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Heuristics for the Traveling Tournament Problem: Scheduling the Brazilian Soccer Championship
Celso C. RIBEIROSebastián URRUTIA
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Summary• Motivation• Formulation• Constructive heuristic• Neighborhoods• Iterated local search• Extended ILS for TTP• Computational results• Concluding remarks
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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
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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.
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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).
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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.
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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.
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4 3
2
1
5
6
Formulation
Example: 1-factorization of K6
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4 3
2
1
5
6
1Formulation
Example: 1-factorization of K6
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4 3
2
1
5
6
2Formulation
Example: 1-factorization of K6
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4 3
2
1
5
6
3Formulation
Example: 1-factorization of K6
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4 3
2
1
5
6
4Formulation
Example: 1-factorization of K6
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4 3
2
1
5
6
5Formulation
Example: 1-factorization of K6
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• 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
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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).
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Constructive heuristic
• 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.
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Constructive heuristic
4 3
2
1
5
6
Example: “polygon method” for n=6
1st round
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Constructive heuristic
3 2
1
5
4
6
Example: “polygon method” for n=6
2nd round
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Constructive heuristic
2 1
5
4
3
6
Example: “polygon method” for n=6
3rd round
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Constructive heuristic
1 5
4
3
2
6
Example: “polygon method” for n=6
4th round
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Constructive heuristic
5 4
3
2
1
6
Example: “polygon method” for n=6
5th round
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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
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Constructive heuristic
• 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.
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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
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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.
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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
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Constructive heuristic
• 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.
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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
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Neighborhoods• Neighborhood “home-away swap”
(HAS): select a game and exchange the stadium where it takes place.
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
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Neighborhoods• Neighborhood “home-away swap”
(HAS): select a game and exchange the stadium where it takes place.
Real teams (n=6)
Round FLU SAN FLA GRE PAL PAY
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
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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)
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
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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)
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
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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)
Round FLU SAN FLA GRE PAL PAY
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
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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
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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
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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.
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Neighborhoods
4 3
2
1
5
6
2
Enforce game 1 vs. 3 at round (factor) 2.
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4 3
2
1
5
6
2Neighborhoods
Teams 1 and 3 are now playing twice in this round.
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4 3
2
1
5
6
2Neighborhoods
Eliminate the other games played by teams 1 and 3 in this round.
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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.
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4 3
2
1
5
6
4Neighborhoods
Consider the factor where game 2 vs. 4 was scheduled.
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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...
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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.
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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
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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
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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
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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
TS HAS PRS HAS until a local optimum
for all neighborhoods is found
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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
TS HAS PRS HAS until a local optimum
for all neighborhoods is found
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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
Apply one GR move randomly selected.
Execute the 3rd step of the constructive
heuristic to ensure feasibility
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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
Initial criterion: At least almost as good as
current solution. Relax the criterion as current
solution is not changing.
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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
Too many iterations without improving best
solution in cycle
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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.
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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).
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Computational results
• 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
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Computational results
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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.
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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.