1 / 92

Metaheuristics in Sports Optimization

Explore the applications of metaheuristics in solving optimization problems in the field of sports. This article discusses various problems and solution methods, including traveling tournament problems, referee assignment, carry-over effect minimization, and more.

dougan
Download Presentation

Metaheuristics in Sports Optimization

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. 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

  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

  5. 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

  6. 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!

  7. 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)

  8. 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

  9. 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.

  10. 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

  11. Results FUTMAX can also be used to follow the situation of each team: Possible points FLUMINENSE Points for guaranteed qualification Points for possible qualification Points accumulated

  12. 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

  13. 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)

  14. 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

  15. 1-factorizations 1 2 5 4 3 Example: 1-factorization of K6 6

  16. 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

  17. 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) Open problem: How many schedules exist for a single round robin tournament with n teams?

  18. 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)

  19. 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

  20. 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

  21. 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

  22. 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

  23. Decomposition methods • Nemhauser and 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 • Henz 2001: CP may work better than IP and complete enumeration for all the tasks

  24. 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

  25. Applications of metaheuristics Traveling Tournament Problem (TTP) and its mirrored version (mTTP)

  26. 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

  27. 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

  28. 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 TTPRibeiro and Urrutia (PATAT 2004, EJOR 2007)

  29. Constructive heuristic • Three steps: • Schedule games using abstract teams: polygon method defines the structure of the tournament • 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) • Select stadium for each game (home/away pattern) in the first phase (mirrored tournament): • Build a feasible assignment of stadiums, starting from a random assignment in the first round • Improve this assignment, using a simple local search algorithm based on home-away swaps

  30. Constructive heuristic 6 Example: “polygon method” for n=6 1 5 2 1st round 3 4

  31. Constructive heuristic 6 Example: “polygon method” for n=6 5 4 1 2nd round 2 3

  32. 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

  33. Partial round swap (PRS) 1 2 1 2 3 8 3 8 7 4 7 4 6 5 6 5

  34. Partial round swap (PRS) 1 2 1 2 3 8 3 8 7 4 7 4 6 5 6 5

  35. 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

  36. Ejection chain: game rotation (GR) 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

  37. Ejection chain: game rotation (GR) 1 1 1 2 5 2 2 5 5 4 4 3 3 4 3 6 6 6 1 1 2 5 2 5 4 3 4 3 6 6 Enforce game (1,3) to be played in round 2

  38. Ejection chain: game rotation (GR) 1 1 1 2 5 2 2 5 5 4 4 3 3 4 3 6 6 6 1 1 2 5 2 5 4 3 4 3 6 6 Enforce game (1,3) to be played in round 2

  39. Ejection chain: game rotation (GR) 1 1 1 2 5 2 2 5 5 4 4 3 3 4 3 6 6 6 1 1 2 5 2 5 4 3 4 3 6 6

  40. Ejection chain: game rotation (GR) 1 1 1 2 5 2 2 5 5 4 4 3 3 4 3 6 6 6 1 1 2 5 2 5 4 3 4 3 6 6

  41. Ejection chain: game rotation (GR) 1 1 1 2 5 2 2 5 5 4 4 3 3 4 3 6 6 6 1 1 2 5 2 5 4 3 4 3 6 6

  42. Ejection chain: game rotation (GR) 1 1 1 2 5 2 2 5 5 4 4 3 3 4 3 6 6 6 1 1 2 5 2 5 4 3 4 3 6 6

  43. Ejection chain: game rotation (GR) 1 1 1 2 5 2 2 5 5 4 4 3 3 4 3 6 6 6 1 1 2 5 2 5 4 3 4 3 6 6

  44. Ejection chain: game rotation (GR) 1 1 1 2 5 2 2 5 5 4 4 3 3 4 3 6 6 6 1 1 2 5 2 5 4 3 4 3 6 6

  45. Ejection chain: game rotation (GR) 1 1 1 2 5 2 2 5 5 4 4 3 3 4 3 6 6 6 1 1 2 5 2 5 4 3 4 3 6 6

  46. 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

  47. 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

  48. 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

  49. Concluding remarks • 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

  50. Applications of metaheuristics Referee Assignment Problem (RAP)

More Related