Heuristics for the MTTPROADEF, February /49 Heuristics for the Mirrored Traveling Tournament Problem Celso C. RIBEIRO Sebastián URRUTIA

Heuristics for the MTTPROADEF, February /49 Summary The Mirrored Traveling Tournament Problem Constructive heuristic Neighborhoods and ejection chains Extended GRASP + ILS heuristic Computational results

Heuristics for the MTTPROADEF, February /49 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.

Heuristics for the MTTPROADEF, February /49 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.

Heuristics for the MTTPROADEF, February /49 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.

Heuristics for the MTTPROADEF, February /49 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.

Heuristics for the MTTPROADEF, February /49 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 G 1 =(V,E 1 ),..., G p =(V,E p ), such that E 1 ...  E p =E. All factors in a 1-factorization of G are 1-factors. Oriented 1-factorization: assign orientations to the edges of a 1- factorization

Heuristics for the MTTPROADEF, February /49 Mirrored tournament: games in the second phase are determined by those in the first. –Each edge of K n represents a game. –Each 1-factor of K n represents a round. –Each ordered oriented 1-factorization of K n represents a feasible schedule for n teams. Example: K 6 1-Factorizations

Heuristics for the MTTPROADEF, February /49 1-Factorizations

Heuristics for the MTTPROADEF, February /49 Three steps: 1.Schedule games using abstract teams: polygon method defines the structure of the tournament 2.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

Heuristics for the MTTPROADEF, February /49 Constructive heuristic Example: “polygon method” for n=6 1 st round

Heuristics for the MTTPROADEF, February /49 Constructive heuristic Example: “polygon method” for n=6 2 nd round

Heuristics for the MTTPROADEF, February /49 Constructive heuristic Example: “polygon method” for n=6 3 rd round

Heuristics for the MTTPROADEF, February /49 Constructive heuristic Example: “polygon method” for n=6 4 th round

Heuristics for the MTTPROADEF, February /49 Constructive heuristic Example: “polygon method” for n=6 5 th round

Heuristics for the MTTPROADEF, February /49 Constructive heuristic Abstract teams (n=6) RoundABCDEF 1/6FEDCBA 2/7DCBAFE 3/8BAEFCD 4/9EDFBAC 5/10CFAEDB

Heuristics for the MTTPROADEF, February /49 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

Heuristics for the MTTPROADEF, February /49 Constructive heuristic ABCDEF A B C D E F443430

Heuristics for the MTTPROADEF, February /49 Constructive heuristic Real teams (n=6) RoundFLUSA N FLAGREPALPAY 1/6PAYPALGR E FLASA N FLU 2/7GR E FLASA N FLUPAYPAL 3/8SA N FLUPALPAYFLAGRE 4/9PALGREPAYSA N FLUFLA 5/10FLAPAYFLUPALGR E SA N

Heuristics for the MTTPROADEF, February /49 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.

Heuristics for the MTTPROADEF, February /49 Constructive heuristic Real teams (n=6) RoundFLUSANFLAGREPALPAY L A U A U L N L E Y U FLA A L N

Heuristics for the MTTPROADEF, February /49 Neighborhood home-away swap (HAS)

Heuristics for the MTTPROADEF, February /49 Neighborhood home-away swap (HAS)

Heuristics for the MTTPROADEF, February /49 Neighborhood team-swap (TS)

Heuristics for the MTTPROADEF, February /49 Neighborhood team-swap (TS)

Heuristics for the MTTPROADEF, February /49 Neighborhood partial round swap (PRS)

Heuristics for the MTTPROADEF, February /49 Neighborhood partial round swap (PRS)

Heuristics for the MTTPROADEF, February /49 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.

Heuristics for the MTTPROADEF, February /49 Ejection chain: game rotation (GR)

Heuristics for the MTTPROADEF, February /49 Ejection chain: game rotation (GR) Enforce game (1,3) to be played in round 2

Heuristics for the MTTPROADEF, February /49 Ejection chain: game rotation (GR) Enforce game (1,3) to be played in round 2

Heuristics for the MTTPROADEF, February /49 Ejection chain: game rotation (GR)

Heuristics for the MTTPROADEF, February /49 Ejection chain: game rotation (GR)

Heuristics for the MTTPROADEF, February /49 Ejection chain: game rotation (GR)

Heuristics for the MTTPROADEF, February /49 Ejection chain: game rotation (GR)

Heuristics for the MTTPROADEF, February /49 Ejection chain: game rotation (GR)

Heuristics for the MTTPROADEF, February /49 Ejection chain: game rotation (GR)

Heuristics for the MTTPROADEF, February /49 Ejection chain: game rotation (GR)

Heuristics for the MTTPROADEF, February /49 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

Heuristics for the MTTPROADEF, February /49 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.

Heuristics for the MTTPROADEF, February /49 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

Heuristics for the MTTPROADEF, February /49 Computational results Circular instances with n = 12,..., 20 teams. MLB instances with n = 12,..., 16 teams. –All available from –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).

Heuristics for the MTTPROADEF, February /49 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.

Heuristics for the MTTPROADEF, February /49 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

Heuristics for the MTTPROADEF, February /49 Computational results Instan ce Best unmirrored Best mirrored gap (%) Time to best (s) circ circ circ circ circ nl nl nl

Heuristics for the MTTPROADEF, February /49 Computational results Instan ce Best unmirrored Best mirrored gap (%) Time to best (s) circ circ circ circ circ nl nl nl

Heuristics for the MTTPROADEF, February /49 Computational results Instan ce Best unmirrored Best mirrored gap (%) Time to best (s) circ circ circ circ circ nl nl nl

Heuristics for the MTTPROADEF, February /49 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

Heuristics for the MTTPROADEF, February /49 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