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.

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

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)

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.

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

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)

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

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

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!

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”)

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!)

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.

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

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)

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

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

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

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

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

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

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

Oriented 1-factorization of K6 2 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

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)

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!

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

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

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

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

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

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

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

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

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

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

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

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)

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

Referee assignment

Referee assignment

Referee assignment

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 41/55

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

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

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

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

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

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

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

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

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

Applications of exact methods (IP) Improved algorithm and formulation for the Chilean soccer tournament 46/55

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. 47/55

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 48/55

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)

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.) 55/55

Perspectives and concluding remarks However, old technologies might also be useful… 56/55

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. 58/55

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. 59/55

La Havana, March 2009