• 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
League Championship Algorithm: A new algorithm for numerical function optimization
By: Dr. A. H. Kashan
Any attempt to design algorithms
work on one or several
or distributed problem-solving
Are inspired by nature’s
neighborhood structure(s)
devices inspired by the collective
capability to evolve living
imposed on the members
beings well adapted to of the search space.behavior of social insect colonies
their environment
Metaheuristicsand other animal societies
Evolutionary
algorithms
Trajectory
methods
Swarm
intelligence
Social, political,
music, sport , etc
Tabu search
Evolution strategies
Ant colony optimization
 Genetic programming Variable neighborhood Particle swarm optimization
search
Genetic algorithm
Artificial bee colony
Bacterial foraging
optimization
Group search optimizer
4
League Championship Algorithm: A new algorithm for numerical function optimization
Harmony search
Society and civilization
Imperialist competitive
algorithm
League championship
algorithm
By: Dr. A. H. Kashan
 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
League Championship Algorithm: A new algorithm for numerical function optimization
By: Dr. 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.
8 League Championship Algorithm: A new algorithm for numerical function optimization
By: A. H. Kashan
Sporting terminology
Evolutionary
(LCA)
terminology
League
Population
week
iteration
Team i
ith member in the
population
formation
solution
playing strength
fitness value
Maximum iterations
Number of seasons
9 League Championship Algorithm: A new algorithm for numerical function optimization
By: A. H. Kashan
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.
10 League Championship Algorithm: A new algorithm for numerical function optimization
By: A. H. Kashan
f ( X  ( x1 , x2 ,..., xn )) : an n dimensional numerical function that should be minimized
over the decision space defined by xdmin  xd  xdmax, d  1,..,n
X it  ( xit1 , xit2 ,..., xint ) : A formation (a potential solution) for team i at week t
f ( X it ) : indicates the fitness/function value resultant from X it
Bit  (bit1 , bit2 ,..., bint ) : the best formation for team i experienced till week t
 To determine Bit , a greedy selection is done at each iteration as follows:
If f ( X it )  f ( Bit 1 )
Bit  X it ;
Else if
Bit  Bit 1 ;
End if
11 League Championship Algorithm: A new algorithm for numerical function optimization
By: A. H. Kashan
Week 1
Week 2
.
.
Week L-1
Team 1
Team 2
Team L
A League schedule is generated
Teams play in pairs based on the league
schedule at week t, and winner/ loser are
determined using a playing strength
based criterion;
Terminate
YES
1. Through an artificial
match analysis, changes
1. t=1
are done in the team
2. initialize team formation (new solution)
formations
2. The playing strength
3. initialize best
along with the resultant
formations
formation is determined
(fitness calculation)
3. current best formation
is updated.
Start
is
t< S×(L-1)
?
NO
NO
Is it the
end of the
season?
YES
Week 1
Week 2
Week L-1
Team 1
Team 2
Team L
Do possible transfers
for each team
13 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
 We assume that a better team can comply with more factors that an
ideal team owns.
t
p
 Consider teams i and j to fight at week t. Define i as the expected
t
{ f ( Bit )}
chance of team i to beat team j at week t and f  imin
1,...,L
idealized rule 1
f (X )  f
 t
t
f (X )  f
pi
t
i
t
j
t
p tj
pit 
f ( X tj )  f t
f ( X tj )  f ( X it )  2 f t
p  p 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
pit team i wins and team j losses; otherwise j wins and i losses
(idealized rule 4).
t
i
t
j
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.
16 League Championship Algorithm: A new algorithm for numerical function optimization
By: A. H. Kashan
No
Could we WIN
the game from
team j at week t
?
Yes
the loss is
directly due to
our WEAKNESSES
the success is
directly due to
our STRENGTHES
Idealized rule 5
Idealized rule 5
the loss is directly due
to the STRENGTHES of
team j
the success is directly
due to the
WEAKNESSES of team j
Artificial match analysis doing by team i (S/W evaluation)
17 League Championship Algorithm: A new algorithm for numerical function optimization
By: A. H. Kashan
Could our
opponent WIN
the game from
team k at week
t?
No
the opponent’s
style of play
might be a direct
OPPORTUNITY
Threats are the
results of their
playing
STRENGTHES
Idealized
rule 5
Focusing on the
WEAKNESSES of
team k, gives us a way
of avoiding the
possible threats
Idealized
rule 5
Focusing on the
STRENGTHES of team
k, gives us a way of
affording the possible
opportunities
the opponent’s
style of play
might be a
direct THREAT
Opportunities
are the results
of their playing
WEAKNESSES
Artificial match analysis doing by team i (O/T evaluation)
18 League Championship Algorithm: A new algorithm for numerical function optimization
By: A. H. Kashan
S/T
strategy
S
W
S/O
strategy
W/T
strategy
W/O
strategy
i was winner
i was winner
i was loser
i was loser
l was winner
l was loser
l was winner
l was loser
Focusing on …
Focusing on …
Focusing on …
Focusing on …
own strengths
own strengths
(or weaknesses of j)
(or weaknesses of j)
-
-
-
-
own weaknesses
own weaknesses
(or strengths of j)
(or strengths of j)
weaknesses of l
O
(or strengths of k)
strengths of l
T
weaknesses of l
-
strengths of l
-
(or weaknesses of k)
(or strengths of k)
(or weaknesses of k)
19 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 “ X kt  X it ” 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 “ X it  X kt ” 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 “ X tj  X it ” or “ X it  X tj ” in a similar manner.
20 League Championship Algorithm: A new algorithm for numerical function optimization
By: A. H. Kashan
If i was winner and l was winner, then
t 1
t
t
t
t
t
t
x

b

y
(

r
(
x

x
)


r
(
x

x
d  1,...,n
(S/T equation): id
id
id
1 1
id
kd
1 2
id
jd ))
Else if i was winner and l was loser, then
(S/O equation): xidt 1  bidt  yidt ( 2 r1 ( xkdt  xidt )  1r2 ( xidt  x tjd )) d  1,...,n
Else if i was loser and l was winner, then
(W/T equation): xidt 1  bidt  yidt ( 1r2 ( xidt  xkdt )   2 r1 ( xtjd  xidt )) d  1,...,n
Else if i was loser and l was loser, then the
(W/O equation): xidt 1  bidt  yidt ( 2 r2 ( xkdt  xidt )   2 r1 ( xtjd  xidt )) d  1,...,n
End if
21 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.
 1 and  2 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  1 , while the intensification is implicitly
controlled by getting “approach” to a solution and by coefficient  2 .
 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
League Championship Algorithm: A new algorithm for numerical function optimization
By: A. H. Kashan
If i was winner and l was winner, then
(S/T equation): xidt 1  bidt  yidt ( 1r1 (bidt  bkdt )  1r2 (bidt  btjd )) d  1,...,n
Else if i was winner and l was loser, then
(S/O equation): xidt 1  bidt  yidt ( 2 r1 (bkdt  bidt )  1r2 (bidt  btjd )) d  1,...,n
Else if i was loser and l was winner, then
(W/T equation): xidt 1  bidt  yidt ( 1r2 (bidt  bkdt )  2 r1 (btjd  bidt )) d  1,...,n
Else if i was loser and l was loser, then the
(W/O equation): xidt 1  bidt  yidt ( 2 r2 (bkdt  bidt )  2 r1 (b tjd  bidt )) d  1,...,n
End if
23 League Championship Algorithm: A new algorithm for numerical function optimization
By: A. H. Kashan
 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 (q 
use a truncated geometric distribution.
t
i

n
d 1
y id ) made in Bit , we
n  q0 1

ln(
1

(
1

(
1

p
)
)r ) 
t
t
c
qi  

q

1
:
q
 {q0 , q0  1,..., n}
 0
i
ln(1  pc )


 Where r is a random number in [0,1] and pc  (0,1) is a control
parameter. q0 is the least number of changes realized during the
artificial match analysis
t
q
 i number of dimensions are selected randomly from Bit and their
value is changed according to one of the Equations
24 League Championship Algorithm: A new algorithm for numerical function optimization
By: A. H. Kashan

f1 ( x ) 


f 2 (x) 


f 3 ( x) 
n
i 1
x
xi  [100 ,100 ]
2
i
n 1
100 ( xi2  xi 1 ) 2  (1  xi ) 2
i 1

n
i 1
( xi2  10 cos(2xi )  10 )


f 4 ( x )  20 exp   0.2 1 .
n


x  
i 1



exp( 1 .
n
n
i 1

f 5 ( x )  418 .9829 n 
25
n
2
i
xi  [2.048 ,2.048 ]
xi  [5.12,5.12]
xi  [32.76 , 32.76 ]
cos(2xi ))  20  e

n
i 1
xi sin( xi )
xi  [512 .03,511 .97 ]
 Comparison is done between LCA and the highly recognized (PSO)
algorithm
L  10
N particles  10
S  1000
N iterations  9000
 1  0.5
w  0.9 linear

 0.1
 2  0.5
c1  2
pC  0.01
c2  2
vmax/ min  xmax/ min
26 League Championship Algorithm: A new algorithm for numerical function optimization
By: A. H. Kashan
27
Mean of best values for
3
10
4
Mean of best values for
3
Schwefel function
10
Rosenbrock function
LCA
PSO
2
10
10
LCA
PSO
2
10
1
10
1
10
0
0
10
f(X)
f(X)
10
-1
10
-1
10
-2
10
-2
10
-3
10
-3
10
-4
10
-4
10
-5
10
-5
10
-6
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10
10000
0
1000
2000
3000
Week/Iteration
Rastrigin function
6000
7000
8000
9000
10000
LCA
PSO
10
Sphere function
LCA
PSO
Ackley function
1
10
0
Mean of best values for
4
10
1
LCA
PSO
2
10
0
10
10
-1
-1
10
0
10
10
-2
f(X)
10
-3
10
-4
10
f(X)
-2
f(X)
Mean of best values for
2
10
10
5000
Week/Iteration
Mean of best values for
2
4000
-3
10
-4
10
-4
10
-5
-2
10
10
-5
10
10
-6
10
-6
-6
10
10
-7
10
28
-7
0
1000
2000
3000
4000
5000
6000
Week/Iteration
7000
8000
9000
10000
10
-8
0
500
1000
1500
Week/Iteration
2000
2500
10
0
500
1000
1500
Week/Iteration
2000
2500
29
Week 1
Week 5
Week 10
Week 20
Week 50
30
Week 100
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
0
4
10
10
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
-1
10
-2
10
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
2
10
0
10
-2
-3
10
-4
10
-5
10
-6
10
-7
10
10
-4
10
-6
10
-8
10
-10
10
-12
10
-8
10
-14
0
200
400
600
800
1000
1200
1400
1600
10
1800
0
50
100
150
Weeks
200
250
300
350
Weeks
4
5
10
10
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
10
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
2
10
8
10
0
10
6
-2
10
10
-4
10
-5
10
4
10
-6
10
-8
10
2
10
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
-10
10
-10
10
-12
10
-15
10
-14
0
1000
2000
3000
4000
Weeks
32
5000
6000
7000
8000
9000
10
0
1000
2000
3000
4000
0
10
5000
Weeks
6000
7000
8000
9000
0
1000
2000
3000
4000
Weeks
5000
6000
7000
8000
9000
Learning from team’s previous game only
If i was winner, then
(S equation): xidt 1  bidt  yidt ( 1r1 (bidt  btjd ))
Else if i was loser, then
(W equation):xidt 1  bidt  yidt ( 2 r1 (btjd  bidt ))
End if
Learning from opponent’s previous game only
If l was winner, then
t 1
t
t
t
t
x

b

y
(

r
(
b

b
(T equation): id
id
id
1 1
id
kd ))
Else if l was loser, then
t
 bidt ))
(O equation):xidt 1  bidt  yidt ( 2 r1 (bkd
End if
33
d  1,...,n
d  1,...,n
d  1,...,n
d  1,...,n
4
0
10
10
LCA/best/Learning from team's previous game only
LCA/best/Learning from opponent's previous game only
LCA/best
2
10
-2
10
0
10
-4
10
-2
10
-6
10
-4
10
-6
10
-8
10
-8
10
-10
10
LCA/best/Learning from
team's previous game only
LCA/best/Learning from
opponent's previous game only
LCA/best
-12
10
-14
10
0
200
400
600
800
-10
10
-12
10
-14
1000
1200
1400
1600
10
1800
0
50
100
150
200
Weeks
250
300
350
400
450
500
Weeks
4
5
10
10
LCA/best/Learning from team's previous game only
LCA/best/Learning from opponent's previous game only
LCA/best
LCA/best/Learning from team's previous game only
LCA/best/Learning from opponent's previous game only
LCA/best
2
10
8
10
0
10
0
10
-2
6
10
10
-4
f(X)
10
-5
10
4
10
-6
10
-8
10
2
10
-10
10
LCA/best/Learning from
team's previous game only
LCA/best/Learning from
opponent's previous game only
LCA/best
-10
10
-12
10
-15
10
-14
0
1000
2000
3000
4000
Weeks
34
5000
6000
7000
8000
10
9000
0
1000
2000
3000
4000
Weeks
0
10
5000
6000
7000
8000
9000
0
1000
2000
3000
4000
Weeks
5000
6000
7000
8000
9000
• 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:
xidt 1  bidt  yidt ( 1r1ui (bidt  bkdt )  2 r2 (1  ui )(bkdt  bidt )  1r3 (bidt  btjd ))
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
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
4
0
10
10
LCA/best/win-loss-tie 1
LCA/best/win-loss-tie 2
LCA/best/win-loss-tie 3
LCA/best/win-loss-tie 4
LCA/best/win-loss-tie 5
LCA/best
-2
10
-4
10
LCA/best/win-loss-tie 1
LCA/best/win-loss-tie 2
LCA/best/win-loss-tie 3
LCA/best/win-loss-tie 4
LCA/best/win-loss-tie 5
LCA/best
2
10
0
10
-2
10
-4
10
-6
10
-6
10
-8
-8
10
10
-10
10
-10
10
-12
10
-12
10
-14
0
200
400
600
800
1000
1200
1400
1600
10
1800
0
50
100
150
Weeks
200
250
Weeks
4
10
4
LCA/best/win-loss-tie 1
LCA/best/win-loss-tie 2
LCA/best/win-loss-tie 3
LCA/best/win-loss-tie 4
LCA/best/win-loss-tie 5
LCA/best
10
2
10
0
10
LCA/best/win-loss-tie 1
LCA/best/win-loss-tie 2
LCA/best/win-loss-tie 3
LCA/best/win-loss-tie 4
LCA/best/win-loss-tie 5
LCA/best
2
10
8
10
0
10
-2
6
10
10
-2
10
-4
10
-4
10
4
10
-6
10
-6
LCA/best/win-loss-tie 1
LCA/best/win-loss-tie 2
LCA/best/win-loss-tie 3
LCA/best/win-loss-tie 4
LCA/best/win-loss-tie 5
LCA/best
10
-8
10
-8
10
-10
10
-10
10
-12
10
-12
2
10
0
10
10
-14
0
1000
2000
3000
4000
Weeks
40
5000
6000
7000
10
0
1000
2000
3000
4000
Weeks
5000
6000
7000
8000
9000
0
1000
2000
3000
4000
Weeks
5000
6000
7000
8000
900
41
0
5
10
10
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
-2
10
0
10
-4
10
-6
10
-5
10
-8
10
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
-10
10
-12
10
-14
10
0
200
400
600
800
1000
Weeks
4
1200
1400
1600
-10
10
-15
1800
10
0
20
40
60
80
4
10
120
140
160
180
200
10
10
2
100
Weeks
10
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
2
10
10
8
0
10
-2
-2
10
10
-4
6
10
-4
10
10
-6
4
10
-6
10
10
-8
-8
10
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
-10
10
-12
10
-14
10
10
0
10
0
42
1000
2000
3000
4000
5000
Weeks
6000
7000
8000
9000
10
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
-10
10
-12
10
-14
10
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
2
10
0
10
0
1000
2000
3000
4000
5000
Weeks
6000
7000
8000
9000