heuristics for the mttproadef, february 2005 1/49 heuristics for the mirrored traveling tournament...
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![Page 1: Heuristics for the MTTPROADEF, February 2005 1/49 Heuristics for the Mirrored Traveling Tournament Problem Celso C. RIBEIRO Sebastián URRUTIA](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649d585503460f94a37a86/html5/thumbnails/1.jpg)
ROADEF, February 2005 1/49 Heuristics for the MTTP
Heuristics for the Mirrored Traveling Tournament Problem
Celso C. RIBEIROSebastián URRUTIA
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ROADEF, February 2005 2/49 Heuristics for the MTTP
Summary
• The Mirrored Traveling Tournament Problem
• Constructive heuristic• Neighborhoods and ejection chains• Extended GRASP + ILS heuristic• Computational results
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ROADEF, February 2005 3/49 Heuristics for the MTTP
Motivation• Game scheduling is a difficult task,
involving different types of constraints, logistic issues, multiple objectives, and several decision makers.
• Total distance traveled is an important variable to be minimized, to reduce traveling costs and to give more time to the players for resting and training.
• Timetabling is the major area of applications of OR in sports.
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ROADEF, February 2005 4/49 Heuristics for the 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.
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ROADEF, February 2005 5/49 Heuristics for the MTTP
Formulation– 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 (home games) or in a road trip (away games).
• Goal: minimize the total distance traveled by all teams.
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ROADEF, February 2005 6/49 Heuristics for the MTTP
Formulation
• Mirrored Traveling Tournament Problem (MTTP):– 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.
• Set of feasible solutions for the MTTP is a subset of the feasible solutions for the TTP.
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ROADEF, February 2005 7/49 Heuristics for the MTTP
1-Factorizations• 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), such that E1...Ep=E.
• All factors in a 1-factorization of G are 1-factors.
• Oriented 1-factorization: assign orientations to the edges of a 1-factorization
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ROADEF, February 2005 8/49 Heuristics for the MTTP
• 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 oriented 1-factorization of Kn represents a feasible schedule for n teams.
• Example: K6
1-Factorizations
4 3
2
1
5
6
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ROADEF, February 2005 9/49 Heuristics for the MTTP
1-Factorizations
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
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ROADEF, February 2005 10/49 Heuristics for the MTTP
• Three steps:1. Schedule games using abstract teams: polygon
method defines the structure of the tournament2. 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 of stadiums in the first round.
2. Improve this assignment of stadiums, using a simple local search algorithm based on home-away swaps.
Constructive heuristic
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ROADEF, February 2005 11/49 Heuristics for the MTTP
Constructive heuristic
4 3
2
1
5
6
Example: “polygon method” for n=6
1st round
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ROADEF, February 2005 12/49 Heuristics for the MTTP
Constructive heuristic
3 2
1
5
4
6
Example: “polygon method” for n=6
2nd round
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ROADEF, February 2005 13/49 Heuristics for the MTTP
Constructive heuristic
2 1
5
4
3
6
Example: “polygon method” for n=6
3rd round
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ROADEF, February 2005 14/49 Heuristics for the MTTP
Constructive heuristic
1 5
4
3
2
6
Example: “polygon method” for n=6
4th round
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ROADEF, February 2005 15/49 Heuristics for the MTTP
Constructive heuristic
5 4
3
2
1
6
Example: “polygon method” for n=6
5th round
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ROADEF, February 2005 16/49 Heuristics for the MTTP
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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ROADEF, February 2005 17/49 Heuristics for the MTTP
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
matrix contains the total number of times in which the other teams play consecutively with X and Y in any order.
– 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: QAP heuristic
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ROADEF, February 2005 18/49 Heuristics for the MTTP
Constructive heuristic
A 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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ROADEF, February 2005 19/49 Heuristics for the MTTP
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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ROADEF, February 2005 20/49 Heuristics for the MTTP
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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ROADEF, February 2005 21/49 Heuristics for the MTTP
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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ROADEF, February 2005 22/49 Heuristics for the MTTP
Neighborhood home-away swap (HAS)
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
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ROADEF, February 2005 23/49 Heuristics for the MTTP
Neighborhood home-away swap (HAS)
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
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ROADEF, February 2005 24/49 Heuristics for the MTTP
Neighborhood team-swap (TS)
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
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ROADEF, February 2005 25/49 Heuristics for the MTTP
Neighborhood team-swap (TS)
4 3
12
5
64 3
1
2
5
64 3
1
2
5
6
4 3
1
2
5
64 3
1
2
5
6
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ROADEF, February 2005 26/49 Heuristics for the MTTP
Neighborhood partial round swap (PRS)
7 4
3
1
8
6
2
5
7 4
3
1
8
6
2
5
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ROADEF, February 2005 27/49 Heuristics for the MTTP
Neighborhood partial round swap (PRS)
7 4
3
1
8
6
2
5
7 4
3
1
8
6
2
5
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ROADEF, February 2005 28/49 Heuristics for the MTTP
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 1-factor of the 1-factorization associated with the current schedule.
– Use an ejection chain to recover a 1-factorization.
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ROADEF, February 2005 29/49 Heuristics for the MTTP
Ejection chain: game rotation (GR)
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
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ROADEF, February 2005 30/49 Heuristics for the MTTP
Ejection chain: game rotation (GR)
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
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ROADEF, February 2005 31/49 Heuristics for the MTTP
Ejection chain: game rotation (GR)
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
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ROADEF, February 2005 32/49 Heuristics for the MTTP
Ejection chain: game rotation (GR)
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
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ROADEF, February 2005 33/49 Heuristics for the MTTP
Ejection chain: game rotation (GR)
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
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ROADEF, February 2005 34/49 Heuristics for the MTTP
Ejection chain: game rotation (GR)
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
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ROADEF, February 2005 35/49 Heuristics for the MTTP
Ejection chain: game rotation (GR)
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
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ROADEF, February 2005 36/49 Heuristics for the MTTP
Ejection chain: game rotation (GR)
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
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ROADEF, February 2005 37/49 Heuristics for the MTTP
Ejection chain: game rotation (GR)
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
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ROADEF, February 2005 38/49 Heuristics for the MTTP
Ejection chain: game rotation (GR)
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
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ROADEF, February 2005 39/49 Heuristics for the MTTP
Neighborhoods• Only movements 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.
• The ejection chain move is able to find solutions that are not reachable through other neighborhoods: escape from local optima
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ROADEF, February 2005 40/49 Heuristics for the MTTP
GRASP + ILS heuristic• Hybrid improvement heuristic for the
MTTP:– Combination of GRASP and ILS metaheuristics.– 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 movement in GR neighborhood.
– Detailed algorithm to appear in EJOR.
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ROADEF, February 2005 41/49 Heuristics for the MTTP
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
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ROADEF, February 2005 42/49 Heuristics for the MTTP
Computational results• Circular instances with n = 12, ..., 20 teams.• MLB instances with n = 12, ..., 16 teams.
– All available from http://mat.gsia.cmu.edu/TOURN/
– Largest instances exactly solved to date: n=6 (sequential), n=8 (parallel)
• Numerical results on a Pentium IV 2.0 MHz.• Comparisons with best known solutions for
the corresponding less constrained not necessarily mirrored instances (TTP).
![Page 43: Heuristics for the MTTPROADEF, February 2005 1/49 Heuristics for the Mirrored Traveling Tournament Problem Celso C. RIBEIRO Sebastián URRUTIA](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649d585503460f94a37a86/html5/thumbnails/43.jpg)
ROADEF, February 2005 43/49 Heuristics for the MTTP
Computational results
• Constructive heuristic:– Very fast
• Instance MLB16: 1000 runs in approximately 1 second
– Average gap is 17.1%– Better solutions than those found after
several days of computations by some metaheuristic aproaches to the not necessarily mirrrored version of the problem.
![Page 44: Heuristics for the MTTPROADEF, February 2005 1/49 Heuristics for the Mirrored Traveling Tournament Problem Celso C. RIBEIRO Sebastián URRUTIA](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649d585503460f94a37a86/html5/thumbnails/44.jpg)
ROADEF, February 2005 44/49 Heuristics for the MTTP
• GRASP + ILS heuristic: time limit is 10 minutes only
• Before this work, times were measured in days.
• Largest gap with respect to the best known solution for the less constrained not necessarily mirrored problem was 9,5%.
Computational results
![Page 45: Heuristics for the MTTPROADEF, February 2005 1/49 Heuristics for the Mirrored Traveling Tournament Problem Celso C. RIBEIRO Sebastián URRUTIA](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649d585503460f94a37a86/html5/thumbnails/45.jpg)
ROADEF, February 2005 45/49 Heuristics for the MTTP
Computational results
Instance
Best unmirrored
Best mirrored
gap (%)
Time to best (s)
circ12 420 456 8.6 8.5
circ14 682 714 4.7 1.1
circ16 976 1004 2.9 115.3
circ18 1420 1364 -3.9 284.2
circ20 1908 1882 -1.4 578.3
nl12 112298 120655 7.4 24.0
nl14 190056 208086 9.5 69.9
nl16 267194 285614 6.9 514.2
![Page 46: Heuristics for the MTTPROADEF, February 2005 1/49 Heuristics for the Mirrored Traveling Tournament Problem Celso C. RIBEIRO Sebastián URRUTIA](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649d585503460f94a37a86/html5/thumbnails/46.jpg)
ROADEF, February 2005 46/49 Heuristics for the MTTP
Computational results
Instance
Best unmirrored
Best mirrored
gap (%)
Time to best (s)
circ12 420 456 8.6 8.5
circ14 682 714 4.7 1.1
circ16 976 1004 2.9 115.3
circ18 1420 1364 -3.9 284.2
circ20 1908 1882 -1.4 578.3
nl12 112298 120655 7.4 24.0
nl14 190056 208086 9.5 69.9
nl16 267194 285614 6.9 514.2
![Page 47: Heuristics for the MTTPROADEF, February 2005 1/49 Heuristics for the Mirrored Traveling Tournament Problem Celso C. RIBEIRO Sebastián URRUTIA](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649d585503460f94a37a86/html5/thumbnails/47.jpg)
ROADEF, February 2005 47/49 Heuristics for the MTTP
Computational results
Instance
Best unmirrored
Best mirrored
gap (%)
Time to best (s)
circ12 420 456 8.6 8.5
circ14 682 714 4.7 1.1
circ16 976 1004 2.9 115.3
circ18 1364 1364 0.0 284.2
circ20 1882 1882 0.0 578.3
nl12 112298 120655 7.4 24.0
nl14 190056 208086 9.5 69.9
nl16 267194 285614 6.9 514.2
![Page 48: Heuristics for the MTTPROADEF, February 2005 1/49 Heuristics for the Mirrored Traveling Tournament Problem Celso C. RIBEIRO Sebastián URRUTIA](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649d585503460f94a37a86/html5/thumbnails/48.jpg)
ROADEF, February 2005 48/49 Heuristics for the MTTP
• New heuristic improved by 3.9% and 1.4% the best known solutions for the corresponding less constrained unmirrored instances circ18 and circ20.
• Computation times are smaller than computation time of other heuristics, e.g. for instance MLB14:– Anagnostopoulos et al. (2003):
approximately five days of computation time
– GRASP + ILS: 10 minutes
Computational results
![Page 49: Heuristics for the MTTPROADEF, February 2005 1/49 Heuristics for the Mirrored Traveling Tournament Problem Celso C. RIBEIRO Sebastián URRUTIA](https://reader035.vdocuments.site/reader035/viewer/2022062407/56649d585503460f94a37a86/html5/thumbnails/49.jpg)
ROADEF, February 2005 49/49 Heuristics for the MTTP
• Constructive heuristic is very fast and effective.
• GRASP+ILS heuristic found very good solutions:– Very fast (10 minutes).– Solutions found for some instances are even
better than those available for the corresponding less constrained not necessarily mirrored instances.
– Optimal solutions for a new class of instances for n = 4, 6, 8, 10,12, and 16 (Urrutia & Ribeiro, “Minimizing travels by maximizing breaks in round robin tournament schedules”, 2004)
• Effectiveness of the ejection chains.
Concluding Remarks