Metaheuristics for optimization problems in sports La Havana, March /91 Celso C. Ribeiro Joint work with S. Urrutia, A. Duarte, and A. Guedes 8th International Workshop on Operations Research Applications of Metaheuristics to Optimization Problems in Sports La Havana, March 2009

Metaheuristics for optimization problems in sports La Havana, March /91 Summary Optimization problems in sports – Motivation – How it started: qualification problems – Problems, applications, and solution methods Applications of metaheuristics – Traveling tournament problem – Referee assignment – Carry-over effect minimization – Brazilian professional basketball tournament Perspectives and concluding remarks

Metaheuristics for optimization problems in sports La Havana, March /91 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)

Metaheuristics for optimization problems in sports La Havana, March /91 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

Metaheuristics for optimization problems in sports La Havana, March /91 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!” Fairness issues: finals of Argentina’s First Division soccer tournament last December: 1) Boca Juniors 2) San Lorenzo de Almagro 3) Tigre Suppose San Lorenzo won Tigre by one goal in the first match, and Boca and San Lorenzo made a tie in the second match. Tigre could not win anymore the tournament and would play the last game without motivation and self interest, maybe not even with the complete main team (Xmas vacations...). Boca could have been clearly benefited. Fair solution: winner of the first match should play the last game with Boca. There is even more: if San Lorenzo have won the first two games, the tournament would have been decided and the third game would have no importance!

Metaheuristics for optimization problems in sports La Havana, March /91 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)

Metaheuristics for optimization problems in sports La Havana, March /91 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

Metaheuristics for optimization problems in sports La Havana, March /91 Optimization problems in sports Examples: – Qualification/elimination problems – Tournament scheduling – Referee assignment – Tournament planning (teams? dates? rules?) – League assignment (which teams in each league?) – Carry-over minimization – Practice assignment –... – Optimal strategies for curling!

Metaheuristics for optimization problems in sports La Havana, March /91 Qualification/elimination problems How all this work it started... 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”)

Metaheuristics for optimization problems in sports La Havana, March /91 FUTMAX in the WWW FALSE !

Metaheuristics for optimization problems in sports La Havana, March /91 Qualification/elimination problems Two basic approaches: click hereclick here – Probabilistic model + simulation (abound in the sports press, journalists love but do not understand: “The probability that Estudiantes win is 14,87%”) – Number of points to qualify: ìnteger programming application, doctorate thesis of Sebastián Urrutia (“easy” only in the last round!)

Metaheuristics for optimization problems in sports La Havana, March /91 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): – IP model determines the maximum number K of points a team can make such as that p other teams can make more than K points. – Team must win K+1 points to qualify.

Metaheuristics for optimization problems in sports La Havana, March /91 Qualification/elimination problems Schwartz 1966: mathematical elimination from play-offs in the Major League Baseball (MLB) solved with maximum flow algorithm Robinson 1991: IP models and further results for the play-offs elimination problem McCormick 2000: elimination from the p-th position is NP-complete. Bernholt et al. 2002: first place elimination is NP-complete under the {(3,0),(1,1)} soccer rule Adler et al. 2003: ILP models for MLB

Metaheuristics for optimization problems in sports La Havana, March /91 Qualification/elimination problems Ribeiro & Urrutia 2005: integer programming for qualification/elimination problems in the Brazilian soccer championship and the World Cup (FUTMAX) Cheng & Steffy 2006: integer programming for qualification/elimination problems in the National Hockey League (spin-off project)

Metaheuristics for optimization problems in sports La Havana, March /91 FUTMAX in the WWW FUTMAX project Results of the games automatically collected from the web (multi-agents system): Noronha, Ribeiro, Urrutia & Lucena 2008 Four IP problems generated for each team Problems solved with CPLEX 9.0 HTML file automatically built from the results Automatic publication in the web: click hereclick here FUTMAX is often able to prove that statements made by the press and administrators are not true

Metaheuristics for optimization problems in sports La Havana, March /91 Results FUTMAX can also be used to follow the situation of each team: Possible points Points for guaranteed qualification Points for possible qualification Points accumulated FLUMINEN SE

Metaheuristics for optimization problems in sports La Havana, March /91 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

Metaheuristics for optimization problems in sports La Havana, March /91 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)

Metaheuristics for optimization problems in sports La Havana, March /91 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 G 1 =(V,E 1 ),..., G p =(V,E p ), such that E 1 ...  E p =E 1-factorization: factorization into 1- factors Oriented factorization: orientations assigned to edges

Metaheuristics for optimization problems in sports La Havana, March / factorizations Example: 1-factorization of K 6

Metaheuristics for optimization problems in sports La Havana, March /91 Oriented 1-factorization of K

Metaheuristics for optimization problems in sports La Havana, March /91 SRR tournament: – Each node of K n represents a team – Each edge of K n represents a game – Each 1-factor of K n represents a round – Each ordered 1-factorization of K n 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 K 12 ” (1995) 1-factorizations

Metaheuristics for optimization problems in sports La Havana, March /91 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)

Metaheuristics for optimization problems in sports La Havana, March /91 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

Metaheuristics for optimization problems in sports La Havana, March /91 Break minimization problems There is a break whenever a team has two consecutive home games (or two consecutive away games) Break minimization: – De Werra 1981: minimum number of breaks is n-2 Every team must have a different home-away pattern (they must play in some round) Only two patterns without breaks: –HAHAHAH... –AHAHAHA... – Constructive algorithm to obtain schedules with exactly n-2 breaks

Metaheuristics for optimization problems in sports La Havana, March /91 Break minimization problems Break minimization is somehow opposed to distance minimization Urrutia & Ribeiro 2006: a special case of the Traveling Tournament Problem (distance minimization) is equivalent to a break maximization problem

Metaheuristics for optimization problems in sports La Havana, March /91 Predefined timetables/venues Given a fixed timetable, find a home- away assignment minimizing breaks/distance: – Metaheuristics, constraint programming, integer programming – Miyashiro & Matsui 2005: polynomial method for break minimization if the minimal number of breaks is smaller than or equal to n Given a fixed venue assignment for each game, find a timetable minimizing breaks/distance: – IP: Melo, Urrutia & Ribeiro 2007 (JoS); Costa, Urrutia & Ribeiro 2008 (PATAT): ILS metaheuristic

Metaheuristics for optimization problems in sports La Havana, March /91 Decomposition methods Nemhauser & Trick 1998: 1. Find home-away patterns 2. Create an schedule for place holders consistent with a subset of home-away patterns 3. Assign teams to place holders Order in which the above tasks are tackled may vary depending on the application Henz 2001: CP may work better than IP and complete enumeration for all the tasks

Metaheuristics for optimization problems in sports La Havana, March /91 Decomposition methods Frequently used in real-life tournaments: – Nemhauser & Trick 1998: Atlantic Coast Conference (basketball) – Bartsch et al. 2006: Austrian and German soccer – Della Croce & Oliveri 2006: Italian soccer – Ribeiro & Urrutia 2006, 2009: Brazilian soccer – Durán, Noronha, Ribeiro, Sourys & Weintraub 2006: Chilean soccer Other applications: voleyball in Argentina, soccer in Japan, NHL, basketball in New Zealand, etc.

Metaheuristics for optimization problems in sports La Havana, March /91 Applications of metaheuristics Traveling Tournament Problem (TTP) and its mirrored version (mTTP)

Metaheuristics for optimization problems in sports La Havana, March /91 Formulation Traveling Tournament Problem (TTP) – n (even) teams take part in a tournament – Each team has its own stadium at its home city – Distances between the stadiums are known – A team playing two consecutive away games goes directly from one city to the other, without returning to its home city

Metaheuristics for optimization problems in sports La Havana, March /91 Formulation – Double round-robin tournament: 2(n-1) rounds, each with n/2 games Each team plays against every other team twice, one at home and the other away – No team can play more than three games in a home stand (home games) or in a road trip (away games) Goal: minimize the distance traveled by all teams, to reduce traveling costs and to give more time to the players to rest and practice

Metaheuristics for optimization problems in sports La Havana, March /91 Formulation Mirrored Traveling Tournament Problem (mTTP): – All teams face each other once in the first phase (n-1 rounds) – In the second phase (n-1 rounds), teams play each other again in the same order, following an inverted home-away pattern – Games in the second phase determined by those in the first Set of feasible solutions to the MTTP is a subset of those to the TTP Ribeiro & Urrutia (PATAT 2004, EJOR 2007)

Metaheuristics for optimization problems in sports La Havana, March /91 Three steps: 1.Schedule games using abstract teams: polygon method defines the structure of the tournament 2.Assign real teams to abstract teams: greedy heuristic to QAP (number of travels between stadiums of the abstract teams x distances between the stadiums of the real teams) 3.Select stadium for each game (home/away pattern) in the first phase (mirrored tournament): 1.Build a feasible assignment of stadiums, starting from a random assignment in the first round 2.Improve this assignment, using a simple local search algorithm based on home-away swaps Constructive heuristic

Metaheuristics for optimization problems in sports La Havana, March /91 Constructive heuristic Example: “polygon method” for n=6 1 st round

Metaheuristics for optimization problems in sports La Havana, March /91 Constructive heuristic Example: “polygon method” for n=6 2 nd round

Metaheuristics for optimization problems in sports La Havana, March /91 Simple neighborhoods Home-away swap (HAS): modify the stadium of a game Team swap (TS): exchange the sequence of opponents of a pair of teams over all rounds

Metaheuristics for optimization problems in sports La Havana, March /91 Partial round swap (PRS)

Metaheuristics for optimization problems in sports La Havana, March /91 Partial round swap (PRS)

Metaheuristics for optimization problems in sports La Havana, March /91 Ejection chain: game rotation (GR) Neigborhood “game rotation” (GR) (ejection chain): – Enforce a game to be played at some round: add a new edge to a given 1- factor of the current 1-factorization (schedule) – Use an ejection chain to recover a 1- factorization

Metaheuristics for optimization problems in sports La Havana, March /91 Ejection chain: game rotation (GR)

Metaheuristics for optimization problems in sports La Havana, March /91 Ejection chain: game rotation (GR) Enforce game (1,3) to be played in round 2

Metaheuristics for optimization problems in sports La Havana, March /91 Ejection chain: game rotation (GR) Enforce game (1,3) to be played in round 2

Metaheuristics for optimization problems in sports La Havana, March /91 Ejection chain: game rotation (GR)

Metaheuristics for optimization problems in sports La Havana, March /91 Ejection chain: game rotation (GR)

Metaheuristics for optimization problems in sports La Havana, March /91 Ejection chain: game rotation (GR)

Metaheuristics for optimization problems in sports La Havana, March /91 Ejection chain: game rotation (GR)

Metaheuristics for optimization problems in sports La Havana, March /91 Ejection chain: game rotation (GR)

Metaheuristics for optimization problems in sports La Havana, March /91 Ejection chain: game rotation (GR)

Metaheuristics for optimization problems in sports La Havana, March /91 Ejection chain: game rotation (GR)

Metaheuristics for optimization problems in sports La Havana, March /91 Neighborhoods Only moves in neighborhoods PRS and GR may change the structure of the initial schedule However, PRS moves not always exist, due to the structure of the solutions built by polygon method e.g. for n = 6, 8, 12, 14, 16, 20, 24 PRS moves may appear after an ejection chain move is made Ejection chain moves may find solutions that are not reachable through other neighborhoods: escape from local optima

Metaheuristics for optimization problems in sports La Havana, March /91 GRASP+ILS heuristic Hybrid improvement heuristic for the MTTP: – Combination of GRASP and ILS – Initial solutions: randomized version of the constructive heuristic – Local search with first improving move: use TS, HAS, PRS and HAS cyclically in this order, until a local optimum for all neighborhoods is found – Perturbation: random move in GR neighborhood

Metaheuristics for optimization problems in sports La Havana, March /91 GRASP+ILS heuristic while.not.StoppingCriterion S  GenerateRandomizedInitialSolution() S  LocalSearch(S) repeat S’  Perturbation(S,history) S’  LocalSearch(S’) S  AceptanceCriterion(S,S’,history) S*  UpdateBestSolution(S,S*) until ReinitializationCriterion end

Metaheuristics for optimization problems in sports La Havana, March /91 Constructive heuristic is very fast and effective GRASP+ILS is very fast and finds very good solutions, even better than the best known for the corresponding (less constrained) not necessarily mirrored instances Effectiveness of the ejection chains Theoretical complexity still open Concluding remarks

Metaheuristics for optimization problems in sports La Havana, March /91 Applications of metaheuristics Referee Assignment Problem (RAP)

Metaheuristics for optimization problems in sports La Havana, March /91 Motivation Regional amateur leagues in the US (baseball, basketball, soccer): hundreds of games every weekend in different divisions In a single league in California there are up to 500 soccer games in a weekend, to be refereed by hundreds of certified referees

Metaheuristics for optimization problems in sports La Havana, March /91 Motivation 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

Metaheuristics for optimization problems in sports La Havana, March /91 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

Metaheuristics for optimization problems in sports La Havana, March /91 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

Metaheuristics for optimization problems in sports La Havana, March /91 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

Metaheuristics for optimization problems in sports La Havana, March /91 Problem statement Games are already scheduled (facility – time slot) Each game has a number of refereeing positions to be assigned to referees Each refereeing position to be filled by a referee is called a refereeing slot S = {s 1, s 2,..., s n }: refereeing slots to be filled by referees R = {r 1, r 2,..., r m }: referees

Metaheuristics for optimization problems in sports La Havana, March /91 Problem statement p i : skill level of referee r i q j : minimum skill level a referee must have to be assigned to refereeing slot s j M i : maximum number of games referee r i can officiate T i : target number of games referee r i is willing to officiate Each referee may choose a set of time slots where he/she is not available to officiate

Metaheuristics for optimization problems in sports La Havana, March /91 Problem statement Problem: assign a referee to each refereeing slot Constraints: – Referees officiate in a single facility on the same day – Minimum skill level requirements – Maximum number of games – Timetabling conflicts and availability Objective: 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 (0-1 integer linear programming model)

Metaheuristics for optimization problems in sports La Havana, March /91 Solution approach Three-phase heuristic approach 1.Greedy constructive heuristic 2.ILS-based repair heuristic to make the initial solution feasible (if necessary): minimization of the number of violations 3.ILS-based procedure to improve a feasible solution

Metaheuristics for optimization problems in sports La Havana, March /91 Solution approach Algorithm RefereeAssignmentHeuristic (MaxIter) 1. S*  BuildGreedyRandomizedSolution (); 2. If not isFeasible (S*) then 3. S*  RepairHeuristic (S*, MaxIter); 4. If isFeasible (S*) then 5. S*  ImprovementHeuristic (S*); 6. Else “infeasible” 7.Return S*

Metaheuristics for optimization problems in sports La Havana, March /91 Numerical results Randomly generated instances following patterns similar to real-life applications Instances with up to 500 games and 1,000 referees – Different number of facilities – Different patterns of the target number of games Five different instances for each configuration MaxIter = 1,000

Metaheuristics for optimization problems in sports La Havana, March /91 Numerical results For each instance, average time and average objective value over ten runs Codes implemented in C Results on a 2.0 GHz Pentium IV processor with 256 Mbytes Initial solutions: – greedy constructive heuristic – random assignments (to test the repair heuristic)

Metaheuristics for optimization problems in sports La Havana, March /91 Numerical results Instance ConstructionRepairImprovement time (s)valuefeas.time (s)valuefeas.time (s)value I1I ——— I2I I3I I4I40.03—— I5I Table 1: Instances with 500 games, 750 referees, and 65 facilities

Metaheuristics for optimization problems in sports La Havana, March /91 Numerical results Instanc e pattern GreedyRandom const. (s) repair (s)feas.repair (s) feas. I1I1 P0P I2I2 P0P I3I3 P0P I4I4 P0P I5I5 P0P I1I1 P1P I2I2 P1P I3I3 P1P I4I4 P1P I5I5 P1P Table 4: Greedy vs. randomly generated initial solutions

Metaheuristics for optimization problems in sports La Havana, March /91 Improvements and extensions Greedy constructive heuristic: – First, assign each referee to a number of refereeing slots as close as possible to his/her target number of games – Second, if there are still unassigned slots, assign more games to each referee Improvement heuristic: – 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!”)

Metaheuristics for optimization problems in sports La Havana, March /91 Numerical results Figure 3: 500 games, 750 referees, 85 facilities, pattern P0 (target = 529)

Metaheuristics for optimization problems in sports La Havana, March /91 Bi-criteria problem (biRAP) Same constraints as in the single objective version Objectives: 1.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 2.minimize the sum over all referees of the total idle time between consecutive games

Metaheuristics for optimization problems in sports La Havana, March /91 Solution approach Exact approach: dichotomic method 50 games and 100 referees

Metaheuristics for optimization problems in sports La Havana, March /91 Solution approach Heuristic approach: – Perform three-phase ILS-based heuristic for a fixed number of search directions – Each search direction represents a set of weights associated with each objective – Directions are chosen as in the dichotomic method – All new potentially efficient solutions found during the search are progressively stored – Former potentially efficient solutions are discarded during the search (quadtree is used) – Perform a post-optimization path-relinking procedure

Metaheuristics for optimization problems in sports La Havana, March /91 Numerical results

Metaheuristics for optimization problems in sports La Havana, March /91 Numerical results

Metaheuristics for optimization problems in sports La Havana, March /91 Numerical results

Metaheuristics for optimization problems in sports La Havana, March /91 Conclusions New optimization problem in sports Effective heuristics: construction, repair, improvement, path relinking Quick procedures to build good initial solutions Bicriteria approach finds good approximations of the Pareto frontier Other constraints and criteria may be considered

Metaheuristics for optimization problems in sports La Havana, March /91 Applications of metaheuristics Carry-over minimization problem

Metaheuristics for optimization problems in sports La Havana, March /91 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

Metaheuristics for optimization problems in sports La Havana, March /91 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 A receives COE due to B Team G receives COE due to D Team A receives COE due to E

Metaheuristics for optimization problems in sports La Havana, March /91 Carry-over effects matrix SRRT and carry-over effects matrix (COEM) RRT COE matrix

Metaheuristics for optimization problems in sports La Havana, March /91 Carry-over effects matrix RRT 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

Metaheuristics for optimization problems in sports La Havana, March /91 Carry-over effects value COE matrix COEM DG = 3 COEM FH = 2

Metaheuristics for optimization problems in sports La Havana, March /91 Carry-over effects value COE Matrix Minimize!!! COEM DG = 3 COEM FH = 2

Metaheuristics for optimization problems in sports La Havana, March /91 Example 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

Metaheuristics for optimization problems in sports La Havana, March /91 Problem statement Find a schedule with minimum COEV – RRT distributing the carry-over effects as evenly as possible among the teams Best solution approaches to date in literature: – Random generation of 1-factors permutations – Constraint Programming – Combinatorial designs

Metaheuristics for optimization problems in sports La Havana, March /91 Solution approach Multi-start + ILS heuristic Solutions represented by 1- factorizations – Canonical factorizations – Binary 1-factorizations Constructive algorithms – Rearrangement of the 1-factors of a solution (TSP-like greedy algorithms) Nearest neighbor Arbitrary insertion

Metaheuristics for optimization problems in sports La Havana, March /91 Solution approach Local search – Rearrangement of the 1-factors of the solution (TSP-like procedures) – Partial Round Swap (PRS) Pertubations – Ejection chain: Game Rotation (GR) neighborhood

Metaheuristics for optimization problems in sports La Havana, March /91 Application to Brazilian national basketball tournament Optimization in sports is a field of increasing interest Very attractive area for Operations Research applications Many interesting applications, often reviewed by the media Student motivation: OR course with sports problems Several problems with interesting theoretical structure Even small instances are hard to solve (e.g., TTP for n=10) Quick construction procedures to build good initial (feasible) solutions are a must Repair procedures Successful applications of metaheuristics Kendall, Knust, Ribeiro & Urrutia (2008): “Scheduling in sports: An annotated bibliography” (200 references) Perspectives and concluding remarks WikiSport: open content project maintained by the Working Group on Operations Research Applied to Sports (UFF and UFMG, Brazil) at Brazilian Soccer Confederation (CBF) announced last month the fixture of its 2009 First Division, which was the first built by an automatic optimization system Next year: all divisions

Metaheuristics for optimization problems in sports La Havana, March /91