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December 2003 Traveling tournament problem1/88
Heuristics for the Traveling Tournament Problem: Scheduling the Brazilian Soccer Championship
Celso C. RIBEIROSebastián URRUTIA
December 2003 Traveling tournament problem2/88
Summary• Motivation• Formulation• Constructive heuristic• Neighborhoods• Iterated local search• Extended ILS: GRASP + ILS
heuristic• Computational results• Concluding remarks
December 2003 Traveling tournament problem3/88
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. – Ronaldo (Real Madrid) scored a goal with
14 seconds on December 2nd with the assistance of Zidane, Roberto Carlos, Raul, and Beckham:
US$ 222 millions!
December 2003 Traveling tournament problem4/88
Motivation• Game scheduling is a difficult task,
involving different types of constraints, logistic issues, multiple objectives to optimize, and several decision makers (officials, managers, TV, etc…).
• The total distance traveled becomes an important variable to be minimized, to reduce traveling costs and to give more time to the players for resting and training.
• Avoid unfair draws!
December 2003 Traveling tournament problem5/88
Motivation• Short stories about unfair draws:
– Latin-American qualification phase for 2006 World Cup• Teams did not want to play two consecutive
games in the highlands (Bolivia and Equator), but were not able to find a consensus: schedule of 2002 World Cup was repeated.
– Argentinian national soccer championship• Boca Juniors played at home all but one
games with major teams at Maradona’s come back.
December 2003 Traveling tournament problem6/88
Motivation• Short stories about unfair draws:
– IRB admitted unfair draw of 2003 Rugby World Cup• The International Rugby Board has admitted the
World Cup draw was unfairly stacked against poorer countries so tournament organizers could maximize their profits: richer nations (e.g. Australia, Wales, New Zealand, South Africa, England, France) were given more time to play their games because of commercial arrangements with broadcasters. "Yes (it's unfair), but that's the way it is," Millar said.
• While all the bigger teams had at least 20 days to play their four pool games, the smaller sides' matches were crammed into a much tighter schedule (e.g. Italy, Argentina, Tonga, Samoa).
December 2003 Traveling tournament problem7/88
Summary• Motivation• Formulation• Constructive heuristic• Neighborhoods• Iterated local search• Extended ILS: GRASP + ILS
heuristic• Computational results• Concluding remarks
December 2003 Traveling tournament problem8/88
Formulation
• 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.
December 2003 Traveling tournament problem9/88
Formulation
• Conditions (cont.):– Tournament is a strict double round-robin
tournament:• There are 2(n-1) rounds, each one 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 or in a road trip (away games).
December 2003 Traveling tournament problem10/88
Formulation• 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.
• Common structure in Latin-American tournaments.
• Goal: minimize the total distance traveled by all teams.
December 2003 Traveling tournament problem11/88
Formulation• Variants:
– single round-robin– no-repeaters– no synchronized rounds– multiple games (more than two, variable)– teams with complementary patterns in the
same city– pre-scheduled games and TV constraints– stadium availability– minimize airfare and hotel costs, etc.
December 2003 Traveling tournament problem12/88
Formulation
• Some references:• Easton, Nemhauser, & Trick, “The traveling
tournament problem: Description and benchmarks” (2001)
• Trick, “Challenge traveling tournament instances”, web page
• Nemhauser & Trick, “Scheduling a major college basketball conference” (1998)
• Thompson, “Kicking timetabling problems into touch”, (1999)
• Anagnostopoulos, Michel, Van Hentenryck, & Vergados, “A simulated annealing approach to the traveling tournament problem” (2003)
December 2003 Traveling tournament problem13/88
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.
• All factors in a 1-factorization of G are 1-factors.
December 2003 Traveling tournament problem14/88
4 3
2
1
5
6
Formulation
Example: 1-factorization of K6
December 2003 Traveling tournament problem15/88
4 3
2
1
5
6
1Formulation
Example: 1-factorization of K6
December 2003 Traveling tournament problem16/88
4 3
2
1
5
6
2Formulation
Example: 1-factorization of K6
December 2003 Traveling tournament problem17/88
4 3
2
1
5
6
3Formulation
Example: 1-factorization of K6
December 2003 Traveling tournament problem18/88
4 3
2
1
5
6
4Formulation
Example: 1-factorization of K6
December 2003 Traveling tournament problem19/88
4 3
2
1
5
6
5Formulation
Example: 1-factorization of K6
December 2003 Traveling tournament problem20/88
• Mirrored tournament: games in the second phase are determined by those in the first.– Each edge of Kn represents a game.– Each 1-factor of Kn represents a round.– Each ordered 1-factorization of Kn is a
schedule.– Without considering the stadiums, there are
(n-1)! times (number of nonisomorphic factors) different “mirrored tournaments”.Dinitz, Garnick, & McKay, “There are 526,915,620 nonisomorphic one-factorizations of K12” (1995)
Formulation
December 2003 Traveling tournament problem21/88
Summary• Motivation• Formulation• Constructive heuristic• Neighborhoods• Iterated local search• Extended ILS: GRASP + ILS
heuristic• Computational results• Concluding remarks
December 2003 Traveling tournament problem22/88
Constructive heuristic• Three steps:
1. Schedule games using abstract teams (structure of the draw).
2. Assign real teams to abstract teams.3. Select stadium for each game
(home/away pattern) in the first phase (mirrored tournament).– Other algorithms first define the
home/away pattern and then assign teams to games.
15’
December 2003 Traveling tournament problem23/88
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.
December 2003 Traveling tournament problem24/88
Constructive heuristic
4 3
2
1
5
6
Example: “polygon method” for n=6
1st round
December 2003 Traveling tournament problem25/88
Constructive heuristic
3 2
1
5
4
6
Example: “polygon method” for n=6
2nd round
December 2003 Traveling tournament problem26/88
Constructive heuristic
2 1
5
4
3
6
Example: “polygon method” for n=6
3rd round
December 2003 Traveling tournament problem27/88
Constructive heuristic
1 5
4
3
2
6
Example: “polygon method” for n=6
4th round
December 2003 Traveling tournament problem28/88
Constructive heuristic
5 4
3
2
1
6
Example: “polygon method” for n=6
5th round
December 2003 Traveling tournament problem29/88
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
December 2003 Traveling tournament problem30/88
Constructive heuristic
• Step 2: assign real teams to abstract teams
– “Polygon method” was used to build a schedule with 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.
December 2003 Traveling tournament problem31/88
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
December 2003 Traveling tournament problem32/88
Constructive heuristic• Step 2: assign real teams to abstract
teams– “Polygon method” was used to build a
schedule with abstract teams.– Build a matrix with the number of consecutive
games for each pair of abstract teams.– Greedily (QAP) 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.
December 2003 Traveling tournament problem33/88
Constructive heuristic 0 4 4 4 4 4 4 3 4 4 3 4 4 4 4 44 0 2 25 0 0 0 0 0 0 0 0 0 0 25 24 2 0 2 25 0 0 0 0 0 0 0 0 0 0 254 25 2 0 2 25 0 0 0 0 0 0 0 0 0 04 0 25 2 0 2 25 0 0 0 0 0 0 0 0 04 0 0 25 2 0 2 25 0 0 0 0 0 0 0 04 0 0 0 25 2 0 2 25 0 0 0 0 0 0 03 0 0 0 0 25 2 0 2 26 0 0 0 0 0 04 0 0 0 0 0 25 2 0 1 26 0 0 0 0 04 0 0 0 0 0 0 26 1 0 2 25 0 0 0 03 0 0 0 0 0 0 0 26 2 0 2 25 0 0 04 0 0 0 0 0 0 0 0 25 2 0 2 25 0 04 0 0 0 0 0 0 0 0 0 25 2 0 2 25 04 0 0 0 0 0 0 0 0 0 0 25 2 0 2 254 25 0 0 0 0 0 0 0 0 0 0 25 2 0 24 2 25 0 0 0 0 0 0 0 0 0 0 25 2 0
n = 16: note the large number of times inwhich two teams are faced consecutively, which is explored by step 2 of the constructiveheuristic.
December 2003 Traveling tournament problem34/88
Constructive heuristic• Teams (n=6):
– FLAMENGO and FLUMINENSE: two teams in the same city (Rio de Janeiro)
– SANTOS and PALMEIRAS: two teams in two very close cities (Santos and São Paulo) 400 kms south of Rio de Janeiro
– GREMIO: team with home city approximately 1100 kms south of Rio de Janeiro
– PAYSANDU: team with home city approximately 2500 kms north of Rio de Janeiro
December 2003 Traveling tournament problem35/88
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
December 2003 Traveling tournament problem36/88
Constructive heuristic• Step 3: select stadium for each game in
the first phase of the tournament– Schedule games in multiple-game trips with
no more than three away games each: face teams with close stadiums in the same road trip.
– 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.
December 2003 Traveling tournament problem37/88
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
December 2003 Traveling tournament problem38/88
Summary• Motivation• Formulation• Constructive heuristic• Neighborhoods• Iterated local search• Extended ILS: GRASP + ILS
heuristic• Computational results• Concluding remarks
December 2003 Traveling tournament problem39/88
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
December 2003 Traveling tournament problem40/88
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
December 2003 Traveling tournament problem41/88
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
December 2003 Traveling tournament problem42/88
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
December 2003 Traveling tournament problem43/88
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
December 2003 Traveling tournament problem44/88
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, not always possible).
Rounds ATM SAP CON FLA FLU INT CRU GRE1/82/9 FLA @INT @ATM SAP3/104/11 @SAP ATM @INT FLA5/126/137/14
30’
December 2003 Traveling tournament problem45/88
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, not always possible).
Rounds ATM SAP CON FLA FLU INT CRU GRE1/82/9 @SAP ATM @INT FLA3/104/11 FLA @INT @ATM SAP5/126/137/14
December 2003 Traveling tournament problem46/88
Neighborhoods
• Neigborhood “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.
December 2003 Traveling tournament problem47/88
Neighborhoods
4 3
2
1
5
6
2
Enforce game 1vs. 3 at round (factor) 2.
December 2003 Traveling tournament problem48/88
4 3
2
1
5
6
2Neighborhoods
Teams 1 and 3 are now playing twice in this round.
December 2003 Traveling tournament problem49/88
4 3
2
1
5
6
2Neighborhoods
Eliminate the other games played by teams 1 and 3 in this round.
December 2003 Traveling tournament problem50/88
4 3
2
1
5
6
2Neighborhoods
Enforce the former oponents of teams 1 and 3 to play each other in this round: new game 2 vs. 4 in this round.
December 2003 Traveling tournament problem51/88
4 3
2
1
5
6
4Neighborhoods
Consider the factor where game 2 vs. 4 was scheduled.
December 2003 Traveling tournament problem52/88
Neighborhoods
4 3
2
1
5
6
4
Enforce game 1 vs. 4 (eliminated from round 2) to be played in this round.
December 2003 Traveling tournament problem53/88
Neighborhoods
4 3
2
1
5
6
4
Eliminate games 2 vs. 4 (enforced in round 2) and 1 vs. 5 (since team 1 cannot play twice).
December 2003 Traveling tournament problem54/88
Neighborhoods
4 3
2
1
5
6
4
Enforce game 2 vs. 5 to be played in this round.
December 2003 Traveling tournament problem55/88
Neighborhoods
4 3
2
1
5
6
1
Consider the factor where game 2 vs. 5 was scheduled.
December 2003 Traveling tournament problem56/88
Neighborhoods
4 3
2
1
5
6
1
Enforce game 1 vs. 5 (eliminated from round 4) to be played in this round.
December 2003 Traveling tournament problem57/88
Neighborhoods
4 3
2
1
5
6
1
Eliminate games 2 vs. 5 (enforced in round 4) and 1 vs. 6 (since team 1 cannot play twice).
December 2003 Traveling tournament problem58/88
Neighborhoods
4 3
2
1
5
6
1
Enforce game 2 vs. 6 to be played in this round.
December 2003 Traveling tournament problem59/88
Neighborhoods
4 3
2
1
5
6
5
Consider the factor where game 2 vs. 6 was scheduled.
40’
December 2003 Traveling tournament problem60/88
Neighborhoods
4 3
2
1
5
6
5
Enforce game 1 vs. 6 (eliminated from round 1)to be played in this round.
December 2003 Traveling tournament problem61/88
Neighborhoods
4 3
2
1
5
6
5
Eliminate games 2 vs. 6 (enforced in round 1) and 1 vs. 3 (since team 1 cannot play twice and this game was enforced in round 2 at the beginning of the ejection chain).
December 2003 Traveling tournament problem62/88
Neighborhoods
4 3
2
1
5
6
5
Finally, enforce game 2 vs. 3 (eliminated from round 2 at the beginning of the ejection chain ) to be played in this round.
December 2003 Traveling tournament problem63/88
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:– The ejection chain move is able to find
solutions that are not reachable through other neighborhoods.
– PRS moves may appear after an ejection chain move is made.
December 2003 Traveling tournament problem64/88
Neighborhoods
• PRS = GR for n=4; PRS GR for n6.• Only neighborhoods PRS and GR are
able to change the structure of the schedule of the initial solution built by the “polygon method”.
• However, PRS cannot always be used, due to the structure of the solutions built by “polygon method” for some values of n.
December 2003 Traveling tournament problem65/88
Neighborhoods
• The length of the ejection chain is variable.
• If ejections chains are not used, one may be stucked at schedules with the structure of the solutions built by the “polygon method”.– n{6,8,12,14,16,20,24}: no PRS moves exist
if “polygon method” is used, but... – ... PRS moves may appear after an ejection
chain move is made.
December 2003 Traveling tournament problem66/88
Summary• Motivation• Formulation• Constructive heuristic• Neighborhoods• Iterated local search• Extended ILS: GRASP + ILS
heuristic• Computational results• Concluding remarks
December 2003 Traveling tournament problem67/88
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
December 2003 Traveling tournament problem68/88
Extended GRASP + ILS heuristic
while .not.StoppingCriterionS GenerateRandomizedInitialSolution() S,S LocalSearch(S) /* S best solution in cycle */ repeat /* S* best overall solution */
S’ Perturbation(S,history)S’ LocalSearch(S’)S AceptanceCriterion(S,S’,history)S* UpdateOverallBestSolution(S,S*)S UpdateCycleBestSolution(S,S)
until ReinitializationCriterionend
December 2003 Traveling tournament problem69/88
Extended GRASP + ILS heuristic
• Initial solutions (for each cycle): – Use the constructive heuristic.– Randomize the second step: team
assignment using the matrix of consecutive games.
– Third step (stadium assignment) was already randomized.
– New cycle is started if ReinitializationCriterion is met.
December 2003 Traveling tournament problem70/88
Extended GRASP + ILS heuristic
• Local search:– First improving strategy.– Multiple neighborhoods are used in this
order:• TS HAS PRS HAS • GR is very costly and is not used during the
local search.• Repeat until a local optimum with respect to
all neighborhoods is found.
– Neighbor solutions are investigated at random for each type of neighborhood.
December 2003 Traveling tournament problem71/88
Extended GRASP + ILS heuristic
• Perturbation:– Obtain S’ by applying one ejection
chain randomly selected GR move to the current solution S.
– The new solution may be infeasible: perform the third step of the constructive heuristic (stadium assignment).
December 2003 Traveling tournament problem72/88
Extended GRASP + ILS heuristic
while .not.StoppingCriterionS GenerateRandomizedInitialSolution() S,S LocalSearch(S) /* S best solution in cycle */ repeat /* S* best overall solution */
S’ Perturbation(S,history)S’ LocalSearch(S’)S AceptanceCriterion(S,S’,history)S* UpdateOverallBestSolution(S,S*)S UpdateCycleBestSolution(S,S)
until ReinitializationCriterionend
December 2003 Traveling tournament problem73/88
Extended GRASP + ILS heuristic
• Acceptance criterion:– Set threshold acceptance criterion at =0.1– Accept new solution if its cost is lower than
(100+)% of the current solution.– If the current solution does not change after
a certain number of iterations, double the value of (i.e., acceptance criterion is relaxed if current solution does not change).
– Reset =0.1 when the current solution changes.
50’
December 2003 Traveling tournament problem74/88
Extended GRASP + ILS heuristic
• Reinitialization criterion:– If n/3 deteriorating moves are accepted
since the last time the best solution in the cycle S was updated, then start a new cycle computing a new initial solution.• Re-initialization occurs if too many iterations
are performed without improving the best solution in a cycle.
• Computations in a cycle are not interrupted if the algorithm is improving the current solution S.
December 2003 Traveling tournament problem75/88
Summary• Motivation• Formulation• Constructive heuristic• Neighborhoods• Iterated local search• Extended ILS: GRASP + ILS
heuristic• Computational results• Concluding remarks
December 2003 Traveling tournament problem76/88
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 national soccer championship with 24 teams.
December 2003 Traveling tournament problem77/88
Computational results• Largest problems solved to optimality
using an integer programming formulation: n = 6 teams– Gaps: 0.6% for n=8; 3.6% for n=10 (MLB)
• Lower bounds: approximate solutions listed at Michael Trick’s home page for the corresponding unmirrored 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).
December 2003 Traveling tournament problem78/88
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
December 2003 Traveling tournament problem79/88
Computational results
December 2003 Traveling tournament problem80/88
Computational results• Methodology used to evaluate and
compare different algorithms for the same problem:
– Probability distribution of time-to-target-solution-value: experimental plotsAiex, Resende, & Ribeiro: “Probability distribution of solution time in GRASP: An experimental investigation”, (2002)Resende & Ribeiro: “GRASP and path-relinking: Recent advances and applications”, (2003)
December 2003 Traveling tournament problem81/88
Computational results• Select an instance and a target
value:– Perform 200 runs using different seeds.– Stopping when a solution value at least
as good as the target is found.– For each run, measure the time-to-
target-value.– Plot the probabilities of finding a
solution at least as good as the target value within some computation time.55’
December 2003 Traveling tournament problem82/88
Computational results
0
10
20
30
40
50
60
70
80
90
100
1 10 100 1000 10000
Cum
ula
tive p
robabili
ty
Time (seconds)
GRASP+ILS
Instance: br24Target: 586,000 kms
December 2003 Traveling tournament problem83/88
Computational results
560000
570000
580000
590000
600000
610000
620000
630000
640000
0 100 200 300 400 500 600
Solu
tion v
alu
e
Time (seconds)
GRASP+ILS solution value
Instance: br24Time: 10 minutes
December 2003 Traveling tournament problem84/88
Computational results
540000
550000
560000
570000
580000
590000
600000
610000
620000
630000
640000
0 3600 7200 10800 14400 18000 21600 25200 28800 32400 36000 39600 43200
Solu
tion v
alu
e
Time (seconds)
GRASP+ILS solution value
Instance: br24Time: 12 hours
December 2003 Traveling tournament problem85/88
Summary• Motivation• Formulation• Constructive heuristic• Neighborhoods• Iterated local search• Extended ILS: GRASP + ILS
heuristic• Computational results• Concluding remarks
December 2003 Traveling tournament problem86/88
Concluding remarks
• Extended GRASP + 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 real instance.
December 2003 Traveling tournament problem87/88
Concluding remarks• Combination of constraint programming
with heuristics to handle difficult constraints.
• Incorporation of additional real-life constraints in progress: TV constraints, pre-scheduled games, complementary teams, airfares + hotel costs, ...
• Talks with Brazilian federations of soccer (CBF) and basketball (CBB) to schedule the 2004 editions of national tournaments.
December 2003 Traveling tournament problem88/88
Slides and publications
• Slides of this talk can be downloaded from: http://www.inf.puc-rio.br/~celso/talks
• This paper (soon) and other papers about GRASP, path-relinking, and their applications available at:http://www.inf.puc-rio.br/~celso/publicacoes
• Materials about applications of OR techniques to problems in sports management and scheduling may be found at: http://www.esportemax.orgPlease let us know about new links and materials!
60’