Since the 1970s that the idea of a general algorithmic framework, which can be applied with...

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


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


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


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.


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


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


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


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


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


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


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


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


ti

tj XX t

jti XX

21



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


12

23

LCA/best: A variant


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


},...,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


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


28

Comparison study


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

)


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

)


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

)


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

)


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


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


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.


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


0 1000 2000 3000 4000 5000 6000 7000 8000 900010

-15

10-10

10-5

100

105

Weeks


0 1000 2000 3000 4000 5000 6000 7000 8000 9000

100

102

104

106

108

Weeks


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


Effect of LCA updating equations


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


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



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





• 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



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


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



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



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



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


0 1000 2000 3000 4000 5000 6000 7000

10-12

10-10

10-8

10-6

10-4

10-2

100

102

104

Weeks


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


0 1000 2000 3000 4000 5000 6000 7000 8000 9000

100

102

104

106

108

Weeks




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



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


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


0 1000 2000 3000 4000 5000 6000 7000 8000 9000

100

102

104

106

108

1010

Weeks




Thanks for your attention!

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Since the 1970s that the idea of a general algorithmic framework, which can be applied with...

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