heuristics for the traveling tournament problem: scheduling the brazilian soccer championship

December 2003 Traveling tournament problem1/57

Heuristics for the Traveling Tournament Problem: Scheduling the Brazilian Soccer Championship

Celso C. RIBEIROSebastián URRUTIA


Summary• Motivation• Formulation• Constructive heuristic• Neighborhoods• Iterated local search• Extended ILS for TTP• Computational results• Concluding remarks


Motivation• Professional sports leagues are a major

economic activity around the world.• Teams and leagues do not want to waste

their investments in players and structure as a consequence of poor schedules of games.

• Game scheduling is a difficult task, involving different types of constraints and many objectives to optimize. – Several decision makers (team managers,

league authorities, TV, etc) with contrary goals.– Economic issues– Logistic issues– Fairness


Formulation (TTP)

• Conditions:– 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 (TTP)

• Constraints:– Tournament is a strict double round-robin

tournament:• Each team plays against every other team

twice, one at home and the other away.• There are 2(n-1) rounds, each one with n/2

games.

– No team can play more than three games in a home stand or in a road trip (away games).


Formulation (TTP)• Conditions (cont.):

– Tournament is mirrored: • All teams face each other once in the first phase

with n-1 rounds.• In the second phase with the last n-1 rounds the

teams play each other again in the same order, following an inverted home/away pattern.

• Games in the second phase are determined by those in the first (simple strict round robin).

• Common structure in Latin-American tournaments.

• Goal: minimize the total distance traveled by all teams.


Formulation

• Given a graph G=(V, E), a factor of G is a graph G’=(V,E’) with E’E.

• G’ is a 1-factor if all its nodes have degree equal to one.

• A factorization of G=(V,E) is a set of edge-disjoint factors G1=(V,E1), ..., Gp=(V,Ep), with E1...Ep=E.

• In a 1-factorization of G all factors are 1-factors.


4 3

2

1

5

6

Formulation

Example: 1-factorization of K6


4 3

2

1

5

6

1Formulation



4 3

2

1

5

6

2Formulation



4 3

2

1

5

6

3Formulation



4 3

2

1

5

6

4Formulation



4 3

2

1

5

6

5Formulation



• For strict single round robin and strict mirrored double round robin tournaments:– Each edge in Kn represents a game.

– Each 1-factor of Kn represents a round.

– Each ordered 1-factorization of Kn represents a schedule.

Formulation


Constructive heuristic

• Three steps:1. Schedule games using abstract

teams (structure of the draw, 1-factorization).

2. Assign real teams to abstract teams.3. Select stadium for each game

(home/away pattern) in the first phase (mirrored tournament).



• Step 1: schedule games using abstract teams

– This phase creates the structure of the tournament.

– “Polygon method” is used.– Tournament structure is fixed and

will not change in the other steps.



4 3

2

1

5

6

Example: “polygon method” for n=6

1st round



3 2

1

5

4

6


2nd round



2 1

5

4

3

6


3rd round



1 5

4

3

2

6


4th round



5 4

3

2

1

6


5th round


Constructive heuristic Abstract teams (n=6)

Round

A B C D E F

1/6 F E D C B A

2/7 D C B A F E

3/8 B A E F C D

4/9 E D F B A C

5/10 C F A E D B



• Step 2: assign real teams to abstract teams

– Build a matrix with the number of consecutive games for each pair of abstract teams:• For each pair of teams X and Y, an entry

in this table contains the total number of times in which the other teams play consecutively with X and Y in any order.


Constructive heuristicA B C D E F

A 0 1 6 5 2 4

B 1 0 2 5 6 4

C 6 2 0 2 5 3

D 5 5 2 0 2 4

E 2 6 5 2 0 3

F 4 4 3 4 3 0


Constructive heuristic• Step 2: assign real teams to

abstract teams– Greedily assign pairs of real teams

with close home cities to pairs of abstract teams with large entries in the matrix with the number of consecutive games.


Constructive heuristic Real teams (n=6)

Round

FLU SAN

FLA GRE

PAL PAY

1/6 PAY PAL GRE

FLA SAN

FLU

2/7 GRE

FLA SAN

FLU PAY PAL

3/8 SAN

FLU PAL PAY FLA GRE

4/9 PAL GRE

PAY SAN

FLU FLA

5/10 FLA PAY FLU PAL GRE

SAN



• Step 3: select stadium for each game in the first phase of the tournament

– Two-part strategy:• Build a feasible assignment of stadiums,

starting from a random assignment in the first round.

• Improve the assignment of stadiums, performing a simple local search algorithm based on home-away swaps.


Constructive heuristic Real teams (n=6)

Round FLU SAN FLA GRE PAL PAY

1/6 PAY@PA

LGRE

@FLA

SAN@FL

U

2/7 GRE@FL

ASAN

@FLU

PAY@PA

L

3/8@SA

NFLU

@PAL

PAY FLA@GR

E

4/9 PAL@GR

E@PA

YSAN

@FLU

FLA

5/10@FL

APAY FLU

@PAL

GRE@SA

N


Neighborhoods• Neighborhood “home-away swap”

(HAS): select a game and exchange the stadium where it takes place.

Real teams (n=6)


1/6 PAY@PA

LGRE

@FLA

SAN@FL

U

2/7 GRE@FL

ASAN

@FLU

PAY@PA

L

3/8@SA

NFLU

@PAL

PAY FLA@GR

E

4/9 PAL@GR

E@PA

YSAN

@FLU

FLA

5/10@FL

APAY FLU

@PAL

GRE@SA

N


Neighborhoods• Neighborhood “home-away swap”

(HAS): select a game and exchange the stadium where it takes place.

Real teams (n=6)


1/6 PAY PAL GRE@FL

A@SA

N@FL

U

2/7 GRE@FL

ASAN

@FLU

PAY@PA

L

3/8@SA

NFLU

@PAL

PAY FLA@GR

E

4/9 PAL@GR

E@PA

YSAN

@FLU

FLA

5/10@FL

APAY FLU

@PAL

GRE@SA

N


Neighborhoods• Neighborhood “team swap” (TS):

select two teams and swap their games, also swap the home-away assignment of their own game. Real teams (n=6)


1/6 PAY@PA

LGRE

@FLA

SAN@FL

U

2/7 GRE@FL

ASAN

@FLU

PAY@PA

L

3/8@SA

NFLU

@PAL

PAY FLA@GR

E

4/9 PAL@GR

E@PA

YSAN

@FLU

FLA

5/10@FL

APAY FLU

@PAL

GRE@SA

N





1/6 PAY@PA

LGRE

@FLA

SAN@FL

U

2/7 GRE@FL

ASAN

@FLU

PAY@PA

L

3/8@SA

NFLU

@PAL

PAY FLA@GR

E

4/9 PAL@GR

E@PA

YSAN

@FLU

FLA

5/10@FL

APAY FLU

@PAL

GRE@SA

N





1/6 PAY@PA

LSAN

@FLA

GRE@FL

U

2/7 GRE@FL

APAY

@FLU

SAN@PA

L

3/8@SA

NFLU PAL PAY

@FLA

@GRE

4/9 PAL@GR

E@FL

USAN

@PAY

FLA

5/10@FL

APAY GRE

@PAL

FLU@SA

N


Neighborhoods

• Neighborhood “partial round swap” (PRS): select two games AxB and CxD from round X and two games AxC and BxD from round Y, and swap their rounds (only for n8 and not always applicable).

Rounds ATM SAP CON FLA FLU INT CRU GRE1/82/9 FLA @INT @ATM SAP3/104/11 @SAP ATM @INT FLA5/126/137/14


Neighborhoods

• Neighborhood “partial round swap” (PRS): select two games AxB and CxD from round X and two games AxC and BxD from round Y, and swap their rounds (only for n8 and not always applicable).

Rounds ATM SAP CON FLA FLU INT CRU GRE1/82/9 @SAP ATM @INT FLA3/104/11 FLA @INT @ATM SAP5/126/137/14


Neighborhoods

• Neighborhood “game rotation” (GR) (ejection chain):– Enforce a game to be played at some

round: add a new edge to a 1-factor of the 1-factorization associated with the current schedule.

– Use an ejection chain to recover a 1-factorization.


Neighborhoods

4 3

2

1

5

6

2

Enforce game 1 vs. 3 at round (factor) 2.


4 3

2

1

5

6

2Neighborhoods

Teams 1 and 3 are now playing twice in this round.


4 3

2

1

5

6

2Neighborhoods

Eliminate the other games played by teams 1 and 3 in this round.


4 3

2

1

5

6

2Neighborhoods

Enforce the former opponents of teams 1 and 3 to play each other in this round: new game 2 vs. 4 in this round.


4 3

2

1

5

6

4Neighborhoods

Consider the factor where game 2 vs. 4 was scheduled.


Neighborhoods

4 3

2

1

5

6

4

Enforce game 1 vs. 4 (eliminated from round 2) to beplayed in this round, and so on...


Neighborhoods

• The ejection chain terminates when the game enforced in the beginning is removed from the round where it was played in the original schedule.

• PRS moves may appear after an ejection chain move is made.


Iterated local searchMartin, Otto, & Felten (1991); Martin & Otto

(1996)

S GenerateInitialSolution() S,S* LocalSearch(S) repeat

S’ Perturbation(S,history)S’ LocalSearch(S’)S AceptanceCriterion(S,S’,history)S* UpdateBestSolution(S,S*)

until StoppingCriterion


Extended ILS heuristicwhile .not.StoppingCriterion

S GenerateRandomizedInitialSolution() S LocalSearch(S)repeat

S’ Perturbation(S,history)S’ LocalSearch(S’)S AceptanceCriterion(S,S’,history)S* UpdateOverallBestSolution(S,S*)

until ReinitializationCriterionend






Constructive heuristicrandomized in the 2nd

step






TS HAS PRS HAS until a local optimum

for all neighborhoods is found






Apply one GR move randomly selected.

Execute the 3rd step of the constructive

heuristic to ensure feasibility






Initial criterion: At least almost as good as

current solution. Relax the criterion as current

solution is not changing.






Too many iterations without improving best

solution in cycle


Computational results

• Benchmark circular instances with n = 8, 10, 12, 14, 16, 18, and 20 teams.

• Harder benchmark MLB instances with n = 8, 10, 12, 14, and 16 teams. – All available from Michael Trick’s web

page.

• 2003 edition of the Brazilian nationalsoccer championship with 24 teams.


Computational results• Largest problems solved to optimality using

an integer programming formulation: n = 6 teams

• Approximate solutions listed at Michael Trick’s home page for the corresponding unmirrored (not necessary mirrored) instances.

• Solutions found for CIRC and MLB instances are even better than those available for the corresponding unmirrored instances in 2002 (e.g. 3.5 hours for NL16 in Pentium IV 2.0 MHz).



• Total distance traveled for the 2003 edition of the Brazilian soccer championship with 24 teams (instance br24) in 12 hours (Pentium IV 2.0 MHz):Realized (official draw): 1, 048,134 kms

Our solution: 549,020 kms (50% reduction)• Approximate corresponding savings in airfares:US$ 1,700,000


Concluding remarks

• Extended ILS heuristic found very good solutions to benchmark instances:– Solutions found for CIRC and MLB instances

are even better than those available for the corresponding unmirrored instances in 2002.

• Effectiveness of ejection chain neighborhood.

• Significant savings in airfares costs and traveled distance in a real instance.


Work in progress• Combination of constraint programming

with heuristics to handle difficult constraints.

• Incorporation of additional real-life constraints: TV constraints, pre-scheduled games, complementary teams, airfares + hotel costs, ...

• Talks with Brazilian federations of soccer(CBF) and basketball (CBB) to schedule the2004 edition of national tournaments.

heuristics for the traveling tournament problem: scheduling the brazilian soccer championship

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