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The League Championship Algorithm: A new algorithm for numerical
function optimization
By: A. H. Kashan
• 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
Introduction
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 League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan
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 programmingGenetic algorithm
Swarm intelligence
Tabu searchVariable neighborhood
search
Ant colony optimizationParticle swarm optimizationArtificial bee colony Bacterial foraging
optimization Group search optimizer
Harmony searchSociety and civilizationImperialist competitive
algorithmLeague championship
algorithm
work on one or severalneighborhood 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
League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan
The league championship algorithm (LCA)
A review on the sporting terminology and required background
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
A review on the sporting terminology and required background
League Championship Algorithm: A new algorithm for numerical function optimization By: Dr. A. H. Kashan
8
LCA as an EA
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.
mutation
recombination
Fitness evaluation
selection
9
Metaphores
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
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
Idealized rules
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
Notations
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
iX
:),...,,( 21
t
in
t
i
t
i
t
i bbbB
:)( t
iXf
t
iB
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
ist< S×(L-1)
?
Week 1 Week 2 . .
Week L-1
Team 1
Team 2
Team L
1. t=12. 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
t +1
t
Start
13
Generating the league schedule
1 2 3 4
5 6 7 8week 1 week 2
1 2 3 4
5 6 7 8week 3
1 5 2 3
6 7 8 4
week 4
1 6 5 2
7 8 4 3week 7
1 3 4 8
2 5 6 7
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
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
Determining winner/loser
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
Determining winner/loser
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 pptt
i
t
j
tt
jt
i fXfXf
fXfp
2)()(
)(
t
ip
t
ip
)}({min,...,1
t
iLi
t Bff
15 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
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.
Setting up a new formationfor team i
16 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
Yes
No 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
Setting up a new formation for team i
Idealized rule 5
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
Setting up a new formation for team i
Artificial match analysis doing by team i (O/T evaluation)18
Could our opponent WIN the game from team k at week
t ?
No
Yes 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
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
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 …
Sown 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)
Tstrengths of l
(or weaknesses of k)-
strengths of l
(or weaknesses of k)-
S/T strategy
S/O strategy
W/T strategy
Setting up a new team formation
19
W/O strategy
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
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.
Setting up a new team formation
t
i
t
k XX
20
t
k
t
i XX
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
ti
tj XX t
jti XX
21
Setting up a new team formation
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
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
Setting up a new team formation
1 2
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
12
23
LCA/best: A variant
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
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
How big would be the number of changes?
t
iB
n
d id
t
i yq1
t
iq
t
iB
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
},...,1,{ : 1)1ln(
)))1(1(1ln(000
10
nqqqqp
rpq t
i
c
qn
ct
i
)1,0(cp0q
26
n
i ixxf1
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 xxxf1
2
3 )10)2cos(10()( ]12.5,12.5[ix
exn
xnxf
n
i i
n
i i
20))2cos(.1exp(
.12.0exp20)(
1
1
24
] 76.32 , 76.32[ix
n
i ii xxnxf15 )sin( 9829.418)(
]97.511,03.512[ix
Test functions
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
27
Parameter settings
Comparison is done between LCA and the highly recognized (PSO) algorithm
LCA PSO10L
1000S
5.01 5.02
01.0Cp
1.0 9.0 linearw
21 c
minmax/minmax/ xv
10particlesN
9000iterationsN
22 c
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
28
Comparison study
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010
-5
10-4
10-3
10-2
10-1
100
101
102
103
Week/Iteration
f(X
)
Mean of best values for
LCAPSO
Rosenbrock function
0 500 1000 1500 2000 250010
-8
10-6
10-4
10-2
100
102
104
Week/Iteration
f(X
)
Mean of best values for
LCAPSO
Sphere function
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
Week/Iteration
f(X
)
Mean of best values for
LCAPSO
Rastrigin function
0 500 1000 1500 2000 250010
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
Week/Iteration
f(X
)
Mean of best values for
LCAPSO
Ackley function
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010
-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
104
Week/Iteration
f(X
)
Mean of best values for
LCAPSO
Schwefel function
Comparison study
29
Visualization on Six-Hump Camelback function
30
x1
x2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x1
x2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Week 1 Week 5
x1
x2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x1
x2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Week 10 Week 20
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
31
x1
x2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x1
x2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Week 50 Week 100
Visualization on Six-Hump Camelback function
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
Effect of LCA updating equations
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.
32 League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
33
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
100
Weeks
LCA/best/omitting S/T equationLCA/best/omitting S/O equationLCA/best/omitting W/T equationLCA/best/omitting W/O equationLCA/best
0 50 100 150 200 250 300 35010
-14
10-12
10-10
10-8
10-6
10-4
10-2
100
102
104
Weeks
LCA/best/omitting S/T equationLCA/best/omitting S/O equationLCA/best/omitting W/T equationLCA/best/omitting W/O equationLCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 900010
-15
10-10
10-5
100
105
Weeks
LCA/best/omitting S/T equationLCA/best/omitting S/O equationLCA/best/omitting W/T equationLCA/best/omitting W/O equationLCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
100
102
104
106
108
Weeks
LCA/best/omitting S/T equationLCA/best/omitting S/O equationLCA/best/omitting W/T equationLCA/best/omitting W/O equationLCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 900010
-14
10-12
10-10
10-8
10-6
10-4
10-2
100
102
104
Weeks
LCA/best/omitting S/T equationLCA/best/omitting S/O equationLCA/best/omitting W/T equationLCA/best/omitting W/O equationLCA/best
Effect of LCA updating equations
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
Learning from team’s previous game onlyIf i was winner, then
(S equation): Else if i was loser, then
(W equation): End if
Learning from opponent’s previous game onlyIf l was winner, then
(T equation): Else if l was loser, then
(O equation): End if
34
11 1( ( ))t t t t t
id id id id id jdx b y r b b nd ,...,1
12 1( ( ))t t t t t
id id id id jd idx b y r b b nd ,...,1
Effect of adopting different learning strategies in the artificial post-match analysis
11 1( ( ))t t t t t
id id id id id kdx b y r b b
12 1( ( ))t t t t t
id id id id kd idx b y r b b
nd ,...,1
nd ,...,1
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
35
0 200 400 600 800 1000 1200 1400 1600 180010
-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Weeks
LCA/best/Learning fromteam's previous game onlyLCA/best/Learning fromopponent's previous game only
LCA/best
0 50 100 150 200 250 300 350 400 450 50010
-14
10-12
10-10
10-8
10-6
10-4
10-2
100
102
104
Weeks
LCA/best/Learning from team's previous game only
LCA/best/Learning from opponent's previous game onlyLCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 900010
-15
10-10
10-5
100
105
Weeks
LCA/best/Learning from team's previous game only
LCA/best/Learning from opponent's previous game onlyLCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 900010
-14
10-12
10-10
10-8
10-6
10-4
10-2
100
102
104
Weeks
f(X
)
LCA/best/Learning fromteam's previous game onlyLCA/best/Learning fromopponent's previous game only
LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
100
102
104
106
108
Weeks
LCA/best/Learning from team's previous game only
LCA/best/Learning from opponent's previous game onlyLCA/best
Effect of adopting different learning strategies in the artificial post-match analysis
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
• 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).
36
Effect of adopting different learning strategies in the artificial post-match analysis
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
Tie outcome is interpreted as the consequent of the strengths/ opportunities and weaknesses/threats (beside the four conditions used in LCA/best the following conditions are also used)
37
Inclusion of the tie outcome
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
Tie outcome is neutral. There is no learning from ties (beside the four conditions used in LCA/best the following conditions are also used)
38
Inclusion of the tie outcome
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
Tie outcome is randomly interpreted as win or lossFor 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 winIf i had won/tied and l had won/tied, then use (S/T) equation to setup a new
formationElse if i had won/tied and l was loser, then use (S/O) equation setup a new
formationElse if i was loser and l had won/tied, then use (W/T) equation to setup a new
formationElse if i was loser and l was loser, then use (W/O) equation to setup a new
formationEnd if
39
11 1 2 2 1 3( ( ) (1 )( ) ( ))t t t t t t t t t
id id id id i id kd id i kd id id id jdx b y r u b b r u b b r b b
Inclusion of the tie outcome
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
Tie outcome is interpreted as lossIf i was winner and l was winner, then use (S/T) equation to setup a new
formationElse if i was winner and l had lost/tied, then use (S/O) equation setup a
new formationElse if i had lost/tied and l was winner, then use (W/T) equation to
setup a new formationElse if i had lost/tied and l had lost/tied, then use (W/O) equation to
setup a new formationEnd if
40
Inclusion of the tie outcome
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
41
0 200 400 600 800 1000 1200 1400 1600 180010
-12
10-10
10-8
10-6
10-4
10-2
100
Weeks
LCA/best/win-loss-tie 1LCA/best/win-loss-tie 2LCA/best/win-loss-tie 3LCA/best/win-loss-tie 4LCA/best/win-loss-tie 5LCA/best
0 50 100 150 200 25010
-14
10-12
10-10
10-8
10-6
10-4
10-2
100
102
104
Weeks
LCA/best/win-loss-tie 1LCA/best/win-loss-tie 2LCA/best/win-loss-tie 3LCA/best/win-loss-tie 4LCA/best/win-loss-tie 5LCA/best
0 1000 2000 3000 4000 5000 6000 7000
10-12
10-10
10-8
10-6
10-4
10-2
100
102
104
Weeks
LCA/best/win-loss-tie 1LCA/best/win-loss-tie 2LCA/best/win-loss-tie 3LCA/best/win-loss-tie 4LCA/best/win-loss-tie 5LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 900010
-14
10-12
10-10
10-8
10-6
10-4
10-2
100
102
104
Weeks
LCA/best/win-loss-tie 1LCA/best/win-loss-tie 2LCA/best/win-loss-tie 3LCA/best/win-loss-tie 4LCA/best/win-loss-tie 5LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
100
102
104
106
108
Weeks
LCA/best/win-loss-tie 1LCA/best/win-loss-tie 2LCA/best/win-loss-tie 3LCA/best/win-loss-tie 4LCA/best/win-loss-tie 5LCA/best
Inclusion of the tie outcome
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
42
Inclusion of the end season transfers
• “transfer” is referred to as the action taken whenever a player moves between clubs.
• Likewise in LCA we can introduce a transfer like operator with the aim of speeding up the convergence of the algorithm.
• At the end of each season transfers are allowed for team i.
• The procedure of the transfer operator is as follows:
43
Inclusion of the end season transfers
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
44
0 200 400 600 800 1000 1200 1400 1600 180010
-14
10-12
10-10
10-8
10-6
10-4
10-2
100
Weeks
LCA/best/Tr=0.1LCA/best/Tr=0.3LCA/best/Tr=0.5LCA/best/Tr=0.7LCA/best/Tr=0.9LCA/best
0 20 40 60 80 100 120 140 160 180 20010
-15
10-10
10-5
100
105
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 900010
-14
10-12
10-10
10-8
10-6
10-4
10-2
100
102
104
Weeks
LCA/best/Tr=0.1LCA/best/Tr=0.3LCA/best/Tr=0.5LCA/best/Tr=0.7LCA/best/Tr=0.9LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 900010
-14
10-12
10-10
10-8
10-6
10-4
10-2
100
102
104
LCA/best/Tr=0.1LCA/best/Tr=0.3LCA/best/Tr=0.5LCA/best/Tr=0.9LCA/best/Tr=0.9LCA/best
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
100
102
104
106
108
1010
Weeks
LCA/best/Tr=0.1LCA/best/Tr=0.3LCA/best/Tr=0.5LCA/best/Tr=0.7LCA/best/Tr=0.9LCA/best
Inclusion of the end season transfers
League Championship Algorithm: A new algorithm for numerical function optimization By: A. H. Kashan
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