International Journal of Scientific & Engineering Research Volume 2, Issue 8, August-2011 1

ISSN 2229-5518

One Half Global Best Position Particle

Swarm Optimization Algorithm

Narinder Singh, S.B. Singh

Abstract-In this paper, a new particle swarm optimization algorithm have been proposed. The algorithm is named as One Half Personal Best Position Particle Swarm Optimizations (OHGBPPSO) and a novel philosophy by modifying the velocity update equation has been presented. The performance of algorithm has been tested through numerical and graphical results. The results obtained are compared with the standard PSO (SPSO) for scalable and non-scalable problems.

Index Terms- Particle Swarm Optimization, One Half Global Best Position Particle Swarm Optimization, Personal Best Position, Global

Best Position, Global optimization, Velocity update equation.

—————————— ——————————

1 INTRODUCTION

TANDARD Particle Swarm Optimization:

Particle swarm optimization (PSO) [1] is a
is represent the how much confidence in itself and second acceleration coefficient is referred the how
stochastic, population-based search method, modeled
after the behavior of bird flocks. A PSO algorithm
maintains a swarm of individuals (called particles),
much confidence in its neighborhood),

r r

1 j 2 j

U (0,1) ,

where each individual (particle) represents a

yij

is the personal best position of particle i and
candidate solution. Particles follow a very simple
dimension j , and

yˆ j

is the neighborhood best
behavior: emulate the success of neighboring particles, and own successes achieved. The position of a particle is therefore influenced by the best particle in a neighborhood, as well as the best solution found by the particle. Particle position xi are adjusted using
position of particle i and dimension j . The t is
represent the rate of change in time.
Eberhart and Shi [15] suggested a more generalized PSO, where a constriction coefficient is applied to both terms of the velocity formula. Clerc and

x (t  1) 

x (t )

v (t  1) t

...(1)

Kennedy [14] showed that the constriction PSO can

i i i

Update Position

Previous Position

New Update Velocity

converge without using Vmax:
v (t 1)   (v (t)  c r
 ( y
x )  c r
 ( yˆ
x ))

where the velocity component, vi (t ) represents the

step size. For the basic PSO,

ij ij

1 1 j ij ij

2 2 j j ij

( y x ) ( yˆ

x )

where the constriction factor was set 0.7289. By using the constriction coefficient, the amplitude of the

  

c r ij ij

c r

j ij

v (t 1)

wv (t)

1 1 2 2

...(2)

ij ij j t

j t

particle‘s oscillation decreases, resulting in its

Update Velocity Current Motion Conitive Component Social Compoent

convergence over time. Kennedy [10] carried out
where w is the inertia weight [12], 1 and

2 are the

some experiments using a PSO variant, which drops the velocity term from the PSO equation.
acceleration coefficients (first acceleration coefficient

Dr S.B. Singh, is Professor and Head in Department of Mathematics, Punjabi University, Patiala, INDIA, Punjab, Pin No- 147002, Email ID:sbsingh69@yahoo.com. The research interests of author are Mathematical Modeling

and Optimization Techniques. He has published 50

If pi and pg were kept constant, a canonical PSO samples the search space following a bell shaped distribution centered exactly between the pi and pg.
This bare bones PSO produces normally distributed

research papers in Journals/Conference Proceedings. He was awarded Khosla Gold Medal by I.I.T. Roorkee for his

random numbers around the mean

pid pgd

2

(for

outstanding research contributions.

each dimension d), with the standard deviation of the

Mr. Narinder Singh is Ph.D. student in Department of

Mathematics, Punjabi University, Patiala, Punjab, INDIA,


Gaussian distribution being

pid pgd .

Pin No- 147002, Email ID: narindersinghgoria

@ymail.com.
Mendes and Kennedy [4] found that von Neumann
topology (north, south, east and west, of each particle

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International Journal of Scientific & Engineering Research Volume 2, Issue 8, August-2011 2

ISSN 2229-5518

placed on a 2 dimensional lattice) seems to be an overall winner among many different communication
topologies.
of particle i and dimension j . The t
the rate of change in time.
is represent
Kennedy [10] also proposed an alternative version of the barebones PSO, where
In the velocity update equation of this new PSO the
first term represents the current velocity of the
particle and can be thought of as a momentum term. The second term is proportional to the

(2 y

yˆ )

v (t  1)  

y if U (0,1) 0.5

ij

y yˆ

...(3)

vector

c r

11 j

( ij j

2

t

x )

ij

, is responsible for the

ij N ( ij ij ,)

otherwise

attractor of particle‘s current position and positive

 2

direction of its own best position (pbest). The third

(2 y

yˆ )

Based on equation (3), there is a 50% chance that the jth dimension of the particle dimension changes to the
term is proportional to the vector

c r (

2 2 j

ij j


2

t

x )

ij ,

corresponding personal best position. This version of
the barebones PSO biases towards exploiting personal best positions.
PSO variants are continually being devised in an attempt to overcome this deficiency, see e.g. [16] [17] [18] [19] [20] [21] [22] [23] [24] for a few recent additions. These PSO variants greatly increase the complexity of the original method and Pedersen and co workers [25, 26] have demonstrated that satisfactory performance can be achieved with the basic PSO if only its parameters are properly tuned.

2 THE NEW PROPOSED ALGORITHM

The motivation behind introducing OHGBPPSO is that in the velocity update equation instead of
is responsible for the attractor of particle‘s current
position.

The algorithm of OHGBPPSO is shown below: ALGORITHM- OHGBPPSO

- Randomly initialize particle position and
Velocities.
- While do not terminate
 Evaluate fitness objective functional value at current position x .

i

 If objective functional value is better than
modifying the Personal Best and Global Best Position.
Personal Best Position ( y

ij

) then update y .

ij

We introduce a new velocity update equation as
follows:
 If objective function value is better than Global

(( y

yˆ j

)  x ) (( y

yˆ j

)  x )

Best Position ( yˆ j ) then update yˆ j .

v (t  1)  wv (t )  c r

ij 2

ij c r

ij 2 ij

ij ij

1 1 j


t 2 2 j t

OR

- For each particle;
 Update velocity v

(t  1) and position

(2 y

yˆ ) (2 y

yˆ ) ij

( ij j

x ) (

ij j

x )

v (t  1)  wv (t )  c r

2 ij c r

2 ij

...(4)

x (t  1)

ij ij

1 1 j t

2 2 j t

ij

ij   ij

c r j

yˆ j

y   x

c r j

yˆ j

y   x

where w is the inertia weight,

c1 and c2

are the

ij 2 ij

t

ij 2 ij

t

acceleration coefficients (first acceleration coefficient

x (t  1)  x

(t )  v

(t  1)

represent the how much confidence in itself and
second acceleration coefficient referred the how much

ij ij ij

confidence in its neighborhood),

r r

1 j 2 j

U (0,1) ,

yij

END OF THE ALGORITHM

is the personal best position of i particle and j
dimension, and

yˆ j is the neighborhood best position

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International Journal of Scientific & Engineering Research Volume 2, Issue 8, August-2011 3

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Figure-I: Comparison of Particle Movement of SPSO and OHGBPPSO by Scalable Problems.

OHGBPPSO

SPSO

Pbes t+0.5 Gbes t

0.5 Gbes t

Pbes t

Current Pos ition of Partic le

-0.5 Gbes t

Figure-II: Comparison Movement of Particle SPSO and OHGBPPSO by Non-Scalable Problems.

SPSO

OHGBPPSO

Pbes t+0.5 Gbes t

0.5 Gbes t

Pbes t

Current Pos ition of Partic le

-0.5 Gbes t

3 TEST PROBLEMS

The relative performance of SPSO and OHGBPPSO is evaluated on two kinds of problem sets. Problem Set
1 consists of 15 scalable problems and Problem Set-II
consists of 13 non-scalable Problems.

4 SCALABLE AND NON-SCALABLE PROBLEMS

4.1 Scalable Problem: In which scalable problem the problem size is increase and decrease according to time.

4.2 Non-Scalable Problem: In which non-scalable problem the problem size is fixed, but the problems have many local as well as global optima.

Table-1: Detail of 15 Scalable Problems SET-I (Continued) (In which Particle size in the swarm increasing and decreasing, no particle sized is fixed).

Min f ( x)   20 exp(0.02

n1

i 1

xi )

between 30  x

i

 30 and

 exp(n1

n

i 1

cos(xi ))  20  e

Min Objective Function Value is

0.

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2. Cosine

Mixture

n n

Min f ( x)  0.1cos(5xi )  xi

In which search space lies

i 1

i 1

between 1  x

i

 1 and Min

3. Exponential

Objective Function Value is 0.1 (n) .

n In which search space lies

Min f ( x)  (0.5 

x2 )

between 1  x

 1 and Min

4. Griewank

1 n

i  1 i

2 n x

i

Objective Function Value is -1.

In which search space lies

Min f ( x)  1 

x

4000 i  1 i

 

i  1

cos( i )

i

between 600  x

i

 600 and

5. Rastrigin

Min Objective Function Value is

0.

n In which search space lies

Min f ( x)  10n

 [ x2 10 cos(2x )]

between 5.12  x

 5.12 and

i  1 i i

i

Min Objective Function Value is

0.

6. Function ‘6’

n 1 2 2 2

In which search space lies

Min f (x) 

 [100( x

1  x )

 ( x

1) ]

between 30  x

 30 and

i  1

i i i

i

Min Objective Function Value is

0.

7. Zakharov’s

Min f (x) 

n x2  [ n

( i ) x ]2  [ n

( i ) x ]4

  

In which search space lies

i  1 i

i  1 2 i

i  1 2 i

between 5.12  x

i

 5.12 and

8. Sphere

Min Objective Function Value is

0.

n In which search space lies

Min f ( x)   x2

i  1 i

between 5.12  x

i

 5.12 and

Min Objective Function Value is

0.

9. Axis parallel

n In which search space lies

hyper ellipsoid

Min f ( x)  

i  1

ix2

i

between 5.12  x

i

 5.12 and

10. Schwefel ‘3’

Min Objective Function Value is

0.




n n In which search space lies

Min f ( x) 

x

i  1 i

  x

i  1 i

between 10  x

i

 10 and

11. Dejong n

Min Objective Function Value is

0.

In which search space lies

Min f ( x) 

i  1

( x4  rand (0,1))

i

between 10  x

i

 10 and

12. Schwefel ‘4’

Min f (x)  Max{ x

,1  i n}

Min Objective Function Value is

0.

In which search space lies

i between 100  x

i

 100 and

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Min Objective Function Value is

0.

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13. Cigar

n In which search space lies

Min f ( x)  x2  100000  x2

between 10  x

 10 and Min

i i  1 i

i

Objective Function Value is 0.

14. Brown ‗3‘

n 1

Min f (x)  ( x )( x

1)  ( x2
1)( x2 1)]

In which search space lies

 [

i  1

i i 1

i 1 i

between 1  x

i

 4 and Min

15. Function ‘15’

Objective Function Value is 0.

n In which search space lies

Min f ( x)  

i  1

ix2

i

between 10  x

i

 10 and

Min Objective Function Value is

0.

Table-2: Detail of 13 Non- Scalable Problems SET-II ((In which Particle size in the swarm is fixed, no particle increasing and decreasing in the swarm).

Problem

No.

Problems Name Problems Range

1. Becker and Lago

Min f (x)  ( x

 5)2  ( x

 5)2

In which search space lies

1 2 between 10  x

i

 10 and Min

2. Bohachevsky ‘1’

Min f ( x)  x2  2x2  0.3 cos(3x )

Objective Function Value is 0. In which search space lies

1 2 1

between 50  x

 50 and Min

x i

3. Bohachevsky ‘2’

0.4 cos(4

Min f ( x)  x2  2x2 

2 ) 0.7

Objective Function Value is 0. In which search space lies

1 2

x x

between 50  x

i

 50 and Min

0.3 cos(3

1) cos(4

2 ) 0.3

Objective Function Value is 0.

4. Branin

Min f ( x)  a( x

bx2  cx

d )2

In which search space lies

2 1 1

between 5 

1  100 ,

g (1 

h) cos( x )  g

1

5  x

2

 15 and Min Objective

a  1, b


5.1 , c  5 , d  6,

Function Value is 0.398.

42

g  10, h  1

5. Eggcrate

8

Min f (x)  x2  x2  25(sin2 x

 sin2 x )

In which search space lies

1 2 1 2

between 2x

i

 2and Min

6. Miele and

Min f ( x)  (exp( x )  x

)4 100( x
x )6

Objective Function Value is 0. In which search space lies

Cantrell

1 4 2 3

between 1  x

 1 and Min

 (tan( x
x ))4  x8 i

7. Modified

Min f (x) 100(x


3 4 1

x2 )2  [6.4( x

 0.5)

Objective Function Value is 0.

In which search space lies

Rosenbrock

2 1 2

between 5  x , x

1 2

 5 and

Min Objective Function Value is 0

 x x

In which search space lies

8. Easom

Min f ( x) cos( 1) cos( 2 )

* exp(( x

)2  ( x

)2 )

between 10  x

i

 10 and Min

1 2 Objective Function Value is -1

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9. Periodic

Min f ( x)  1  sin2 x

 sin2 x

In which search space lies

1 2 between 10  x

i

 10 and Min

 0.1exp( x2  x2 )

1 2

Objective Function Value is 0.9

10. Powell’s

Min f ( x)  ( x

 10x

)2  5( x

x )2

In which search space lies

1 2 3 4

between 10  x

 10 and Min

 ( x

 2x

)4  10( x x )4 i

11. Camel back-3

2 3 1 4

Min f ( x)  2x2  1.05x4  1 x6

Objective Function Value is 0

In which search space lies

1 1 6 1

between 5  x , x

1 2

 5 and

x x

x2

Min Objective Function Value is 0

12. Camel back-6

1 2 2

Min f ( x)  4x2  2.1x4  1 x6

In which search space lies

1 1 3 1

between 5  x , x

1 2

 5 and

x x

 4x2  4x4

Min Objective Function Value is -

13. Aluffi-Pentini’s

1 2 2 2

Min f ( x)  0.25x4  0.5x4  0.5x2

1.0316

In which search space lies

1 1 1

between 10  x

i

 10 and Min

 0.1x

 0.5x2

Objective Function Value is 0.352

1 2

5. PARAMETER SETTING AND ANALYSIS OF RESULTS

5.1 Parameter Setting: The maximum number of function evaluations is fixed to be 30,000.The swarm size is fixed to 20 and dim is 30. The inertia weight is 0.7 and the acceleration coefficients for SPSO and OHGBPPSO are set to

c c .

robustness. In observing Table 4, it can be seen that OHGBPPSO gives a better quality of solutions as compared to SPSO. Thus, for the non- scalable problems OHGBPPSO outperforms SPSO with respect to efficiency, reliability, cost and
robustness, Table 3.
be 1 2

1.5

It is observed that SPSO could not solve two

5.2 Results Analysis: In observing Table 3, it can be

seen that OHGBPPSO gives a better quality of solutions as compared to SPSO. Thus, for the scalable problems OHGBPPSO outperforms SPSO with respect to efficiency, reliability, cost and
problems with 100% success, whereas OHGBPPSO solved all the problems with 100% success.

Table-3 Comparison of SPSO and OHGBPPSO by Scalable Problems Set-I

Problem

No.

Minimum Function

Value

Mean Function Value

Standard

Deviation

Rate of Success

SPSO

OHGBPPSO

SPSO

OHGBP

PSO

SPSO

OHGBP

PSO

SPSO

OHGBPPS

O

1

0.674207

0.524363

14699.6000

3835.60000

0.323300

0.089916

68.00%

100%

2

0.683359

0.415349

902.400000

812.400000

0.051324

0.109088

100%

100%

3

0.000000

0.000000

40.000000

40.000000

0.000569

0.000483

100%

100%

4

0.770386

0.680129

7038.80000

4505.20000

0.024076

0.053167

100%

100%

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5

20.89413

0.222967

13000.0000

5724.00000

16.219312

0.393235

0.00%

100%

6

0.007444

0.002482

140.400000

130.000000

0.263601

0.280204

100%

100%

7

0.001967

0.000010

52.800000

64.400000

0.251745

0.218062

100%

100%

8

0.000000

0.000000

40.000000

40.000000

0.029778

0.025274

100%

100%

9

0.000005

0.000004

40.400000

44.400000

0.135186

0.253899

100%

100%

10

0.001648

0.001330

40.400000

42.000000

0.141846

0.225886

100%

100%

11

0.635139

0.172111

5653.20000

4446.40000

0.065082

0.189287

100%

100%

12

0.012020

0.011124

65.200000

74.000000

0.256712

0.235492

100%

100%

13

0.057514

0.047514

1196.00000

1587.00000

0.216409

0.246410

100%

100%

14

0.002002

0.002000

40.000000

40.000000

0.129936

0.147956

100%

100%

15

0.000000

0.000000

40.000000

40.000000

0.011661

0.013672

100%

100%

Table-4 Comparison of SPSO and OHGBPPSO by Non-Scalable Problems Set-II

Problem

No.

Minimum Function

Value

Mean Function Value

Standard Deviation

Success of Rate

SPSO

OHGBP

PSO

SPSO

OHGBP

PSO

SPSO

OHGBP

PSO

SPSO

OHGBPPSO

1

0.500000

0.500104

41.600000

49.600000

0.088031

0.089541

100%

100%

2

0.004665

0.015410

48.800000

50.000000

0.279521

0.251598

100%

100%

3

0.002060

0.009141

54.400000

61.200000

0.240645

0.239715

100%

100%

4

0.003335

0.006881

81.600000

86.400000

0.257595

0.258021

100%

100%

5

0.002562

0.034907

60.400000

59.600000

0.249756

0.251556

100%

100%

6

73046.5964

73046.59648

30000.000

30000.000

0.000000

0.000000

0.00%

0.00%

7

14.541432

26.900800

30000.000

30000.000

0.000000

0.000000

0.00%

0.00%

8

0.009239

0.091784

59.200000

96.800000

0.286622

0.230102

100%

100%

9

0.480964

0.480470

40.000000

40.000000

0.037841

0.033635

100%

100%

10

0.075842

0.034330

578.80000

546.80000

0.233322

0.245201

100%

100%

11

0.006245

0.006541

46.800000

49.600000

0.206116

0.236984

100%

100%

12

0.017029

0.003487

51.200000

56.800000

0.262820

0.229284

100%

100%

13

0.010104

0.012869

45.600000

54.400000

0.217098

0.210105

100%

100%

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Figure A: Comparing the SPSO and OHGBPPSO with the help of 15 Scalable Problems SET-I.

Figure B: Comparing the SPSO and OHGBPPSO with the help of 13 Non-Scalable

Problems SET-II

Note: x-axis represented the scalable and non-scalable problems and y-axis denoted the Minimimum Objective

Function Values

6 CONCLUSIONS In this paper, a new PSO approach One Half Global

Best Position Particle Swarm Optimization is

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ISSN 2229-5518

presented. The algorithm is tested on scalable problems (increasing or decreasing particle in the swarm) and non-scalable problems (swarm size is fixed). The results show that when the particle size is increasing and decreasing in the swarm, the proposed algorithm outperforms the Standard Particle Swarm Optimization. But in the case when the particle size is fixed and no particle enters/leaves the swarm the Standard Particle Swarm Algorithm is better than the proposed one.

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