Parallel Cut Generation for Service Cost Allocation in Transmission Systems

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Celso C. Ribeiro Sebastián Urrutia
FUTMAX Play-offs elimination in soccer
tournaments
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Summary

• Motivation
• Related work
• Problem definition
• Formulation
• FUTMAX in the WWW
• FUTMAX in the press
• Results

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Motivation

• Guaranteed Qualification Problem
• How many points a team has to make in a
  tournament to be able to be qualified for the
  finals (or play-off games)?
• Brazilian Soccer Championship
• Press broadcasts already at the first round the
  chances of qualification for the play-offs,
  based in very obscure statistics and estimates.
• Predictions loudly announced are often wrong!

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Motivation

• Interesting application of Operations Research
  and Integer Programming in sports.
• Soccer is the most popular sports in most
  countries.
• Soccer can even make researchers from Argentina
  and Brazil to work together, instead of trying to
  prove which of them is the best...
• or arguing about who is the second best player
  ever!

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Motivation

• Three teams fight for two places in the
  play-offs.
• Each of them has two games to play Flamengo
  against Vasco and Bahia, Cruzeiro against Grêmio
  and Santos, and Bahia against Flamengo e
  Fluminense.
• How many points Cruzeiro has to make to be sure
  of being qualified?
• 4 points!

Flamengo 37
Cruzeiro 37
Bahia 36
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Motivation what if 26 teams are playing?
São Paulo 34 Ponte Preta 26 Fluminense 21
São Caetano 32 Figuerense 24 Vasco 20
Santos 32 Atlético-PR 24 Goiás 20
Juventude 31 Grêmio 24 Portuguesa 20
Corinthians 31 Cruzeiro 23 Paysandu 19
Guarani 28 Gama 22 Flamengo 19
Vitória 27 Internacional 22 Palmeiras 19
Coritiba 27 Botafogo 22 Paraná 17
Atlético-MG 27 Bahia 21
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Motivation

• Brazilian soccer championship (2002 edition)
• 26 teams
• First phase qualification
• Each team plays every other team exactly once.
• The eight teams in the first positions are
  qualified.
• The four teams in the last positions do not play
  next year.
• Second phase play-offs
• Eliminatory games
• Points
• win 3 points tie 1 point loss 0 point

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Motivation

• Brazilian championship
• 29 rounds from August to December
• The press publishes since the first round (!) the
  probabilities of qualification of each team.
• The press also makes use of very simple
  statistics from obscure sources to announce very
  early that a team with a certain number of points
  (41 in 2002) can be sure to be qualified (often
  based on estimates or average results).
• These predictions are loudly announced and often
  false!
• Trainers also very often make erroneous
  predictions.

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Related work

• Schwartz 1966 mathematical elimination from
  play-offs in the Major League Baseball (MLB)
  solved with maximum flow algorithm
• Robinson 1991 integer programming models and
  further results for the play-offs elimination
  problem
• McCormick 2000 elimination from the p-th
  position is NP-complete.
• Adler et al. 2003 ILP models for MLB
• Bernholt et al. 1970 first place elimination is
  NP-complete under the (3,0),(1,1) soccer
  rule.

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Related work

• Berkeleys RIOT project posts the number of wins
  each team needs to have a chance of qualification
  in the MLB
• No ties
• Easier (1,0) rule
• Small groups with 5 or 6 teams
• A team has to finish in the first position within
  its group to be qualified.
• Each team plays up to 160 games!
• Problems in each group are easily solvable.

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Problem definition

• Play-offs qualification problems
• How many points a team should make to
• be sure of finishing among the p8 teams in the
  first positions? (sufficient condition for
  play-offs qualification)
• have a chance of finishing among the p8 teams
  in the first positions? (necessary condition for
  play-offs qualification)

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Problem definition

• Elimination problems
• How many points a team should make to
• be sure of finishing among the p22 teams in
  the first positions? (sufficient condition for
  non-elimination)
• have a chance of finishing among the p22 teams
  in the first positions? (necessary condition for
  non-elimination)

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Problem definition

• In this talk, we consider only the Guaranteed
  Qualification Problem How many points a team k
  should make to be sure of finishing among the p8
  in the first positions?
• Instead, we compute the maximum number of points
  a team can make and still not be qualified in the
  p8 first positions.
• Then, we add 1 to the above number to determine
  the minimum number of points for guaranteed
  qualification GQS(k)

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Formulation

• n number of teams 26
• M maximum difference of points between any two
  teams (all wins vs. all losses) 3(n-1) 75
• k team under consideration

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Formulation
?
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Formulation
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Formulation

• Ties in the number of points are broken in favor
  of teams with more wins.
• In the previous model, add up a very small amount
  (necessarily smaller than one) to the number of
  points
• Use e.g.
• With a similar model, we can also compute
  PQS(k) minimum number of points for possible
  qualification

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Formulation

• When is team k mathematically qualified for the
  play-offs?
• Team k is mathematically qualified if the
  previous problem is infeasible!
• When does a team k depend only on itself to be
  qualifed?
• Team k depends only on itself if GQS(k) is less
  than or equal to the total number of points it
  can make.
• When is team k mathematically eliminated from
  the play-offs?
• Team k is mathematically eliminated if PQS(k) is
  less than or equal to the total number of points
  it can make.

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FUTMAX in the WWW

• FUTMAX project
• Results of the games are collected from the web.
• Model are generated (four problems for each
  team).
• All problems are solved with CPLEX 9.0
• HTML file is automatically built from the
  results.
• Automatic publication in the web
• http//www.futmax.org 2006 World Cup

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FUTMAX in the press

• September 24, 2002 FUTMAX web site launched
• September 29 round table interview to Radio
  Globo (major sports radio station, program
  Enquanto a bola não rola)
• September 30 article in the Internet section of
  Jornal do Brasil
• October 24 interview for TV Campus
• December 18 article in Jornal da PUC
• October 2003 talks with SPORTV (TV Globo) and
  computations of chances of qualification (live)

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Results

• Already at the 11th round, some teams did not
  depend only on themselves to be qualified.
• At this time, Vasco was in a difficult situation
  and fans were complaining.
• The trainer of Vasco met the press and said that
  if we win the next ten games, then we will be
  qualified.
• FUTMAX was able to prove to the press that this
  was not true.

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Results

• October 31, 2002 São Paulo won Ponte Preta and
  made a total of 43 points.
• The press (Folha de São Paulo) announced that
  São Paulo was mathematically qualified for the
  play-offs.
• FUTMAX was able to prove to the press that this
  was not true.
• November 3, 2002 São Caetano made a total of 42
  points and the press also announced it was
  qualified.
• Again, FUTMAX was able to prove to the press that
  this was not true.

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Results
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Results
FUTMAX can be used to follow the situation of
each team
Possible points
FLUMINENSE
Points for guaranteed qualification
Points for possible qualification
Points accumulated
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Results

• Spin-offs followed by HockeyPlex project (same
  idea for National Hockey League, USA) based on
  FUTMAX
• Motivation for students
• Activities of our research group made public
• New web page http//www.esportemax.org
• Slides available at
• http//www.inf.puc-rio.br/celso/talks.htm
• Papers (OR/MS Today 2004 ITOR to appear)
  available at
• http//www.inf.puc-rio.br/celso/publicacoes.htm