PVC routing algorithms

1
Heuristics for the Mirrored Traveling Tournament
Problem
Celso C. RIBEIRO Sebastián URRUTIA
2
Summary

• The Mirrored Traveling Tournament Problem
• Constructive heuristic
• Neighborhoods and ejection chains
• Extended GRASP ILS heuristic
• Computational results

3
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.

4
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.

5
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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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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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?...?EpE.
• 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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1-Factorizations

• 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

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1-Factorizations
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Constructive heuristic

• Three steps
• Schedule games using abstract teams polygon
  method defines the structure of the tournament
• 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)
• Select stadium for each game (home/away pattern)
  in the first phase (mirrored tournament)
• Build a feasible assignment of stadiums, starting
  from a random assignment of stadiums in the first
  round.
• Improve this assignment of stadiums, using a
  simple local search algorithm based on home-away
  swaps.

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Constructive heuristic
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Example polygon method for n6
1
5
2
1st round
3
4
12
Constructive heuristic
6
Example polygon method for n6
5
4
1
2nd round
2
3
13
Constructive heuristic
6
Example polygon method for n6
4
3
5
3rd round
1
2
14
Constructive heuristic
6
Example polygon method for n6
3
2
4
4th round
5
1
15
Constructive heuristic
6
Example polygon method for n6
2
1
3
5th round
4
5
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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
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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
  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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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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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
20
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.

21
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
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Neighborhood home-away swap (HAS)
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Neighborhood home-away swap (HAS)
1
2
5
4
3
6
24
Neighborhood team-swap (TS)
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Neighborhood team-swap (TS)
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Neighborhood partial round swap (PRS)
1
2
1
2
3
8
3
8
7
4
7
4
6
5
6
5
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Neighborhood partial round swap (PRS)
1
2
1
2
3
8
3
8
7
4
7
4
6
5
6
5
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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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Ejection chain game rotation (GR)
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Ejection chain game rotation (GR)
1
1
1
2
5
2
5
2
5
4
4
3
3
4
3
6
6
6
1
1
2
5
2
5
4
3
4
3
6
6
Enforce game (1,3) to be played in round 2
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Ejection chain game rotation (GR)
1
1
1
2
5
2
5
2
5
4
4
3
3
4
3
6
6
6
1
1
2
5
2
5
4
3
4
3
6
6
Enforce game (1,3) to be played in round 2
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Ejection chain game rotation (GR)
1
1
1
2
5
2
5
2
5
4
4
3
3
4
3
6
6
6
1
1
2
5
2
5
4
3
4
3
6
6
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Ejection chain game rotation (GR)
1
1
1
2
5
2
5
2
5
4
4
3
3
4
3
6
6
6
1
1
2
5
2
5
4
3
4
3
6
6
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Ejection chain game rotation (GR)
1
1
1
2
5
2
5
2
5
4
4
3
3
4
3
6
6
6
1
1
2
5
2
5
4
3
4
3
6
6
35
Ejection chain game rotation (GR)
1
1
1
2
5
2
5
2
5
4
4
3
3
4
3
6
6
6
1
1
2
5
2
5
4
3
4
3
6
6
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Ejection chain game rotation (GR)
1
1
1
2
5
2
5
2
5
4
4
3
3
4
3
6
6
6
1
1
2
5
2
5
4
3
4
3
6
6
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Ejection chain game rotation (GR)
1
1
1
2
5
2
5
2
5
4
4
3
3
4
3
6
6
6
1
1
2
5
2
5
4
3
4
3
6
6
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Ejection chain game rotation (GR)
1
1
1
2
5
2
5
2
5
4
4
3
3
4
3
6
6
6
1
1
2
5
2
5
4
3
4
3
6
6
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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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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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GRASP ILS heuristic

• while .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 ReinitializationCriterion
• end

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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
  n6 (sequential), n8 (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).

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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.

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Computational results

• 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.

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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
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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
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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
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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 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

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Concluding Remarks

• Constructive heuristic is very fast and
  effective.
• GRASPILS 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.