Metaheuristics for optimization problems in sports META’08, October 2008 1/92 Celso C. Ribeiro Joint work with S. Urrutia, A. Duarte, and A. Guedes 2nd.

Metaheuristics for optimization problems in sports META’08, October 2008 1/92 Celso C. Ribeiro Joint work with S. Urrutia, A. Duarte, and A. Guedes 2nd International Conference on Metaheuristics and Nature Inspired Computing (META’08) Applications of Metaheuristics to Optimization Problems in Sports Hammamet, October 2008

Metaheuristics for optimization problems in sports META’08, October 2008 2/92 Summary Optimization problems in sports – Motivation – 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 META’08, October 2008 3/92 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 META’08, October 2008 4/92 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 META’08, October 2008 5/92 Motivation Amateur sports: – 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 META’08, October 2008 6/92 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... – Optimal strategies for curling!

Metaheuristics for optimization problems in sports META’08, October 2008 7/92 Qualification/elimination problems 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 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)

Metaheuristics for optimization problems in sports META’08, October 2008 8/92 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 META’08, October 2008 9/92 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.

Metaheuristics for optimization problems in sports META’08, October 2008 10/92 FUTMAX in the WWW FUTMAX project Results of the games automatically collected from the web (multi-agents) Models generated (four problems for each team) Problems solved with CPLEX 9.0 HTML file automatically built from the results Automatic publication in the web FUTMAX is often able to prove that statements made by the Press and administrators are not true

Metaheuristics for optimization problems in sports META’08, October 2008 11/92 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 META’08, October 2008 12/92 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 META’08, October 2008 13/92 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 META’08, October 2008 14/92 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 META’08, October 2008 15/92 4 3 2 1 5 6 1-factorizations Example: 1-factorization of K 6

Metaheuristics for optimization problems in sports META’08, October 2008 16/92 Oriented 1-factorization of K 6 4 3 2 1 5 6 4 3 2 1 5 6 4 3 2 1 5 6 4 3 2 1 5 6 4 3 2 1 5 6 12 3 4 5

Metaheuristics for optimization problems in sports META’08, October 2008 17/92 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 Open problem: How many schedules exist for a single round robin tournament with n teams?

Metaheuristics for optimization problems in sports META’08, October 2008 18/92 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 META’08, October 2008 19/92 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 META’08, October 2008 20/92 Break minimization problems There is a break whenever a team has two consecutive home games (or two 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 META’08, October 2008 21/92 Break minimization problems Break minimization is somehow opposed to distance minimization Urrutia & Ribeiro 2006: a special case of the Traveling Tournament Problem is equivalent to a break maximization problem

Metaheuristics for optimization problems in sports META’08, October 2008 22/92 Fixed 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: – Melo, Urrutia & Ribeiro 2007: integer programming

Metaheuristics for optimization problems in sports META’08, October 2008 23/92 Decomposition methods Nemhauser and 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 META’08, October 2008 24/92 Decomposition methods Frequently used for scheduling real 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: Brazilian soccer – Durán, Noronha, Ribeiro, Sourys, Weintraub 2006: Chilean soccer

Metaheuristics for optimization problems in sports META’08, October 2008 25/92 Applications of metaheuristics Traveling Tournament Problem (TTP) and its mirrored version (mTTP)

Metaheuristics for optimization problems in sports META’08, October 2008 26/92 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 META’08, October 2008 27/92 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 META’08, October 2008 28/92 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 and Urrutia (PATAT 2004, EJOR 2007)

Metaheuristics for optimization problems in sports META’08, October 2008 29/92 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 META’08, October 2008 30/92 Constructive heuristic 4 3 2 1 5 6 Example: “polygon method” for n=6 1 st round

Metaheuristics for optimization problems in sports META’08, October 2008 31/92 Constructive heuristic 3 2 1 5 4 6 Example: “polygon method” for n=6 2 nd round

Metaheuristics for optimization problems in sports META’08, October 2008 32/92 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 META’08, October 2008 33/92 Partial round swap (PRS) 7 4 3 1 8 6 2 5 7 4 3 1 8 6 2 5

Metaheuristics for optimization problems in sports META’08, October 2008 34/92 Partial round swap (PRS) 7 4 3 1 8 6 2 5 7 4 3 1 8 6 2 5

Metaheuristics for optimization problems in sports META’08, October 2008 35/92 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 META’08, October 2008 36/92 Ejection chain: game rotation (GR) 4 3 2 1 5 6 4 3 2 1 5 6 4 3 2 1 5 6 4 3 2 1 5 6 4 3 2 1 5 6

Metaheuristics for optimization problems in sports META’08, October 2008 37/92 Ejection chain: game rotation (GR) 4 3 2 1 5 6 4 3 2 1 5 6 4 3 2 1 5 6 4 3 2 1 5 6 4 3 2 1 5 6 Enforce game (1,3) to be played in round 2

Metaheuristics for optimization problems in sports META’08, October 2008 38/92 Ejection chain: game rotation (GR) 4 3 2 1 5 6 4 3 2 1 5 6 4 3 2 1 5 6 4 3 2 1 5 6 4 3 2 1 5 6 Enforce game (1,3) to be played in round 2

Metaheuristics for optimization problems in sports META’08, October 2008 46/92 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 META’08, October 2008 47/92 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 META’08, October 2008 48/92 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 META’08, October 2008 49/92 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 Lower bounds: – Independent lower bound: Easton et al. 2001 – MNTLB (improvement over ILB): Urrutia et al. 2007 – Benders decomposition: Trick & Rasmussen 2007 Concluding remarks

Metaheuristics for optimization problems in sports META’08, October 2008 50/92 Applications of metaheuristics Referee Assignment Problem (RAP)

Metaheuristics for optimization problems in sports META’08, October 2008 51/92 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 META’08, October 2008 52/92 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 META’08, October 2008 53/92 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 META’08, October 2008 54/92 Referee assignment Possible constraints (cont.): – Referee groups: cliques of referees that request to be assigned to the same games (relatives, car pools) 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 META’08, October 2008 55/92 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 META’08, October 2008 56/92 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 META’08, October 2008 57/92 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 META’08, October 2008 58/92 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 META’08, October 2008 59/92 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 META’08, October 2008 60/92 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 META’08, October 2008 61/92 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 META’08, October 2008 62/92 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 META’08, October 2008 63/92 Numerical results Instance ConstructionRepairImprovement time (s)valuefeas.time (s)valuefeas.time (s)value I1I10.021286.2010———32.34619.60 I2I20.021360.0050.471338.001031.81623.40 I3I30.021269.0020.601247.001033.87621.60 I4I40.03——1.141303.201030.28627.20 I5I50.031302.0031.4012591.141033.73654.00 Table 1: Instances with 500 games, 750 referees, and 65 facilities

Metaheuristics for optimization problems in sports META’08, October 2008 64/92 Numerical results Instanc e pattern GreedyRandom const. (s) repair (s)feas.repair (s) feas. I1I1 P0P0 0.0311.271079.809 I2I2 P0P0 0.036.691080.8010 I3I3 P0P0 0.0311.331086.208 I4I4 P0P0 0.034.611030.6010 I5I5 P0P0 0.033.391029.1010 I1I1 P1P1 0.032.751033.5010 I2I2 P1P1 0.0219.2910134.602 I3I3 P1P1 0.0314.7710135.108 I4I4 P1P1 0.031.221038.0010 I5I5 P1P1 0.032.691032.9010 Table 4: Greedy vs. randomly generated initial solutions

Metaheuristics for optimization problems in sports META’08, October 2008 65/92 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 for 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 META’08, October 2008 66/92 Numerical results Figure 3: 500 games, 750 referees, 85 facilities, pattern P0 (target = 529)

Metaheuristics for optimization problems in sports META’08, October 2008 67/92 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 META’08, October 2008 68/92 Bi-criteria problem (biRAP) Formulation: bi-criteria set partitioning problem Variables: possible “routes” for each referee (stops correspond to refereeing positions) Each “route” has at most 4 to 5 stops: number of variables is limited Each refereeing position has to be filled by exactly one qualified refere Each referee must perform exactly one “route” Goal: find the set of (potentially) efficient solutions

Metaheuristics for optimization problems in sports META’08, October 2008 69/92 Solution approach Exact approach: dichotomic method 50 games and 100 referees

Metaheuristics for optimization problems in sports META’08, October 2008 70/92 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 META’08, October 2008 71/92 Numerical results

Metaheuristics for optimization problems in sports META’08, October 2008 74/92 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 META’08, October 2008 75/92 Applications of metaheuristics Carry-over minimization problem

Metaheuristics for optimization problems in sports META’08, October 2008 76/92 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 META’08, October 2008 77/92 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 META’08, October 2008 78/92 Carry-over effects matrix SRRT and carry-over effects matrix (COEM) RRT COE matrix

Metaheuristics for optimization problems in sports META’08, October 2008 79/92 Carry-over effects matrix RRT and carry-over effects matrix (COEM) RRT COE Matrix

Metaheuristics for optimization problems in sports META’08, October 2008 80/92 Carry-over effects value COE matrix COEM DG = 3 COEM FH = 2

Metaheuristics for optimization problems in sports META’08, October 2008 81/92 Carry-over effects value COE Matrix Minimize!!! COEM DG = 3 COEM FH = 2

Metaheuristics for optimization problems in sports META’08, October 2008 82/92 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 META’08, October 2008 83/92 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 META’08, October 2008 84/92 Solution approach Multi-start + ILS heuristic Solutions represented by 1- factorizations – Canonical factorizations – Binary 1-factorizations Constructive algorithms – Rearragment of the 1-factors of a solution (TSP-like greedy algorithms) Nearest neighbor Arbitrary insertion

Metaheuristics for optimization problems in sports META’08, October 2008 85/92 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)

Metaheuristics for optimization problems in sports META’08, October 2008 86/92 Multi-start + ILS heuristic Multi-start phase: generation of 10,000 solutions – 50% based on canonical 1-factorizations – 50% based on binary 1-factorizations (whenever possible) – Constructive methods applied to the 1- factors of the 1-factorizations – Local search Best solution of the multi-start phase is the input for the ILS algorithm

Metaheuristics for optimization problems in sports META’08, October 2008 87/92 Multi-start + ILS heuristic For try = 1 to 10000 Do S ← Initial_Solution(); S ← Local_Search(S); S* ← Update_Best_Solution(S, S*); End-For; S ← S*; While Not Stopping-Criterion Do S' ← Pertubation(S); S' ← Local_Search(S’); S ← Acceptance_Criterion(S, S'); S* ← Update_Best_Solution(S, S*); End-While;

Metaheuristics for optimization problems in sports META’08, October 2008 88/92 Results Literature: instances with up to 20 teams

Metaheuristics for optimization problems in sports META’08, October 2008 89/92 Future research Weighted COEV minimization problem Weighted COEV min-max problem

Metaheuristics for optimization problems in sports META’08, October 2008 90/92 Applications of metaheuristics Scheduling the Brazilian basketball tournament

Metaheuristics for optimization problems in sports META’08, October 2008 91/92 Perspectives and concluding remarks Optimization in sports is a field of increasing interest Sports management and scheduling are very attractive areas for applications of Operations Research Many interesting applications, often reviewed by the media 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

Metaheuristics for optimization problems in sports META’08, October 2008 92/92

Metaheuristics for optimization problems in sports META’08, October 2008 1/92 Celso C. Ribeiro Joint work with S. Urrutia, A. Duarte, and A. Guedes 2nd.

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Metaheuristics for optimization problems in sports META’08, October 2008 1/92 Celso C. Ribeiro Joint work with S. Urrutia, A. Duarte, and A. Guedes 2nd.

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