League championship algorithm

• Since the 1970s that the idea of a general algorithmic framework, which
can be applied with relatively few modifications to different optimization
problems, emerged.
• Metaheuristics: methods that combine rules and randomness while
imitating natural phenomena.
• These methods are from now on regularly employed in all the sectors of
business, industry, engineering.
• besides all of the interest necessary to application of metaheuristics,
occasionally a new metaheuristic algorithm is introduced that uses a
novel metaphor as guide for solving optimization problems.
2
League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan

Some examples
• particle swarm optimization algorithm (PSO): models the flocking
behavior of birds;
• harmony search (HS): models the musical process of searching for
a perfect state of harmony;
• bacterial foraging optimization algorithm (BFOA): models
foraging as an optimization process where an animal seeks to
maximize energy per unit time spent for foraging;
• artificial bee colony (ABC): models the intelligent behavior of
honey bee swarms;
• central force optimization (CFO): models the motion of masses
moving under the influence of gravity;
• imperialist competitive algorithm (ICA): models the imperialistic
competition between countries;
• fire fly algorithm (FA): performs based on the idealization of the
flashing characteristics of fireflies.
3

4
Metaheuristics
Evolutionary
algorithms
Trajectory
methods
Social, political,
music, sport , etc
Are inspired by nature’s
capability to evolve living
beings well adapted to
their environment
Evolution strategies
 Genetic programming
Genetic algorithm
Swarm
intelligence
Tabu search
Variable neighborhood
search
Ant colony optimization
Particle swarm optimization
Artificial bee colony
Bacterial foraging
optimization
Group search optimizer
Harmony search
Society and civilization
Imperialist competitive
algorithm
League championship
algorithm
work on one or several
neighborhood structure(s)
imposed on the members
of the search space.
Any attempt to design algorithms
or distributed problem-solving
devices inspired by the collective
behavior of social insect colonies
and other animal societies

 A sports league is an organization that exists to provide a regulated competition for a
number of teams to compete in a specific sport.
 Formations are a method of positioning players on the pitch to allow a team to play
according to its pre-set tactics.
 The main aim of match analysis is:
to identify strengths (S) which can then be further built upon,
to identify weaknesses (W) which suggest areas for improvement,
to use data to try to counter opposing strengths (threats (T)) and exploit
weaknesses (opportunities (O))
 This kind of analysis is typically known as strengths/weaknesses/opportunities/
threats (SWOT) analysis
 The SWOT analysis, explicitly links internal (S/W) and external factors (O/T).
 Identification of SWOTs is essential because subsequent steps in the process of
planning for achievement of the selected objective may be derived from the SWOTs.
6 League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan

 In strategic planning there are four basic categories of matches for which
strategic alternatives can be considered:
S/T matches show the strengths in light of major threats from
competitors. The team should use its strengths to avoid or defuse threats.
S/O matches show the strengths and opportunities. Essentially, the team
should attempt to use its strengths to exploit opportunities.
W/T matches show the weaknesses against existing threats. Essentially,
the team must attempt to minimize its weaknesses and avoid threats.
These strategy alternatives are generally defensive.
W/O matches illustrate the weaknesses coupled with major
opportunities. The team should try to overcome its weaknesses by taking
advantage of opportunities.
 The SWOT analysis provides a structured approach to conduct the gap
analysis. A gap is “the space between where we are and where we want
to be”.
 A transfer is the action taken whenever a player moves between clubs.
7

8 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
 LCA, is a population based algorithmic framework for global
optimization over a continuous search space.
 A common feature among all population based algorithms is that
they attempt to move a population of possible solutions to
promising areas of the search space, in terms of the problem’s
objective, during seeking the optimum.

Sporting terminology
(LCA)
League
week
Team i
formation
playing strength
Maximum iterations
Evolutionary
terminology
Population
iteration
ith member in the
population
solution
fitness value
Number of seasons

10
1) It is more likely that a team with better playing strength wins the game.
2) The outcome of a game is not predictable given known the teams’ playing
strength perfectly. It is not unlikely that the world leading FC BARCELONA loses
the game to ZORRAT-KARANE-PARS-ABAD from Iranian 3rd soccer division.
3) The probability that team i beats team j is assumed equal from both teams point
of view.
4) The outcome of the game is only win or loss (We will later break this rule).
5) Any strength helped team i to win from team j has a dual weakness caused j to
lose. In other words, any weakness is a lack of a particular strength.
6) Teams only focus on their upcoming match without regards of the other future
matches. Formation settings are done just based on the previous week events.
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan

11
an n dimensional numerical function that should be minimized
over the decision space defined by
A formation (a potential solution) for team i at week t
indicates the fitness/function value resultant from
the best formation for team i experienced till week t
 To determine , a greedy selection is done at each iteration as follows:
:)),...,,(( 21 nxxxXf 
ndxxx ddd ,..,1,maxmin

:),...,,( 21
t
in
t
i
t
i
t
i
xxxX 
ifEnd
BB
ifElse
XB
BfXfIf
t
i
t
i
t
i
t
i
t
i
t
i
;
;
)()(
1
1





t
i
X
:),...,,( 21
t
in
t
i
t
i
t
i
bbbB 
:)( t
i
Xf
t
iB

is
t< S×(L-1)
?
Week 1 Week 2 .
.
Week L-1
Team 1
Team 2
Team L
1. t=1
2. initialize team
formations
3. initialize best
formations
A League schedule is generated
1. Through an artificial
match analysis, changes
are done in the team
formation (new solution)
2. The playing strength
along with the resultant
formation is determined
(fitness calculation)
3. current best formation
is updated.
Teams play in pairs based on the league
schedule at week t, and winner/ loser are
determined using a playing strength
based criterion;
Is it the
end of the
season?
YES
Do possible transfers
for each team
Terminate
NO
Week 1 Week 2 Week L-1
Team 1
Team 2
Team L
NO
YES
Start

 In an ideal league environment we can assume a linear
relationship between the team’s playing strength and the outcome
of its game.
 proportional to its playing strength, each team may have a chance
to win (idealized rule 2)
 we determine the winner/loser in a stochastic manner by allowing
teams to have their chance of win based on their degree of fit
 The degree of fit is proportional to the team’s playing strength and
is measured based on the distance with an ideal reference point.
14

 We assume that a better team can comply with more factors that an
ideal team owns.
 Consider teams i and j to fight at week t. Define as the expected
chance of team i to beat team j at week t and
idealized rule 1
idealized rule 3
Since teams are evaluated based on their distance with a common
reference, the ratio of distances determines the winning portions.
A random number in [0,1] is generated, if it is less than or equal to
team i wins and team j losses; otherwise j wins and i losses
(idealized rule 4).
t
i
t
j
tt
j
tt
i
p
p
fXf
fXf



)(
)(
1 t
j
t
i pp
tt
i
t
j
tt
jt
i
fXfXf
fXf
p
2)()(
)(



t
ip
t
ip
)}({min
,...,1
t
iLi
t
Bff



l= Index of the team that will play with team i based on the league
schedule at week t+1.
j= Index of the team that has played with team i based on the
league schedule at week t.
k= Index of the team that has played with team l based on the
league schedule at week t.

YesNo
Could we WIN
the game from
team j at week t
?
Idealized rule 5
the loss is
directly due to
our WEAKNESSES
the success is directly
due to the
WEAKNESSES of team j
the success is
directly due to
our STRENGTHES
the loss is directly due
to the STRENGTHES of
team j
Artificial match analysis doing by team i (S/W evaluation)
17
Idealized rule 5

Artificial match analysis doing by team i (O/T evaluation)
18
Could our
opponent WIN
the game from
team k at week
t ?
No
the opponent’s
style of play
might be a
direct THREAT
the opponent’s
style of play
might be a direct
OPPORTUNITY
Threats are the
results of their
playing
STRENGTHES
Opportunities
are the results
of their playing
WEAKNESSES
Focusing on the
STRENGTHES of team
k, gives us a way of
affording the possible
opportunities
Focusing on the
WEAKNESSES of
team k, gives us a way
of avoiding the
possible threats
Idealized
rule 5
Idealized
rule 5

i was winner
l was winner
Focusing on …
i was winner
l was loser
Focusing on …
i was loser
l was winner
Focusing on …
i was loser
l was loser
Focusing on …
S
own strengths
(or weaknesses of j)
own strengths
(or weaknesses of j)
- -
W - -
own weaknesses
(or strengths of j)
own weaknesses
(or strengths of j)
O -
weaknesses of l
(or strengths of k)
-
weaknesses of l
(or strengths of k)
T
strengths of l
(or weaknesses of k)
-
strengths of l
(or weaknesses of k)
-
S/T
strategy
S/O
strategy
W/T
strategy
19
W/O
strategy

 Assume that team k has won the game from team l. To beat l, it is
reasonable that team i devises a playing style rather similar to that
was adopted by team k at week t .
 By “ ” we address the gap between the playing style of
team i and team k, sensed via “focusing on the strengths of team k”.
 In a similar way we can interpret “ ” when “focusing on the
weaknesses of team k”.
 In other words, it may be reasonable to avoid a playing style rather
similar to that was adopted by team k.
 We can interpret “ ” or “ ” in a similar manner.
t
i
t
k
XX 
20
t
k
t
i
XX 
t
i
t
j XX  t
j
t
i XX 

21
If i was winner and l was winner, then
(S/T equation):
Else if i was winner and l was loser, then
(S/O equation):
Else if i was loser and l was winner, then
(W/T equation):
Else if i was loser and l was loser, then the
(W/O equation):
End if
))()(( 2111
1 t
jd
t
id
t
kd
t
id
t
id
t
id
t
id xxrxxrybx 

))()(( 2112
1 t
jd
t
id
t
id
t
kd
t
id
t
id
t

))()(( 1221
1 t
id
t
jd
t
kd
t
id
t
id
t
id
t

))()(( 1222
1 t
id
t
jd
t
id
t
kd
t
id
t
id
t

nd ,...,1
nd ,...,1
nd ,...,1
nd ,...,1

 In above formulas we rely upon the fact that normally teams play based
on their current best formation (that found it suitable over the times),
while preparing the required changes recommended by the match
analysis.
 and are constant coefficients used to scale the contribution of
“retreat” or “approach” components, respectively.
 the diversification is controlled by allowing to “retreat” from a solution
and also by coefficient , while the intensification is implicitly
controlled by getting “approach” to a solution and by coefficient .
 We refer the above system of updating equations as LCA/recent since
they use the teams’ most recent formation as a basis to determine the
new formations.
22
1 2
1
2

23
If i was winner and l was winner, then
(S/T equation):
Else if i was winner and l was loser, then
(S/O equation):
Else if i was loser and l was winner, then
(W/T equation):
Else if i was loser and l was loser, then the
(W/O equation):
End if
))()(( 2111
1 t
jd
t
id
t
kd
t
id
t
id
t
id
t
id bbrbbrybx 

))()(( 2112
1 t
jd
t
id
t
id
t
kd
t
id
t
id
t

))()(( 1221
1 t
id
t
jd
t
kd
t
id
t
id
t
id
t

))()(( 1222
1 t
id
t
jd
t
id
t
kd
t
id
t
id
t

nd ,...,1
nd ,...,1
nd ,...,1
nd ,...,1

24
 It is unusual that coaches do changes in all or many aspects of the
team. normally a few number of changes are devised.
 To simulate the number of changes ( ) made in , we
use a truncated geometric distribution.
 Where r is a random number in [0,1] and is a control
parameter. is the least number of changes realized during the
artificial match analysis
 number of dimensions are selected randomly from and their
value is changed according to one of the Equations
t
iB


n
d
id
t
i
yq
1
t
iq
t
iB
},...,1,{:1
)1ln(
)))1(1(1ln(
000
10
nqqqq
p
rp
q t
i
c
qn
ct
i 









)1,0(cp
0q

25


n
i
i
xxf
1
2
1
)(

]100,100[ix





1
1
22
1
2
2
)1()(100)(
n
i
iii
xxxxf

]048.2,048.2[ix


n
i
ii
xxxf
1
2
3
)10)2cos(10()( 

]12.5,12.5[ix
ex
n
x
n
xf
n
i i
n
i i












20))2cos(.1exp(
.12.0exp20)(
1
1
2
4


]76.32,76.32[ix


n
i
ii
xxnxf
1
5
)sin(9829.418)(

]97.511,03.512[ix

26
 Comparison is done between LCA and the highly recognized (PSO)
algorithm
10L
1000S
5.01 
5.02 
01.0Cp
1.09.0  linear
w
21 c
minmax/minmax/ xv 
10particles
N
9000iterationsN
22 c

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
Week/Iteration
f(X)
Mean of best values for
LCA
PSO
Rosenbrock function
0 500 1000 1500 2000 2500
10
-8
10
-6
10
-4
10
-2
10
0
10
2
10
4
Week/Iteration
f(X)
LCA
PSO
Sphere function
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
Week/Iteration
f(X)
LCA
PSO
Rastrigin function
0 500 1000 1500 2000 2500
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
Week/Iteration
f(X)
LCA
PSO
Ackley function
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
Week/Iteration
f(X)
LCA
PSO
Schwefel function
28

29
Week 1 Week 5
Week 10 Week 20

In order to see that whether each of S/T, S/O, W/T and
W/O updating equations has a significant effect on the
performance of LCA, we sequentially omit the possible
effect that each equation might have on the evolution
of the solutions.
31

32
0 200 400 600 800 1000 1200 1400 1600 1800
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Weeks
LCA/best/omitting S/T equation
LCA/best/omitting S/O equation
LCA/best/omitting W/T equation
LCA/best/omitting W/O equation
LCA/best
0 50 100 150 200 250 300 350
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
10
2
10
4
Weeks
LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
10
-15
10
-10
10
-5
10
0
10
5
Weeks
LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
10
0
10
2
10
4
10
6
10
8
Weeks
LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
10
2
10
4
Weeks
LCA/best

Learning from team’s previous game only
If i was winner, then
(S equation):
Else if i was loser, then
(W equation):
End if
Learning from opponent’s previous game only
If l was winner, then
(T equation):
Else if l was loser, then
(O equation):
End if
33
))(( 11
1 t
jd
t
id
t
id
t
id
t
id bbrybx 
 nd ,...,1
))(( 12
1 t
id
t
jd
t
id
t
id
t
 nd ,...,1
))(( 11
1 t
kd
t
id
t
id
t
id
t
id
bbrybx 

))(( 12
1 t
id
t
kd
t
id
t
id
t

nd ,...,1
nd ,...,1

34
0 200 400 600 800 1000 1200 1400 1600 1800
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
Weeks
LCA/best/Learning from
team's previous game only
opponent's previous game only
LCA/best
0 50 100 150 200 250 300 350 400 450 500
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
10
2
10
4
Weeks
LCA/best/Learning from team's previous game only
LCA/best/Learning from opponent's previous game only
LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
10
-15
10
-10
10
-5
10
0
10
5
Weeks
LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
10
2
10
4
Weeks
f(X)
team's previous game only
opponent's previous game only
LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
10
0
10
2
10
4
10
6
10
8
Weeks
LCA/best

• Interestingly, these empirical results are in accordance with the
business reality.
• In business strategy there are two schools of thought, the
“environmental (external)” and the “resource based (internal)”.
• Through 1970s and 80s, the dominant school was the
environmental school which dictates that a firm should analyze
the forces present within the environment in order to asses the
profit potential of the industry.
• Nevertheless, above average performance is more likely to be
the result of core capabilities inherent in a firm’s resources
(internal view) than its competitive positioning in its industry
(external view).
35

Tie outcome is interpreted as the consequent of the strengths/
opportunities and weaknesses/threats
36

Tie outcome is neutral. There is no learning from ties
37

Tie outcome is randomly interpreted as win or loss
For example, in this situation, under the case of “Else if i was winner
and l had tied” the new formation is set up as follows:
Tie outcome is interpreted as win
If i had won/tied and l had won/tied, then use (S/T) equation to setup a
new formation
Else if i had won/tied and l was loser, then use (S/O) equation setup a
new formation
Else if i was loser and l had won/tied, then use (W/T) equation to setup
a new formation
Else if i was loser and l was loser, then use (W/O) equation to setup a
new formation
End if
38
))())(1()(( 312211
1 t
jd
t
id
t
id
t
kdi
t
kd
t
idi
t
id
t
id
t
id bbrbburbburybx 


Tie outcome is interpreted as loss
If i was winner and l was winner, then use (S/T) equation to setup a new
formation
Else if i was winner and l had lost/tied, then use (S/O) equation setup a
new formation
Else if i had lost/tied and l was winner, then use (W/T) equation to
setup a new formation
Else if i had lost/tied and l had lost/tied, then use (W/O) equation to
setup a new formation
End if
39

40
0 200 400 600 800 1000 1200 1400 1600 1800
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
Weeks
LCA/best/win-loss-tie 1
LCA/best
0 50 100 150 200 250
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
10
2
10
4
Weeks
LCA/best
0 1000 2000 3000 4000 5000 6000 7000
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
10
2
10
4
Weeks
LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
10
2
10
4
Weeks
LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 900
10
0
10
2
10
4
10
6
10
8
Weeks
LCA/best

42
0 200 400 600 800 1000 1200 1400 1600 1800
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
Weeks
LCA/best/Tr=0.1
LCA/best/Tr=0.3
LCA/best/Tr=0.5
LCA/best/Tr=0.7
LCA/best/Tr=0.9
LCA/best
0 20 40 60 80 100 120 140 160 180 200
10
-15
10
-10
10
-5
10
0
10
5
Weeks
LCA/best/Tr=0.1
LCA/best/Tr=0.3
LCA/best/Tr=0.5
LCA/best/Tr=0.7
LCA/best/Tr=0.9
LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
10
2
10
4
Weeks
LCA/best/Tr=0.1
LCA/best/Tr=0.3
LCA/best/Tr=0.5
LCA/best/Tr=0.7
LCA/best/Tr=0.9
LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
10
2
10
4
LCA/best/Tr=0.1
LCA/best/Tr=0.3
LCA/best/Tr=0.5
LCA/best/Tr=0.9
LCA/best/Tr=0.9
LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
10
0
10
2
10
4
10
6
10
8
10
10
Weeks
LCA/best/Tr=0.1
LCA/best/Tr=0.3
LCA/best/Tr=0.5
LCA/best/Tr=0.7
LCA/best/Tr=0.9
LCA/best

Refrence
• Husseinzadeh Kashan, A., "League
Championship Algorithm (LCA): An algorithm
for global optimization inspired by sport
championships,“, 2014.
• Drkashan.ir
43

League championship algorithm

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