Heuristics for the Mirrored Traveling Tournament Problem

ROADEF, February 2005 1/49 Heuristics for the MTTP

Heuristics for the Mirrored Traveling Tournament Problem

Celso C. RIBEIROSebastián URRUTIA


Summary

• The Mirrored Traveling Tournament Problem

• Constructive heuristic• Neighborhoods and ejection chains• Extended GRASP + ILS heuristic• Computational results


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.


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– 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.


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.


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


• 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


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


• 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



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



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


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



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


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


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.


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


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


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


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


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


Neighborhood partial round swap (PRS)

7 4

3

1

8

6

2

5

7 4

3

1

8

6

2

5


Neighborhood 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 1-factor of the 1-factorization associated with the current 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 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



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



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



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


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


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.


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


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).


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.


• 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%.




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



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



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


• 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



• 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

Documents

Heuristics for the Mirrored Traveling Tournament Problem