International Journal of Intelligence Science, 2014, 4, 7-16
Published Online January 2014 (http://www.scirp.org/journal/ijis)
http://dx.doi.org/10.4236/ijis.2014.41002
OPEN ACCESS IJIS
Soccer League Competition Algorithm, a New Method for
Solving Systems of Nonlinear Equations
Naser Moosavian, Babak Kasaee Roodsari
Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
Received October 9, 2013; revised November 9, 2013; accepted November 15, 2013
Copyright © 2014 Naser Moosavian, Babak Kasaee Roodsari. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited. In accordance of the Creative Commons Attribution License all Copyrights © 2014 are reserved for SCIRP and the
owner of the intellectual property Naser Moosavian, Babak Kasaee Roodsari. All Copyright © 2014 are guarded by law and by
SCIRP as a guardian.
ABSTRACT
This paper introduces Soccer League Competition (SLC) algorithm as a new optimization technique for solving
nonlinear systems of equations. Fundamental ideas of the method are inspired from soccer leagues and based on
the competitions among teams and players. Like other meta-heuristic methods, the proposed technique starts
with an initial population. Population individuals called players are in two types: fixed players and substitutes
that all together form some teams. The competition among teams to take the possession of the top ranked posi-
tions in the league table and the internal competitions between players in each team for personal improvements
results in the convergence of population individuals to the global optimum. Results of applying the proposed al-
gorithm in solving nonlinear systems of equations demonstrate that SLC converges to the answer more accu-
rately and rapidly in comparison with other Meta-heuristic and Newton-type methods.
KEYWORDS
Soccer League Competition; Nonlinear Equations; Meta-Heuristic Algorithm
1. Introduction
Solving systems of nonlinear equations is one of the
main concerns in a diverse range of engineering applica-
tions such as computational mechanics, weather forecast,
hydraulic analysis of water distribution systems, aircraft
control and petroleum geological prospecting. Many pre-
vious efforts have been made to find a solution for sys-
tems of nonlinear equations. Results of these studies
comprise some theories and algorithms [1-4]. Among
such approaches, Newton’s method is one of the most
powerful numerical methods and an important basic me-
thod which has a quadratic convergence if the function F
is continuously differentiable and if a good initial guess
x0 is provided [5]. Frontini and Sormani [6] proposed a
third-order method based on a quadrature formula to
solve systems of nonlinear equations. Cordero and Tor-
regrosa [7] developed some variants of Newton’s method
based on trapezoidal and midpoint rules of quadrature.
Also, Darvishi and Barati [8-10] presented some high
order iterative methods and Babajee et al. [11] proposed
a fourth-order iterative technique. Luo et al. [12] solved a
system of nonlinear equations using a combination of
chaos search and Newton-type methods. More recently,
Mo et al. [5] presented a combination of the conjugate
direction method (CD) and particle swarm optimization
(PSO) for solving systems of nonlinear equations.
The convergence and performance characteristics of
Newton-type methods are highly sensitive to the initial
guess of the solution supplied to the methods and the
algorithm would fail if the initial guess of the solution is
improper. However, it is difficult to select a good initial
guess for most systems of nonlinear equations [13]. The
system of nonlinear equations is considered as follows:




112
212
12
,,, 0
,,, 0
,,, 0.
n
n
nn
fxx x
fxx x
fxxx











(1)
Applying the global optimization methods, the system

N. MOOSAVIAN, B. K. ROODSARI
8
of Equation (1) is transformed to an optimization prob-
lem. This is achieved by using the auxiliary function:
 

2
12
1
min,, ,,
n
i
i
n
F
fxx


xxx x

(2)
Global minimum of above formulation is zero and
is a root for the corresponding system of equations if
. This paper presents a new meta-heuristic
algorithm, called Soccer League Competitions (SLC), for
solving Equation (2).
*
x

*0Fx 
In Section 2, the basic concepts of SLC are defined. In
Section 3, the performance and effectiveness of SLC are
validated by some examples. Finally, the conclusions are
presented in Section 4.
2. Soccer League Competitions
Level one soccer league consists of teams (clubs) com-
peting each other during a season. In this environment,
some stronger teams aim to sit in the first positions of the
league table while some weaker teams plan to survive in
the level one league in order to prevent a crash out to the
second level league. During the course of a season, each
team plays the others twice, once at their home stadium
and once at that of their opponents. Teams receive 3
points for a win and no point is awarded for a lost. Teams
are weekly ranked by total points and the club with the
most point is crowned champion at the end of each sea-
son. The number of matches in each season depends on
the team numbers. For instance, in a league consisting of
M teams the total number of matches is calculated as
follows:
 
Total Match12MM (3)
In this league, each team participates in 1
M

inde-
pendent matches, and totally,
   
12MM compe-
titions are being held during a season.
There is always an intense competition between the
teams at the bottom of the league table. As a rule, the two
bottom table teams are crashed out to the second level
soccer league (relegations spots) at the end of the season.
In return, two first table teams of the second level league
(promotions spots) replaced with the relegated teams.
Generally, promotions spots import new players to the
league which may have potential of being a future star.
Each team consists of 11 fixed players (FP) and some
substitutes (S). A team’s power depends on the power of
its players. Moreover, powerful teams have a higher
chance of winning their matches. However, it is not pos-
sible to predict the exact winner of a specified match
before the game ends.
As well as the league competitions among teams, there
is an internal competition in each team. Players compete
with each other to attract the head coach’s attention by
improving their performance. This internal competition
leads to a growth in the quality and power of a team.
In each team, there is a key player which is called Star
Player (SP). SP has the best performance among other
players in the team. Moreover, there is a unique player in
each league which is called the Super Star Player (SSP).
SSP is defined as the most powerful player in the league.
After every match, players included in winner and
loser teams of each match adopt different strategies for
improving their future performance. When a team wins a
match, fix players try to imitate the team’s SP, and the
SSP of the league (this strategy is simulated by Imitation
Operator in this study). They aim to experience a promo-
tion to the SP or, optimistically, occupy the place of SSP
in the league. But, the main provocation of winner’s sub-
stitutes is being a fixed player in the team. For this pur-
pose, they try to have a performance approximately equal
to the average level of fixed players in the team (this
tendency is described by the Provocation Operator in this
study). In other words, higher provocation for advance-
ment gives them more chance of being a fixed player in
the future.
On the other hand, loser teams seek for ways of im-
proving their performance for reaching better results in
future matches. For this reason, fixed players of these
teams have to revise their playing style. This revision
may include a change in some aspects of their older hab-
its (this strategy is defined as the Mutation Operator in
this study). In addition, head coach usually considers
new combinations of substitutes in order to stop the fail-
ures in the future (this change is performed by the Sub-
stitution Operator in this study).
Above mentioned strategies improve the overall per-
formance of the teams after each match. Therefore,
team’s powers progressively increase while all teams
play much better at the end of the season. Obviously,
players with a noticeable progression increase the win-
ning chance of their team.
As the first rank teams of each league have better fi-
nancial affordance, they are able to recruit powerful
players of other teams. This intensifies their power for
future seasons. In the next section, the solving style of an
optimization problem using the Soccer League Competi-
tion (SLC) algorithm is discussed.
2.1. Soccer League Competition (SLC)
Algorithm
Competitions between teams in a soccer league for
reaching success, and among players for being a SP or
SSP can be simulated for solving optimization problems.
Similar to a soccer league in which every player desires
to be the best (SSP), in an optimization problem each
solution vector seeks for the global optimum position.
Therefore, each player in a league, Star Player (SP) in
each team, and the Super Star Player (SSP) can be as-
sumed as a solution vector, a local optimum, and the
OPEN ACCESS IJIS

N. MOOSAVIAN, B. K. ROODSARI 9
global optimum, respectively.
Each team consists of 11 fixed players (defined by
principal solution vectors in the SLC algorithm) and
some substitutes (described by reserved solution vectors
in SLC algorithm). For each player, an objective function
is calculated which stands for the power of its corre-
sponding player. In a minimization problem smaller val-
ues of objective functions (cost function) illustrate pow-
erful players (PP). The total power of a team is defined
as the average power value of its players including fixed
and substitute. The following formula shows how a
Team’s Power (TP) is calculated.
 
nPlayer
1
1nPlayer ,
j
TP iPP ij




(4)
nPlayer is the total number of players in the ith team.
is the power of jth player in the ith team

,PP ij






,1cost,ijPPij . In each match, the team with
more power has a higher chance of winning. The prob-
ability of victory for each team in a match is given by:
   
Pv kTPkTP iTP k
(5)
 
Pv iTP iTP iTPk
(6)
Pv stands for the probability of victory. It should be
noted that the sum of Pv(k) and Pv(i) equals 1.
After each match, the winner and the loser are noticed
and some players (solution vectors), including fixed and
substitute, experience changes. These changes, which are
aimed to improve performance of both players and teams,
are simulated with the following operators:
-Imitation Operator
-Provocation Operator
-Mutation Operator
-Substitution Operator
In the next part, detailed description of operators is de-
fined.
2.1.1. Imitation Operator
Fixed players (FP) of the winner team, imitate both the
Star Player (SP) in their own team and the Super Star
Player (SSP) in the league to improve their future activi-
ties. Similarly, solution vectors relating to the fixed
players in the winner team move toward the best solution
of the own team and the best solution vector of the
league. In the SLC algorithm, Imitation is performed by
the following formulas:
  
 

11
2
,,
,
,
F
PijFPijSSP FPij
SP iFP ij



 (7)
  
 
21
2
,,
,
where


1~,U


,

2~0,U


,


1~0,2U

, and


2 are random numbers with uniform distri-
bution.
~0,2U 


,
F
Pij stands for the jth fixed player of the
ith team, and


SP i is the star player of the ith team. It
is also proposed that: 12


, 01

 .
First, solution vector of fixed players (FP) in the win-
ner team experiences a big move toward the resultant
vector direction of SP and SSP (Equation (7)). If the
newly generated solution vector at this new position was
better than the older solution vector, it is replaced with
the old one. Otherwise, the solution vector experiences a
medium move toward the resultant vector (Equation (8)).
If this solution was better than the older one, it is re-
placed with the old vector. In the case that none of the
discussed movements gave a better solution vector, the
player is kept in its position with no change.
2.1.2. Pro vocation Oper ator
Substitutes of a winner team (S) have to prove a per-
formance equal to the average performance level value of
the fixed players in their team in order to be a fixed
player. This process, which is performed by the Provoca-
tion Operator in SLC algorithm, is described by



 
1
,,SijCiCi Sij   

(9)



 
2
,,SijCiSij Ci  

(10)
where


1~0.9,1U

, are random
numbers with uniform distribution, and

2~0.4, 0.6U




Ci is the
average value of fixed player’s solution vectors in the ith
team. S(i,j) is the jth substitute of the ith team.
Firstly, solution vector of the weakest substitute player
in the winner team experiences a forward move toward
the gravity center of fixed players (Equation (9)). If the
newly generated solution vector relating to this new posi-
tion was better than the last one, it is replaced by its old
vector. Otherwise, mentioned player will experience a
backward movement toward the gravity center (Equation
(10)). If this solution was better than the weakest solution,
this vector is replaced with the old one. In the situation
that none of the discussed movements gave a better solu-
tion vector for improving the weakest solution, a new
vector is generated randomly and replaced with the old
one. In an overall view, provocation operator acts on the
weakest substitutes of winners. If advancement was evi-
dent in their performance, they are kept in the team. Oth-
erwise, they are exported from the team while new ran-
dom players (solution vectors) are entered for future
games.
2.1.3. Mu tation Operato r

,
F
P ijFP ijSSPFP ij
SP iFP ij



 (8)
Fixed players of loser team in a match should revise their
activity in order to prevent failure in future games. To
perform this operation, the positions of some players are
randomly changed. This mechanism is similar to muta-
OPEN ACCESS IJIS

N. MOOSAVIAN, B. K. ROODSARI
10
tion process in Genetic Algorithm (GA) for creating di-
versification in solutions.
2.1.4. Sub stitution Operator
The head coach usually considers new combinations of
substitutes for future games. Similarly, a random-based
approach is applied to reflect the head coach impact in
this algorithm. To do this, a pair of new substitute vec-
tors is being tested. If a suitable answer was obtained,
this effective pair is entered to the team. This process,
which is performed by the Substitution Operator in SLC
algorithm, is described by
 
,,1
NEW
SijSijSik   

,
,


n
(11)
 
,,1
NEW
Sik SikSij    (12)

~0,1U 
is a random vector with uniform distribu-
tion. The number of new examined pairs is proposed to
be equal to the number of team substitutes.
In an overall view, 4 described operators have the fol-
lowing effects in the algorithm:
The Imitation Operator expedites the searching capa-
bility of the algorithm.
The Provocation Operator provides high accurate solu-
tions to the complex optimizations problems.
The Mutation and Substitution Operators help the
proposed algorithm to escape from local minimums and
plateaus.
After each game, 4 discussed operators act on the
players (solution vectors) and team’s powers are updated
according to the new solutions. Obviously, powerful
teams are more likely to be successful in their future
matches. This process continues to the end of the season
and the Super Star Player (SSP) of the league yields the
Global Optimum (best solution) for the optimization
problem. After each season, players are arranged taking
into account their updated power. Before commencing a
new league, top players are devoted to the best teams,
medium players are allocated to the teams with an aver-
age performance, and the weakest players are transferred
to the bottom teams in the league table.
In the next section, the steps and flowchart of SLC al-
gorithm are presented.
2.2. Steps and Flowchart of SLC Algorithm
Step 1. Initialize the problem and algorithm pa-
rameters
In this step, the optimization problem is specified as
follows:
 

2
12
1
Min,, , ,
n
i
i
F
fxx


xxx x
(9)
where F(x) is an objective function; x is a set of each
decision variable xi; n is the number of decision variables;
and Xi is the set of possible range of values for each deci-
sion variable. Then, the number of seasons (nSeason), the
number of teams included in the league (nTeam), the
number of fixed players (nFixedPlayer), and the number
of substitutes (nSubstitute) are determined.
Step 2. Generate samples
The total number of players in a league is calculated
by the following formula:


nPlayersnTeamnFixedPlayer nSubstitute
In most problems, it is suggested that
3 nTeam5,


nFixedPlayer 11,
nSubstitute 11
In this step, randomly solution vectors are generated as
many as the number of players in the league and each
vector is devoted to a specified player. Hence the matrix
TEAM which is generated randomly is given as:
1
2
nFixedPlayer
1
2
nSubstitute
11 1
12
22 2
12
nFixedPlayer nFixedPlayernFixedPlayer
12
11 1
12
22 2
12
nSubstitute nSubst
12
TEAM
N
N
N
N
N
FP
FP
FP
RP
RP
RP
xF xFxF
xF xFxF
xF xFxF
xR xRxR
xR xRxR
xR xR


















 



 
itute nSubstitute
N
xR



























(10)
Next, an objective function relating to each solution
vector (player’s power) is calculated.
Step 3. Teams assessment
In this step, all players are arranged according to their
calculated power and are devoted to teams. Each team’s
power is equal to the average power of its players.
Step 4. Start the league
In this step, competitions are started between all pos-
sible pairs of teams in the league, the winner and the
loser of every match are determined, the Imitation Op-
erator acts on fixed players in winner teams, the Provo-
cation Operator acts on substitutes of winner teams, the
Mutation Operator acts on 3 out of 11 from fixed players
of loser teams, and the Substitution Operator acts on re-
served players in the loser teams. Then, player’s powers
OPEN ACCESS IJIS

N. MOOSAVIAN, B. K. ROODSARI 11
and the team’s powers are updated. This process is con-
tinued by the end of the season.
Step 5. Relegation and promotion
In this step, the worst team (relegation spot) is ex-
ported from the first level league, and in return, a new
team (promotion spot) is imported to this league. It
should be noted that this step is only applied for complex
optimization problems.
Step 6. Check the stopping criterion
In this section, Steps 3, 4, and 5 are repeated until the
termination criterion (nSeason) is satisfied.
Figure 1 illustrates the flowchart of SLC procedure
for solving optimization problems.
3. Numerical Results
In this section, the solutions for some systems of nonlin-
ear equations are described. All of computations were
executed in MATLAB programming language environ-
ment using five independent nonlinear systems with an
Intel(R) Core(TM) 2Duo CPU P8700 @ 2.53 GHz and
4.00 GB RAM. For the examples 3 to 5, the stopping
criterion is considered to be


3
10
n
F x.
In all problems it is considered that β = 1 and θ = 0.7.
Case study 1. Geometry size of thin wall rectangle
girder section:
 


 
 
1
3
3
2
22
3
22165
22
9369
12 12
26835,
2
fxbhbtht
btht
bh
fx
htbt
fx hb t
 

 



Figure 1. SLC procedure for solving optimization problems.
where h is the height, b is the width and t is the thickness
of the section. Mo et al. [5] solved this system using
conjugate direction particle swarm optimization. In an-
other study, Luo et al. [12] presented a solution for men-
tioned system using a hybridization of chaos search and
Newton-type methods. Also, Jaberipour et al. [13] used a
new version of particle swarm optimization (PPSO) for
solving this system. In this research, the SLC algorithm
is applied to solve above-mentioned problem. The bound
variables were set between 0 and 30 m. As shown in
Table 1, different intervals of χ1, χ2 are examined to find
the best performance of SLC algorithm. As it can be seen
in this problem, if χ1 = 1 and χ2 = 0.5, SLC reaches to the
average accuracy level of 1024 after 100000 function
evaluations and 20 independent runs. Therefore, we set χ1
= 1 and χ2 = 0.5. According to the Table 2, the best solu-
tion equals zero that is the global optimum of this prob-
lem. In Table 2, values of the best, worst, mean, standard
deviation and number of function evaluations are consid-
ered after 20 different runs. It is assumed that number of
teams, fixed players, and substitutes equal 3, 6 and 6,
respectively. In case A, all operators are taken into ac-
count in SLC algorithm while in case B mutation and
substitution operators are exempted from the algorithm.
As can be seen in case A, the best solution equals zero
which is the exact value of the global optimum, but in
case B, the best solution equals 1.29e26.
To verify the performance of the proposed algorithm,
Particle Swarm Optimization (PSO) and Differential
Evolution (DE) algorithms are applied to solve this sys-
tem. As it can be seen in Table 3, the best solution of DE
after 100,000 function evaluations equals 0.639, and PSO
reaches to 267.594 and 1.89e25 after 10,000 and 100,000
function evaluations, respectively. It should be noted that
PSO algorithm can converge to the global optimum in
one out of 20 runs. The initial population in DE and PSO
are considered to be 20, and 300, respectively.
Table 4 presents the solution obtained from SLC and
previous studies. According to the results, the SLC pro-
Table 1. Comparison results (Objective Functions) of SLC
for different values of χ1 and χ2 (Number of function
evaluations = 100,000).
χ1 χ2 best worst mean Stda
0.5 - 1.50 - 1 0.1781 48.6241 9.9851 11.5017
0.7 - 1.20.3 - 0.74.567E−09b 17614b 916.7857b3931.8
0.8 - 1.10.4 - 0.61.26E−21 4.63E−08 2.48E−09 1.03E−08
0.9 - 10.45 - 0.554.35E−241.19E−11 5.94E−13 2.66E−12
1 0.5 0 7.91E−24 1.38E−24 1.87E−24
aStandard Deviation. bValue of objective function.
OPEN ACCESS IJIS

N. MOOSAVIAN, B. K. ROODSARI
12
Table 2. Reviewing effects of different parameters for SLC
in cases A and B.
Case A
SLC best worst mea n std FCN
3a
6b
6c
0d 7.91E−24 1.38E−24 1.87E−24 10,000
Case A
best worst mean std FCN
3
6
6
0 1.87E−24 4.76E−25 4.91E−25 100,000
Case B
best worst mean std FCN
3
6
6
1.29E−26 9.25E−25 4.86E−25 3.18E−25 10,000
Case B
best worst mean std FCN
3
6
6
5.17E−26 3.32E−24 6.02E−25 7.28E−25 100,000
aNumber of teams. bNumber of fixed players. cNumber of substitutes. dValue
of objective function.
Table 3. Results of PSO and DE for case study 1.
best worst mean std FCN
DE 0.639a 3023.6a 663.8763a 949.29 100000
PSO 267.594 10000 9513.4 2176.2 10000
PSO 1.89E−25 10000 8500 3663.5 100000
aValue of objective function.
Table 4. Comparison of SLC solutions with other methods
in case study 1.
Methods B h t f
1(x) f 2(x) f 3(x)
SLC 22.057 20.294 2.1705 165 9369 6835
PPSO [8] 43.156 10.129 12.944 709.24 9369 528.04
Mo et al. [11] 8.9431 23.271 12.913 165 9369 529.32
Luo et al. [10] 12.566 22.895 2.7898 166.72 9544.32585.5
vides exact solution and outperforms other discussed
methods.
Case study 2. Consider








12
,,,
m
Ffxfxfxx
with






 
1112
11
1010 109
3512 0
3 5120,2,,9
351 0,
iiiii
fxxxx
fxxxx xi
fxxxx


 
 

The above system has ten unknown variables and ten
equations.
SLC was used for solving this system. Maximum and
minimum of decision variables were set between −1 and
0. In Tables 5 and 6, performance of SLC is verified for
different number of players and teams. Values of χ1, χ2
are assumed based on explanations of section 2.1.2 of
this paper. Due to the results of both case A and B, the
best performance is reached when the number of teams,
fixed players, and substitutes are assumed to be 5, 10,
Table 5. Reviewing effects of different parameters for SLC
in case A.
Case Aa
SLC Best worst mean std FCN
3
10
10
3.63E−25 1.89E−05 1.49E−06 4.51E−06 10,000
Best worst mean std FCN
5
10
10
9.40E−13 1.06E−06 5.29E−07 2.36E−07 10,000
aSLC with all operators.
Table 6. Reviewing effects of different parameters for SLC
in case B.
Case Ba
SLC best worst me a n std FCN
3
10
10
7.64E−31 1.66E+00 1.67E−01 5.25E−01 10000
best worst mean std FCN
5
10
10
4.70E−20 4.76E−12 5.08E−13 1.21E−12 10000
aSLC without mutation and substitution operators.
OPEN ACCESS IJIS

N. MOOSAVIAN, B. K. ROODSARI
OPEN ACCESS IJIS
13
vious from Table 8 that SLC finds the exact solution
more accurately comparing with other discussed meth-
ods.
and 10, respectively. In other words, better solutions are
reached when the number of fixed players and substitutes
equals the number of decision variables. PSO and DE
algorithms are also used to solve this system. As shown in
Table 7, the mean solution value of DE (considering in-
itial population = 20) equals 7.12e−12, while PSO reaches
to 2.72e−4 and 1.22e−7 when the initial population equals
100 and 1000, respectively. It should be mentioned that
the mean solution of SLC equals 5.08e−13 (Table 6).
Case study 3. Consider








12
,,,
m
Ffxfxfxx
with



1
1
1,1, 2,,1
1,
iii
mm
fx xxim
fx xx


 


Table 8 demonstrates the best solution obtained from
SLC algorithm, and compares this solution with results
of Differential Equation (DE) and Particle Swarm Algo-
rithm (PSO). It should be noted that the number of func-
tion evaluation in all methods is equal 10,000. It is ob-
When m is odd, the exact zeros of F(x) are (1, 1, ···,1)
and (1, 1, ···,1). Shin et al. [14] solved the above
system for various values of m using Newton-Like
methods. They set the initial guess to be (0.5, 0.5, ···,0.5)
for all methods.
Table 7. Results of DE and PSO for case study 2.
best worst mean std FCN
DE 2.89E−13 4.13E−11 7.12E−12 1.23E−11 10,000
PSOa 2.76E−08 0.0025 2.72E−04 6.16E−04 10,000
PSOb 6.61E−08 7.27E−07 1.22E−07 1.54E−07 10,000
aPopulation = 100. bPopulation = 1000.
Table 8. Comparison of SLC solutions with other methods in case study 2.
Methods Mo et al. [11] DE PSO SLC
x1 0.91555 −0.382084413 −0.382101391 −0.382084304
x2 −0.22226 −0.438097433 −0.438106147 −0.438097493
x3 −0.41465 −0.445927406 −0.445938190 −0.445927622
x4 −0.43925 −0.446971289 −0.446966223 −0.446971297
x5 0.42089 −0.446951503 −0.446961182 −0.446951485
x6 −0.35459 −0.446355699 −0.446377379 −0.446355653
x7 −0.13577 −0.444141223 −0.444154249 −0.444141159
x8 0.42756 −0.436187399 −0.436180807 −0.436187334
x9 0.75220 −0.407859019 −0.407836044 −0.407858897
x10 −0.44070 −0.309566750 −0.309547753 −0.309566879
f1(x) −3.17E−06 −8.6491E−07 −9.9243E−05 2.2204E−16
f2(x) 3.52E−07 1.2254E−07 −2.5652E−05 0.0000E+00
f3(x) −1.70E−06 1.5398E−06 −8.0323E−05 8.8818E−16
f4(x) 1.77E−06 −1.2364E−07 6.7860E−05 1.2212E−15
f5(x) −1.68E+00 −5.1561E−08 −3.4056E−05 −3.2196E−15
f6(x) 2.53E+00 −1.9788E−07 −1.2628E−04 −1.2212E−15
f7(x) −8.42E−01 −3.0441E−07 −8.8738E−05 −2.7756E−15
f8(x) −3.91E−07 −1.6868E−07 1.5435E−05 −2.2204E−16
f9(x) 6.81E−07 −1.0542E−06 1.1699E−04 0.0000E+00
f10(x) 2.34E−07 9.0578E−07 9.3726E−05 −6.8834E−15

N. MOOSAVIAN, B. K. ROODSARI
14
To verify the performance of SLC algorithm above-
mentioned problem will be analyzed for 13, 71, 151, and
201 dimensions. We assume zero and one as the upper
and lower bounds of unknown variables in SLC algo-
rithm . Values of χ1, χ2 are assumed
based on explanations of section 2.1.2 of this paper. For
each considered dimension value (m), the number of
substitute players equal the number of decision variables
while the number of fixed players in 3 cases equals the
substitute players and in one case equals half of this val-
ue. Calculation results for 20 independent runs are pre-
sented in

0.5 1.5
i
x

Tables 9-12. Similar to the previous examples,
analysis is performed for 2 cases of A and B. In case A,
all operators are taken into account in SLC algorithm
while in case B mutation and substitution operators are
exempted from the algorithm. According to Tables 9 and
Table 9. Review of parameter variation effect in SLC algo-
rithm for m = 13.
Case A Case B Case A Case B
3 5
13 13
13
1631a 1057a
13
3294a 2226a
aNumber of function evaluations.
Table 10. Review of parameter variation effect in SLC al-
gorithm for m = 71.
Case A Case B Case A Case B
9 3
71 35
71
70,024 63,377
71
23,827 10,602
aNumber of function evaluations.
Table 11. Review of parameter variation effect in SLC al-
gorithm for m = 151.
Case A Case B Case A Case B
9 3
151 75
151
191,370 253,330
151
75,137 100,460
Table 12. Review of parameter variation effect in SLC al-
gorithm for m = 201.
Case A Case B Case A Case B
9 5
201 100
201
278,920 404,265
201
104,370 151,860
10, the performance of SLC algorithm in case B is better
than case A when the number of decision variables are m
= 13 (number of function evaluations = 1057) and m = 71
(number of function evaluations = 10,602). In contrast,
SLC algorithm in case A has a better performance when
the number of decision variables are m = 151 (number of
function evaluations = 75,137) and m = 201(number of
function evaluations = 104,370).
To verify the performance of SLC algorithm, PSO and
DE algorithms are applied to solve this system of
nonlinear equations. As shown in Table 13, SLC has
better convergence accuracy in comparison with PSO
considering all different problem dimensions. It is also
found that DE is not a good rival for SLC and PSO at all.
For instance, this algorithm has not yet converged to the
solution after more than 1 million function evaluations for
m = 151 and m = 201. To reach the best performance for
PSO and DE, the initial population in DE algorithm equals
half of its number of decision variables and in PSO algo-
rithm this value is assumed to be 100* number of decision
variables. The best solution of DE after 100,000 function
evaluations equals 0.639, and PSO reaches to 267.594 and
1.89e−25 after 10,000 and 100,000 function evaluations,
respectively. It should be noted that PSO algorithm can
converge to the global optimum in one out of 20 runs. The
initial population in DE and PSO are considered to be 20,
and 300, respectively.
The CPU time results are provided in Table 14. As
shown in Table 14, the convergence rate in Newton-like
methods dramatically increases as the problem’s dimen-
sion (number of unknown variables) rises. In contrast,
the convergence time in SLC has no dependency with the
dimension parameter.
It should be mentioned that each linear system of
equations should be solved in alliterations of the solving
process [14]. The main reasons in less convergence time
in SLC method is that it does not require solving linear
system of equations and it only calls the optimization
function during each season.
Considering the discussed examples in this article,
SLC can properly solve huge system of nonlinear equa-
tions. To sum up, the following suggestions are worth to
mention after detail preview of the problems:
1) Choose the values of χ1 and χ2 close to 1 and 0.5,
respectively.
2) The number of teams should be selected between 3
and 5 for problems with dimension of lower than 200.
3) The number of fixed players should be selected be-
tween the number of decision variables and half of
this value.
4) The number of substitute players should be equal to
the number of decision variables.
5) To decrease the number of function evaluation in
problems with small dimensions, Mutation and Subs-
OPEN ACCESS IJIS

N. MOOSAVIAN, B. K. ROODSARI 15
Table 13. Results of DE and PSO for case study 3.
DE m = 13 m = 71 m = 151 m = 201
FCN 70,400 269,600 1,000,000 1,000,000
F(x) 0.001 0.001 0.79828 8.2763
PSO m = 13 m = 71 m = 151 m = 201
FCN 4290 58,220 135,900 261,300
F(x) 0.001 0.001 0.001 0.001
Table 14. Comparison results of CPU time for SLC with other methods in case study 3.
m = 13 m = 71 m = 151 m = 201
Method
CPU time(s) CPU time(s) CPU time(s) CPU time(s)
N-K [14] 0.484375 7.828125 75.03125 272.546875
Ned1 [14] 0.5 11.890625 121.578125 461.8125
Ned2 [14] 0.53125 14.46875 172.046875 588.90625
Ned3 [14] 0.375 10.28125 166.71875 431.984375
mNm [14] 0.546875 12.109375 138.28125 790.21875
FM [14] 0.640625 13.75 199.046875 802.640625
CL [14] 0.421875 8.609375 109.09375 501.546875
SLC 0. 3121 1.9216 9.0952 16.9021
titution operators can be neglected.
To find a solution, SLC algorithm rapidly reaches the
local optimums and considers the best of them as the
global optimum solution using Imitation operator. Next,
the local optimum solution is precisely approximated by
the provocation operator. During the above-mentioned
operations, both Mutation and Substitution operators
check the skewed points to prevent the ignorance of any
other possible local or global optimum points in the do-
main. Combination of 4 discussed operators together
with the team ranking procedures in leagues, make SLC
algorithm as an incomparable optimizer among many
other meta-heuristic and mathematical methods.
4. Conclusion
In this article, a new meta-heuristic algorithm was intro-
duced, entitling Soccer League Competitions (SLC), to
solve nonlinear systems of equations. This algorithm
seeks for the answer using 4 independent operators and
rapidly converges to the results. Due to the comparison
results between the proposed algorithm with other
meta-heuristic and Newton-like methods, SLC provides
more accurate answers in a considerably smaller time.
Furthermore, SLC has the lowest sensitivity to the prob-
lem’s dimensions. In conclusion, the proposed algorithm
is recommended for huge and complex optimization
problems and nonlinear systems of equations specifically
when the running time is considered as an important fac-
tor.
REFERENCES
[1] W. R. J. Ortega, “Iterative Solution of Nonlinear Equation
in Several Variable,” Academic Press, New York, 1970.
[2] S. Krzyworzcka, “Extension of the Lanczos and CGS
Methods to Systems of Nonlinear Equations,” Journal of
Computational and Applied Mathematics, Vol. 69, No. 1,
1996, pp. 181-190.
http://dx.doi.org/10.1016/0377-0427(95)00032-1
[3] S. J. J. Nocedal, “Numerical Optimization,” Spring Science
+ Business Media Inc., Berlin, 1999.
[4] D. G. Huang and Y. Ma, “Nonlinear Numerical Analysis,”
Wuhan University Press, Wuhan, 2000.
[5] Y. Mo, H. Liu and Q. Wang, “Conjugate Direction Par-
ticle Swarm Optimization Solving Systems of Nonlinear
Equations,” Computers & Mathematics with Applications,
Vol. 57, No. 11-12, 2009, pp. 1877-1882.
http://dx.doi.org/10.1016/j.camwa.2008.10.005
[6] M. Frontini and E. Sormani, “Third-Order Methods from
Quadrature Formulae for Solving Systems of Nonlinear
Equations,” Applied Mathematics and Computation, Vol.
149, No. 3, 2004, pp. 771-782.
http://dx.doi.org/10.1016/S0096-3003(03)00178-4
[7] A. Cordero and J. R. Torregrosa, “Variants of Newton’s
OPEN ACCESS IJIS

N. MOOSAVIAN, B. K. ROODSARI
16
Method for Functions of Several Variables,” Applied Ma-
thematics and Computation, Vol. 183, No. 1, 2006, pp.
199-208. http://dx.doi.org/10.1016/j.amc.2006.05.062
[8] M. T. Darvishi and A. Barati, “Super Cubic Iterative Me-
thods to Solve Systems of Nonlinear Equations,” Applied
Mathematics and Computation, Vol. 188, No. 2, 2007, pp.
1678-1685. http://dx.doi.org/10.1016/j.amc.2006.11.022
[9] M. T. Darvishi and A. Barati, “A Fourth-Order Method
from Quadrature Formulae to Solve Systems of Nonlinear
Equations,” Applied Mathematics and Computation, Vol.
188, No. 1, 2007, pp. 257-261.
http://dx.doi.org/10.1016/j.amc.2006.09.115
[10] M. T. Darvishi and A. Barati, “A Third-Order Newton-
Type Method to Solve Systems of Nonlinear Equations,”
Applied Mathematics and Computation, Vol. 187, No. 2,
2007, pp. 630-635.
http://dx.doi.org/10.1016/j.amc.2006.08.080
[11] D. K. R. Babajee, et al., “A Note on the Local Conver-
gence of Iterative Methods Based on Adomian Decompo-
sition Method and 3-Node Quadrature Rule,” Applied
Mathematics and Computation, Vol. 200, No. 1, 2008, pp.
452-458. http://dx.doi.org/10.1016/j.amc.2007.11.009
[12] Y.-Z. Luo, G.-J. Tang and L.-N. Zhou, “Hybrid Approach
for Solving Systems of Nonlinear Equationsusing Chaos
Optimization and Quasi-Newton Method,” Applied Soft
Computing, Vol. 8, No. 2, 2008, pp. 1068-1073.
http://dx.doi.org/10.1016/j.asoc.2007.05.013
[13] M. Jaberipour, E. Khorram and B. Karimi, “Particle
Swarm Algorithm for Solving Systems of Nonlinear Equ-
ations,” Computers & Mathematics with Applications,
Vol. 62, No. 2, 2011, pp. 566-576.
http://dx.doi.org/10.1016/j.camwa.2011.05.031
[14] B. C. Shin, M. T. Darvishi and C.-H. Kim, “A Compari-
son of the Newton-Krylov Method with High Order New-
ton-Like Methods to Solve Nonlinear Systems,” Applied
Mathematics and Computation, Vol. 217, No. 7, 2010, pp.
3190-3198. http://dx.doi.org/10.1016/j.amc.2010.08.051
OPEN ACCESS IJIS