More than the CocaCola formula Scheduling the Argentinas

More than the Coca-Cola formula: Scheduling the Argentina’s Football Superliga Guillermo Durán Mario Guajardo EURO 2019, Dublin Denis Sauré Gonzalo Zamorano

OUR “MATHSPORT” RESEARCH GROUP • 15 years scheduling football leagues 2004 2015 Jun-2018 Nov-2018 | | Chile South American Qualifiers FIFA World Cup Argentina Ecuador • Sports scheduling in professional, amateur and youth leagues of football, volleyball, basketball • Also: referee assignment, data analysis, score prediction. . .

Relatively traditional application (implemented in practice) of sports scheduling, with a couple of perhaps interesting particular features. Outline • Background • Modelling and solution approach • Results

BACKGROUND • 2017: Major re-structuring of Argentinean football leagues. • 2018 -19: second season that SAF is in charge of the First Division tournament. • Our collaboration with SAF started in 2017 with the scheduling of the Youth Divisions for the 2018 season and the assignment of specific days and times for the games of the second part of the First Division Superliga. • 2018: kick-off project to schedule the First Division Superliga, which currently consists of 26 teams. - Scheduling matches to rounds - Scheduling matches to specific days and times | 10 - August -2018 | 25 rounds 09 - April - 2019

26 TEAMS Argentina is the 8 th largest country in the world

BACKGROUND Single round robin tournament, set of teams varying from season to season (relegation, promotion, managerial decisions), and poor scheduling implied: • Potentially large differences in the distances travelled by the teams over the course of the tournament. • Some matches took place at the same venue consecutively in several tournaments. For example: HOME AWAY River Plate vs Banfield 6 consecutive times Banfield vs Colón 5 consecutive times Boca Juniors vs Newell’s Old Boys Lanús vs Boca Juniors 5 consecutive times . . . 146 pairs of teams with home-away condition repeated 2 to 6 consecutive times Could we find a schedule that minimizes home-away repetitions while balancing travelling distances (and satisfying many other conditions)?

LITERATURE: SPORTS SCHEDULING https: //mat. tepper. cmu. edu/TOURN/

PRACTICE LITERATURE: SPORTS SCHEDULING FOOTBALL • Holland (Schreuder 1992) • Germany and Austria (Bartsch et al. 2006) • Chile (Durán et al. 2007) • Denmark (Rasmussen 2008) • Belgium (Goossens and Spieksma 2009) • Norway (Flatberg et al. 2009) • Honduras (Fiallos et al. 2010) • Brazil (Ribeiro and Urrutia 2011) • Ecuador (Recalde et al. 2013) • FIFA World Cup Qualifiers in South America (Durán et al. 2017)

Outline • Background • Modelling and solution approach • Results

INTEGER PROGRAMMING MODEL Decision variables 1 if team i plays at home against team j in round k (i≠j) 0 otherwise i ∈ I (set of teams), j ∈ I (i ≠ j), k ∈ K (set of rounds) 1 if team i plays away in rounds k and k+1 i ∈ I, k ∈ K (k < |K|) 0 otherwise 1 if team i plays at home in rounds k and k+1 i ∈ I, k ∈ K (k < |K|) 0 otherwise

INTEGER PROGRAMMING MODEL Objective function ri, j : Penalty for repeating team i home game against team j (equal to 6, 5, 4, 3/2, 2/2 for matches repeated 6, 5, 4, 3, 2 consecutive times in previous tournaments; 0 otherwise) s. t. Many constraints …

INTEGER PROGRAMMING MODEL • Single round robin constraint. Every team plays against every other team exactly one game. • Compactness. Every team plays exactly one game per round. • Total home games. Pre-defined from previous tournament.

INTEGER PROGRAMMING MODEL • Break constraints. No breaks in the start and the end. • Logical constraints. Relationships between variables.

MORE CONSTRAINTS • More break constraints (upper bounds) • Game constraints (e. g. 10 “derby” games, TV wishes) • Place constraints (e. g. stadium not available) • Top team constraints • Complementary constraints 10 pairs of teams with opposite Home-Away status, e. g. H-A-A… A-H-H…

GEOGRAPHICAL CONSTRAINTS • Teams grouped into four clusters Cluster 1 • For a team of cluster c: Cluster 2 Cluster 3 Cluster 4 - lower and upper bounds on the number of games away against teams from cluster d (d ≠ c) - lower and upper bounds on the number of games away against teams from cluster c - at most one game in a far cluster in an Away-Away break - consideration of international games

SOLUTION APPROACH • ~17, 500 variables; 5, 500 constraints. • Mathematical programming solvers CPLEX and Gurobi (first tests, not rigorous, revealed Gurobi performed better). • Run about 55 “instances”. • Some take too long (hours, days) and no feasible solution. • Common heuristic approach: decomposition into subproblems. (Rasmussen & Trick 2008) H-A-A… H-A-H…

SOLUTION APPROACH • It did not work as expected. • “Geographical patterns” (i, k) ∊ Tc : team i plays away in cluster c in round k Then for all cluster c we formulate: H-c 1 -H-c 2 -c 1 -H-c 3 -H-c 4 -H… • Provided solutions quickly (seconds, minutes) and of good quality (even optimal, which we know from the LP relaxation without fixed patterns).

Outline • Background • Modelling and solution approach • Results

RESULTS Our schedule announced in July 2018. The tournament finished in April 2019.

RESULTS • 114 out of the 146 repeated matches now invert the homeaway status. • 100% of matches repeated 6, 5, or 4 consecutive times in previous tournaments now do not repeat. • 75% of matches repeated 2 or 3 consecutive times in previous tournaments now do not repeat. • Balanced travels: 75% less standard deviation in km travelled

CONCLUDING REMARKS • Very nice application of Mathsport in practice with a huge impact (SAF is one of the most important football league in the world) • 26 teams, 10 complementary pairs, 4 geographical clusters • Geographical patterns were crucial to find good solutions quickly • We are working now on the scheduling of 2019 -2020 season.

CONCLUDING REMARKS • Some other projects with SAF: • Scheduling youth leagues • Assignment of days and times for the games of SAF using a mathematical model. • A web page to predict results in different competitions: SAF, World Cup, America Cup (301060. exactas. uba. ar) • Homework: why 301060?

More than the Coca-Cola formula: Scheduling the Argentina’s Football Superliga Guillermo Durán Mario Guajardo EURO 2019, Dublin Denis Sauré Gonzalo Zamorano