The University of Nottingham
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1
2 Balancing Conflicting Objectives for UK Football Graham Kendall
3 The University of Nottingham Graham Kendall, 20th Jan 2009, UNMC Sports Scheduling
4 Contents Problem Definition Previous results (JORS) CPLEX vs DFS Multi-objective (clashes vs distance) Suggesting maximum distance Future Work Charles Darwin "A man who dares to waste an hour of life has not discovered the value of life"
5 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
6 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
7 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)
8 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)
9 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 Number of actual clashes from previous seasons
10 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)
11 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 giving Manchester United the biggest League Cup Final victory in the competition's history (2006)
12 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)
13 Maps of Premier League Clubs
14 Division 1:
15 405 Miles (Plymouth to Sunderland) 153 Miles (Plymouth to Cardiff) Division 1:
16 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
17 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
18 Results (JORS) Published Fixtures Generated Fixtures Season Boxing Day Total Distance Clashes New Years Day Total Distance Clashes Season Boxing Day Total Distance Clashes New Years Day Total Distance Clashes Season Boxing Day Total Distance Clashes 8 8 New Years Day Total Distance Clashes Season Boxing Day Total Distance Clashes New Years Day Total Distance Clashes 8 8 Nottingham Forest Football Ground
19 Season Results (JORS) Total Distance for Published Fixtures Total Distance for Generated Fixtures Distance Saving (34.94%) (29.95%) (22.15%) (22.47%) Number of Pair Clashes for Published Fixtures Number of Pair Clashes for Generated Fixtures Kendall G. Scheduling English Football Fixtures Over Holiday Periods. (2008) Journal of the Operational Research Society, 59(6), pages ( Bobby Moore in 1966
20 Contents Problem Definition Previous results (JORS) CPLEX vs DFS Multi-objective (clashes vs distance) Suggesting maximum distance Future Work Are you blind ref?
21 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
22 Revised Methodology Search Phase Season Division Stage 1a Stage 1b Local Search Barclays Premiership 5 <1 < Coca-Cola Championship 823 <1 <5 Coca-Cola League One 149 <1 <5 Coca-Cola League <1 <5 Barclays Premiership 4 <1 < Coca-Cola Championship 993 <1 <5 Coca-Cola Lauge One 222 <1 <5 Coca-Cola League <1 <5 Barclays Premiership 8 <1 < Coca-Cola Championship 891 <1 <5 Coca-Cola Lauge One 157 <1 <5 Coca-Cola League <1 <5 Barclays Premiership 3 <1 < 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
23 Revised Methodology Two-stage algorithm CPLEX Simulated Annealing Barclays Premiers hip Coca- Cola Champi onship Coca- Cola League One Coca- Cola League / / / / / / / / / / / / / / / / 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
24 Revised Methodology The University of Nottingham Two-stage algorithm CPLEX Simulated Annealing Season Simulated Annealing Local Search Comparison of run times for SA and Local Search (seconds) Gary Lineker scoring against Paraguay in 1986
25 Season Published Fixtures Revised Methodology Kendall, 2008 Best Result Number of Pair Clashes for Published Fixtures Number of Pair Clashes for Generated Fixtures
26 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
27 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 Number of actual clashes from previous seasons
28 The University of Nottingham Min Min Clashes Min
29 The University of Nottingham Min Min Clashes Min
30 The University of Nottingham Min Min Clashes Min
31 Min Min Clashes Min
32 The University of Nottingham Min Min Clashes Min
33 Min Min Clashes Min
34 The University of Nottingham Min Min Clashes Min
35 t-test Analysis (Statistically Different?) 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 Min Clashes Min Min
36 Min Min Clashes Min
37 t-test Analysis (Statistically Different?) X No No Yes Yes Yes No No Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes Yes X No Yes Yes Yes No No Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes Yes 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 X No Yes Yes Yes No No Yes Yes Yes Yes Yes Yes Yes Yes X Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes Yes 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
38 Min Min t-test Analysis (Statistically Different?) 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
39 Min t-test Analysis (Statistically Different?) 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 X
40 Contents Problem Definition Previous results (JORS) CPLEX vs DFS Multi-objective (clashes vs distance) Suggesting maximum distance Future Work Maradonna
41 Season League Miles 153 Miles
42 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
43 (weight = 4) Maximum Distance Miles
44 (weight = 20) Maximum Distance Miles
45 (weight = 100) Maximum Distance Miles
46 (weight = 5000) Maximum Distance Miles
47 153 Miles Season League Miles 10 th Feb 2007: Darius Henderson scores from the penalty spot
48 Season 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
49 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
50 Questions/Discussion
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