the university of nottinghampszgxk/gxk1/presentations/20012009unmcpgche.pdfgeographically close...

Graham Kendall, 20th Jan 2009, UNMC – Sports Scheduling

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niversity of Nottingham

© Graham Kendall (not to be distributed or used without permission)


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Balancing Conflicting Objectives for UK Football

Graham Kendall


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"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

Charles Darwin 1809 - 1882


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Contents

Paul Gascoigne gets booked, ruling him out of a potential world cup final



• CPLEX vs DFS



• Future Work


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George Best won 37 caps for Northern Ireland, including this match against England in May

1971

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), 30th December 2005, pp 68


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Michael Owen returns to the line up at Newcastle after injury (29th April

2006)

League Structure

Four DivisionsBarclays Premiership (20 teams)Coca-Cola Championship (24 teams)Coca-Cola League One (24 teams)Coca-Cola League Two (24 teams)

Number of Matches380 (Premiership), 552 for other divisions. Total 2036

Matches to be ScheduledEach league plays a double round robin tournament. Therefore, there are no matches played between different divisions, but…..


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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)


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Pairing System





Number of actual clashes from previous seasons

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


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Chelsea win the Premiership for the second consecutive season (April

2006)

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


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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 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 otherIn fact this is violated in the published fixtures, but we treat it as a hard constraint

The aim is to minimise the distance travelled


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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 otherIn 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)


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2002-2003 2003-2004

2005-20062004-2005

Maps of Premier League Clubs


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Division 1: 2004-2005


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Division 1: 2004-2005

153 Miles (Plymouth to

Cardiff)

405 Miles(Plymouth to Sunderland)


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Contents



• CPLEX vs DFS



• Future WorkGordon Banks saves a “certain”

headed goal from Pele


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Bobby Charlton: Then and Now

Solution Methodology (JORS)

Two-stage algorithm

Depth-first searchProduces two fixtures for each division, one for Boxing Day and one for New Years Day (i.e. eight sets of fixtures)

Local SearchOperates across divisions in order to resolve clashes


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Nottingham Forest Football Ground

Results (JORS)Published Fixtures Generated Fixtures

Season 2005-2006

Boxing DayTotal Distance 4295 3049

Clashes 12 10

New Years DayTotal Distance 6336 3868

Clashes 14 10

Season 2004-2005


Clashes 10 10


Clashes 10 10

Season 2003-2004


Clashes 8 8


Clashes 14 14

Season 2002-2003


Clashes 10 10


Clashes 8 8


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Bobby Moore in 1966

Results (JORS)Season Total

Distance for

Published Fixtures

Total Distance for Generated Fixtures

Distance Saving Number of Pair Clashes for Published Fixtures

Number of Pair Clashes for Generated Fixtures

2005-2006 10631 6917 3714 (34.94%)

26 20

2004-2005 9685 6784 2901 (29.95%)

20 20

2003-2004 8168 6359 1809 (22.15%)

22 22

2002-2003 7791 6020 1751 (22.47%)

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)


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Contents



• CPLEX vs DFS



• Future Work

“Are you blind ref?”


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Revised Methodology

Stamford Bridge: Chelsea’s home ground

Two-stage algorithm

CPLEXProduces Boxing Day fixtures and New Years Day fixtures at the same time.

Simulated AnnealingOperates across divisions in order to resolve clashes


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Revised MethodologySearch Phase

Season Division Stage 1a Stage 1b Local Search

2005-06

Barclays Premiership 5 <1 <5

Coca-Cola Championship 823 <1 <5

Coca-Cola League One 149 <1 <5

Coca-Cola League 2 148 <1 <5

2004-05



Coca-Cola Lauge One 222 <1 <5


2003-04





2002-03





• Times in minutes

• Large differences, but generally expensive


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Revised Methodology

Two-stage algorithmCPLEXSimulated Annealing

• 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.

CPLEX

Barclays Premiers

hip

Coca-Cola

Championship

Coca-Cola

League One

Coca-Cola

League 2

2005-06 677 / 4 948 / 5 1156 / 4 1369 / 42004-05 888 / 6 1104 / 5 1394 / 4 1357 / 52003-04 917 / 4 1331 / 4 1277 / 4 1224 / 52002-03 737 / 3 1301 / 3 1297 / 4 1217 / 6

Objective value / time for CPLEX in seconds


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Revised MethodologyTwo-stage algorithm

CPLEXSimulated Annealing

Comparison of run times for SA and Local Search (seconds)

Gary Lineker scoring against Paraguay in 1986

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


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Revised Methodology

Season Published Fixtures

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


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Contents



• CPLEX vs DFS



• Future Work20th Dec 2007: Sunderland double

their lead and Kenwyne Jones celebrates


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Pairing System





Number of actual clashes from previous seasons

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


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2002-2003

5550

5600

5650

5700

5750

5800

5850

0 5 10 15 20 25

Min

Min

Clashes Min20 556418 557416 573618 558116 579016 5809


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2003-2004

5300

5400

5500

5600

5700

5800

5900

6000

6100

0 5 10 15 20 25

Min

Min

Clashes Min22 546420 541218 551116 588720 555018 555916 589818 559216 588616 6028


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2004-2005

5300

5400

5500

5600

5700

5800

5900

0 5 10 15 20 25

Min

Min

Clashes Min20 536518 534516 581218 544316 564516 5810


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2005-2006

5000

5200

5400

5600

5800

6000

6200

0 5 10 15 20 25 30

Min

Min

Clashes Min26 523424 533522 524020 533418 548116 604124 517122 530820 546018 559516 602722 533520 533418 560816 614620 550018 552816 588418 571316 603216 6084


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2006-2007

5600

5700

5800

5900

6000

6100

6200

6300

0 5 10 15 20 25

Min

Min

Clashes Min22 571320 573518 587216 611020 572118 571416 619518 576216 589416 6071


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2007-2008

52005300540055005600570058005900600061006200

0 5 10 15 20 25 30

Min

Min

Clashes Min24 536622 540320 542518 569016 590522 537020 532118 562516 596120 534018 561616 610118 553616 609116 5951


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2008-2009

5550

5600

5650

5700

5750

5800

5850

0 5 10 15 20 25

Min

Min

Clashes Min20 556418 557416 573618 558116 579016 5809


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2004-2005

5300

5400

5500

5600

5700

5800

5900

0 5 10 15 20 25

Min

Min

Clashes Min20 536518 534516 581218 544316 564516 5810

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


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2005-2006

5000

5200

5400

5600

5800

6000

6200

0 5 10 15 20 25 30

Min

Min

Clashes Min26 523424 533522 524020 533418 548116 604124 517122 530820 546018 559516 602722 533520 533418 560816 614620 550018 552816 588418 571316 603216 6084


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2005-2006t-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


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2006-2007

5600

5700

5800

5900

6000

6100

6200

6300

0 5 10 15 20 25

Min

Min

Clashes Min22 571320 573518 587216 611020 572118 571416 619518 576216 589416 6071

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 Yes14-6 X Yes Yes No No Yes No Yes Yes14-4 X Yes Yes No Yes No Yes Yes14-2 X Yes Yes No Yes No Yes12-8 X Yes Yes No Yes Yes12-6 X Yes No Yes Yes12-4 X Yes No Yes10-8 X Yes Yes10-6 X No8-8 X


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2008-2009

5550

5600

5650

5700

5750

5800

5850

0 5 10 15 20 25

Min

Min

Clashes Min20 556418 557416 573618 558116 579016 5809

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

6-10 X


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Contents



• CPLEX vs DFS



• Future Work

Maradonna


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Season 2002-2003

League 1

246 Miles

153 Miles


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Experimental Setup

• Simulated Annealing (as before)

• Suggested Maximum Distance•{500, 250, 180, 160, 140, 120, 100}

• Weighted•{4, 20, 100, 5000}

22nd Dec 2007: Nicklas Bendtner heads in the winner seconds after

coming on as a substitute


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2002-2003 (weight = 4)

5200

5300

5400

5500

5600

5700

5800

5900

6000

6100

6200

0 100 200 300 400 500 600

Maximum Distance Miles

500 6106250 5566200 5463180 5683160 5356140 5432120 5326100 5459


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2002-2003 (weight = 20)

5000

5200

5400

5600

5800

6000

6200

6400

6600

0 100 200 300 400 500 600

Maximum Distance Miles

500 6382250 5659200 5866180 5450160 5271140 5341120 5204100


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2002-2003 (weight = 100)Maximum Distance Miles

500 5516250 6242200 5434180 5599160 5491140 5370120 5250100

5200

5400

5600

5800

6000

6200

6400

0 100 200 300 400 500 600


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2002-2003 (weight = 5000)Maximum Distance Miles

500 5495250 5949200 5600180 5352160 5357140 5423120100

5300

5400

5500

5600

5700

5800

5900

6000

0 100 200 300 400 500 600


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Season 2002-2003

League 1153 Miles

119 Miles

10th Feb 2007: Darius Henderson scores from the penalty spot


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Season 2002-2003

The only stats so far

•30 runs

•Maximum Miles = 160

•Weights = {4, 5000}

•Results are statistically significant

27th Oct 2007: Wayne Rooney scores for the eighth time in seven games for

club and country


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Future Research Directions

Carry out further analysisRun full set of experiments for all divisions/seasonsContact FA/Police

Produce a 4-fixture scheduleUse this as a start for generating fixtures for the complete season.

Wembley Stadium


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Questions/Discussion

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