Celso C. Ribeiro Joint work with S. Urrutia, Duarte, T. Noronha, E.H. Haeusler, R. Melo,

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 2/55

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!”

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) 8/55

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 2002... • 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 K6 1 1 1 1 1 2 2 2 2 2 5 5 5 5 5 4 4 4 4 4 3 3 3 3 3 6 6 6 6 6 1 2 3 4 5

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 gamesDuarte, 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 • http://www.ic.uff.br/matheuristics2012

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. 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 Applications of metaheuristics 41/55

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) COE matrix RRT

45. Carry-over effects • SRRT and carry-over effects matrix (COEM) COE Matrix RRT 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) 50/55