Transcript PVC routing algorithms

Applications of Metaheuristics to
Optimization Problems in Sports
Celso C. Ribeiro
Joint work with S. Urrutia,
A. Duarte, and A. Guedes
Hammamet, October 2008
2nd International Conference on Metaheuristics and Nature Inspired
Computing (META’08)
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Metaheuristics for optimization problems in sports
Summary
• Optimization problems in sports
– Motivation
– Problems, applications, and solution methods
• Applications of metaheuristics
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Traveling tournament problem
Referee assignment
Carry-over effect minimization
Brazilian professional basketball tournament
• Perspectives and concluding remarks
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Metaheuristics for optimization problems in sports
Motivation
• Sports competitions involve many economic
and logistic issues
• Multiple decision makers: federations, TV,
teams, security authorities, ...
• Conflicting objectives:
– Maximize revenue (attractive games in specific
days)
– Minimize costs (traveled distance)
– Maximize athlete performance (time to rest)
– Fairness (avoid playing all strong teams in a row)
– Avoid conflicts (teams with a history of conflicts
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playing
Motivation
• Professional sports:
– Millions of fans
– Multiple agents: organizers, media, fans,
players, security forces, ...
– Big investments:
• Belgacom TV: €12 million per year for soccer
broadcasting rights
• Baseball US: > US$ 500 millions
• Basketball US: > US$ 600 millions
– Main problems: maximize revenues, optimize
logistic, maximize fairness, minimize conflicts
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Metaheuristics for optimization problems in sports
Motivation
• Amateur sports:
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Thousands of athletes
Athletes pay for playing
Large number of simultaneous events
Amateur leagues do not involve as much money
as professional leagues but, on the other hand,
amateur competitions abound
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Metaheuristics for optimization problems in sports
Optimization problems in sports
• Examples:
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Qualification/elimination problems
Tournament scheduling
Referee assignment
Tournament planning (teams? dates? rules?)
League assignment (which teams in each
league?)
– Carry-over minimization
...
– Optimal strategies for curling!
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Metaheuristics for optimization problems in sports
Qualification/elimination problems
• Team managers, players, fans and the press
are often eager to know the chances of a
team to be qualified for the playoffs of a
given competition
How many points a team should make to:
• … be sure of finishing among the p teams in
the first positions? (sufficient condition for
play-offs qualification)
• … have a chance of finishing among the p
teams in the first positions? (necessary
condition
for
play-offs
qualification)
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Qualification/elimination problems
• Schwartz 1966: mathematical elimination
from play-offs in the Major League Baseball
(MLB) solved with maximum flow algorithm
• Robinson 1991: IP models and further results
for the play-offs elimination problem
• McCormick 2000: elimination from the p-th
position is NP-complete.
• Bernholt et al. 2002: first place elimination is
NP-complete under the {(3,0),(1,1)} soccer
rule
• Adler et al. 2003: ILP models for MLB
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Metaheuristics for optimization problems in sports
Qualification/elimination problems
• Ribeiro & Urrutia 2005: integer
programming for qualification/elimination
problems in the Brazilian soccer
championship and the World Cup
(FUTMAX)
• Cheng & Steffy 2006: integer programming
for qualification/elimination problems in the
National Hockey League.
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Metaheuristics for optimization problems in sports
FUTMAX in the WWW
• FUTMAX project
• Results of the games automatically collected
from the web (multi-agents)
• Models generated (four problems for each
team)
• Problems solved with CPLEX 9.0
• HTML file automatically built from the results
• Automatic publication in the web
• FUTMAX is often able to prove that
statements
made 10/92
by the
Pressforand
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Metaheuristics
optimization problems in sports
Results
FUTMAX can also be used to follow the situation of each team:
Possible
points
Points for guaranteed
qualification
Points for possible
qualification
FLUMINENSE
Points accumulated
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Metaheuristics for optimization problems in sports
Tournament scheduling
• Timetabling is the major area of
applications: game scheduling is a difficult
task, involving different types of constraints,
logistic issues, multiple objectives, and
several decision makers
• Round Robin schedules:
– Every team plays each other a fixed number of
times
– Every team plays once in each round
– Single (SRR) or double (DRR) round robin
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Metaheuristics for optimization problems in sports
Tournament scheduling
• Problems:
– Minimize distance (costs)
– Minimize breaks (fairness and equilibrium, every
two rounds there is a game in the city)
– Balanced tournaments (even distribution of fields
used by the teams: n teams, n/2 fields, SRR with n1 games/team, 2 games/team in n/2-1 fields and
1 in the other)
– Minimize carry over effect (maximize fairness,
polygon method)
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Metaheuristics for optimization problems in sports
1-factorizations
• Factor of a graph G=(V, E): subgraph
G’=(V,E’) with E’E
• 1-factor: all nodes have degree equal to 1
• Factorization of G=(V,E): set of edge-disjoint
factors G1=(V,E1), ..., Gp=(V,Ep), such that
E1...Ep=E
• 1-factorization: factorization into 1-factors
• Oriented factorization: orientations
assigned to edges
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Metaheuristics for optimization problems in sports
1-factorizations
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Example:
1-factorization of K6
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Metaheuristics for optimization problems in sports
Oriented 1-factorization of K6
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Metaheuristics for optimization problems in sports
1-factorizations
• SRR tournament:
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Each node of Kn represents a team
Each edge of Kn represents a game
Each 1-factor of Kn represents a round
Each ordered 1-factorization of Kn represents a
feasible schedule for n teams
– Edge orientations define teams playing at
home
– Dinitz, Garnick & McKay, “There are
526,915,620 nonisomorphic one-factorizations
Open
of Kproblem:
12” (1995)
How many
schedules
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round robin tournament with n teams?
Metaheuristics for optimization problems in sports
Distance minimization problems
• Whenever a team plays two consecutive
games away, it travels directly from the
facility of the first opponent to that of the
second
• Maximum number of consecutive games
away (or at home) is often constrained
• Minimize the total distance traveled (or the
maximum distance traveled by any team)
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Metaheuristics for optimization problems in sports
Distance minimization problems
• Methods:
– Metaheuristics: simulated annealing, iterated
local search, hill climbing, tabu search, GRASP,
genetic algorithms, ant colonies
– Integer programming
– Constraint programming
– IP/CP column generation
– CP with local search
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Metaheuristics for optimization problems in sports
Break minimization problems
• There is a break whenever a team has two
consecutive home games (or two away
games)
• Break minimization:
– De Werra 1981: minimum number of breaks is
n-2
• Every team must have a different home-away pattern
(they must play in some round)
• Only two patterns without breaks:
– HAHAHAH...
– AHAHAHA...
– Constructive algorithm to obtain schedules with
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exactly
Break minimization problems
• Break minimization is somehow opposed to
distance minimization
• Urrutia & Ribeiro 2006: a special case of the
Traveling Tournament Problem is equivalent
to a break maximization problem
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Metaheuristics for optimization problems in sports
Fixed timetables/venues
• Given a fixed timetable, find a home-away
assignment minimizing breaks/distance:
– Metaheuristics, constraint programming,
integer programming
– Miyashiro & Matsui 2005: polynomial method
for break minimization if the minimal number of
breaks is smaller than or equal to n
• Given a fixed venue assignment for each
game, find a timetable minimizing
breaks/distance:
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– Melo, Urrutia & Ribeiro
2007: integer
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Decomposition methods
•
Nemhauser and Trick 1998:
1. Find home-away patterns
2. Create an schedule for place holders
consistent with a subset of home-away
patterns
3. Assign teams to place holders
•
•
Order in which the above tasks are tackled
may vary depending on the application
Henz 2001: CP may work better than IP
and complete enumeration for all the tasks
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Metaheuristics for optimization problems in sports
Decomposition methods
• Frequently used for scheduling real
tournaments:
– Nemhauser & Trick 1998: Atlantic Coast
Conference (basketball)
– Bartsch et al. 2006: Austrian and German soccer
– Della Croce & Oliveri 2006: Italian soccer
– Ribeiro & Urrutia 2006: Brazilian soccer
– Durán, Noronha, Ribeiro, Sourys, Weintraub
2006: Chilean soccer
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Metaheuristics for optimization problems in sports
Applications of metaheuristics
Traveling Tournament Problem (TTP)
and its mirrored version (mTTP)
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Metaheuristics for optimization problems in sports
Formulation
• Traveling Tournament Problem (TTP)
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n (even) teams take part in a tournament
Each team has its own stadium at its home city
Distances between the stadiums are known
A team playing two consecutive away games
goes directly from one city to the other, without
returning to its home city
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Metaheuristics for optimization problems in sports
Formulation
– Double round-robin tournament:
• 2(n-1) rounds, each with n/2 games
• Each team plays against every other team twice, one
at home and the other away
– No team can play more than three games in a
home stand (home games) or in a road trip
(away games)
• Goal: minimize the distance traveled by all
teams, to reduce traveling costs and to give
more time to the players to rest and practice
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Metaheuristics for optimization problems in sports
Formulation
• Mirrored Traveling Tournament Problem
(mTTP):
– All teams face each other once in the first phase
(n-1 rounds)
– In the second phase (n-1 rounds), teams play
each other again in the same order, following an
inverted home-away pattern
– Games in the second phase determined by those
in the first
• Set of feasible solutions to the MTTP is a
subset of those to the TTP
Ribeiro
and Urrutia
2004,
EJOR
2007)
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problems
in sports
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•
Constructive heuristic
Three steps:
1. Schedule games using abstract teams: polygon
method defines the structure of the tournament
2. Assign real teams to abstract teams: greedy
heuristic to QAP (number of travels between
stadiums of the abstract teams x distances between
the stadiums of the real teams)
3. Select stadium for each game (home/away pattern)
in the first phase (mirrored tournament):
1. Build a feasible assignment of stadiums, starting from a
random assignment in the first round
2. Improve this assignment, using a simple local search
algorithm based on home-away swaps
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Metaheuristics for optimization problems in sports
Constructive heuristic
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Example:
“polygon method” for n=6
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1st round
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Metaheuristics for optimization problems in sports
Constructive heuristic
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Example:
“polygon method” for n=6
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1
2nd round
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Metaheuristics for optimization problems in sports
Simple neighborhoods
• Home-away swap (HAS): modify the stadium
of a game
• Team swap (TS): exchange the sequence of
opponents of a pair of teams over all rounds
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Metaheuristics for optimization problems in sports
Partial round swap (PRS)
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Metaheuristics for optimization problems in sports
Partial round swap (PRS)
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Metaheuristics for optimization problems in sports
Ejection chain: game rotation (GR)
• Neigborhood “game rotation” (GR) (ejection
chain):
– Enforce a game to be played at some round: add
a new edge to a given 1-factor of the current 1factorization (schedule)
– Use an ejection chain to recover a 1factorization
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Metaheuristics for optimization problems in sports
Ejection chain: game rotation (GR)
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Metaheuristics for optimization problems in sports
Ejection chain: game rotation (GR)
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Enforce game (1,3) to
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Metaheuristics for optimization problems in sports
Ejection chain: game rotation (GR)
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Enforce game (1,3) to
be played in round 2
Metaheuristics for optimization problems in sports
Ejection chain: game rotation (GR)
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Metaheuristics for optimization problems in sports
Ejection chain: game rotation (GR)
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Metaheuristics for optimization problems in sports
Ejection chain: game rotation (GR)
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Metaheuristics for optimization problems in sports
Ejection chain: game rotation (GR)
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Metaheuristics for optimization problems in sports
Ejection chain: game rotation (GR)
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Metaheuristics for optimization problems in sports
Ejection chain: game rotation (GR)
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Metaheuristics for optimization problems in sports
Ejection chain: game rotation (GR)
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Metaheuristics for optimization problems in sports
Neighborhoods
• Only moves in neighborhoods PRS and GR may
change the structure of the initial schedule
• However, PRS moves not always exist, due to
the structure of the solutions built by polygon
method e.g. for n = 6, 8, 12, 14, 16, 20, 24
• PRS moves may appear after an ejection chain
move is made
• Ejection chain moves may find solutions that
are not reachable through other
neighborhoods: escape from local optima
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Metaheuristics for optimization problems in sports
GRASP+ILS heuristic
• Hybrid improvement heuristic for the MTTP:
– Combination of GRASP and ILS
– Initial solutions: randomized version of the
constructive heuristic
– Local search with first improving move: use TS,
HAS, PRS and HAS cyclically in this order, until a
local optimum for all neighborhoods is found
– Perturbation: random move in GR
neighborhood
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Metaheuristics for optimization problems in sports
GRASP+ILS heuristic
while .not.StoppingCriterion
S  GenerateRandomizedInitialSolution()
S  LocalSearch(S)
repeat
S’  Perturbation(S,history)
S’  LocalSearch(S’)
S  AceptanceCriterion(S,S’,history)
S*  UpdateBestSolution(S,S*)
until ReinitializationCriterion
end
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Metaheuristics for optimization problems in sports
Concluding remarks
• Constructive heuristic is very fast and
effective
• GRASP+ILS is very fast and finds very good
solutions, even better than the best known
for the corresponding (less constrained) not
necessarily mirrored instances
• Effectiveness of the ejection chains
• Theoretical complexity still open
• Lower bounds:
– Independent lower bound: Easton et al. 2001
– MNTLB (improvement over ILB): Urrutia et al.
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20072008
Applications of metaheuristics
Referee Assignment Problem (RAP)
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Metaheuristics for optimization problems in sports
Motivation
• Regional amateur leagues in the US
(baseball, basketball, soccer): hundreds of
games every weekend in different divisions
• In a single league in California there are up
to 500 soccer games in a weekend, to be
refereed by hundreds of certified referees
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Metaheuristics for optimization problems in sports
Motivation
• MOSA (Monmouth & Ocean Counties Soccer
Association) League (NJ): boys & girls, ages 818, six divisions per age/gender group, six
teams per division: 396 games every Sunday
(US$ 40 per referee; U$ 20 per linesman, two
linesmen)
• Problem: assign referees to games
Duarte, Ribeiro & Urrutia (PATAT 2006, LNCS
2007)
• Referee assignment involves many constraints
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and multiple objectives
Referee assignment
• Possible constraints:
– Different number of referees may be necessary
for each game
– Games require referees with different levels of
certification: higher division games require
referees with higher skills
– A referee cannot be assigned to a game where
he/she is a player
– Timetabling conflicts and traveling times
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Metaheuristics for optimization problems in sports
Referee assignment
• Possible constraints (cont.):
– Referee groups: cliques of referees that request
to be assigned to the same games (relatives, car
pools)
• Hard links
• Soft links
– Number of games a referee is willing to referee
– Traveling constraints
– Referees that can officiate games only at a
certain location or period of the day
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Metaheuristics for optimization problems in sports
Referee assignment
• Possible objectives:
– Difference between the target number of games a
referee is willing to referee and the number of
games he/she is assigned to
– Traveling/idle time between consecutive games
– Number of inter-facility travels
– Number of games assigned outside his/her
preferred time-slots or facilities
– Number of violated soft links
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Metaheuristics for optimization problems in sports
Problem statement
• Games are already scheduled (facility – time
slot)
• Each game has a number of refereeing
positions to be assigned to referees
• Each refereeing position to be filled by a
referee is called a refereeing slot
• S = {s1, s2,..., sn}: refereeing slots to be filled
by referees
• R = {r1, r2,..., rm}: referees
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Metaheuristics for optimization problems in sports
Problem statement
• pi: skill level of referee ri
• qj: minimum skill level a referee must have
to be assigned to refereeing slot sj
• Mi: maximum number of games referee ri
can officiate
• Ti: target number of games referee ri is
willing to officiate
• Each referee may choose a set of time slots
where he/she is not available to officiate
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Metaheuristics for optimization problems in sports
Problem statement
• Problem: assign a referee to each refereeing
slot
• Constraints:
– Referees officiate in a single facility on the same
day
– Minimum skill level requirements
– Maximum number of games
– Timetabling conflicts and availability
• Objective: minimize the sum over all referees
of the absolute value of the difference between
theOctober
target
actual
number
of games
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Metaheuristics
for optimization
problems in sports
Solution approach
• Three-phase heuristic approach
1. Greedy constructive heuristic
2. ILS-based repair heuristic to make the initial
solution feasible (if necessary):
minimization of the number of violations
3. ILS-based procedure to improve a feasible
solution
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Metaheuristics for optimization problems in sports
Solution approach
Algorithm RefereeAssignmentHeuristic (MaxIter)
1. S*  BuildGreedyRandomizedSolution ();
2. If not isFeasible (S*) then
3. S*  RepairHeuristic (S*, MaxIter);
4. If isFeasible (S*) then
5. S*  ImprovementHeuristic (S*);
6. Else “infeasible”
7. Return S*
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Metaheuristics for optimization problems in sports
Numerical results
• Randomly generated instances following
patterns similar to real-life applications
• Instances with up to 500 games and 1,000
referees
– Different number of facilities
– Different patterns of the target number of
games
• Five different instances for each
configuration
• MaxIter = 1,000 61/92 Metaheuristics for optimization problems in sports
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Numerical results
• For each instance, average time and average
objective value over ten runs
• Codes implemented in C
• Results on a 2.0 GHz Pentium IV processor with
256 Mbytes
• Initial solutions:
– greedy constructive heuristic
– random assignments (to test the repair heuristic)
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Metaheuristics for optimization problems in sports
Numerical results
Instance
Construction
time (s)
value
Repair
feas.
time (s)
Improvement
value
feas.
time (s)
value
I1
0.02
1286.20
10
—
—
—
32.34
619.60
I2
0.02
1360.00
5
0.47
1338.00
10
31.81
623.40
I3
0.02
1269.00
2
0.60
1247.00
10
33.87
621.60
I4
0.03
—
—
1.14
1303.20
10
30.28
627.20
I5
0.03
1302.00
3
1.40 12591.14
10
33.73
654.00
Table 1: Instances with 500 games, 750 referees, and 65 facilities
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Metaheuristics for optimization problems in sports
Numerical results
Greedy
Random
Instance
pattern
I1
P0
0.03
11.27
10
79.80
9
I2
P0
0.03
6.69
10
80.80
10
I3
P0
0.03
11.33
10
86.20
8
I4
P0
0.03
4.61
10
30.60
10
I5
P0
0.03
3.39
10
29.10
10
I1
P1
0.03
2.75
10
33.50
10
I2
P1
0.02
19.29
10
134.60
2
I3
P1
0.03
14.77
10
135.10
8
I4
P1
0.03
1.22
10
38.00
10
I5
P1
0.03
2.69
10
32.90
10
const. (s)
repair (s)
feas.
repair (s)
feas.
Table 4: Greedy vs. randomly generated initial solutions
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Metaheuristics for optimization problems in sports
Improvements and extensions
• Greedy constructive heuristic:
– First, assign each referee to a number of refereeing
slots as close as possible to his/her target number
of games
– Second, if there are still unassigned slots, assign
more games to each referee
• Improvement heuristic:
– After each perturbation, instead of applying a local
search for both facilities involved in this
perturbation, solve a MIP model associated with
the subproblem considering all refereeing slots
and referees corresponding to these facilities
META’08,(“MIP
October 2008
Metaheuristics for optimization problems in sports
it!”)
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Numerical results
Figure 3: 500 games, 750 referees, 85 facilities, pattern P0 (target =
529)
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Metaheuristics for optimization problems in sports
Bi-criteria problem (biRAP)
•
•
Same constraints as in the single objective
version
Objectives:
1. minimize the sum over all referees of the absolute
value of the difference between the target and the
actual number of games assigned to each referee
2. minimize the sum over all referees of the total idle
time between consecutive games
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Metaheuristics for optimization problems in sports
Bi-criteria problem (biRAP)
• Formulation: bi-criteria set partitioning
problem
• Variables: possible “routes” for each referee
(stops correspond to refereeing positions)
• Each “route” has at most 4 to 5 stops:
number of variables is limited
• Each refereeing position has to be filled by
exactly one qualified refere
• Each referee must perform exactly one
“route”
• Goal: find the set of (potentially)
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Metaheuristics for optimization problems in sports
Solution approach
• Exact approach: dichotomic method
50 games and
100 referees
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Metaheuristics for optimization problems in sports
Solution approach
• Heuristic approach:
– Perform three-phase ILS-based heuristic for a
fixed number of search directions
– Each search direction represents a set of weights
associated with each objective
– Directions are chosen as in the dichotomic method
– All new potentially efficient solutions found during
the search are progressively stored
– Former potentially efficient solutions are
discarded during the search (quadtree is used)
– Perform a post-optimization path-relinking
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2008
Metaheuristics for optimization problems in sports
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procedure
Numerical results
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Metaheuristics for optimization problems in sports
Numerical results
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Metaheuristics for optimization problems in sports
Numerical results
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Metaheuristics for optimization problems in sports
Conclusions
• New optimization problem in sports
• Effective heuristics:
construction, repair, improvement, path
relinking
• Quick procedures to build good initial solutions
• Bicriteria approach finds good approximations
of the Pareto frontier
• Other constraints and criteria may be considered
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Metaheuristics for optimization problems in sports
Applications of metaheuristics
Carry-over minimization problem
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Metaheuristics for optimization problems in sports
Carry-over effects
• Team B receives a carry-over effect (COE)
due to team A if there is a team X that plays
A in round r and B in round r+1
A
B
C
D
E
F
G
H
META’08, October 2008
1
H
C
B
E
D
G
F
A
2
C
D
A
B
G
H
E
F
3
D
E
F
A
B
C
H
G
4
E
F
H
G
A
B
D
C
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5
F
G
E
H
C
A
B
D
6
G
H
D
C
F
E
A
B
7
B
A
G
F
H
D
C
E
Metaheuristics for optimization problems in sports
Carry-over effects
• Team B receives a carry-over effect (COE)
due to team A if there is a team X that plays
A in round r and B in round r+1
Team A
receives COE
due to B
META’08, October 2008
A
B
C
D
E
F
G
H
1
H
C
B
E
D
G
F
A
2
C
D
A
B
G
H
E
F
3
D
E
F
A
B
C
H
G
4
E
F
H
G
A
B
D
C
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5
F
G
E
H
C
A
B
D
6
G
H
D
C
F
E
A
B
7
B
A
G
F
H
D
C
E
Team G
receives COE
due to D
Team A
receives COE
due to E
Metaheuristics for optimization problems in sports
Carry-over effects matrix
• SRRT and carry-over effects matrix (COEM)
A
B
C
D
E
F
G
H
1
H
C
B
E
D
G
F
A
2
C
D
A
B
G
H
E
F
3
D
E
F
A
B
C
H
G
4
E
F
H
G
A
B
D
C
5
F
G
E
H
C
A
B
D
6
G
H
D
C
F
E
A
B
7
B
A
G
F
H
D
C
E
B
0
0
1
2
1
0
3
0
C
3
0
0
0
0
0
1
3
D
0
0
3
0
2
0
0
2
E
1
1
0
2
0
2
0
1
F
2
0
3
0
2
0
0
0
G
1
0
0
3
0
3
0
0
H
0
1
0
0
1
2
3
0
COE matrix
RRT
META’08, October 2008
A
B
C
D
E
F
G
H
A
0
5
0
0
1
0
0
1
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Metaheuristics for optimization problems in sports
Carry-over effects matrix
• RRT and carry-over effects matrix (COEM)
A
B
C
D
E
F
G
H
1
H
C
B
E
D
G
F
A
2
C
D
A
B
G
H
E
F
3
D
E
F
A
B
C
H
G
4
E
F
H
G
A
B
D
C
5
F
G
E
H
C
A
B
D
6
G
H
D
C
F
E
A
B
7
B
A
G
F
H
D
C
E
B
0
0
1
2
1
0
3
0
C
3
0
0
0
0
0
1
3
D
0
0
3
0
2
0
0
2
E
1
1
0
2
0
2
0
1
F
2
0
3
0
2
0
0
0
G
1
0
0
3
0
3
0
0
H
0
1
0
0
1
2
3
0
COE Matrix
RRT
META’08, October 2008
A
B
C
D
E
F
G
H
A
0
5
0
0
1
0
0
1
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Metaheuristics for optimization problems in sports
Carry-over effects value
A
B
C
D
E
F
G
H
A
0
5
0
0
1
0
0
1
B
0
0
1
2
1
0
3
0
C
3
0
0
0
0
0
1
3
D
0
0
3
0
2
0
0
2
E
1
1
0
2
0
2
0
1
F
2
0
3
0
2
0
0
0
G
1
0
0
3
0
3
0
0
H
0
1
0
0
1
2
3
0
COEMDG = 3
COEMFH = 2
COE matrix
H
H
COEV   (COEMij ) 2
i A j A
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Metaheuristics for optimization problems in sports
Carry-over effects value
A
B
C
D
E
F
G
H
A
0
5
0
0
1
0
0
1
B
0
0
1
2
1
0
3
0
C
3
0
0
0
0
0
1
3
D
0
0
3
0
2
0
0
2
E
1
1
0
2
0
2
0
1
F
2
0
3
0
2
0
0
0
G
1
0
0
3
0
3
0
0
H
0
1
0
0
1
2
3
0
COEMDG = 3
COEMFH = 2
COE Matrix
Minimize!!!
H
H
COEV   (COEMij ) 2
i A j A
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Metaheuristics for optimization problems in sports
Example
• Karate-Do competitions
• Groups playing round-robin tournaments
– Pan-american Karate-Do championship
– Brazilian classification for World Karate-Do
championship
• Open weight categories
– Physically strong contestants may fight weak
ones
– One should avoid that a competitor benefits
from fighting (physically) tired opponents
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Metaheuristics for optimization problems in sports
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coming from matches
against
strong athletes
Problem statement
• Find a schedule with minimum COEV
– RRT distributing the carry-over effects as evenly
as possible among the teams
• Best solution approaches to date in
literature:
– Random generation of 1-factors permutations
– Constraint Programming
– Combinatorial designs
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Metaheuristics for optimization problems in sports
Solution approach
• Multi-start + ILS heuristic
• Solutions represented by 1-factorizations
– Canonical factorizations
– Binary 1-factorizations
• Constructive algorithms
– Rearragment of the 1-factors of a solution (TSPlike greedy algorithms)
• Nearest neighbor
• Arbitrary insertion
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Metaheuristics for optimization problems in sports
Solution approach
• Local search
– Rearrangement of the 1-factors of the solution
(TSP-like procedures)
– Partial Round Swap (PRS)
• Pertubations
– Ejection chain: Game Rotation (GR)
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Metaheuristics for optimization problems in sports
Multi-start + ILS heuristic
• Multi-start phase: generation of 10,000
solutions
– 50% based on canonical 1-factorizations
– 50% based on binary 1-factorizations (whenever
possible)
– Constructive methods applied to the 1-factors of
the 1-factorizations
– Local search
• Best solution of the multi-start phase is the
input for the ILS algorithm
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Metaheuristics for optimization problems in sports
Multi-start + ILS heuristic
For try = 1 to 10000 Do
S ← Initial_Solution();
S ← Local_Search(S);
S* ← Update_Best_Solution(S, S*);
End-For;
S ← S*;
While Not Stopping-Criterion Do
S' ← Pertubation(S);
S' ← Local_Search(S’);
S ← Acceptance_Criterion(S, S');
S* ← Update_Best_Solution(S, S*);
End-While;
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Metaheuristics for optimization problems in sports
Results
• Literature: instances with up to 20 teams
Teams Best results Our results
4
12
12
6
60
60
8
56
56
10
108
108
12
176
160
14
234
254
16
240
240
18
340
400
20
380
486
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Metaheuristics for optimization problems in sports
Future research
• Weighted COEV minimization problem
• Weighted COEV min-max problem
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Metaheuristics for optimization problems in sports
Applications of metaheuristics
Scheduling the Brazilian
basketball tournament
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Metaheuristics for optimization problems in sports
Perspectives and concluding remarks
• Optimization in sports is a field of increasing interest
• Sports management and scheduling are very attractive
areas for applications of Operations Research
• Many interesting applications, often reviewed by the
media
• Several problems with interesting theoretical structure
• Even small instances are hard to solve (e.g., TTP for
n=10)
• Quick construction procedures to build good initial
(feasible) solutions are a must
• Repair procedures
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Metaheuristics for optimization problems in sports
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• Successful applications of metaheuristics
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Metaheuristics for optimization problems in sports