metaheuristics for optimization problems in sports meta’08, october 2008 1/92 celso c. ribeiro...

META’08, October 2008 1/92 Metaheuristics for optimization problems in sports

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


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


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


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


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!


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)


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


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.


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


ResultsFUTMAX can also be used to follow the situation of each team:

Possible points

Points for guaranteed qualification

Points for possible qualification

Points accumulated

FLUMINENSE


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


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)


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

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


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)


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 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


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


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


Decomposition methods• Nemhauser and 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

• Henz 2001: CP may work better than IP and complete enumeration for all the tasks


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


Applications of metaheuristics

Traveling Tournament Problem (TTP) and its mirrored version (mTTP)


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


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


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)


• 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 round2. Improve this assignment, using a simple local

search algorithm based on home-away swaps

Constructive heuristic



4 3

2

1

5

6

Example: “polygon method” for n=6

1st round



3 2

1

5

4

6

Example: “polygon method” for n=6

2nd round


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


Partial round swap (PRS)

7 4

3

1

8

6

2

5

7 4

3

1

8

6

2

5


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



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



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 Enforce game (1,3) tobe played in round 2



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


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


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


GRASP+ILS heuristicwhile .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 ReinitializationCriterionend


• 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



Referee Assignment Problem (RAP)


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


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 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)• 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


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 = {s1, s2,..., sn}: refereeing slots to be filled by referees

• R = {r1, r2,..., rm}: referees


Problem statement• pi: skill level of referee ri • qj: minimum skill level a referee must

have to be assigned to refereeing slot sj

• Mi: maximum number of games referee ri can officiate

• Ti: target number of games referee ri is willing to officiate

• Each referee may choose a set of time slots where he/she is not available to officiate


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)


Solution approach

• 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


Solution approachAlgorithm RefereeAssignmentHeuristic (MaxIter)1. S* BuildGreedyRandomizedSolution ();2. If not isFeasible (S*) then3. S* RepairHeuristic (S*, MaxIter);4. If isFeasible (S*) then5. S* ImprovementHeuristic (S*);6. Else “infeasible”7.Return S*


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


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)


Numerical results

Instance

Construction Repair Improvement

time

(s)value feas. time (s) value feas. time (s) value

I1 0.02 1286.20 10 — — — 32.34 619.60

I2 0.02 1360.00 5 0.47 1338.00 10 31.81 623.40

I3 0.02 1269.00 2 0.60 1247.00 10 33.87 621.60

I4 0.03 — — 1.14 1303.20 10 30.28 627.20

I5 0.03 1302.00 3 1.40 12591.14

10 33.73 654.00

Table 1: Instances with 500 games, 750 referees, and 65 facilities


Numerical resultsInstan

cepattern

Greedy Random

const.

(s)repair

(s)feas. repair

(s)feas.

I1 P00.03 11.27 10 79.80 9

I2 P00.03 6.69 10 80.80 10

I3 P00.03 11.33 10 86.20 8

I4 P00.03 4.61 10 30.60 10

I5 P00.03 3.39 10 29.10 10

I1 P10.03 2.75 10 33.50 10

I2 P10.02 19.29 10 134.60 2

I3 P10.03 14.77 10 135.10 8

I4 P10.03 1.22 10 38.00 10

I5 P10.03 2.69 10 32.90 10

Table 4: Greedy vs. randomly generated initial solutions


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!”)


Numerical results

Figure 3: 500 games, 750 referees, 85 facilities, pattern P0 (target = 529)


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


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


Solution approach

• Exact approach: dichotomic method

50 games and 100 referees


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


Numerical results


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



Carry-over minimization problem


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 matrix

• 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 matrix

• RRT 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 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


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


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


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


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)


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


Multi-start + ILS heuristicFor 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;


Results

• Literature: instances with up to 20 teams

Teams Best results Our results4 12 126 60 608 56 56

10 108 10812 176 16014 234 25416 240 24018 340 40020 380 486


Future research

• Weighted COEV minimization problem

• Weighted COEV min-max problem



Scheduling the Brazilian basketball tournament


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 1/92 celso c. ribeiro...

Documents

sports meta08

sports motivation problems

summary optimization

sports examples

sports hammamet

motivation professional

motivation amateur sports

millions main problems