Hybrid Butterfly Based Particle Swarm Optimization for Optimization Problems

8/10/2019 Hybrid Butterfly Based Particle Swarm Optimization for Optimization Problems

1/6

Hybrid Butterfly Based Particle Swarm Optimization

for Optimization Problems

Aashish Kumar Bohre1 Dr. Ganga Agnihotri

2Dr. Manisha Dubey

3

Electrical Engineering Department1, 2, 3

MANIT, Bhopal, INDIA

1, 2, 3

Abstract- One of the superior optimization algorithms amongst all

earlier introduced algorithms is the particle swarm optimization

algorithms. This paper introduces Butterfly Particle swarm

optimization (BF-PSO) with some novel control parameters such

as sensitivity of butterfly towards nectar by different means of

communication and probability of the nectar presence. This new

hybrid algorithm is based on the intelligent characteristics and

behavior of butterfly during the process of food (nectar) search

and mimics their intelligent network structures. Sensitivity and

the probability of nectar, according to the degree of nodes is

calculated using this new algorithm. By adding the effect of these

modifications in the standard Particle Swarm Optimization

(PSO), the algorithm performance and the ability to search

optimum value of the Particle Swarm Optimization is improved.

Finally, the results of applying the BF-PSO on benchmark

functions are shown. The overall improvement in performances

of the BF-PSO on the basis of the sensitivity of butterfly and

probability of nectar source.

Keywords- Particle Swarm Optimization (PSO), BF-PSO

(ButterflyPSO), sensitivity of butterfly (s), probability of nectar (p).

I. INTRODUCTION

The development of artificial intelligence systems doesn'tnecessitate the entire copy of the natural system; it requires

exploration of ideas, model, and behaviour for animplementation. It is potential (or intelligence) of individual

insect to identify the various complexity of the tasks and playorperform for them. Some of the best examples include theforging process, communication of information between

insects, collection and processing (feeding) of food, matingprocess between male and female insects and various other

interesting behaviours, which are quite commonly seen in thenatural environment.

Wilson [1] outlined biological communication as anatural process on the theatrical character of one cell (or

organism) that changes the probability pattern of characters(or behaviour) in associate to other another cell (or organism)

in the way of adoption.

Tristram D. Wyatt, [2] describes the behaviour of

insects and animals. The importance of chemicalcommunication is described with some examples from a

varied variety of insects, animals, moreover as humans,butterfly or moths, marine copepods, Caenorhabditis elegans,

Drosophila, goldfish, snakes, mice and elephants. The

ecology, evolution and behavior, offers an associate degree

introduction to the fast progress in our understanding ofolfaction at the molecular and neurologic level.

Eberhart, R. C., and Kennedy, J. [3-5] reported the

optimization of nonlinear functions; exploitation particleswarm methodology is delineated.

The intelligence characteristics and behavior of

artificial systems are aging, dying, fight, think, learn,collaborate and perish. Frequently, when introduce the

different Artificial intelligence researchers' attempts for

illustrates complicated behaviors of artificial systems. This isgained from relatively simple principles. The evolution of theDarwins theory and the concept of the survival of the fittest

for the artificial systems were initiated in the earlier decades(such as genetic algorithms, evolutionary programming and

etc.) [7-11, 21].

In this paper optimization technique based on the butterflyis reported. The motivation towards the butterfly based swarmoptimization is searching of food processing, intelligence and

behavior. The searching process of butterflies basicallyconcentrated on the food source that is nectar sources. The

butterflies have the natural sensitivity to sense the nectarprobability. The butterfly develops an interactive intelligent

system with high communication to find the optimal solutions[12- 19].

II. THEPARTICLESWARMOPTIMIZATION

TECHNIQUE

The Particle Swarm optimization comes under the class ofSwarm Intelligence techniques [3-5]. The particle swarm

optimization was suggested in 1995, by Kennedy and Eberhart[22,23]. Its a popular and more effective optimization

technique based on the population search. The PSO is astochastic method of optimizing simulated the intelligent

swarms behaviour to solve the different nonlinear problemsin optimization based on population. The PSO concept

inspired by mimicking the foraging process (social behaviour),the swarms like as fishes, birds, etc. By putting a randomvelocity to each particle, it acts through the search space

corresponding to the target part, limited by problem specificconstraints. Each particle is evaluated for its fitness basedon the output of the objective function. The particle is

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attracted to the best fitness location attained by it as well as

that attained across the population so far. The each and everyparticle follows the particular velocity throughout the

specified region of the problem. The velocity and its direction

updates by the previous best experiences of particles and

memorized, its the best experience by the experiences ofneighboring (social-influence) [20, 22, 23]. Hence the way in

which, the particle has a leaning to fly towards a more optimalregion within the hunting field or space. [3, 4, 5, 9, 11, 20].

The position of particle updates based on the

updating of the velocity of each particle within the current

iteration. In the next iteration the each particle position is

calculated by summing the velocity and current position. Let

us assume that the particles position and velocityrespectively in the dimension space d are

and . Finally theeach particle velocity is updated by the equations given below

(1) and (2) [3, 4, 5, 20, 22, 23].

( 1) 1 1

2 2

* ( )

( ) (1)

k k k

k

v w v c r pbest currentposition

c r gbest currentposition

+ = +

+

( 1 ) 1 (2 )k k kx x v+ += +

Where

N: total size of swarms.

t: the iteration number.

w: is the inertia weight.

rl, r2: two random numbers between (0, 1).

Cl, C2: the cognitive and social scaling parameters.

III. THE BUTTERFLIES SWARM BEHAVIOR AND

INTELLIGENCE NETWORK

The butterflies as an advanced insects have a complete life

cycle. Butterflies pass through a life cycle. There are basicallyfour different phases. The first beginning phase is the eggs.This is the phase in which eggs lays down by girl butterfly.

The butterfly places the eggs on leaves. After that, the second

phase start, its the caterpillar. This is the phase in which the

eggs incubate [21, -23]. It holds approximately five days for

the eggs to incubate, by completing this process, caterpillarcomes out. At this time, the caterpillar eats all the time. The

growth of the caterpillar is very fast. Once its grown all theway, then the third stage starts. It is like the chrysalis stage of

the butterfly [12-15]. The caterpillar forms a chrysalis. The

caterpillar is inside the chrysalis. Within the chrysalis, itbegins to change. It fast changes into a butterfly. At one time

the caterpillar has changed into a butterfly, the fourth stage

takes up. The fourth stage also last phase of butterfly lifecycle. The last phase of life cycle is only the butterfly. The

butterfly comes out of the chrysalis. Immediately they can

learn to fly, and also can get a mate. When they find mates,

they lay eggs. Hence the life cycle process begins again, so

that first phase comes again and the life cycle process

repeated. The adult stage is what most people think of when

they think of butterflies. The caterpillar and larva seems reallydifferent from each other. The caterpillars have stubby

branches, very short antennae and little tiny eyes, etc. Theadult butterflies have long branches, long antennae, and

compound eyes. They can too fly by using their large and

colorful wings [16-17].

The behavior of adult butterflies during the food

search process is very realistic for finding the food plant. In

this process the butterflies find nectar sources or food plants

(flowers). The butterfly have different sensors such antenna,eyes, etc., to find out the food plant. They can communicate or

exchange information between them and neighbors also byvarious ways, such as dancing, color, chemicals, sound, and

physical actions, etc. By this butterfly shows the collective

intelligence behavior on the butterfly network.

The butterfly graphs were originally brought out as

the underlying graph of FFT networks, which can execute the

fast Fourier transform (FFT) very efficiently. The butterflywing structure is implemented by graph theory, then from

butterfly graph. The connection between two or more

butterflies graph form butterfly network. The butterfly

intelligence network (based on wing structure) is used to

represent the linear network also [25-27].

Fig. 1. Butterfly network structure [25-27]

This butterfly network having different nodes A, B, C and D

as shown in figure 1. These vertex or node can be assumed ascommunication nodes of butterfly network, which used to

exchange or transmit information between butterflies by

different means.

IV.

THE BUTTERFLY SWARM BASED SEARCHPROCESS

In this section butterfly, intelligent network is formed whenthe sensitivity of butterfly meets the probability of the nectar.

In the sensitive region with minimum sensitivity of butterflymatches minimum probability of nectar. In this region, all

2014 First International Conference on Networks & Soft Computing 173


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food sources are called in active state, while in the outer

region present inactive state. The outer region wheresensitivity of a butterfly is not able to detect the probability of

nectar in this region. The active and inactive regions are

separated by the boundary which is from when the minimum

sensitivity of butterfly meets to the minimum probability of

nectar. A network consists of N nodes having sourced ofnectar. Each node (food source) is selected by their degree,when the node does not have nectar than the degree of a node

is considered zero, and when there is the nectar source thanaccording to sensing ability of butterflies connectivity will

increase the degree (higher degree) of a node in the sensitive

region because all the nodes outside the region will have alower degree. Depending of the degree we can classify or

arrange the nodes of the best nectar source. The optimalsolution in the sensitive region (probabilistic region) will

depend on the degree of the nodes. In the butterfly network

node (flower) having more amounts of nectar will attract more

and more butterflies by natural sensitivity; hence it willincrease the degree of the node (flower) because of that node

has a higher probability of nectar. The butterflies exchange the

information not only by dancing, but also they have naturalintelligence for sensing nectar through color, insect, chemical,

sound and physical action so it develops better communicationnetwork compared to others. This intelligent communicationsystems, that is the information exchange between the each

and every adult butterflies contributes and perform the

"collective intelligence" behavior within the butterfly network

[12-15]. The structure of the butterfly network consisting N

nodes at instant t, the butterfly communication network states

of regions is given in figure 2, which can be defined as:

(3)

(4)

Fig. 2. Active and inactive regions concept of BF-PSO

In the food search process, butterfly finds the optimal location

depending upon the sensitivity of the flower and probability ofnectar, after finding the solution, it communicates directly or

indirectly from the others by different mean of communicationintelligence. The representation for BF-PSO search process is

given in figure 3. Dancing mechanism is not understood inbees, but here assumption is that the recruitment amongst theentire bee is the function of good-quality food, butterfly

develops a good communication network by several

parameters such as color, chemicals, sound, and physical

actions. They can it chooses the flower of maximum nectar

probability randomly search the optimal solution and before

terminating the respective flight (iteration) it finds a nextoptimal solution. Hence the loop continues and also

propagates with the time, so in less time butterflies are able tochoose several local best (lbest) them according to

arrangement of degree of nodes, they choose selects global

best (gbest). During the forging process in new search regiondecision making process butterfly interact with the

substantially different sensory system. So in case of butterfly

the search has somewhat become sequential, so it limits thearea, hence it takes less time compared to others. It searches

for nectar near the plant where it lays eggs. Butterflies don't

make nests like bees or wasps; usually they live on same foodsource until it finds a new food source [15-19]. The algorithm

termination criteria will be the maximum number of flights

(iteration) with improvement in fitness of respective objective

function. Improvement of butterfly-PSO according to

ingeniously and due to antenna on their mouth by whichflowers attract butterfly more compare to others there is a verygood relation of butterfly and surrounding atmosphere to

provide the best solution. After butterfly relinquishes the food,

butterfly propagate in the direction of the new nectar source or

gets information from others.

Fig. 3. BF-PSO search process representation

V. THE BUTTERFLY PARTICLE SWARM BASED

OPTIMIZATION TECHNIQUES (BF-PSO)

The butterfly PSO algorithm is essentially based on nectar

probability factor and sound communication system means

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sensitivity. BF-PSO consists of intelligent behavior of the

butterfly to find out the nectar in maximum quantity.

By taking the effect of sensitivity and nectar

probability we have modified the standard PSO with some

necessary variation in optimization parameters. The sensitivityis more nearer to the nectar source, and the minimum

sensitivity is required to search the next nectar sources, so the

range of the sensitivity and probability from 0.1 to 1.

The parameters and equations for BF-PSO are given as:

The sensitivity is:

S = exp(-(itera_max-itera)/itera_max) (5)

The Probability in an experiment or assignment for an event

evaluates how likely an event is. In other words, probabilitymeasure the chance for that likely event occurs as a result of

an experiment [6]. The results of an experiment which leads to

chance or possibility of an event is called favorable outcomes.

The probability of an event is number of likely observations ofthe events divide by the total number of observations in the

assignment. In any event, the probability that an event willoccur is lies between 0 and 1. The probability value 1 denotes

sure event, occurs every time in an experiment and it's

repeated. The probability value 0 denotes impossible event,

the event never occurs in an experiment.

The probability of an eventP(E)can be given as:

( )totalnumberof favourableoutcomes fortheeventinanExperiment

P Etotalnumberof possibleoutcomesinanExperiment

=

(6)

In the similar way the probability of global best butterfly

swam is given as:

1

N

i

global best fitness of BF swarm

local best fitness of all BF swarmsP

=

=

(7)

Then modified equations are the basic equations of BF-PSOtechnique given below for the velocity and position updating:

( 1) 1 1

2 2

' * (1 ) ( )

( ) (8)

k k k k k

k k

v w v s p c r lbest currentpop

p c r gbest currentpop

+ = +

+

( 1 ) 1' ( 9 )k k kx x v+ += +

VI. RESULTS

The implemented results to validate the proposed BF-PSO

algorithm on the several well known functions have been

chosen. The standard benchmark functions as given in the

table I (the functions F1, F2, F3 and F4). The comparativeresults for functions using various optimization methods such

as genetic algorithm (GA), S-PSO and BF-PSO given belowin figures.4, 5, 6 and figure 7. Also the optimum values for

each function with different dimensions are given in table II

and table II.

Fig. 4. Comparative results for function F1




0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

6

7

8x 10

4

no. of iterations

Fitnessvalue

SPSOBFPSO

GA

0 100 200 300 400 500 600 700 800 900 10000 1000

1

2

3

4

5

6

7

8x 10

11

No. of iteration

FitnessValue

SPSO

BFPSO

GA

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

80

90

No. of iteration

Fitnessvalue

SPSO

BFPSO

GA

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3x 10

8

No. of itrration

Fitness

value

SPSO

BFPSO

GA



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TABLE I. THE BENCHMARK FUNCTION AND RANGES

Name Equation Range

F1

[-100, 100 ]

F2

[-10, 10]

F3

[-100, 100 ]

F4

[-30, 30]

TABLE II. THE COMPARISON OF RESULTS FOR BENCHMARKFUNCTIONS (1000 ITERATION AND 20 TRIALS)

Algorithm

Function

Dime

nsion

GA S-PSO BF-PSO

Mean

Best

Mean

Best

Mean

Best

F1 30 7.50808e+003

2.23034e-006

1.85086e-010

F2 30 26.52993 0.06236 0.00337

F3 30 40.96571 0.84339 0.21226

F4 30 2.08842e+006

43.45286 25.2002

VII. CONCLUSIONS

The effect of sensitivity of butterfly and probability of nectar

in BF-PSO proposed algorithm, for the various test functions

are improved results as compared to other methods (such asGA, PSO etc). The search operation has been increased to a

larger extent. BF-PSO shows the good convergence rate and at

each time the algorithm provides more optimum output, hencein BF-PSO results found with sound accuracy as well good

convergence.

In the existing work presented by Magnus Erik, and

Hvass Pedersen and Hvass Laboratories [24] for the

benchmark functions F1 (sphere), F2 (rosenbrock), F3

(schwefel2-22), F4(schwefel2-21). They have compared test

function results with PSO (Particle Swarm optimization) andMOL (Many Optimizing Liaisons). The results given in thetable II gives the better results as compared to existing one. So

that due to introducing the two new parameters the sensitivity

of butterfly and probability of nectar in BF-PSO give the best

optimized values; hence there is a higher convergence rate in

the BF-PSO then other method. The convergence of fitnesstowards minimum value with respect to iteration for BF-PSOcompared with S-PSO and GA is shown in figures from 4 to 7.However the mean value of BF-PSO is somewhat higher

compared to PSO, so diversity increases, and consists of high

accuracy with higher convergence rate.

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Particle%20Swarm/PSO%20parameters.pdf

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Aashish Kumar Bohre1 received BE degree in Electrical and Electronics

engineering from UIT- RGPV Bhopal, (2009), and M-Tech degree in Power

System (2011) from MANIT, Bhopal. At the moment he is Pd.D. scholar atMANIT, Bhopal, India. Email: [email protected],

[email protected]

Dr. Ganga Agnihotri2 received BE degree in Electrical engineering fromMACT, Bhopal (1972), the ME degree (1974) and PhD degree (1989) from

University of Roorkee, India. Since 1976 she is with Maulana Azad College

of Technology, Bhopal in various positions. Currently she is professor. Herresearch interest includes Power System Analysis, Power System

Optimization and Distribution Operation. Email: [email protected]

Dr. Manisha Dubey3was born in Jabalpur in India on 15th December 1968.

She received her B.E (Electrical), M.Tech. (Power Systems) and Ph.D

(Electrical Engg.) in 1990, 1997 and 2006 respectively. She is working as

Professor at the Department of Electrical Engineering, National Institute ofTechnology, Bhopal, India. Her research interests include power systems,

Genetic Algorithms, Fuzzy Logic systems and application of Soft Computing

Techniques in power system dynamics and control. Email:[email protected]

TABLE III. THE RESULTS FOR BENCHMARK FUNCTIONS (100000 ITERATION AND 100 TRIALS)

Algorithm

Function

Dimensi

on

S-PSO BF-PSO

Best Mean Std. dev. Best Mean Std. dev.

F1

2 0 0 0 0 0 0

5 0 0 0 0 0 0

10 2.6322e-243 7.8328e-228 0 0 0 0

20 1.4948e-107 2.5578e-093 2.5410e-092 3.6856e-149 8.4820e-095 8.4820e-094

30 6.9550e-060 2.2670e-057 4.8193e-050 1.3530e-066 4.5480e-058 1.3459e-052

F2

2 0 0 0 0 0 0

5 9.6438e-241 9.1180e-230 0 0 010 2.5704e-136 5.3000 8.7145 5.7095e-242 2.4000 5.8810

20 1.6966e-062 6.2000 13.9102 5.2522e-197 1 3.1623

30 4.2881e-037 18 26.9680 1.9579e-041 18 25.7337

F3

2 0 0 0 0 0 0

5 7.4581e-187 1.5974e-175 0 0 3.0287e-317 0

10 1.0425e-059 1.2085e-052 5.8852e-052 3.2675e-135 1.3719e-118 1.3719e-117

20 8.5961e-010 6.0935e-007 3.8596e-006 7.7867e-024 2.5853e-016 1.6592e-015

30 0.0046 0.0883 0.1935 1.2286e-007 6.0233e-005 1.5112e-004

F4

2 0 0.2004 2.0044 0 0 0

5 3.7325e-008 1.2020 4.8593 5.4141e-012 0.7439 3.8281

10 8.3947e-008 3.8031e+003 3.2663e+004 4.9854e-011 1.6312e+003 1.7722e+004

20 0.0145 1.9121e+003 1.2659e+004 1.4295e-008 9.6327e+002 9.00e+003

30 0.0711 3.7969e+003 1.7968e+004 5.6642e-004 3.6306e+003 1.7721e+004


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Hybrid Butterfly Based Particle Swarm Optimization for Optimization Problems