International Journal of Computer Applications (0975 – 8887)
Volume 14– No.1, January 2011
19
Particle Swarm Optimization: Technique, System and
Challenges
Dian Palupi Rini
Faculty of Computer Science,
Sriwijaya University, Indonesia
Faculty of Computer Science &
Information System,
Universiti Teknologi Malaysia
Siti Mariyam Shamsuddin
Faculty of Computer Science &
Information System,
Universiti Teknologi Malaysia
Siti Sophiyati Yuhaniz
Faculty of Computer Science &
Information System,
Universiti Teknologi Malaysia
ABSTRACT
Particle Swarm Optimization (PSO) is a biologically inspired
computational search and optimization method developed in 1995
by Eberhart and Kennedy based on the social behaviors of birds
flocking or fish schooling. A number of basic variations have been
developed due to improve speed of convergence and quality of
solution found by the PSO. On the other hand, basic PSO is more
appropriate to process static, simple optimization problem.
Modification PSO is developed for solving the basic PSO problem.
The observation and review 46 related studies in the period
between 2002 and 2010 focusing on function of PSO, advantages
and disadvantages of PSO, the basic variant of PSO, Modification
of PSO and applications that have implemented using PSO. The
application can show which one the modified or variant PSO that
haven’t been made and which one the modified or variant PSO that
will be developed.
Keywords
Particle Swarm Optimization (PSO), Variant PSO, Modification
PSO, Basic PSO problem, Bird Flocking, Evolutionary
Optimization, biologically inspired computational search.
1. INTRODUCTION
Theory of particle swarm optimization (PSO) has been growing
rapidly. PSO has been used by many applications of several
problems. The algorithm of PSO emulates from behavior of
animals societies that don’t have any leader in their group or
swarm, such as bird flocking and fish schooling. Typically, a flock
of animals that have no leaders will find food by random, follow
one of the members of the group that has the closest position with
a food source (potential solution). The flocks achieve their best
condition simultaneously through communication among members
who already have a better situation. Animal which has a better
condition will inform it to its flocks and the others will move
simultaneously to that place. This would happen repeatedly until
the best conditions or a food source discovered. The process of
PSO algorithm in finding optimal values follows the work of this
animal society. Particle swarm optimization consists of a swarm of
particles, where particle represent a potential solution.
Recently, there are several modifications from original PSO. It
modifies to accelerate the achieving of the best conditions. The
development will provide new advantages and also the diversity of
problems to be resolved. Study on the development of PSO is
necessary to do to know how far its development, its advantages
and disadvantages and how much use this method to settle a
problem. Tutorial and theoretical of PSO has made about what is
PSO [1], [2], those describe about what PSO is, simple data tested,
and comparison with others evolutionary computations.
This paper will describe what for the modifications, advantages
and disadvantages each modification of PSO and make a
conclusion from those. In section 2 describes about basic PSO,
basic variation of PSO, and modification of PSO, and section 3
observation, and the last section 4 describe about summary and
future work.
2. VARIANT OF PSO
Exploration is the ability of a search algorithm to explore different
region of the search space in order to locate a good optimum.
Exploitation, on the other hand, is the ability to concentrate the
search around a promising area in order to refine a candidate
solution[3].With their exploration and exploitation, the particle of
the swarm fly through hyperspace and have two essential
reasoning capabilities: their memory of their own best position -
local best (lb) and knowledge of the global or their neighborhood's
best - global best (gb).
Position of the particle is influenced by velocity. Let denote
the position of particle in the search space at time step ; unless
otherwise stated, t denotes discrete time steps. The position of the
particle is changed by adding a velocity, to the current
position [1]:
………………….…………….. (1)
where [2] :
………………………………(2)
International Journal of Computer Applications (0975 – 8887)
Volume 14– No.1, January 2011
20
with , acceleration coefficient and
and random vector and . Simple example of PSO, there is a
function [3]:
where
Denote as a lower limit and as an upper limit. So, PSO
procedure can be described by the following steps: First, Assume
that the size of the group of particle is N. It is necessary that the
size N is not too large, but also not too small, so that there are
many possible positions toward the best solution or optimal.
Second, generate initial population with range and by
random order to get the . It is necessary if the overall
value of the particle is uniformly in the search area.
After that, the particle and the velocity at iteration are denoted
as and thus, these initial particles will
be . Vector , ( ) is called
a particle or vector coordinates of the particle. (Such as:
chromosomes in genetic algorithms). Evaluation of the objective
function value for each particle and expressed by
Then calculate the speed of all particles. All particles move
towards the optimal point with a velocity. Initially all of the
particle velocity is assumed to be zero. Set iteration
At the iteration, find the two important parameters for each
particle that is:
a. The best value of (the coordinates of particle at
iteration ) and declare as , with the lowest value of
objective function (minimization case), which
found a particle at all previous iteration. The best value for
all particles which found up to the iteration,
with the value function the smallest goal / minimum
among all particles for all the previous iterations, .
b. Calculate the velocity of particle j at iteration i using the
following formula using formula (2):
Where and , respectively, are learning rates for
individual ability (cognitive) and social influence (group), and
and uniformly random numbers are distributed in the
interval 0 and 1. So the parameters and represent weight
of memory (position) of a particle towards memory (position)
of the groups (swarm). The value of and is usually 2, so
multiply and ensure that the particles will approach
the target about half of the difference
c. Calculate the position or coordinates of particle at the
iteration by :
Evaluation of the objective function value for each particle and
expressed as:
The last step, check whether the current solution is convergent. If
the positions of all particles leading to an equal value, then this is
called convergence. If not convergent then step 4 is repeated by
updating iterations , by calculating new values from
and . This iteration process continues until all
particles convergence the same solution. Usually be determined by
the termination criteria (Stopping criterion), for example the
amount of the excess solution with a solution now previously been
very small.
If the current solution is convergent, then the iteration will stop.
We do not know whether the final value is the best value. Below
are the stopping criteria conditions for the iteration: First,
terminate when a maximum number of iterations, or FEs, has been
exceeded. Second, Terminate when an acceptable solution has been
found, Third, Terminate when no improvement is observed over a
number of iteration. Fourth, terminate when the normalized swarm
radius is close to zero. Fifth, terminate when the objective function
slope is approximately zero. Although the particle has stopped, we
do not know whether the particle will pitch on local optima, local
minima, global optima or global optima.
In the original particle swarm optimization, there has also a lack of
solution, because it is very easy to move to local optima. In certain
circumstances, where a new position of the particle equal to global
best and local best then the particle will not change its position. If
that particle is the global best of the entire swarm then all the other
particles will tend to move in the direction of this particle. The end
of result is the swarm converging prematurely to a local optimum.
If the new position of the particle pretty far from global best and
local best then the velocity will changing quickly turned into a
great value. This will directly affect the particle's position in the
next step. For now the particle will have an updated position of
great value, as a result, the particle may be out of bounds the
search area.
In analysis, PSO has advantages and disadvantages [4].
Advantages of the basic particle swarm optimization algorithm:
PSO is based on the intelligence. It can be applied into both
scientific research and engineering use. Then PSO have no
overlapping and mutation calculation. The search can be carried
out by the speed of the particle. During the development of several
generations, only the most optimist particle can transmit
information onto the other particles, and the speed of the
researching is very fast. After that the calculation in PSO is very
simple. Compared with the other developing calculations, it
occupies the bigger optimization ability and it can be completed
easily. The last one is PSO adopts the real number code, and it is
decided directly by the solution. The number of the dimension is
equal to the constant of the solution.
On the other hands, disadvantages of the basic particle swarm
optimization algorithm are the method easily suffers from the
partial optimism, which causes the less exact at the regulation of its
speed and the direction. Then the method cannot work out the
problems of scattering and optimization and the method cannot
work out the problems of non-coordinate system, such as the
solution to the energy field and the moving rules of the particles in
the energy field.
2.1 Basic Variants of PSO
The lacks of PSO have been reduced with a variation of PSO.
Many variations have been developed to improve speed of
convergence and quality of solution found by the PSO. The
variation is influenced by a number of control parameters, namely
the dimension of the problem, the number of particles (swarm
size), acceleration coefficients (The acceleration coefficient, and
together with random vector r1 and r2, control the stochastic
influence), inertia weight, neighborhood size, number of iteration,
and the random values which scale the contribution of the
cognitive and social component. Below are the basic variations of
particle swarm optimization:
International Journal of Computer Applications (0975 – 8887)
Volume 14– No.1, January 2011
21
a. Velocity clamping
Velocity clamping will control the global exploration of the
particle. If the velocity of a particle exceeds the maximum
allowed speed limit, it will set a maximum value of velocity
( ). So that , j indicates the maximum allowable speed
for a particle in the dimension. Speed (velocity) of the particle
is adjusted using the equation [2]:
………………...(3)
High value of will cause global exploration, whereas
lower values result in local exploration. will control the
movement of the particle and aspect of exploration and
exploitation. Velocity clamping did not influence the position of
the particle. This only reduces the size of the step velocity.
Changes in the search direction not only can make a particle to
perform a better exploration but also has negative effects and the
optimum value cannot be found.
The following equation [2] is used to initialize the max and min
velocity to the solution:
……………………………(4)
……………………………(5)
Where as and are the minimum and maximum
positions of the particle in the dimension. is a constant factor
and is taken from 0 until 1. The problem is if all the velocity
becomes equal to the particle will continue to conduct
searches within a hypercube and will probably remain in the
optima but will not converge in the local area.
There are some researchers that have develop velocity
clamping method, such as : [5], [6]
b. Inertia weight
It is a mechanism to control an exploration and exploitation
abilities of the swarm, and as mechanism to eliminate the need of
velocity clamping. The inertia weight, , controls the momentum
of the particle by weighing the contribution of the previous
velocity – basically controlling how much memory of the previous
flight direction will influence the new velocity. For the
PSO, the velocity equation [5] changes from equation:
…………………….………………. (6)
A similar change is made from the- PSO. Inertia weight
presenting how much the amount of memory from the previous
flight direction will affect the new velocity. If > 1, then the
velocity will decrease with time, the particle will accelerate to
maximum velocity and the swarm will be divergent. If < 1, then
the velocity of particle will decrease until it reaches zero. The
larger value of will facilitate an exploration, rather small values
will promote the exploitation.There are some researchers that have
develop inertia weight application, such as :[7], [8], [9], [10]
c. Constriction Coefficient
Velocity update equation that using constriction coefficient
changes to:
…………………………………………….…… (7)
Where
With
Equation above is used under the constraints that
. The constriction approach was developed as a natural,
dynamic way to ensure convergence to a stable point, without the
need for velocity clamping. Condition and of the
swarm is guaranteed to convergence.
There are some researchers that have develop constriction
coefficient , such as : [11], [12].
d. Synchronous Versus Asynchronous Updates
Synchronous Updates [13] are done separately from the particle
(personal best and neighborhood bests) position updates, only
given one feedback per iteration update, slower feedback and
better for . While asynchronous is better for , updates
calculate the new best positions after each particle position update
and have the advantage that immediate feedback is given about the
best region of search space. There are some researchers that have
develop this method, such as : [14], [15], [16], [17], [18], [19],
[20].
2.2. Modification of PSO
The modification in PSO consists of three categories: extension of
field searching space, adjustment the parameters, and hybrid with
another technique. The modifications of PSO can enhance its
performance.
a. Single Solution PSO
A large number of PSO variations can be found to locate single
solutions. These PSO implementations were specially developed to
obtain single solutions to continuous-valued, unconstrained, static,
single-objective, optimization problem, most of these algorithm
can also be applied to other problem types.
b. Niching with PSO
In the EC field, algorithms that locate multiple solutions are refers
to as niching algorithm. The process of finding multiple solution or
niche is generally referred to as speciation. Niching algorithms
model yet another natural process, where large numbers of
individuals compete for the use of limited resources on physical
environment.
Nieces are partitions of an environment while species are partitions
of computational optimization, a niece represents one solutions to
the problem, while a species refers to the group of individuals
(particle in the context of PSO) that convergence on a single niece.
c. Constraint Optimization using PSO
Constraint reduces the feasible space where in solution to the
problem can be found. Optimization algorithms need to ensure that
a feasible solution is found. That is the optimization algorithm
should find a solution that both optimizes the objective function
satisfies all constraints. If it is not possible to satisfy all constrains,
the algorithm has to balance the trades off between optimal
objective function value and number of constrain violated.
International Journal of Computer Applications (0975 – 8887)
Volume 14– No.1, January 2011
22
d. Multi-objective optimization with PSO
Many real world optimization problems require the simultaneous
optimization of a number of objectives (multi-objectives). Using
the notation, the multi-objectives optimization problem is defined
as:
…………………….. (8)
The main objective of MOO algorithms is to find a set of solution
which optimally balance the trade-offs among the objective of a
MOP. It is different with the basic PSO that return only one
solution.
e. Dynamic Environment With PSO
In dynamic Environments, PSO should be fast to allow quick re-
optimization. It is desirable to find a good solution before the next
environment change. In original PSO, it is impossible to
convergence to an equilibrium state in its first goal to locate the
optimum.
There are several solutions for dynamic environment. Such as: a.
Environment change detection, It is to allow timeous and efficient
tracking of optimum, b. Response to environment changes, c.
Changing the inertia weight update, d. Reinitialize Particle
Solution, e. Limit Memory, f. Local Search, g. Split adaptive PSO,
h. Fine-Grained, i. charged Swarm, The changed PSO charges the
velocity equation by adding a particle acceleration, , to the
standard equation, That is:
Where: …...…………………...…….(9)
f. Discrete PSO
PSO was originally developed for continuous-valued spaces. Many
problems are, however, defined for discrete value. Fortunately, the
PSO is easily adaptable to discrete-value spaces.
i. Binary PSO
For the binary PSO [1], particle represents position in binary space.
Formally, element of a particle’s position ( can form as:
on . A natural normalization of velocities is
obtained by using sigmoid function, that is:
……………………..….. (10)
The position update changes to:
…………..... (11)
Where . Many applications have used binary PSO
to solve their problem.
ii. General Discrete
Clerk defines these operators for participles that represent a
permutation of the valid discrete values with a strong ordering
implied between dimensions. In general velocity and position
equation change to:
………………………..………..(12)
Position update: first, all velocities are normalized to the range
[0,1] by dividing the velocities by the maximum range of the
corresponding dimensions,
………………..…………(13)
Then, each position determines if there is a swap with probability
. Last, if a swap has to be executed, the affected position of the
particle change to that of the global best (or local best) position.
3. OBSERVATION AND REVIEW
Particle swarm optimization (PSO) is a biologically inspired
computational search and optimization method developed in 1995
by Eberhart and Kennedy based on the social behaviors of birds
flocking or fish schooling. Recently, there are many variants of
PSO, and it may always grow rapidly. Figure 1 describes the
variants of particle swarm.
Particle Swarm
Optimization
Modification
variant of PSO
Basic Variant
of PSO
Velocity
Clamping
Synchronous Vs
ASynchronous
Constriction
Coefficient
Inertia Weight
Discrete PSO
Dynamic
Environment
of PSO
Multi-objective
optimization
Constraint
Optimization
using PSO
Niching with PSO
Single Solution
of PSO
Fig 1: Variant of Particle Swarm Optimization
We have considered that velocity clamping, inertia weight,
constriction coefficient, synchronous and asynchronous updates
are the basic variations of PSO that have been developed to
improve speed of convergence and quality of solution found by the
PSO. Figure 2 presents distribution of articles in terms of basic
variant of PSO. Regarding on this inertia weight has the largest
number of literatures between 2006 and 2010. Due to the progress
of variant PSO is rather new, so there is only a few articles that has
made.
Every basic variant of PSO has utility that will cover shortfall
of PSO. In addition they also have advantages and disadvantages
as shown in the table below:
International Journal of Computer Applications (0975 – 8887)
Volume 14– No.1, January 2011
23
Table1. The Basic Variant of PSO
Basic
Variant
Function
Advantages
Disadvantages
Velocity Clamping
Control the global exploration of the
particle
Reduces the size of the step velocity, so
that the particles remain in the search
area, but it cannot change the search
direction of the particle
VC reduces the size of the
step velocity so it will
control the movement of
the particle
If all the velocity becomes equal to
the particle will continue to
conduct searches within a hypercube
and will probably remain in the optima
but will not converge in the local area.
Inertia Weight
Controls the momentum of the particle
by weighing the contribution of the
previous velocity,
A larger inertia weight in
the end of search will foster
the convergence ability.
Achieve optimality convergence
strongly influenced by the inertia weight
Constriction
Coefficient
To ensure the stable convergence of the
PSO algorithm [21]
Similar with inertia weight
when the algorithm converges,
the fixed values of the parameters might
cause the unnecessary fluctuation of
particles
Synchronous and
Asynchronous
Updates
Optimization in parallel processing
Improved convergence rate
Higher throughput:
More sophisticated finite element
formulations
Higher accuracy (mesh densities)
Fig 2.Distribution of articles for Basic variant of PSO
In this paper we have know that originally, particle swarm
optimization is used to solve statics problem. For solving another
form of problem, many researchers have developed variant PSO,
such as: Single Solution, Niching with PSO, Constraint
Optimization using PSO, Multi-objective optimization, Dynamic
Environment and Discrete PSO. Every variant of PSO have
different form and function. Each of them also has variety methods
to solve their problem. Table 2 describes every characteristics of
basic variant of PSO. There are many researchers that have
develop many application using modification PSO. Figure 3
presents distribution of articles in terms of modification of Particle
Swarm Optimization. The number of papers using single solution
PSO yields a peak in 2007 and decreases gradually after that.
Niching with PSO is only used by some of researchers. From the
figure below, dynamic environment of PSO and multi-objective
optimization are the bigger numbers of literatures between 2006
and 2010. But a number of article of dynamic environment
decrease in 2010. On the others hands, the used of multi-objective
optimization increase from time to time. This method has a
challenge to increase caused of it can optimized multi-purposes of
problems.
Fig 3.Distribution of articles in the term of modification of
PSO
0
0.5
1
1.5
2
2.5
2006
2007
2008
2009
2010
No. of articles
Year
Velocity
Clamping
Constrain
Coefisient
Inertia
Weight
Synchronous
&Asynchrono
us
0
1
2
3
4
5
6
7
8
9
02-03
04-05
06-07
08-09
10
No. of articles
Year
Single
Solution
Niching with
PSO
Constrain
Optimation
using PSO
Multi
Objective
Optimization
Dynamic
Environment
of PSO
Discrete PSO
