An Improved Particle Swarm Optimization Based on
Repulsion Factor
Zhang Jie, Fan Chaozan, Liu Bo
College of Science North China University of Technology,
NCUT
Beijing, China
Shi Fugui
College of Mathematics Beijing Institute of Technology,
BIT
Beijing China
Abstract—In this paper, through the research of advantages and disadvantages of the particle swarm optimization algorithm,
we get a new improved particle swarm optimization algorithm based on repulsion radius and repulsive factor. And a lot of test
function experimental results show that the algorithm can effectively overcome the PSO algorithm precocious defect. PSO has
significant improvement.
Keywords- PSO; RPSO; repulsion radius; repulsive factor
1. Introduction
Particle Swarm Optimization (PSO) was originally
introduced by Kennedy and Eberhart. It is a population
intelligence algorithm based on researching birds and fish
populations behavior in 1995[1]. The thought is based on
artificial life and evolutionary computation theory. Through
imitating birds flying and foraging behavior, the population
uses collective cooperation to achieve the optimal value.
PSO algorithm is a branch of evolutionary computation. It
based on iterative searches to get the optimal value. Its
advantages are fast convergence, the few parameters set and
easy implement etc. The algorithm can effectively solve the
complex optimization problem. Now it is used in a lot of field.
But PSO algorithm has flaws. Its local search ability is poor,
the performance of the search is dependent on parameters, and
especially it is easy to fall into the local optimum. Therefore,
many researchers research the traditional PSO from the
directions including parameters set, convergence, topology
structure, and combine with other algorithm. They pointed out
the deficiency then improved PSO in order to raise algorithm
performance.
In order to overcome the premature drawback of PSO, we
puts forward an improved algorithm based on the standard
particle swarm optimization, that introducing repulsion radius
and repulsive factor. And a lot of experimental data show that
the algorithm can effectively overcome the PSO algorithm
precocious defect. This algorithm has significant improvement
to PSO.
2. Preparation
Traditional PSO imitates behavior of bird flocking and fish
schooling in finding food. In the solving optimization problem,
the potential solutions of problem correspond to positions of
the birds in the search space (D dimension space). These birds
called particles, each one has its own position
),...,,( 21 iDiii xxxx

and velocity ),...,,( 21 iDiii vvvv  (decide
flight direction and distance). And there is a fitness namely
experience for every one decided by the optimization function
(Fit). In the search process, one particle updates the position
and velocity in flight according to its own experience and
experience of nearby individuals (Pbest and Gbest). This
process has implied parallelism.
Thereafter, the inertial weight w was first time introduced
to the PSO update velocity formula by Shi. Y and Eberhart in
[2], and they pointed out that the self-adapting inertia effect is
one of the important parameters for arithmetic performance.
The bigger w is more conducive to population to search into
the broader space, and the smaller w ensures that the population
eventually will converge to the optimal position. They put
forward the improved evolution model with linear diminishing
w [3]:
GGwwww max/*)( minmaxmax 


In the formula, maxG is maximum iterating time, G is the
current iteration time.
Standard particle swarm optimization formulas with the
inertial weight w are as follows:
)(**2
)(**1
)(*)1(
ibest
ibest
ii
xGrandc
xPrandc
tvwtv





(2.1)
)1()()1(


tvtxtx iii (2.2)
In the formulas, c1,c2 are acceleration coefficients; Pbest is
the current best position of the particle; Gbest is the best
particle position of the population; rand is the random number
between [0, 1]; and we give maximum velocity and the
maximal displacement at the same time [3].
Shi. Y etc. discussed and researched other parameters
including acceleration constant, maximum velocity limit, and
population size etc. in [2] [4].
This work was supported by the National Natural Science Foundation of China(61074151) and (10971242), the Beijing Natural Science Foundation Program
(1102016), Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality
（PHR201108055）, the Beijing College Scientific Research and entrepreneurship Plan Project, the College of Mathematics Beijing Institute of Technology and the
College Scientific and technological Activities of North China University of Technology.
Open Journal of Applied Sciences
Supplement：2012 world Congress on Engineering and Technology
112

Clerc M etc. first started to study the mathematical
foundation and the convergence of the PSO algorithm. After
analyzing PSO working mechanism and its convergence， the
compression factor k was introduced to ensure convergence of
the algorithm by them. And compression factor k could be
regarded as a special example of inertia weight [7].
Krink T etc. put forward the spatial extension. If two
particles will collide, they will go back the previous way. The
improved algorithm used that to overcome premature
convergence in iterative optimization [6] .
Among the researchers of the combination of PSO and
other algorithms, Ratnaweem introduced one mutation operator,
that if test distance between particles and the current optimal is
less than certain value than let the particle random initialization,
into PSO to ensure the diversity of the population. [8].
Pant M etc. introduced a new diversity guided particle
swarm optimizer in [11]. They put the population into three
parts with the diversity of the population. The three parts used
different evolution models. The model of middle part was
similar to the way of Krink T.
In [12], the standard PSO formula (2.1) is divided into three
parts, they are "inertia", "cognitive" and "social". And a kind of
PSO improvement strategies was introduced. According to the
Gaussian distribution, we can use the population statistics
information to updates different types of particles with
different evolution model. Its purpose is to accelerate the
convergence.
3. Repulsion Particle Swarm
Optimization
Before The PSO is proved to be convergent by Clerc M and
other researchers. In experiments, we also can get that velocity
of all particles gradually tend to be 0 in PSO. At the same time,
all the particles are convergent around one point that generally
the global optimal or local optimal (this is "premature"). When
the particles converge to the optimal position, "cognitive" part
basically is 0. At this time, the main influence of the particle
movement is "social" factor, and "inertia" factor is mainly
decided by the previous "social" factors. In order to avoid
"premature" convergence and keeping particle population
activity, we should mainly adjust the relationship between
particles and the optimal position.
We consider that the repulsion factor between microscopic
particles in physic can be a reference for avoiding "premature".
That means when the distance between the particle and optimal
position is smaller than a certain radius r, the Repulsive force
outweigh the gravitational force, the particle will not tend to
optimal particle, but to the opposite direction. This purpose is
to keep population diversity.
This improvement uses the following principles to update
the position and velocity:
(1) when the distance between particle and the optimal
position is equal or more than repulsion radius r, the iteration
process still uses the formula (2.1) (2.2) to adjust;
(2) otherwise, use the following formula adjustment:
)(**3
)(**1
)(*)1(
ibest
ibest
ii
xGrandc
xPrandc
tvwtv





(3.1)
)1()()1(


tvtxtx iii (3.2)
The parameter is the same as (2.1) (2.2), the repulsive
factor c3, should be bigger than c2.
But we find that the bigger radius and c3 in the early period
of the process are more conducive to jumping out premature
points, and the smaller radius and c3 in the later period will
ensure that the population eventually are not far away from the
optimal position. This is similar to the inertial weight w. So we
think that the radius will be exponential decrease and c3 linear
decrease. And the lower limit of c3 finally should be c2 to
insure the convergence of this algorithm.
The improved algorithm is repulsion particle swarm
optimization (RPSO), algorithm procedure is as follows:
(1) Initialization, including population size N, the position
and velocity of each particle, inertial weights, and accelerated
factor, repulsion radius and repulsive factor;
(2) Calculate the fitness value of each particle, Fit;
(3)Individual optimal position of the particle, for each
particle, with its fitness value of Fit and previous individual
optimal value Pbest comparison, if Fit > Pbest, Fit replace
Pbest, else keep Pbest;
(4)Optimal position of population, each particle’s fitness
value, Fit, and previous global optimal value Gbest, if Fit >
Gbest, Fit replace Gbest, else keep Gbest;
(5)Updated velocity and position of the particles; each
particle, calculating the distance between the particles and the
best, if the distance is equal or greater than rejection radius,
update velocity and position of the particles according to the
formula (2.1) (2.2); Otherwise, update of particles velocity and
position according to the formula (3.1) (3.2)
(6) if meet the end conditions (to achieve maximum
iterating times), exit, or back to (2).
4. Data Test
The experiments are used to compare standard particle
swarm optimization and repulsion particle swarm optimization.
Use the following five standard test functions to test
algorithm performance:
（1） Sphere function:

n
i
xxf
1
2
1)(
100100),,...,,( 21


in xxxxx
（2） Rastrigrin function
113

Fig.5
Fig.3
Fig.4
Fig.2
Fig.1
 n ii xxxf
1
2
2]10)2cos(10[)(

12.512.5),,...,,( 21

 in xxxxx ；
（3） Rosenbrock function
2
1
1
2
13 )1()(100)( i
n
iixxxxf  


3030),,...,,( 21  in xxxxx ；
（4） Schaffer function
2222
222
4))(0001.01(
5.0)(sin
5.0)( yx
yx
xf


x，y both real number；
（5） Griewank function
1)cos(
4000
1
)(
1
2
5  n
i
i
n
ii
x
xxf
100100),,...,,( 21

 in xxxxx 。
The best results are as follow:
0)0(,0)0(,0)1(,0)0(,0)0( 54321  fffff
Process parameters are: maximum of inertial weight w is
0.9, minimum 0.4; accelerated factor c1=c2=2; repulsive factor
c3 takes 50 that decrease progressively to 2; square (for easy
operation) of repulsion radius r takes 1e-4 that decrease
progressively to 1 e-45. The population takes 30, maximum
interations is 1500, dimension is 10, each algorithm is 20 time.
We get the all mean, best and worst results. The experimental
results see Table 1 and Fig. 1 ~ Fig. 5.
TABLE I. TEST FUNCTION
Test
Functi
on
Standard PSO RPSO
Best Media Worst Best Media Worst
1
f
1.44E-
42
6.97E-
39
2.50E-
38
1.91E-
41
4.85E-
39
4.16E-
38
2
f 1.990 3.188 4.975 0 3.382 6.965
3
f 5.83E-
15 5.942 17.71 1.82E- 19 7.009 20.56
4
f 0 0.0029 0.0097 0 1.77E-
05
8.12E-
05
5
f 0 0.0014 0.0236 0 5.01E-
18
1.11E-
16
In Fig. 1~Fig. 5, blue line is PSO, red RPSO. All abscissa is 1-1500. All
ordinate are index form.
From the table and curve images, we can find that RPSO
has a very good performance relative to standard PSO in
jumping out "premature" and it keeps the good performance of
standard PSO even better. In function tests, the improvement
effect especially is proved. Particles escape completely from
the local optimal. But we can see that the improved algorithm
has defects in the accuracy and velocity of convergence. The
problem is mainly from repulsion radius and repulsive factor
selection. In the experiments, different choices of radius and
factor can produce unstable results. The best relation between
114

repulsion radius and repulsive factor is unknown. The
modification research still needs to be further explored.”.
5. Conclusion
Through the repulsion concept of microscopic particle in
physic, this paper introduces the concept of repulsion radius
and repulsive factor, and gives a new improved particle swarm
optimization algorithm. The results of test functions have good
performance, especially in Schaffer and Griewank function,
both that local optimum is dense exist around the global
optimal. The improve effect is obvious; this optimization has
the value of further investigation.
6. Acknowledgment
This work was supported by the National Natural Science
Foundation of China(61074151) and (10971242), the Beijing
Natural Science Foundation Program (1102016), Funding
Project for Academic Human Resources Development in
Institutions of Higher Learning Under the Jurisdiction of
Beijing Municipality （PHR201108055）, the Beijing College
Scientific Research and entrepreneurship Plan Project, the
College of Mathematics Beijing Institute of Technology and
the College Scientific and technological Activities of North
China University of Technology.
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