 2
1st round 3 4

16. Polygon method 6
Example:
“polygon method” for n=6 5 4 1
2nd round 2 3

17. Polygon method 6
Example:
“polygon method” for n=6 4 3 5
3rd round 1 2

18. Polygon method 6
Example:
“polygon method” for n=6 3 2 4
4th round 5 1

19. Polygon method 6
Example:
“polygon method” for n=6 2 1 3
5th round 4 5

20. 1-factorizations
Factor of a graph G=(V, E): subgraph G’=(V,E’) with E’E
1-factor: all nodes have degree equal to 1
Factorization of G=(V,E): set of edge-disjoint factors G1=(V,E1), ..., Gp=(V,Ep), such that E1...Ep=E
1-factorization: factorization into 1-factors
Oriented factorization: orientations assigned to edges

21. 1-factorizations 1 2 5 4 3
Example:
1-factorization of K6 6

22. Oriented 1-factorization of K62 3 4 3 2 1 5 6 4 3 2 1 5 6 4 3 2 1 5 6 4 5 4 3 2 1 5 6 4 3 2 1 5 6

23. 1-factorizations SRR tournament: Each node of Kn represents a team
Each edge of Kn represents a game
Each 1-factor of Kn represents a round
Each ordered 1-factorization of Kn represents a feasible schedule for n teams
Edge orientations define teams playing at home
Dinitz, Garnick & McKay, “There are 526,915,620 nonisomorphic one-factorizations of K12” (1995)

24. Distance minimization problems
Whenever a team plays two consecutive games away, it travels directly from the facility of the first opponent to that of the second
Maximum number of consecutive games away (or at home) is often constrained
Minimize the total distance traveled (or the maximum distance traveled by any team)
This was never the Brazilian problem!

25. Distance minimization problems
Methods:
Metaheuristics: simulated annealing, iterated local search, hill climbing, tabu search, GRASP, genetic algorithms, ant colonies
Integer programming
Constraint programming
IP/CP column generation
CP with local search

26. Break minimization problems
There is a break whenever a team has two consecutive home games (or two consecutive away games)
Break minimization is somehow opposed to distance minimization

27. Predefined timetables/venues
Given a fixed timetable, find a home-away assignment minimizing breaks/distance:
Metaheuristics, constraint programming, integer programming
Given a fixed venue assignment for each game, find a timetable minimizing breaks/distance:
Melo, Urrutia & Ribeiro 2007 (JoS); Costa, Urrutia & Ribeiro 2008 (PATAT): ILS metaheuristic
Chilean soccer tournament
Table tennis in Germany

28. Decomposition methods
Nemhauser & Trick 1998:
Find home-away patterns
Create an schedule for place holders consistent with a subset of home-away patterns
Assign teams to place holders
Order in which the above tasks are tackled may vary depending on the application

29. Applications of metaheuristics
Mirrored traveling tournament problem
Typical structure of LA soccer tournaments
GRASP+ILS heuristic
Best benchmark results for some time
Brazilian professional basketball tournament
“Nova liga” (Oscar e Hortênsia)
Referee assignment in amateur leagues
Bi-objective problem

30. Referee assignment
MOSA (Monmouth & Ocean Counties Soccer Association) League (NJ): boys & girls, ages 8-18, six divisions per age/gender group, six teams per division: 396 games every Sunday (US$ 40 per referee; U$ 20 per linesman, two linesmen)
Problem: assign referees to games Duarte, Ribeiro & Urrutia (PATAT 2006, LNCS 2007)
Referee assignment involves many constraints and multiple objectives

31. Referee assignment Possible constraints:
Different number of referees may be necessary for each game
Games require referees with different levels of certification: higher division games require referees with higher skills
A referee cannot be assigned to a game where he/she is a player
Timetabling conflicts and traveling times

32. Referee assignment Possible constraints (cont.):
Referee groups: cliques of referees that request to be assigned to the same games (relatives, car pools, no driver’s licence)
Hard links
Soft links
Number of games a referee is willing to referee
Traveling constraints
Referees that can officiate games only at a certain location or period of the day

33. Referee assignment Possible objectives:
Difference between the target number of games a referee is willing to referee and the number of games he/she is assigned to
Traveling/idle time between consecutive games
Number of inter-facility travels
Number of games assigned outside his/her preferred time-slots or facilities
Number of violated soft links

34. Referee assignment Three-phase heuristic approach
Greedy constructive heuristic
ILS-based repair heuristic to make the initial solution feasible (if necessary): minimization of the number of violations
ILS-based procedure to improve a feasible solution

35. Referee assignment
Improvement heuristic (hybridization of exact and approximate algorithms):
After each perturbation, instead of applying a local search to both facilities involved in this perturbation, solve a MIP model associated with the subproblem considering all refereeing slots and referees corresponding to these facilities (“MIP it!”)
Matheuristics’2012: Fourth International Workshop on Model-Based Metaheuristics
Angra dos Reis, September 16-21, 2012

36. Referee assignment Bi-criteria version Objectives:
minimize the sum over all referees of the absolute value of the difference between the target and the actual number of games assigned to each referee
minimize the sum over all referees of the total idle time between consecutive games
Metaheuristics for multi-criteria combinatorial optimization problems:
Relatively new field with many applications
Search for Pareto frontier (efficient solutions)

37. Referee assignment Exact approach: dichotomic method 50 games and
100 referees

38. Referee assignment

39. Referee assignment

40. Referee assignment

41. Applications of metaheuristics
Mirrored traveling tournament problem
Typical structure of LA soccer tournaments
Brazilian professional basketball tournament
Nova liga (Oscar e Hortênsia)
Referee assignment in amateur leagues
Bi-objective problem
Practice assignment (R. Capua’s doctorate thesis)
Carry-over minimization problem
Hard non-linear optimization integer problem
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42. Carry-over effects
Team B receives a carry-over effect (COE) due to team A if there is a team X that plays A in round r and B in round r+1

43. Carry-over effects
Team B receives a carry-over effect (COE) due to team A if there is a team X that plays A in round r and B in round r+1
Team G receives COE due to D
Team A receives COE due to B
Team A receives COE due to E

44. Carry-over effects SRRT and carry-over effects matrix (COEM) RRT
COE matrix

45. Carry-over effects SRRT and carry-over effects matrix (COEM) RRT
COE Matrix
Suppose B is a very strong competitor: then, five times A will play an
opponent that is tired or wounded due to meeting B before

46. Carry-over effects value
COEMDG = 3
COEMFH = 2
COE matrix

47. Carry-over effects value
COEMDG = 3
COEMFH = 2
COE Matrix
Minimize!!!

48. Carry-over effects Karate-Do competitions
Groups playing round-robin tournaments
Pan-american Karate-Do championship
Brazilian classification for World Karate-Do championship
Open weight categories
Physically strong contestants may fight weak ones
One should avoid that a competitor benefits from fighting (physically) tired opponents coming from matches against strong athletes

49. Carry-over effects value
Find a schedule with minimum COEV
RRT distributing the carry-over effects as evenly as possible among the teams
New problem: weighted COEV
minimization problem
min-max problem
Non-linear integer optimization problem
Hard for IP approaches

50. Carry-over effects
In case you don’t believe in the relevance of the problem, google “Alanzinho” and check Youtube and Wikipedia:
Played at Flamengo, America RJ, Gama, Stabaek (Norway), Trabzonspor (Turkey)
Also relevant in tournaments with an odd number of teams, to avoid that the same team always play against another team coming from a bye (residual effect of polygon method) (College Basketball in Alabama)
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51. Applications of exact methods (IP)
Improved algorithm and formulation for the Chilean soccer tournament
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52. Applications of exact methods (IP)
Projeto CBF
Tabela do Campeonato Brasileiro de Futebol (A e B)
Projeto inicialmente desenvolvido com a TV Globo (2006) e posteriormente encampado pela Diretoria de Competições da CBF (2008)
40 critérios (restrições) diferentes: aspectos esportivos, técnicos, geográficos, financeiros, logísticos, de equilíbrio e de segurança
Otimizar público, audiência e renda de publicidade; otimizar clássicos em finais de semana; etc.
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53. Applications of exact methods (IP)
Projeto CBF
Solução exata por programação inteira
Tabelas oficiais adotadas em 2009, 2010 e 2011 vêm sendo extremamente equilibradas
Usuário escolhe entre diversas tabelas e refina sucessivamente a melhor solução
A cada ano, novas exigências:
Calendário apertado em 2010 levou à maximização dos clássicos aos domingos
Efeito “mala branca” levou a marcar os clássicos regionais para as rodadas finais em 2011
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54. Some recent references
Kendall, Knust, Ribeiro & Urrutia (C&OR, 2010): “Scheduling in sports: An annotated bibliography”
Nurmi, Goossens, Bartsch, Bonomo, Briskorn, Duran, Kyngäs, Marenco, Ribeiro, Spieksma, Urrutia & Wolf-Yadlin (IAENG Transactions, 2010): “A framework for scheduling professional sports leagues”
Ribeiro & Urrutia (Interfaces, to appear): “Scheduling the Brazilian soccer tournament: Solution approach and practice”
Ribeiro (ITOR, to appear): “Sports scheduling: Problems and applications”
ITOR is now indexed by ISI (will appear in JCR in 2012)

55. Perspectives and concluding remarks
Sports timetabling in professional leagues worldwide:
Basketball: USA (College), New Zealand
Baseball: MLB (USA) (also referee assignment)
Soccer: Austria, Costa Rica, Germany, Brazil, Chile, Denmark, The Netherlands, Japan
Volleyball: Argentina
Rugby: World Cup
Hockey: NHL (USA/Canada), Finland
“American” football: NFL (USA) (unclear status in some cases: real-life vs real-data apps.)
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56. Perspectives and concluding remarks
However, old technologies might also be useful…
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58. Perspectives and concluding remarks
Fair and balanced schedules for all teams are a major issue for attractiveness and confidence in the outcome of any tournament.
Few professional leagues have adopted optimization software to date:
Due not only to the difficulty of the problem and to some fuzzy requirements that can hardly be described and formulated, but also to the resistance of teams and leagues that are often afraid of using new tools that break with the past and introduce modern techniques in sports management.
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59. Perspectives and concluding remarks
Operations Research has certainly proved its usefulness in sports management:
Besides the quality of the schedules found, the main advantages of the optimization systems are its ease of use and the construction of alternative schedules, making it possible for the organizers to compare and select the most attractive schedule from among different alternatives, contemplating distinct goals.
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60. La Havana, March 2009