Balancing Conflicting Objectives for UK Football Graham Kendall
The University of Nottingham Graham Kendall, 20th Jan 2009, UNMC Sports Scheduling
Contents Problem Definition Previous results (JORS) CPLEX vs DFS Multi-objective (clashes vs distance) Suggesting maximum distance Future Work Charles Darwin 1809-1882 "A man who dares to waste an hour of life has not discovered the value of life"
Contents Problem Definition Previous results (JORS) CPLEX vs DFS Multi-objective (clashes vs distance) Suggesting maximum distance Future Work Paul Gascoigne gets booked, ruling him out of a potential world cup final
Problem Description Football supporters and motoring organisations have called for changes in the scheduling of fixtures [over the Christmas holiday period] to prevent supporters from having to drive home from matches well into the early hours of the morning. The Times (Sport, Football), 30 th December 2005, pp 68 George Best won 37 caps for Northern Ireland, including this match against England in May 1971
League Structure Four Divisions Barclays Premiership (20 teams) Coca-Cola Championship (24 teams) Coca-Cola League One (24 teams) Coca-Cola League Two (24 teams) Number of Matches 380 (Premiership), 552 for other divisions. Total 2036 Matches to be Scheduled Each league plays a double round robin tournament. Therefore, there are no matches played between different divisions, but.. Michael Owen returns to the line up at Newcastle after injury (29 th April 2006)
Pairing System Teams are paired which means they cannot normally play at home on the same day Typically this is due to teams which are geographically close (e.g. Manchester United/Manchester City, Sunderland/Newcastle, Liverpool/Everton/Tranmere) Note that the pairing system operates across division boundaries We are allowed a given number of pairing violations Alan Shearer breaks Milburn's record, scoring his 201st Newcastle goal (against Portsmouth)
Pairing System Teams are paired which means they cannot normally play at home on the same day Typically this is due to teams which are geographically close (e.g. Manchester United/Manchester City, Sunderland/Newcastle, Liverpool/Everton/Tranmere) Note that the pairing system operates across division boundaries We are allowed a given number of pairing violations Season Boxing Day New Years Day 2008-2009 10 10 2007-2008 14 10 2006-2007 14 8 2005-2006 12 14 2004-2005 10 10 2003-2004 8 14 2002-2003 10 8 Number of actual clashes from previous seasons
Geographic Constraints There are other geographical constraints which must be adhered to The number of London based clubs that can play at home on the same day = 6 The number of London based Premier clubs that can play at home on the same day = 3 The number of Manchester based clubs that can play at home on a the same day = 4 Chelsea win the Premiership for the second consecutive season (April 2006)
Date Specific Constraints On New Years Day and Boxing Day each team is required to play If a team plays at home on Boxing Day they must play away on New Years Day The same teams cannot play each other on New Years Day and Boxing Day Paired teams cannot play each other In fact this is violated in the published fixtures, but we treat it as a hard constraint The aim is to minimise the distance travelled Wayne Rooney makes it 4-0 - giving Manchester United the biggest League Cup Final victory in the competition's history (2006)
Date Specific Constraints On New Years Day and Boxing Day each team is required to play If a team plays at home on Boxing it must play away on New Years Day The same teams cannot play each other on New Years Day and Boxing Day Paired teams cannot play each other In fact this is violated in the published fixtures, but we treat it as a hard constraint The aim is to minimise the distance travelled Analysis of previous seasons fixtures and meetings with the Football League have shown that this is the case Not like TSP or TTP as we cannot minimise the overall distance travelled (throughout the entire season)
2002-2003 2003-2004 2004-2005 2005-2006 Maps of Premier League Clubs
Division 1: 2004-2005
405 Miles (Plymouth to Sunderland) 153 Miles (Plymouth to Cardiff) Division 1: 2004-2005
Contents Problem Definition Previous results (JORS) CPLEX vs DFS Multi-objective (clashes vs distance) Suggesting maximum distance Future Work Gordon Banks saves a certain headed goal from Pele
Solution Methodology (JORS) Two-stage algorithm Depth-first search Produces two fixtures for each division, one for Boxing Day and one for New Years Day (i.e. eight sets of fixtures) Local Search Operates across divisions in order to resolve clashes Bobby Charlton: Then and Now
Results (JORS) Published Fixtures Generated Fixtures Season 2005-2006 Boxing Day Total Distance 4295 3049 Clashes 12 10 New Years Day Total Distance 6336 3868 Clashes 14 10 Season 2004-2005 Boxing Day Total Distance 4563 3725 Clashes 10 10 New Years Day Total Distance 5122 3059 Clashes 10 10 Season 2003-2004 Boxing Day Total Distance 3872 2859 Clashes 8 8 New Years Day Total Distance 4296 3500 Clashes 14 14 Season 2002-2003 Boxing Day Total Distance 3827 2839 Clashes 10 10 New Years Day Total Distance 3964 3201 Clashes 8 8 Nottingham Forest Football Ground
Season Results (JORS) Total Distance for Published Fixtures Total Distance for Generated Fixtures Distance Saving 2005-2006 10631 6917 3714 (34.94%) 2004-2005 9685 6784 2901 (29.95%) 2003-2004 8168 6359 1809 (22.15%) 2002-2003 7791 6020 1751 (22.47%) Number of Pair Clashes for Published Fixtures Number of Pair Clashes for Generated Fixtures 26 20 20 20 22 22 18 18 Kendall G. Scheduling English Football Fixtures Over Holiday Periods. (2008) Journal of the Operational Research Society, 59(6), pages 743-755 (http://dx.doi.org/10.1057/palgrave.jors.2602382) Bobby Moore in 1966
Contents Problem Definition Previous results (JORS) CPLEX vs DFS Multi-objective (clashes vs distance) Suggesting maximum distance Future Work Are you blind ref?
Revised Methodology Two-stage algorithm CPLEX Produces Boxing Day fixtures and New Years Day fixtures at the same time. Simulated Annealing Operates across divisions in order to resolve clashes Stamford Bridge: Chelsea s home ground
Revised Methodology Search Phase Season Division Stage 1a Stage 1b Local Search Barclays Premiership 5 <1 <5 2005-06 Coca-Cola Championship 823 <1 <5 Coca-Cola League One 149 <1 <5 Coca-Cola League 2 148 <1 <5 Barclays Premiership 4 <1 <5 2004-05 Coca-Cola Championship 993 <1 <5 Coca-Cola Lauge One 222 <1 <5 Coca-Cola League 2 401 <1 <5 Barclays Premiership 8 <1 <5 2003-04 Coca-Cola Championship 891 <1 <5 Coca-Cola Lauge One 157 <1 <5 Coca-Cola League 2 218 <1 <5 Barclays Premiership 3 <1 <5 2002-03 Coca-Cola Championship 1023 <1 <5 Coca-Cola Lauge One 874 <1 <5 Coca-Cola League 2 26 <1 <5 Times in minutes Large differences, but generally expensive
Revised Methodology Two-stage algorithm CPLEX Simulated Annealing Barclays Premiers hip Coca- Cola Champi onship Coca- Cola League One Coca- Cola League 2 2005-06 677 / 4 948 / 5 1156 / 4 1369 / 4 2004-05 888 / 6 1104 / 5 1394 / 4 1357 / 5 2003-04 917 / 4 1331 / 4 1277 / 4 1224 / 5 2002-03 737 / 3 1301 / 3 1297 / 4 1217 / 6 CPLEX Produces optimal solutions for the relaxed problem for each division. We create fixtures for boxing day and news years day at the same time (cf DFS, where we ran the DFS for each day). Significant improvements on speed. Objective value / time for CPLEX in seconds
Revised Methodology The University of Nottingham Two-stage algorithm CPLEX Simulated Annealing Season Simulated Annealing Local Search 2008-2009 195 34 2007-2008 222 28 2006-2007 205 33 2005-2006 210 32 2004-2005 196 31 2003-2004 201 29 2002-2003 207 33 Comparison of run times for SA and Local Search (seconds) Gary Lineker scoring against Paraguay in 1986
Season Published Fixtures Revised Methodology Kendall, 2008 Best Result Number of Pair Clashes for Published Fixtures Number of Pair Clashes for Generated Fixtures 2008-09 9312-5564 20 20 2007-08 8544-5366 24 22 2006-07 8439-5713 22 22 2005-06 10631 6917 6020 26 24 2004-05 9685 6784 5838 20 20 2003-04 8168 6359 6006 22 22 2002-03 7791 6040 5547 18 18
Contents Problem Definition Previous results (JORS) CPLEX vs DFS Multi-objective (clashes vs distance) Suggesting maximum distance Future Work 20 th Dec 2007: Sunderland double their lead and Kenwyne Jones celebrates
Pairing System Teams are paired which means they cannot normally play at home on the same day Typically this is due to teams which are geographically close (e.g. Manchester United/Manchester City, Sunderland/Newcastle, Liverpool/Everton/Tranmere) Note that the pairing system operates across division boundaries We are allowed a given number of pairing violations Season Boxing Day New Years Day 2008-2009 10 10 2007-2008 14 10 2006-2007 14 8 2005-2006 12 14 2004-2005 10 10 2003-2004 8 14 2002-2003 10 8 Number of actual clashes from previous seasons
2002-2003 The University of Nottingham 5850 5800 5750 5700 5650 5600 5550 Min 0 5 10 15 20 25 Min Clashes Min 20 5564 18 5574 16 5736 18 5581 16 5790 16 5809
2003-2004 The University of Nottingham 6100 6000 5900 5800 5700 5600 5500 5400 5300 Min 0 5 10 15 20 25 Min Clashes Min 22 5464 20 5412 18 5511 16 5887 20 5550 18 5559 16 5898 18 5592 16 5886 16 6028
2004-2005 The University of Nottingham 5900 5800 5700 5600 5500 5400 5300 Min 0 5 10 15 20 25 Min Clashes Min 20 5365 18 5345 16 5812 18 5443 16 5645 16 5810
6200 6000 5800 5600 5400 5200 5000 2005-2006 Min 0 5 10 15 20 25 30 Min Clashes Min 26 5234 24 5335 22 5240 20 5334 18 5481 16 6041 24 5171 22 5308 20 5460 18 5595 16 6027 22 5335 20 5334 18 5608 16 6146 20 5500 18 5528 16 5884 18 5713 16 6032 16 6084
2006-2007 The University of Nottingham 6300 6200 6100 6000 5900 5800 5700 5600 Min 0 5 10 15 20 25 Min Clashes Min 22 5713 20 5735 18 5872 16 6110 20 5721 18 5714 16 6195 18 5762 16 5894 16 6071
6200 6100 6000 5900 5800 5700 5600 5500 5400 5300 5200 2007-2008 Min 0 5 10 15 20 25 30 Min Clashes Min 24 5366 22 5403 20 5425 18 5690 16 5905 22 5370 20 5321 18 5625 16 5961 20 5340 18 5616 16 6101 18 5536 16 6091 16 5951
2008-2009 The University of Nottingham 5850 5800 5750 5700 5650 5600 5550 Min 0 5 10 15 20 25 Min Clashes Min 20 5564 18 5574 16 5736 18 5581 16 5790 16 5809
2004-2005 t-test Analysis (Statistically Different?) 10-10 10-8 10-6 8-10 8-8 6-10 10-10 X No Yes Yes Yes Yes 10-8 X Yes No Yes Yes 10-6 X Yes No No 8-10 X Yes Yes 8-8 X No 6-10 X 5900 5800 5700 5600 5500 5400 5300 Min 0 5 10 15 20 25 Clashes Min 20 5365 18 5345 16 5812 18 5443 16 5645 Min 16 5810
6200 6000 5800 5600 5400 5200 5000 2005-2006 Min 0 5 10 15 20 25 30 Min Clashes Min 26 5234 24 5335 22 5240 20 5334 18 5481 16 6041 24 5171 22 5308 20 5460 18 5595 16 6027 22 5335 20 5334 18 5608 16 6146 20 5500 18 5528 16 5884 18 5713 16 6032 16 6084
2005-2006 t-test Analysis (Statistically Different?) 12-14 12-12 12-10 12-8 12-6 12-4 10-14 10-12 10-10 10-8 10-6 8-14 8-12 8-10 8-8 6-14 6-12 6-10 4-14 4-12 2-14 12-14 X No No Yes Yes Yes No No Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes Yes 12-12 X No Yes Yes Yes No No Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes Yes 12-10 X No Yes Yes No No Yes Yes Yes No No Yes Yes Yes Yes Yes Yes Yes Yes 12-8 X Yes Yes Yes Yes No Yes Yes No No Yes Yes No Yes Yes Yes Yes Yes 12-6 X Yes Yes Yes No No Yes Yes Yes No Yes Yes No Yes No Yes Yes 12-4 X Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes No Yes Yes Yes 10-14 X No Yes Yes Yes No No Yes Yes Yes Yes Yes Yes Yes Yes 10-12 X Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes Yes 10-10 X No Yes No No No Yes No No Yes Yes Yes Yes 10-8 X Yes Yes Yes No Yes No No Yes Yes Yes Yes 10-6 X Yes Yes Yes No Yes Yes No Yes No No 8-14 X No Yes Yes Yes Yes Yes Yes Yes Yes 8-12 X Yes Yes Yes Yes Yes Yes Yes Yes 8-10 X Yes No No Yes Yes Yes Yes 8-8 X Yes Yes No Yes No No 6-14 X No Yes Yes Yes Yes 6-12 X Yes Yes Yes Yes 6-10 X Yes No No 4-14 X Yes Yes 4-12 X No 2-14 X
2006-2007 Min 6300 6200 6100 6000 5900 Min 5800 5700 5600 0 5 10 15 20 25 t-test Analysis (Statistically Different?) 14-8 14-6 14-4 14-2 12-8 12-6 12-4 10-8 10-6 8-8 14-8 X No Yes Yes No No Yes No Yes Yes 14-6 X Yes Yes No No Yes No Yes Yes 14-4 X Yes Yes No Yes No Yes Yes 14-2 X Yes Yes No Yes No Yes 12-8 X Yes Yes No Yes Yes 12-6 X Yes No Yes Yes 12-4 X Yes No Yes 10-8 X Yes Yes 10-6 X No 8-8 X Clashes Min 22 5713 20 5735 18 5872 16 6110 20 5721 18 5714 16 6195 18 5762 16 5894 16 6071
2008-2009 Min 5850 5800 5750 5700 5650 5600 5550 0 5 10 15 20 25 t-test Analysis (Statistically Different?) 10-10 10-8 10-6 8-10 8-8 6-10 10-10 X No Yes No Yes Yes 10-8 X Yes No Yes Yes 10-6 X Yes No No 8-10 X Yes Yes 8-8 X No Min Clashes Min 20 5564 18 5574 16 5736 18 5581 16 5790 16 5809 6-10 X
Contents Problem Definition Previous results (JORS) CPLEX vs DFS Multi-objective (clashes vs distance) Suggesting maximum distance Future Work Maradonna
Season 2002-2003 League 1 246 Miles 153 Miles
Experimental Setup Simulated Annealing (as before) Suggested Maximum Distance {500, 250, 180, 160, 140, 120, 100} Weighted {4, 20, 100, 5000} 22 nd Dec 2007: Nicklas Bendtner heads in the winner seconds after coming on as a substitute
6200 6100 6000 5900 5800 5700 5600 5500 5400 5300 5200 2002-2003 (weight = 4) 0 100 200 300 400 500 600 Maximum Distance Miles 500 6106 250 5566 200 5463 180 5683 160 5356 140 5432 120 5326 100 5459
6600 6400 6200 6000 5800 5600 5400 5200 5000 2002-2003 (weight = 20) 0 100 200 300 400 500 600 Maximum Distance Miles 500 6382 250 5659 200 5866 180 5450 160 5271 140 5341 120 5204 100
2002-2003 (weight = 100) 6400 6200 6000 5800 5600 5400 5200 0 100 200 300 400 500 600 Maximum Distance Miles 500 5516 250 6242 200 5434 180 5599 160 5491 140 5370 120 5250 100
2002-2003 (weight = 5000) 6000 5900 5800 5700 5600 5500 5400 5300 0 100 200 300 400 500 600 Maximum Distance Miles 500 5495 250 5949 200 5600 180 5352 160 5357 140 5423 120 100
153 Miles Season 2002-2003 League 1 119 Miles 10 th Feb 2007: Darius Henderson scores from the penalty spot
Season 2002-2003 The University of Nottingham The only stats so far 30 runs Maximum Miles = 160 Weights = {4, 5000} Results are statistically significant 27 th Oct 2007: Wayne Rooney scores for the eighth time in seven games for club and country
Future Research Directions The University of Nottingham Carry out further analysis Run full set of experiments for all divisions/seasons Contact FA/Police Produce a 4-fixture schedule Use this as a start for generating fixtures for the complete season. Wembley Stadium
Questions/Discussion