# American Institute of Mathematical Sciences

October  2019, 15(4): 1565-1577. doi: 10.3934/jimo.2018110

## Fair-fixture: Minimizing carry-over effects in football leagues

 1 Department of Industrial Engineering, Özyeğin University, Istanbul, 34794, Turkey 2 BSH Home Appliances Corporation, Istanbul, 34771, Turkey

* Corresponding author: Dilek Günneç

Received  December 2016 Revised  January 2018 Published  July 2018

We study a sports scheduling problem with the objective of minimizing carry-over effects in round robin tournaments.In the first part, focusing on tournaments that allow minimum number of breaks (at most one) for each team, we formulate an integer programming model and provide an efficient heuristic algorithm to solve this computationally expensive problem. We apply the algorithm to the current Turkish Professional Football League and present an alternative scheduling template. In the second part, we discuss how the carry-over effects can be further decreased if the number of breaks is allowed to be of slightly larger value and numerically represent this trade-off.

Citation: Dilek Günneç, Ezgi Demir. Fair-fixture: Minimizing carry-over effects in football leagues. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1565-1577. doi: 10.3934/jimo.2018110
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##### References:
Pattern set for a league of 8 teams
 Week 1 2 3 4 5 6 7 Pattern 1 A H A H A H A 2 A H H A H A H 3 A H A A H A H 4 A H A H H A H 5 A H A H A A H 6 H A H A H A H 7 H A A H A H A 8 H A H H A H A 9 H A H A A H A 10 H A H A H H A
 Week 1 2 3 4 5 6 7 Pattern 1 A H A H A H A 2 A H H A H A H 3 A H A A H A H 4 A H A H H A H 5 A H A H A A H 6 H A H A H A H 7 H A A H A H A 8 H A H H A H A 9 H A H A A H A 10 H A H A H H A
Coe values of TPFL between 2012 and 2016
 Season Coe 2012-13 3666 2013-14 3818 2014-15 3707 2015-16 3820 Average 3752.75
 Season Coe 2012-13 3666 2013-14 3818 2014-15 3707 2015-16 3820 Average 3752.75
Results of our heuristic
 Number of teams Coe Value (1-Break) Time (min) 8 104 0.20 10 192 0.44 12 318 2.73 14 446 2.73 16 626 6.30 18 944 23.90
 Number of teams Coe Value (1-Break) Time (min) 8 104 0.20 10 192 0.44 12 318 2.73 14 446 2.73 16 626 6.30 18 944 23.90
Coe values and number of breaks for the heuristic with the BRK model. ($n = 18$)
 Break (Each) Random Canonical TPFL Coe Break Coe Break Coe Break $x=1$ 944 16 3876 16 3707 16 $1\leq x \leq2$ 650 31 698 29 580 32 $2\leq x \leq3$ 544 45 502 42 466 45 $2\leq x \leq4$ 544 45 502 42 466 45
 Break (Each) Random Canonical TPFL Coe Break Coe Break Coe Break $x=1$ 944 16 3876 16 3707 16 $1\leq x \leq2$ 650 31 698 29 580 32 $2\leq x \leq3$ 544 45 502 42 466 45 $2\leq x \leq4$ 544 45 502 42 466 45
Coe values and number of breaks when break distribution among teams is not considered
 Break (Each) 10 Teams 12 Teams 14 Teams 16 Teams 18 Teams Coe Break Coe Break Coe Break Coe Break Coe Break $x \leq 1$ 192 8 318 10 446 12 626 14 944 16 $x \leq 2$ 144 12 212 18 344 26 472 30 646 30 $x \leq 3$ 144 12 212 16 302 24 396 34 556 40
 Break (Each) 10 Teams 12 Teams 14 Teams 16 Teams 18 Teams Coe Break Coe Break Coe Break Coe Break Coe Break $x \leq 1$ 192 8 318 10 446 12 626 14 944 16 $x \leq 2$ 144 12 212 18 344 26 472 30 646 30 $x \leq 3$ 144 12 212 16 302 24 396 34 556 40
Comparison of our heuristic with three real league schedules for season 2015-16
 Num. of teams Coe Breaks (Each) Break Coe (H) Break (H) Decrease (%) Czech Rep. 16 742 $x=1$ 16 626 14 15.63 Germany 18 1056 $0\leq x \leq1$ 16 944 16 10.60 France 20 719 $0\leq x \leq4$ 45 530 56 26.28
 Num. of teams Coe Breaks (Each) Break Coe (H) Break (H) Decrease (%) Czech Rep. 16 742 $x=1$ 16 626 14 15.63 Germany 18 1056 $0\leq x \leq1$ 16 944 16 10.60 France 20 719 $0\leq x \leq4$ 45 530 56 26.28
Template for an 18-team league
 Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Week 8 Week 9 1-11 18-1 1-8 12-1 1-14 2-1 1-17 4-1 1-5 6-2 2-7 16-2 2-9 3-2 8-3 15-2 2-11 18-2 3-14 12-3 3-13 10-3 7-4 4-6 3-4 5-3 3-17 17-4 4-5 9-4 4-16 6-5 5-9 16-5 7-6 15-4 5-10 13-6 17-5 5-7 9-8 17-7 6-8 8-18 6-9 7-13 8-17 6-12 17-6 15-10 10-11 9-7 17-9 16-7 16-8 10-9 7-10 8-15 11-13 12-15 18-10 10-16 11-8 9-12 11-15 11-18 14-11 18-12 13-16 11-12 12-14 14-10 15-18 14-16 15-14 13-18 16-17 14-18 14-13 13-15 12-13 Week 10 Week 11 Week 12 Week 13 Week 14 Week 15 Week 16 Week 17 9-1 1-7 16-1 1-6 3-1 15-1 1-10 13-1 2-13 14-2 2-12 10-2 2-8 2-5 17-2 2-4 7-3 3-6 9-3 3-16 4-13 11-3 3-15 18-3 4-18 11-4 4-14 12-4 5-12 10-4 4-8 8-5 5-11 15-5 5-18 14-5 6-14 18-6 5-13 10-6 6-16 12-8 6-15 18-7 7-15 8-7 6-11 14-7 8-14 16-9 7-11 13-8 9-18 13-9 7-12 11-9 10-12 13-10 8-10 15-9 17-10 12-16 9-14 12-17 17-15 18-17 17-13 11-17 16-11 14-17 16-18 15-16
 Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Week 8 Week 9 1-11 18-1 1-8 12-1 1-14 2-1 1-17 4-1 1-5 6-2 2-7 16-2 2-9 3-2 8-3 15-2 2-11 18-2 3-14 12-3 3-13 10-3 7-4 4-6 3-4 5-3 3-17 17-4 4-5 9-4 4-16 6-5 5-9 16-5 7-6 15-4 5-10 13-6 17-5 5-7 9-8 17-7 6-8 8-18 6-9 7-13 8-17 6-12 17-6 15-10 10-11 9-7 17-9 16-7 16-8 10-9 7-10 8-15 11-13 12-15 18-10 10-16 11-8 9-12 11-15 11-18 14-11 18-12 13-16 11-12 12-14 14-10 15-18 14-16 15-14 13-18 16-17 14-18 14-13 13-15 12-13 Week 10 Week 11 Week 12 Week 13 Week 14 Week 15 Week 16 Week 17 9-1 1-7 16-1 1-6 3-1 15-1 1-10 13-1 2-13 14-2 2-12 10-2 2-8 2-5 17-2 2-4 7-3 3-6 9-3 3-16 4-13 11-3 3-15 18-3 4-18 11-4 4-14 12-4 5-12 10-4 4-8 8-5 5-11 15-5 5-18 14-5 6-14 18-6 5-13 10-6 6-16 12-8 6-15 18-7 7-15 8-7 6-11 14-7 8-14 16-9 7-11 13-8 9-18 13-9 7-12 11-9 10-12 13-10 8-10 15-9 17-10 12-16 9-14 12-17 17-15 18-17 17-13 11-17 16-11 14-17 16-18 15-16
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