Celso C. Ribeiro Joint work with S. Urrutia, Duarte, T. Noronha, E.H. Haeusler, R. Melo,

Optimization problems in sports1/55

Celso C. Ribeiro

Joint work with S. Urrutia,

A.Duarte, T. Noronha,

E.H. Haeusler, R. Melo,

A.Guedes, F. Costa,S. Martins, R. Capua

et al.

Optimization Problems in Sports

Optimization problems in sports

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

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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.

Optimization problems in sportsLa Havana, March 2009 5/91

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!”

If San Lorenzo would have won the first two games, the tournament would have been decided and the third game would have no importance!


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)

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• 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”)



• 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!)



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

4 3

2

1

5

6

Example: “polygon method” for n=6

1st round


3 2

1

5

4

6


2nd round

Polygon method


2 1

5

4

3

6


3rd round

Polygon method


1 5

4

3

2

6


4th round

Polygon method


5 4

3

2

1

6


5th round

Polygon method


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


4 3

2

1

5

6

1-factorizations

Example: 1-factorization of K6


Oriented 1-factorization of K6

4 3

21

5

64 3

2

1

5

64 3

2

1

5

6

4 3

2

1

5

64 3

2

1

5

6

1 2 3

4 5


• 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)

1-factorizations


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 1998:1. Find home-away patterns2. 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


• 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

Applications of metaheuristics


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 gamesDuarte, 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 1. Greedy constructive heuristic2. 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


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:

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 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


• 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

Applications of metaheuristics


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

1 2 3 4 5 6 7A H C D E F G BB C D E F G H AC B A F H E D GD E B A G H C FE D G B A C F HF G H C B A E DG F E H D B A CH A F G C D B E


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


Carry-over effects

• SRRT and carry-over effects matrix (COEM)

A B C D E F G HA 0 0 3 0 1 2 1 0B 5 0 0 0 1 0 0 1C 0 1 0 3 0 3 0 0D 0 2 0 0 2 0 3 0E 1 1 0 2 0 2 0 1F 0 0 0 0 2 0 3 2G 0 3 1 0 0 0 0 3H 1 0 3 2 1 0 0 0


RRT COE matrix


Carry-over effects

• SRRT and carry-over effects matrix (COEM)

A B C D E F G HA 0 0 3 0 1 2 1 0B 5 0 0 0 1 0 0 1C 0 1 0 3 0 3 0 0D 0 2 0 0 2 0 3 0E 1 1 0 2 0 2 0 1F 0 0 0 0 2 0 3 2G 0 3 1 0 0 0 0 3H 1 0 3 2 1 0 0 0


RRT COE MatrixSuppose 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 valueA B C D E F G H

A 0 0 3 0 1 2 1 0B 5 0 0 0 1 0 0 1C 0 1 0 3 0 3 0 0D 0 2 0 0 2 0 3 0E 1 1 0 2 0 2 0 1F 0 0 0 0 2 0 3 2G 0 3 1 0 0 0 0 3H 1 0 3 2 1 0 0 0

COE matrix

COEMDG = 3

COEMFH = 2

H

Ai

H

AjijCOEMCOEV 2)(


Carry-over effects valueA B C D E F G H

A 0 0 3 0 1 2 1 0B 5 0 0 0 1 0 0 1C 0 1 0 3 0 3 0 0D 0 2 0 0 2 0 3 0E 1 1 0 2 0 2 0 1F 0 0 0 0 2 0 3 2G 0 3 1 0 0 0 0 3H 1 0 3 2 1 0 0 0

COE Matrix

H

Ai

H

AjijCOEMCOEV 2)(

Minimize!!!

COEMDG = 3

COEMFH = 2


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)


Applications of exact methods (IP)

• Improved algorithm and formulation for the Chilean soccer tournament

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• 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.

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• 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

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• 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)

Some recent references


• 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.)

Perspectives and concluding remarks


• However, old technologies might also be useful…



• 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.



• 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.


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Celso C. Ribeiro Joint work with S. Urrutia, Duarte, T. Noronha, E.H. Haeusler, R. Melo,

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