Presentation on theme: "Optimization Problems in Sports"— Presentation transcript:
1 Optimization Problems in Sports Celso C. RibeiroJoint 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 problemPlayoff eliminationTournament schedulingReferee assignmentRegional amateur leagues in the US (baseball, basketball, soccer): hundreds of games every weekend in different divisions2/55
3 MotivationSports competitions involve many economic and logistic issuesMultiple 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 rightsBaseball US: > US$ 500 millionsBasketball US: > US$ 600 millionsMain 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 athletesAthletes pay for playingLarge number of simultaneous eventsAmateur leagues do not involve as much money as professional leagues but, on the other hand, amateur competitions abound
8 MotivationIn a single league in California there might be up to 500 soccer games in a weekend, to be refereed by hundreds of certified refereesMOSA (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 problemsTournament schedulingReferee assignmentTournament planning (teams? dates? rules?)League assignment (which teams in each league?)Player selectionCarry-over minimizationPractice assignment...Optimal strategies for curling!
10 Qualification/elimination problems How all this work started inTeam 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 competitionPress 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 makersRound robin schedules:Every team plays each other a fixed number of timesEvery team plays once in each roundSingle (SRR) or double (DRR) round robinMirrored 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 method6Example:“polygon method” for n=61521st round34
16 Polygon method6Example:“polygon method” for n=65412nd round23
17 Polygon method6Example:“polygon method” for n=64353rd round12
18 Polygon method6Example:“polygon method” for n=63244th round51
19 Polygon method6Example:“polygon method” for n=62135th round45
20 1-factorizationsFactor of a graph G=(V, E): subgraph G’=(V,E’) with E’E1-factor: all nodes have degree equal to 1Factorization of G=(V,E): set of edge-disjoint factors G1=(V,E1), ..., Gp=(V,Ep), such that E1...Ep=E1-factorization: factorization into 1-factorsOriented factorization: orientations assigned to edges
21 1-factorizations12543Example:1-factorization of K66
22 Oriented 1-factorization of K6 2343215643215643215645432156432156
23 1-factorizations SRR tournament: Each node of Kn represents a team Each edge of Kn represents a gameEach 1-factor of Kn represents a roundEach ordered 1-factorization of Kn represents a feasible schedule for n teamsEdge orientations define teams playing at homeDinitz, 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 secondMaximum number of consecutive games away (or at home) is often constrainedMinimize 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 coloniesInteger programmingConstraint programmingIP/CP column generationCP 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 programmingGiven a fixed venue assignment for each game, find a timetable minimizing breaks/distance:Melo, Urrutia & Ribeiro 2007 (JoS); Costa, Urrutia & Ribeiro 2008 (PATAT): ILS metaheuristicChilean soccer tournamentTable tennis in Germany
28 Decomposition methods Nemhauser & Trick 1998:Find home-away patternsCreate an schedule for place holders consistent with a subset of home-away patternsAssign teams to place holdersOrder in which the above tasks are tackled may vary depending on the application
29 Applications of metaheuristics Mirrored traveling tournament problemTypical structure of LA soccer tournamentsGRASP+ILS heuristicBest benchmark results for some timeBrazilian professional basketball tournament“Nova liga” (Oscar e Hortênsia)Referee assignment in amateur leaguesBi-objective problem
30 Referee assignmentMOSA (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 gameGames require referees with different levels of certification: higher division games require referees with higher skillsA referee cannot be assigned to a game where he/she is a playerTimetabling 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 linksSoft linksNumber of games a referee is willing to refereeTraveling constraintsReferees 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 toTraveling/idle time between consecutive gamesNumber of inter-facility travelsNumber of games assigned outside his/her preferred time-slots or facilitiesNumber of violated soft links
34 Referee assignment Three-phase heuristic approach Greedy constructive heuristicILS-based repair heuristic to make the initial solution feasible (if necessary): minimization of the number of violationsILS-based procedure to improve a feasible solution
35 Referee assignmentImprovement 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 MetaheuristicsAngra 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 refereeminimize the sum over all referees of the total idle time between consecutive gamesMetaheuristics for multi-criteria combinatorial optimization problems:Relatively new field with many applicationsSearch for Pareto frontier (efficient solutions)
37 Referee assignment Exact approach: dichotomic method 50 games and 100 referees
41 Applications of metaheuristics Mirrored traveling tournament problemTypical structure of LA soccer tournamentsBrazilian professional basketball tournamentNova liga (Oscar e Hortênsia)Referee assignment in amateur leaguesBi-objective problemPractice assignment (R. Capua’s doctorate thesis)Carry-over minimization problemHard non-linear optimization integer problem41/55
42 Carry-over effectsTeam 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 effectsTeam 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+1Team G receives COE due to DTeam A receives COE due to BTeam A receives COE due to E
45 Carry-over effects SRRT and carry-over effects matrix (COEM) RRT COE MatrixSuppose B is a very strong competitor: then, five times A will play anopponent that is tired or wounded due to meeting B before
46 Carry-over effects value COEMDG = 3COEMFH = 2COE matrix
47 Carry-over effects value COEMDG = 3COEMFH = 2COE MatrixMinimize!!!
48 Carry-over effects Karate-Do competitions Groups playing round-robin tournamentsPan-american Karate-Do championshipBrazilian classification for World Karate-Do championshipOpen weight categoriesPhysically strong contestants may fight weak onesOne 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 COEVRRT distributing the carry-over effects as evenly as possible among the teamsNew problem: weighted COEVminimization problemmin-max problemNon-linear integer optimization problemHard for IP approaches
50 Carry-over effectsIn 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
51 Applications of exact methods (IP) Improved algorithm and formulation for the Chilean soccer tournament46/55
52 Applications of exact methods (IP) Projeto CBFTabela 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 segurançaOtimizar público, audiência e renda de publicidade; otimizar clássicos em finais de semana; etc.47/55
53 Applications of exact methods (IP) Projeto CBFSolução exata por programação inteiraTabelas oficiais adotadas em 2009, 2010 e 2011 vêm sendo extremamente equilibradasUsuário escolhe entre diversas tabelas e refina sucessivamente a melhor soluçãoA cada ano, novas exigências:Calendário apertado em 2010 levou à maximização dos clássicos aos domingosEfeito “mala branca” levou a marcar os clássicos regionais para as rodadas finais em 201148/55
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 ZealandBaseball: MLB (USA) (also referee assignment)Soccer: Austria, Costa Rica, Germany, Brazil, Chile, Denmark, The Netherlands, JapanVolleyball: ArgentinaRugby: World CupHockey: NHL (USA/Canada), Finland“American” football: NFL (USA) (unclear status in some cases: real-life vs real-data apps.)55/55
56 Perspectives and concluding remarks However, old technologies might also be useful…56/55
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.58/55
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.59/55
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