PVC routing algorithms

1
Heuristics for the Traveling Tournament Problem
Scheduling the Brazilian Soccer Championship
Celso C. RIBEIRO Sebastián URRUTIA
2
Summary

• Formulation
• Constructive heuristic
• Neighborhoods
• Extended GRASP ILS heuristic
• Computational results
• Concluding remarks

3
Formulation

• 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.

4
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.

5
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).

6
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.

7
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.

8
Formulation

• Some references
• Easton, Nemhauser, Trick, Scheduling in
  sports, Handbook of Scheduling (2004)
• 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)

9
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?...?EpE.
• All factors in a 1-factorization of G are
  1-factors.

10
Formulation
1
2
5
4
3
Example 1-factorization of K6
6
11
Formulation
1
1
2
5
4
3
Example 1-factorization of K6
6
12
Formulation
2
1
2
5
4
3
Example 1-factorization of K6
6
13
Formulation
3
1
2
5
4
3
Example 1-factorization of K6
6
14
Formulation
4
1
2
5
4
3
Example 1-factorization of K6
6
15
Formulation
5
1
2
5
4
3
Example 1-factorization of K6
6
16
Formulation

• 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)

17
Summary

• Formulation
• Constructive heuristic
• Neighborhoods
• Extended GRASP ILS heuristic
• Computational results
• Concluding remarks

18
Constructive heuristic

• Three steps
• Schedule games using abstract teams (structure of
  the draw).
• Assign real teams to abstract teams.
• 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.

19
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.

20
Constructive heuristic
6
Example polygon method for n6
1
5
2
1st round
3
4
21
Constructive heuristic
6
Example polygon method for n6
5
4
1
2nd round
2
3
22
Constructive heuristic
6
Example polygon method for n6
4
3
5
3rd round
1
2
23
Constructive heuristic
6
Example polygon method for n6
3
2
4
4th round
5
1
24
Constructive heuristic
6
Example polygon method for n6
2
1
3
5th round
4
5
25
Constructive heuristic
Abstract teams (n6) Abstract teams (n6) Abstract teams (n6) Abstract teams (n6) Abstract teams (n6) Abstract teams (n6)
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
26
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.

27
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
28
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.

29
Constructive heuristic
n 16 note the large number of times in which
two teams are faced consecutively, which is
explored by step 2 of the constructive heuristic.
30
Constructive heuristic

• Teams (n6)
• 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

31
Constructive heuristic
Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6)
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
32
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.

33
Constructive heuristic
Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6)
Round FLU SAN FLA GRE PAL PAY
1/6 PAY _at_PAL GRE _at_FLA SAN _at_FLU
2/7 GRE _at_FLA SAN _at_FLU PAY _at_PAL
3/8 _at_SAN FLU _at_PAL PAY FLA _at_GRE
4/9 PAL _at_GRE _at_PAY SAN _at_FLU FLA
5/10 _at_FLA PAY FLU _at_PAL GRE _at_SAN
34
Summary

• Formulation
• Constructive heuristic
• Neighborhoods
• Extended GRASP ILS heuristic
• Computational results
• Concluding remarks

35
Neighborhoods

• Neighborhood home-away swap (HAS) select a
  game and exchange the stadium where it takes
  place.

Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6)
Round FLU SAN FLA GRE PAL PAY
1/6 PAY _at_PAL GRE _at_FLA SAN _at_FLU
2/7 GRE _at_FLA SAN _at_FLU PAY _at_PAL
3/8 _at_SAN FLU _at_PAL PAY FLA _at_GRE
4/9 PAL _at_GRE _at_PAY SAN _at_FLU FLA
5/10 _at_FLA PAY FLU _at_PAL GRE _at_SAN
36
Neighborhoods

• Neighborhood home-away swap (HAS) select a
  game and exchange the stadium where it takes
  place.

Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6)
Round FLU SAN FLA GRE PAL PAY
1/6 PAY PAL GRE _at_FLA _at_SAN _at_FLU
2/7 GRE _at_FLA SAN _at_FLU PAY _at_PAL
3/8 _at_SAN FLU _at_PAL PAY FLA _at_GRE
4/9 PAL _at_GRE _at_PAY SAN _at_FLU FLA
5/10 _at_FLA PAY FLU _at_PAL GRE _at_SAN
37
Neighborhoods

• Neighborhood team swap (TS) select two teams
  and swap their games, also swap the home-away
  assignment of their own game.

Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6)
Round FLU SAN FLA GRE PAL PAY
1/6 PAY _at_PAL GRE _at_FLA SAN _at_FLU
2/7 GRE _at_FLA SAN _at_FLU PAY _at_PAL
3/8 _at_SAN FLU _at_PAL PAY FLA _at_GRE
4/9 PAL _at_GRE _at_PAY SAN _at_FLU FLA
5/10 _at_FLA PAY FLU _at_PAL GRE _at_SAN
38
Neighborhoods

• Neighborhood team swap (TS) select two teams
  and swap their games also swap the home-away
  assignment of their own game.

Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6)
Round FLU SAN FLA GRE PAL PAY
1/6 PAY _at_PAL GRE _at_FLA SAN _at_FLU
2/7 GRE _at_FLA SAN _at_FLU PAY _at_PAL
3/8 _at_SAN FLU _at_PAL PAY FLA _at_GRE
4/9 PAL _at_GRE _at_PAY SAN _at_FLU FLA
5/10 _at_FLA PAY FLU _at_PAL GRE _at_SAN
39
Neighborhoods

• Neighborhood team swap (TS) select two teams
  and swap their games, also swap the home-away
  assignment of their own game.

Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6) Real teams (n6)
Round FLU SAN FLA GRE PAL PAY
1/6 PAY _at_PAL SAN _at_FLA GRE _at_FLU
2/7 GRE _at_FLA PAY _at_FLU SAN _at_PAL
3/8 _at_SAN FLU PAL PAY _at_FLA _at_GRE
4/9 PAL _at_GRE _at_FLU SAN _at_PAY FLA
5/10 _at_FLA PAY GRE _at_PAL FLU _at_SAN
40
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 n?8, not always possible).

41
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 n?8, not always possible).

42
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.

43
Neighborhoods
2
1
2
5
4
3
6
Enforce game 1vs. 3 at round (factor) 2.
44
Neighborhoods
2
1
2
5
4
3
6
Teams 1 and 3 are now playing twice in this round.
45
Neighborhoods
2
1
2
5
4
3
6
Eliminate the other games played by teams 1 and 3
in this round.
46
Neighborhoods
2
1
2
5
4
3
6
Enforce the former oponents of teams 1 and 3 to
play each other in this round new game 2 vs. 4
in this round.
47
Neighborhoods
4
1
2
5
4
3
6
Consider the factor where game 2 vs. 4 was
scheduled.
48
Neighborhoods
4
1
2
5
4
3
6
Enforce game 1 vs. 4 (eliminated from round 2) to
be played in this round.
49
Neighborhoods
4
1
2
5
4
3
6
Eliminate games 2 vs. 4 (enforced in round 2) and
1 vs. 5 (since team 1 cannot play twice).
50
Neighborhoods
4
1
2
5
4
3
6
Enforce game 2 vs. 5 to be played in this round.
51
Neighborhoods
1
1
2
5
4
3
6
Consider the factor where game 2 vs. 5 was
scheduled.
52
Neighborhoods
1
1
2
5
4
3
6
Enforce game 1 vs. 5 (eliminated from round 4)
to be played in this round.
53
Neighborhoods
1
1
2
5
4
3
6
Eliminate games 2 vs. 5 (enforced in round 4) and
1 vs. 6 (since team 1 cannot play twice).
54
Neighborhoods
1
1
2
5
4
3
6
Enforce game 2 vs. 6 to be played in this round.
55
Neighborhoods
5
1
2
5
4
3
6
Consider the factor where game 2 vs. 6 was
scheduled.
56
Neighborhoods
5
1
2
5
4
3
6
Enforce game 1 vs. 6 (eliminated from round 1) to
be played in this round.
57
Neighborhoods
5
1
2
5
4
3
6
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).
58
Neighborhoods
5
1
2
5
4
3
6
Finally, enforce game 2 vs. 3 (eliminated from
round 2 at the beginning of the ejection chain )
to be played in this round.
59
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.

60
Neighborhoods

• PRS GR for n4 PRS ? GR for n?6.
• 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.

61
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.

62
Summary

• Formulation
• Constructive heuristic
• Neighborhoods
• Extended GRASP ILS heuristic
• Computational results
• Concluding remarks

63
Extended GRASP ILS heuristic

• while .not.StoppingCriterion
• S ? 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 ReinitializationCriterion
• end

64
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.

65
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.

66
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 regarding
  stadium assignment (more than three consecutive
  home or away games).
• Perform a fast (very few infeasibilities in the
  solution obtained after the GR move) tabu search
  heuristic using the home-away swap neighborhood.
• Tabu search stops when all infeasibilities have
  been eliminated.

67
Extended GRASP ILS heuristic

• while .not.StoppingCriterion
• S ? 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 ReinitializationCriterion
• end

68
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.

69
Extended GRASP ILS heuristic

• Reinitialization criterion
• If 50 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.

70
Summary

• Formulation
• Constructive heuristic
• Neighborhoods
• Extended GRASP ILS heuristic
• Computational results
• Concluding remarks

71
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 Tricks web page.
• 2003 edition of the Brazilian national
  soccer championship with 24 teams.

72
Computational results

• All numerical results on a Pentium IV 2.0 MHz
  machine
• Lower bounds best known approximate solutions
  for the corresponding less constrained unmirrored
  instances.
• Largest problems solved to optimality using an
  integer programming formulation n 6 teams
• Gaps 0.6 for n8 3.6 for n10 (MLB)

73
Computational results

• Constructive heuristic
• Very fast
• Instance nl16 1000 runs in approximately 1
  second
• Average gap is 17.1, but the constructive
  heuristic is several orders of magnitude faster
  than the best heuristics.
• Better solutions than those found after several
  days of computations by the ant colony with
  backtracking and local search heuristic of
  Crauwells Van Oudheusden (2003)

74
Computational results

• GRASP ILS heuristic time limit is 10 minutes
  only
• Very robust largest gap with respect to the best
  known solution for the less constrained
  unmirrored problem was only 11.2

(before this work, times were measured in days!)
75
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
76
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
77
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
78
Computational results

• 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 compare favorably with other
  heuristics, e.g. for instance nl14
• Anagnostopoulos et al. (2003) approximately five
  days of computation time
• GRASP ILS 10 minutes (only 0.2)

79
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 506,433 kms (52 reduction)

• Approximate corresponding potential savings
  in airfaresUS 1,700,000

80
Summary

• Formulation
• Constructive heuristic
• Neighborhoods
• Extended GRASP ILS heuristic
• Computational results
• Concluding remarks

81
Concluding remarks

• Constructive heuristic is very fast and
  effective.
• Extended GRASP ILS heuristic found very good
  solutions to benchmark instances
• Very fast (10 minutes)
• Solutions found for some instances are even
  better than those available for the corresponding
  less constrained unmirrored instances.
• Effectiveness of ejection chain neighborhood.
• Significant savings in airfares costs and
  traveled distance in real instance.

82
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
  athttp//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!

83
Computational results

• Methodology used to evaluate and compare
  different algorithms for the same problem
• Probability distribution of time-to-target-solutio
  n-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)

84
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.

85
Computational results
Instance br24 Target 586,000 kms
86
Computational results
Instance br24 Time 10 minutes
87
Computational results
Instance br24 Time 12 hours
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
  national tournaments.