Self-Adaptive Spider Monkey Optimization Algorithm for Engineering Optimization Problems

International Journal of Information, Communication and Computing Technology

Jagan Institute of Management Studies, New Delhi

_________________________________________________________________________________________________________ 1 Assistant Professor, Department of Computer Science & IT, Jagannath University, Jaipur

2 Professor, Department of Mathematics, Jagannath University, Jaipur

3 Assistant Professor, Department of Computer Science, St. Xavier’s College, Jaipur

Email: 1 [email protected] ,

2 [email protected],

3 [email protected]

Copyright ©IJICCT, Vol II, Issue II (July-Dec2014): ISSN 2347-7202 96

Self-Adaptive Spider Monkey Optimization Algorithm for Engineering Optimization Problems

1Sandeep Kumar,

2Vivek Kumar Sharma,

3Rajani Kumari

ABSTRACT

Algorithms inspired by intelligent behavior of simple

agents are very popular now a day among researchers. A

comparatively young algorithm motivated by extraordinary

behavior of Spider Monkeys is Spider Monkey Optimization

(SMO) algorithm. SMO algorithm is very successful

algorithm to get to the bottom of optimization problems. This

work presents a self-adaptive Spider Monkey optimization

(SaSMO) algorithm for optimization problems. The

proposed strategy is self-adaptive in nature and therefore no

manual parameter setting is required. The proposed

technique is named as Self-Adaptive Spider Monkey

optimization (SaSMO) algorithm. SaSMO gives better

results for considered problems. Results are compared with

basic SMO and its recent variant MPU-SMO.

KEYWORDS

Spider Monkey Optimization Algorithm, Swarm

intelligence, Engineering optimization problems, Nature

Inspired Algorithms.

1. INTRODUCTION

Intelligent food foraging behavior of Spider Monkeys

inspired J. C Bansal and his colleagues to develop a new

algorithm. J. C. Bansal et al. [15] proposed Spider Monkey

Optimization (SMO) algorithm in year 2013. It is a recent

popular algorithm motivated by intelligent behavior of

simple natural agents. This algorithm is stimulated by means

of the extra ordinary behavior of Spider Monkeys while

searching for food. This algorithm based on fission-fusion

social structure (FFSS). It falls into category of Nature

Inspired Algorithms (NIA) that is inspired by some natural

phenomenon or extraordinary behavior of intelligent insects.

NIAs include Evolutionary algorithms, Immune algorithms,

neural algorithm, Physical algorithms, Probabilistic

algorithms, stochastic algorithms and Swarm algorithms

based on their source of inspiration and motivation. In last

decade a number of new algorithms are developed by

researchers that are inspired by natural phenomenon. Some

recent development includes Water Wave Optimization

(WWO) algorithm [2]. WWO algorithm is motivated by the

shallow water wave theory. It mimics the eye catching

phenomenon of water waves, such as propagation,

refraction, and breaking. A. Brabazon et al. [3] proposed a

new population based strategy based on social roosting and

foraging behavior of one species of bird and named it Raven

Roosting Optimization algorithm. Based on the living

behaviors of microalgae, photosynthetic species an

algorithm was developed by SA Uymaz [4] named as

Artificial Algae Algorithm (AAA). S. Mirjalili et al. [6]

proposed a novel nature inspired strategy for continuous

optimization problems namely Grey Wolf Optimizer

(GWO). GWO algorithm mimics the leadership hierarchy

and hunting mechanism of gray wolves. A. Askarzadeh

incepted Bird Mating Optimizer (BMO) [7] for designing

most favorable searching techniques. This algorithm

emulates the behavior of bird species figuratively to breed

broods with superior genes. X. Li et al. [11] developed

Animal Migration Optimization (AMO) Algorithm that

mimics migration activities of animals.

SMO algorithm consists of population of budding solutions

like other population based algorithms. In this algorithm

budding solutions are represented by food sources of spider

monkeys. The superiority of a food source is decided by

calculating its fitness. The SMO algorithm is comparatively

an easy, rapid and population based stochastic search

strategy. While searching for optimal solution this algorithm

need to maintain balance between two basic activities named

the assortment process, which make sure the exploitation of

the preceding knowledge and the adaptation process, which

empowers exploring diverse fields of the search space.

However, it has been observed that SMO algorithm is very

good in exploration of local search reason and exploitation

of best feasible solutions in its immediacy [1] [5]. Therefore,

to solve a complex problem like parameter estimation for

frequency-modulated sound wave this paper uses a new

variant of SMO algorithm. The proposed algorithm is

adaptive in nature as it automatically modifies the radius of

search area during local leader phase and local leader

decision phase in order to update position along with fitness

based position update. Fitness of a solution decides its

quality. There are number of methods to calculate fitness of

a function but it must include value of function.

There are very few research papers on SMO algorithm in

literature as it is very young algorithm. Recently S. Kumar et

Self-Adaptive Spider Monkey Optimization for Engineering Optimization Problems


al. proposed a couple of variants of SMO algorithm. First is

Modified Position Update in Spider Monkey Optimization

Algorithm (MPU-SMO) [5]. MPU-SMO tries to enhance

exploration and exploitation activities together by modifying

local leader phase and global leader phase with the help of

golden section search (GSS) [22] technique stimulated by

memetic search in ABC [16]. Memetic strategy inspired by

GSS strategy recently used in various algorithms like

RMABC [8], IMeABC [9], EnABC [10], MSDE [12] and

IoABC [13]. Second is Fitness Based Position Update on

Spider Monkey Optimization Algorithm (FPSMO) [1].

Fitness of a solution play very important role in population

based stochastic algorithms. It denotes quality of solution,

that how much they are suitable for next iteration. FPSMO

update position based on the individual’s fitness. It assumes

that best fitted solution has good neighbors and modifies the

step size according to its fitness. It takes a large step for high

fitted solution and a small step for low fitted solutions.

FPSMO move away from bad solutions and keep searching

in proximity of good solutions.

Formation of the paper is as follows: Major steps of SMO

algorithm are explained in section 2. Section 3 describes

newly anticipated variant of SMO algorithm. In section 4,

some engineering optimization problems are discussed in

detail. The solution of selected problems and performance of

the proposed strategy is analyzed in section 5 and 6. At last,

in section 7, paper is concluded followed by references.

2. SPIDER MONKEY OPTIMIZATION

ALGORITHM

Extraordinary food foraging conduct of spider monkeys

motivated J. C. Bansal et al. [15] to develop a new

population based metaheuristics. They named it Spider

Monkey Optimization algorithm. The original SMO

algorithm given by J. C. Bansal et al. [15] consists of seven

phases.

Population initialization

Local Leader Phase (LLP)

Global Leader Phase (GLP)

Global Leader Learning (GLL) phase

Local Leader Learning (LLL) phase

Local Leader Decision (LLD) phase

Global Leader Decision (GLD) phase

Spider Monkey Optimization Algorithm is not considered as

a swarm intelligence algorithm as it elects leader or

coordinator to control the groups. It is stochastic in nature as

there are some random components in each step.

2.1 Phases of Spider Monkey Optimization (SMO)

Algorithm

SMO algorithm is a population based iterative approach. It

has seven key steps including initialization of population.

The complete depiction of every phase is summarized in

next few subsections.

Population initialization

At first, a population of N spider monkeys initialized. Initial

population denoted by a D-dimensional vector SMOi, where

i = 1, 2,..., N. Every spider monkey SMO represents a

feasible solution for the consideration problem. Every SMOi

is initialized using Eq. (1).

min max min( )

(0,1)

ij j j jSMO SMO SMO SMO

where

(1)

Here SMOmin j and SMOmax j indicate lower and upper bounds

of SMOi in jth

direction correspondingly.

Local Leader Phase (LLP)

The subsequent phase is Local Leader Phase. Based on the

experience of local leader and group members SMO

modernize its present location. It compares fitness new

location and current location and applies greedy selection.

The ith SMO that also belongs to kth

local group update its

location using Eq. (2).

[0,1] ( )

[ 1,1] ( )

newij ij kj ij

rj ij

SMO SMO rand LL SMO

rand SMO SMO

(2)

Where SMOij denote ith

SMO in jth

dimension, LLkj

correspond to the kth

local group leader location in jth

dimension. SMOrj is the rth

SMO which is arbitrarily selected

from kth

group such that r≠ i in jth

dimension.

Global Leader Phase (GLP)

The Global Leader phase (GLP) starts just after finishing the

LLP. Based on experience of Global Leader and members of

local group SMO modernize their position using Eq. (3).

[0,1] ( )

[ 1,1] ( )

newij ij j ij

rj ij

SMO SMO rand GL SMO

rand SMO SMO

(3)

International Journal of Information, Communication and Computing Technology (IJICCT)


Where GLj stands for the global leader’s position in jth

dimension and j ∈ {1, 2, ...,D} denotes a randomly selected

index.

The SMOi updates their locations with the help of

probabilities pi’s. Probability of a particular solution

calculated using its fitness. There are number of different

methods for computing fitness and probability, here pi

computed using Eq. (4).

max

0.9 0.1ii

fitnessp

fitness

(4)

Global Leader Learning (GLL) phase

Now global leader modify its location with the help of some

greedy approaches. Highly fitted solution in current swarm

chosen as global leader. It also perform a check that the

position of global leader is modernize or not and modify

Global Limit Count accordingly.

Local Leader Learning (LLL) phase

Now local leader modify its location with the help of some

greedy approaches. Highly fitted solution in current swarm

within a group chosen as local leader. It also perform a

check that the position of local leader is modernize or not

and modify Local Limit Count accordingly.

Local Leader Decision (LLD) phase

In this phase decision taken about position of Local Leader,

if it is not modernized up to a threshold a.k.a. Local Leader

Limit (LLlimit). In case of no change it randomly initializes

position of LL. Position of LL may be decided with the help

of Eq. (5).

[0,1] ( )

[0,1] ( )

newij ij j ij

ij kj

SMO SMO rand GL SM

rand SM LL

(5)

Global Leader Decision (GLD) phase

In this phase decision taken about position of Global Leader,

if it is not modernized up to a threshold a.k.a. Global Leader

Limit (GLlimit), and then she creates subgroups of small size.

Number of subgroups has upper bound named as maximum

number of groups (MG). During this phase, local leaders are

decided for newly created subgroups using LLL process.

The SMO algorithm is very simple and it is something like

artificial bee colony algorithm. It has very few control

parameters. The SMO algorithm has four control parameters

named Local leader limit, Global leader limit, maximum

number of group and perturbation rate. If we have swarm of

size N then maximum number of groups should be N/10.

Local leader limit should be D*N, with dimension-D. Global

leader limit should be in range [N/2, 2N] and perturbation

rate should be in range [0.1, 0.9]. These values of parameters

are not hard bound they may vary as per requirement of

problems but for best results above suggested values are

most suitable.

3. SELF-ADAPTIVE SPIDER MONKEY

OPTIMIZATION (SaSMO) ALGORITHM

In order to get rid of complex optimization problem this

paper presents a novel and efficient approach using Spider

Monkey Optimization algorithm. The newly proposed

strategy is self adaptive in nature as it modify the position of

local leader based on its current position. It uses cognitive

learning to update current position of agents. Its location

based on present location linearly decline from 100 percent

to 50 percent in every iteration. Based on the assumption

that the solution of local leader phase will be far from the

optimal solution in the 1st iteration and it will converge

intimately to the optimal solution in afterward iterations and

dynamically adjust position of local leader. The proposed

algorithm also modifies the process of position update in

global leader. It use probability of selection of each

individual to update modify the position.

It engender the new locality for the entire group members

using Eq. (9) during local leader phase and Apply the greedy

selection mechanism between existing position and newly

computed position.

[0,1] ( )

( ) ( )

newij ij kj ij

i rj ij

SM SM rand LL SM

p SM SM

(9)

Here Pi is probability of selection. The probability pi (pi

denote probability of ith

solution) for every group member.

Probability is calculated using fitness of individuals as per

Eq. (4). During global leader phase it engenders new

positions for the every member of group using Eq. (10).

[0,1] ( )

( ) ( )

newij ij j ij

i rj ij

SM SM rand GL SM

p SM SM

(10)

In order to modernize the position of local leader it use

algorithm 1.



The value of max and min are preset to 1 and 0.5, in that

order. The local leader’s location based on existing location

linearly dwindles from 100 percent to 50 percent in every

round of experiment. This value decided after a series of

experiments.

Position of global leader updated in same manner as basic

SMO algorithm. The proposed algorithm summarized in

algorithm 2. The proposed algorithm relies on the idea that

the solution of local leader phase will be distant from the

best feasible solution in the 1st iteration and it will converge

intimately to the most favorable solution in subsequent

iterations, algorithm 2 will with dynamism regulate the

location of local leader by allowing a spider monkey in the

1st iteration to stroll with a large step size in the search area.

The step size for the wandering of spider monkey will

decrease with increment in the number of the iteration. It is

based on the concept that initial solutions are not good.

These solutions get refined with increasing number of

iterations. Here it is assumed that better solutions lies in

proximity of best fitted solution. In order to take benefit of

this step size automatically reduces with iteration.

4. OPTIMIZATION PROBLEMS

Selection of best suitable solution from a large set of

available solutions with some constraints matters in field of

Engineering, Management and Science. Optimization is a

complex process of finding best feasible for particular

objective function with a set of constraints including

different types of objective functions like uni-model, multi-

model, differential, non differential and discontinuous

functions etc. The performance of SaSMO algorithm tested

on various standard benchmark functions ( Table 1) and

some real world problems. Description of these problems is

given in next subsections.

4.1 Pressure Vessel Design Problem

The problem of minimizing total cost of the substance,

forming and welding of a cylindrical vessel [19]. In case of

pressure vessel design in general four design variables are

measured: shell thickness (x1), spherical head thickness (x2),

radius of cylindrical shell (x3) and shell length (x4). Simple

mathematical illustration of this problem is as follow:

2 2 243 1 3 4 2 3 1 4 1 3( ) 0.6224 1.7781 3.1611 19.84f x x x x x x x x x x

Algorithm 1: Position update of local leader

If position of a Local leader is not updating after

LLLimit then apply following steps.

if U(0,1) > pr

min max min( )

(0,1)

newij j j jSM SM SM SM

Where

else

max max min

[0,1] ( )

( ) ( ( ))max_

newij ij j ij

ij kj

SM SM rand GL SM

iterSM LL

iter

Algorithm 2: Self-Adaptive Spider Monkey

Optimization (SaSMO) Algorithm

Initialize all parameters

Calculate fitness

Choose leaders (global and local both)

Repeat till the extermination criterion is not fulfilled.

Generate the fresh locations for the entire group

members

[0,1] ( )

( ) ( )

newij ij kj ij

i rj ij i

SM SM rand LL SM

p SM SM Here p is probability

Apply the greedy selection mechanism between existing

position and newly computed position.

Compute the probability pi (pi denote probability of ith

solution) for every group member.

max

0.9 0.1ii

fitnessp

fitness

Engender new positions for the every member of group.

[0,1] ( )

( ) ( )

newij ij j ij

i rj ij

SM SM rand GL SM

p SM SM

Modernize local and global leader’s position.

If position of a Local leader is not updating after

LLLimit then apply following steps.

if (U(0,1) > pr)

min max min[0,1] ( )newij j j jSM SM rand SM SM

else

max max min

[0,1] ( )

( ) ( ( ))max_

newij ij j ij

ij kj

SM SM rand GL SM

iterSM LL

iter

If position of Global Leader is not updating after

GLLimit then apply following steps.

if (Global_Limit_Count > GLLimit )

then

Set Global_Limit_Count = 0

if (Number of groups < MG )

then

Divide the population into groups.

else

Merge all the groups into single large group.

Bring up to date position of Local Leader.



Subject to

1 3 1 2 3 2

23 3 4 3

0.0193 , 0.00954 ,

4750*1728 ( )

3

g x x x g x x x

g x x x x

The search boundaries for the variables are

1.125 ≤ x1 ≤ 12.5, 0.625 ≤ x2 ≤ 12.5,

1.0*10-8

≤ x3 ≤ 240 and 1.0*10-8

≤ x4 ≤ 240.

The best ever identified global optimum solution is f(1.125,

0.625, 55.8592, 57.7315) = 7197.729 [19]. The tolerable

error for considered problem is 1.0E-05.

4.2 Lennard-Jones Problem

The function is to minimize the potential energy of a set of N

atoms. The location Xi of the atom i has three coordinates

and therefore the dimension of the search space in 3N.

Actually, the coordinates of a point x are concatenation of

the ones of the Xi. In short it can be written as x=(X1, X2,

........... XN) and function can be written as

1

44 2

1 1

1 1( ) ( )

N N

i j i i j i j

f x

X X X X

Here search space is considered as [-2, 2] and N=5, α=6 [6].

4.3 Parameter estimation for frequency-modulated sound

wave

Frequency-Modulated (FM) sound wave amalgamation has a

significant role in more than a few contemporary music

systems. The parameter optimization of an FM synthesizer is

an optimization problem with six dimension where the

vector to be optimized is X = {a1, w1, a2, w2, a3, w3} of the

sound wave given in equation (11). The problem is to

produce a sound (11) analogous to target (12). This problem

is a exceedingly intricate multimodal one having strong

epistasis, with minimum value f(X) = 0. The expressions for

the anticipated sound and the target sound waves are

specified as:

1 1 2 2 3 3( ) .sin( . . .sin( . . .sin( . . )))y t a t a t a t (11)

0 ( ) (1.0).sin((5.0. . (1.5).sin((4.8). .(2.0).sin((4.9). . )))

y t t tt

(12)

Respectively where θ = 2π/100 and the parameters are

defined in the range [-6.4, 6.35]. The fitness function is the

summation of square errors between the estimated wave (11)

and the target wave (12) as follows:

1002

45 0

0

( ) ( ( ) ( ))

i

f x y t y t

(13)

Acceptable error for this problem is 1.0E-05, i.e. an

algorithm is considered successful if it finds the error less

than acceptable error in a given number of generations.

Parameter Estimation for Frequency Modulated Sound

Waves Problem [15] was solved by number of researchers

with the help of various algorithms. Recently S Das et al.

[20] used Differential evolution using a neighborhood-based

mutation operator to tackle this problem. A Rajshekhar et al.

[19] make use of Levy mutated Artificial Bee Colony

algorithm for global optimization to get rid of this problem.

A novel approach namely Particle Swarm Optimization with

Dynamic Local Search for Frequency Modulation Parameter

Identification proposed by Y Zhang et al. [18] in 2012. Y.

Lai et al. [21] proposed a new approach to mechanize the

optimization of the parameters of a FM synthesizer with the

help of genetic algorithm. S Ghorai et al. [17] developed a

faster DE algorithm to automate process of parameter

calibration for Frequency Modulated Sound Waves. S.

Kumar et al. [12] gives a memetic approach in DE to find

parameters in Frequency Modulated Sound Waves. Recently

S. Kumar et al. [14] developed a new strategy using

opposition based learning method and applied it to solve

Parameter Estimation for FM Sound Waves Problem.

Literature has large number of techniques to estimate

parameters for Frequency Modulated Sound Waves.

4.4 Compression Spring Problem

The compression spring problem [6] minimizes the weight

of a compression spring that is subjected to constraints of

shear stress, surge frequency, minimum deflection and limits

on outside diameter and on design variables with three

design variables considered: The diameter of wire(x1), mean

coil diameter (x2) and count of active coils (x3). Simple

mathematical representation of this problem is as follow:

1 1 2

3 0.001

{1,2,3,....,70} , [0.6;3],

[0.207;0.5]

x granularity x

x granularity

And four constraints

max 21 2 max3

3

8: 0, : 0,

ff

c F xg S g l l

x



3 4

3 3max

2 3 2

max1 3 max

46 3

31 2

: 0, : 0

Where

1 0.75 0.615 , 1000, 189000,

1.05( 2) , 14, , 6,

300, 11.5 10 , 1.258

m pp pm w

f

pf p pm

p w

F ax Fg g

K

x xc F S

x x xFF

l x x lK K

xF K

x x

And the function to be minimized is

22 2 3 1

46

( 2)( )

4

x x xf X

The finest ever identified solution is (7, 1.386599591,

0.292), which gives the fitness value f =2.6254 and 1.0E-04

is tolerable error for compression spring problem.

Table 1: Test Problems

Test Problem Objective Function Search Range Optimum Value D Acceptable

Error

Sphere 2

11

( )n

ii

f x x

[-5.12, 5.12] f(0) = 0 30 1.0E-05

De-Jong’s 4

21

( ) ( )n

ii

f x i x

[-5.12, 5.12] f(0) = 0 30 1.0E-05

Griewank 2

31 1

1( ) ( ) cos 1

4000

DDi

ii i

xf x x

i

[-600, 600] f(0) = 0 30 1.0E-05

Rosenbrock 12 2 2

14

1

100( )( 1 )) (

i D

i i

i

ix x xf x

[-30, 30] f(0) = 0 30 1.0E-01

Rastrigin 2

5

1

( ) 10cos(2 ) 10

D

i i

i

f x x x

[-5.12, 5.12] f(0) = 0 30 1.0E-05

Alpine

61

( ) sin 0.1n

i i ii

f x x x x

[-10, 10] f(0) =0 30 1.0E-05

Michalewicz 220

71

( ) sin (sin( ) )D

ii

i

ixf x x

[0, π] fmin = -9.66015 10 1.0E-05

Cosine Mixture 2

81

1

( ) 0.1( cos(5 )) 0.1

DD

i ii

i

f x x x D

[-1, 1] f(0) = -D*0.1 30 1.0E-05

Exponential Problem

29

1( ) exp( 0.5 )

n

ii

f x x

[-1, 1] f(0) = 1 10 1.0E-05

Brown 3 2 21

1 2( ) 1 2 110 1

1( ) ( )i i

D x xii

if x x x

[-1, 4] f(0) = 0 30 1.0E-05

Schewel

111 1

( )DD

i ii i

f x x x

[-10, 10] f(0) = 0 30 1.0E-05

Salomon Problem 212

1( ) 1 cos(2 ) 0.1 , ,

D

ii

f x p p where p x

[-100, 100] f(0) = 0 30 1.0E-01

Axis Parallel hyper-ellipsoid

213

1( )

D

ii

f x ix

[-5.12, 5.12] f(0) = 0 30 1.0E-05



Sum of different

powers

1

141

( )iD

ii

f x x

[-1, 1] f(0) = 0 30 1.0E-05

Step Function 2

151

( ) ( 0.5 )D

ii

f x x

[-100, 100] f(-0.5≤x≤0.5) = 0 30 1.0E-05

Inverted Cosine

wave function

2 211 1

161

2 21 1

( 0.5( ) (exp( ,

8

. cos(4 0.5 )

Di i i i

i

i i i i

x x x xf x

Where I x x x x

[-5. 5] f(0) = -D+1 10 1.0E-05

Neumaier 3 Problem

(NF3) 2

17 11 2

( ) ( 1)D D

i i ii i

f x x x x

[-100, 100] f(0) = -210 10 1.0E-01

Rotated Hypere-

ellipsoid 218

11

( )

Di

jj

i

f x x

[-65.536,

65.536]

f(0) = 0 30 1.0E-05

Levy montalvo -1 12 2 2 2

19 1 1

1

( ) (10sin ( ) ( 1) (1 10sin ( )) ( 1) ),

11 ( 1)

4

D

i i D

i

i i

f x y y y yD

Where y x

[-10, 10] f(-1) = 0 30 1.0E-05

Levy montalvo -2 12 2 2

20 1 11

2 2

( ) 0.1(sin (3 ) ( 1) (1 sin (3 ))

( 1) (1 sin (2 ))

D

i ii

D D

f x x x x

x x

[-5, 5] f(1) = 0 30 1.0E-05

Ellipsoidal 2

211

( ) ( 1)D

ii

f x x

[-D, D] f(1, 2,..,D) = 0 30 1.0E-05

Beale function 2 2 222 1 2 1 2

3 21 2

( ) (1.5 (1 )) (2.25 (1 ))

(2.625 (1 ))

f x x x x x

x x

[-4.5, 4.5] f(3. 0.5) = 0 2 1.0E-05

Colville function 2 2 2 2 223 2 1 1 4 3

2 2 23 2 4 2 4

( ) 100( ) (1 ) 90( )

(1 ) 10.1[( 1) ( 1) ] 19.8( 1)( 1)

f x x x x x x

x x x x x

[-10, 10] f(1) = 0 4 1.0E-05

Braninss Function 2 224 2 1 1 1( ) ( ) (1 )cosf x a x bx cx d e f x e

1

2

[ 5,10],

[0,15]

x

x

f(-π, 12.275) =

0.3979 2 1.0E-05

Kowalik function

21121 2

25 213 4

( )( ) ( )i i

ii

i i

x b b xf x a

b b x x

[-5, 5]

f(0.1928, 0.1908,

0.1231, 0.1357)

= 3.07E-04

4 1.0E-05

2D Tripod function 26 2 1 1 2 1 2 2( ) ( )(1 ( )) ( 50 ( )(1 2 ( ))) ( 50(1 2 ( )))f x p x p x x p x p x x p x [-100, 100] f(0, -50)=0 2 1.0E-04

Shifted Rosenbrock

12 2 2

27 11

1 2 1 2

( ) (100( ) ( 1) ,

1, [ , ,... ], [ , ,.. ]

D

i i i biasi

D D

f x z z z f

z x o x x x x o o o o

[-100, 100] f(o)=fbias=390 10 1.0E-01

Shifted Sphere 2

28 1 2 1 21

( ) , ( ), [ , ,.. ], [ , ,... ]D

i bias D Di

f x z f z x o x x x x o o o o

[-100, 100] f(o)=fbias=-450 10 1.0E-05

Shifted Rastrigin

229

1

1, 2 1 2

( ) ( 10cos(2 ) 10) ,

( ), [ ,... ], [ , ,....... ]

D

i i biasi

D D

f x z z f


[-5, 5] f(o)=fbias=-330 10 1.0E-02

Shifted Schwefel

230

1 1

1, 2 1 2

( ) ( ) , ( ),

[ ,... ], [ , ,....... ]

D i

j biasi j

D D

f x z f z x o

x x x x o o o o

[-100, 100] f(o)=fbias=-450 10 1.0E-05



Shifted Griewank

2

311 1

1, 2 1 2

( ) cos( ) 1 ,4000

( ), [ ,... ], [ , ,....... ]

DDi i

biasi i

D D

z zf x f

i


[-600, 600] f(o)=fbias=-180 10 1.0E-05

Shifted Ackley 2

321 1

1 2 1 2

1 1( ) 20exp( 0.2 ) exp( cos(2 ))

20 , ( ), [ , ,... ], [ , ,... ]

D D

i ii i

bias D D

f x z zD D

e f z x o x x x x o o o o

[-32,32] f(o)=fbias=-140 10 1.0E-05

Goldstein-Price

2 2 233 1 2 1 1 2 1 2 2

2 2 21 2 1 1 2 1 2 2

( ) (1 ( 1) (19 14 3 14 6 3 ))

(30 (2 3 ) (18 32 12 48 36 27 ))

f x x x x x x x x x

x x x x x x x x

[-2, 2] f(0, -1)=3 2 1.0E-14

Six-hump camel

back 2 4 2 2 2

34 1 1 1 1 2 2 21

( ) (4 2.1 ) ( 4 4 )3

f x x x x x x x x [-5, 5] f(-0.0898, 0.7126) =

-1.0316 2 1.0E-05

Easom’s function 2 2

1 2( ( ) ( ) )35 1 2( ) cos cos

x xf x x x e

[-10, 10] f(π, π) = -1 2 1.0E-13

Dekkkers and Aarts 5 2 2 2 2 2 5 2 2 436 1 2 1 2 1 2( ) 10 ( ) 10 ( )f x x x x x x x [-20, 20]

f(0,15)=f(0, -15)= -24777

2 5.0E-01

Hosaki Problem 2 3 4 237 1 1 1 1 2 2

7 1( ) (1 8 7 ) exp( )

3 4f x x x x x x x

1

2

[0,5],

[0,6]

x

x

-2.3458 2 1.0E-06

McCormick 238 1 2 1 2 1 2

3 5( ) sin( ) ( ) 1

2 2f x x x x x x x

1

2

1.5 4,

3 3,

x

x

f(-0.547, -1.547) =-

1.9133 30 1.0E-04

Meyer and Roth Problem

521 3

391 1 2

( ) ( )1

ii

i i i

x x tf x y

x t x v

[-10, 10]

f(3.13, 15.16,0.78) = 0.4E-04

3 1.0E-03

Shubert 5 5

40 1 21 1

( ) cos(( 1) 1) cos(( 1) 1)t i

f x i i x i i x

[-10, 10] f(7.0835, 4.8580)= -

186.7309 2 1.0E-05

Sinusoidal 411 1

( ) [ sin( ) sin( ( ))], 2.5, 5, 30n n

i ii i

f x A x z B x z A B z

[-10, 10] f(90+z)=-(A+1) 10 1.00E-02

Moved axis parallel hyper-ellipsoid

242

1( ) 5

D

ii

f x ix

[-5.12, 5.12]

f(x) =0; x(i) 5i, i=1:D

30 1.0E-15

5. EXPERIMENTAL SETUP

This paper validated the performance of the planned Self-

Adaptive Spider Monkey Optimization algorithm with the

original technique in forty two benchmark problems and

four real worl problems Pressure Vessel Design problem,

Lennard-Jones problem, Parameter Estimation for FM

Sound Waves Problem and Compression Spring problem.

The performance of newly proposed algorithm is compared

with Basic SMO algorithm [15] and SMO’s recent version

MPU-SMO [5]. The performance compared based on

standard deviation (SD), mean error (ME), average function

evaluation (AFE) and success rate (SR).

Experiments are performed in C programming language with

following experimental setup for SaSMO.

The size of swarm N = 50 ( Number of Spider Monkeys

at the time of initialization)

MG = 5 ( Maximum group limiting maximum number of

spider monkeys in a group as MG = N/10)

Global Leader Limit (GLlimit)=50,

Local Leader Limit (LLlimit)=1500,

pr ∈ [0.1, 0.4], linearly growing over iterations,

Remaining all parameters is similar to basic SMO algorithm

[15].



Table 2: Evaluation of the outcomes of test problems

Test Problem SaSMO MPU-SMO SMO

SD ME AFE SR SD ME AFE SR SD ME AFE SR

Sphere 1.60E-06 8.25E-06 14597.25 100 6.57E-07 9.54E-06 44435.12 100 9.33E-07 8.89E-06 16219.17 100

De-Jong’s 3.13E-06 5.77E-06 11563.4 100 6.45E-07 9.47E-06 27691.99 100 1.22E-06 8.31E-06 12848.22 100

Griewank 9.80E-04 1.05E-04 28207.11 99 3.42E-03 1.12E-03 87401.67 89 1.45E-06 8.84E-06 31341.23 100

Rosenbrock 1.27E+01 6.43E+00 177433 11 5.15E+01 5.42E+01 201808.6 4 4.81E+01 5.41E+01 203397.1 1

Rastrigin 2.76E-06 6.90E-06 81293.64 100 1.99E-06 7.81E-06 91623.6 100 1.27E-06 8.60E-06 90326.27 100

Alpine 1.33E-05 1.37E-05 75277.59 70 9.63E-06 1.17E-05 173750.6 81 5.06E-07 9.54E-06 72530.65 100

Michalewicz 3.53E-06 4.34E-06 32553.34 100 3.14E-06 5.64E-06 48314.99 100 3.53E-06 5.55E-06 36170.38 100

Cosine Mixture 1.85E-06 8.07E-06 17603.42 100 5.54E-07 9.53E-06 34828.34 100 1.47E-02 1.49E-03 19559.36 99

Exponential

Problem 1.38E-06 8.65E-06 11182.94 100 3.12E-07 9.70E-06 31575.63 100 9.39E-07 8.91E-06 12425.49 100

Brown 3 1.84E-06 8.08E-06 14545.58 100 4.82E-07 9.50E-06 53797.81 100 9.23E-07 8.79E-06 16161.75 100

Schewel 7.39E-07 9.30E-06 26217.68 100 7.61E-07 9.60E-06 67482.73 100 4.78E-07 9.43E-06 29130.75 100

Salomon

Problem 1.49E-06 8.40E-06 16916.93 100 4.23E-02 9.38E-01 77315.53 89 4.15E-02 9.24E-01 18796.59 100

Axis Parallel

hyper-ellipsoid 4.96E-01 2.58E+00 160839.9 0 3.39E-07 9.62E-06 53227.79 100 6.81E-07 9.20E-06 18711 100

Sum of different

powers 2.71E-06 6.17E-06 5301.45 100 2.03E-06 7.54E-06 7658.64 100 1.96E-06 7.34E-06 5890.5 100

Step Function 0.00E+00 0.00E+00 10033.55 100 9.95E-02 1.00E-02 27487.95 99 0.00E+00 0.00E+00 11148.39 100

Inverted Cosine

wave function 2.25E-06 7.38E-06 53613.46 100 2.02E-06 8.13E-06 90650.07 100 1.66E-06 7.92E-06 59570.51 100

Neumaier 3

Problem (NF3) 7.58E-01 3.96E-01 86206.15 62 5.79E-02 9.84E-02 42451.28 98 5.44E-03 9.80E-02 79455.2 100

Rotated Hypere-

ellipsoid 1.64E-06 8.23E-06 21396.47 100 4.29E-07 9.57E-06 63116.97 100 8.91E-07 8.87E-06 23773.86 100

Levy montalvo -

1 1.67E-06 8.10E-06 13486.18 100 5.83E-07 9.44E-06 27431.67 100 8.69E-07 8.95E-06 14984.64 100

Levy montalvo -

2 1.72E-06 7.95E-06 13605.57 100 6.55E-07 9.43E-06 35504.89 100 9.58E-07 8.89E-06 15117.3 100

Ellipsoidal 1.69E-06 8.42E-06 17563.39 100 8.64E-07 9.53E-06 65693.17 100 7.78E-07 8.99E-06 19514.88 100

Beale function 2.89E-06 5.74E-06 2898.423 100 2.77E-06 5.28E-06 4414.41 100 3.03E-06 5.01E-06 3220.47 100

Colville

function 2.00E-03 8.17E-03 17195.96 100 2.19E-03 7.31E-03 19106.62 100 2.31E-03 7.34E-03 21878.6 100

Braninss

Function 6.88E-06 6.22E-06 18496.32 88 7.09E-06 6.08E-06 31362.01 86 6.34E-06 5.37E-06 20551.47 91

Kowalik

function 1.45E-04 1.16E-04 75072.94 95 1.58E-05 8.68E-05 56696.51 100 1.41E-05 8.57E-05 50081.04 100

2D Tripod

function 2.32E-05 6.23E-05 5202.549 100 2.42E-05 6.05E-05 5780.61 100 2.55E-05 6.46E-05 9623.38 100



Shifted

Rosenbrock 2.16E-01 1.41E-01 98742.48 83 7.05E-02 9.94E-02 98602.75 93 4.04E+00 1.17E+00 141195.5 70

Shifted Sphere 2.12E-06 7.34E-06 6980.094 100 1.21E-06 8.26E-06 14371.83 100 1.71E-06 7.71E-06 7755.66 100

Shifted

Rastrigin 1.64E+01 1.16E+02 186736.1 0 1.54E+01 1.16E+02 207583.9 0 1.74E+01 1.08E+02 207484.5 0

Shifted

Schwefel 5.39E+03 2.06E+04 186807.6 0 6.57E+03 2.21E+04 207600.7 0 5.56E+03 1.94E+04 207564 0

Shifted

Griewank 1.59E-03 2.77E-04 69724.47 96 3.55E-03 1.54E-03 103522.6 79 2.92E-06 5.53E-06 68582.74 100

Shifted Ackley 1.37E-06 8.51E-06 10824.76 100 1.01E-06 8.99E-06 24075.81 100 1.00E-06 8.83E-06 12027.51 100

Goldstein-Price 4.08E-15 4.95E-15 4885.353 100 4.55E-15 5.73E-15 8595.18 100 4.52E-15 4.67E-15 5428.17 100

Six-hump camel

back 1.49E-05 1.64E-05 94389.39 50 1.49E-05 1.66E-05 104877.1 50 1.42E-05 1.80E-05 117140.3 44

Easom’s

function 3.03E-14 4.45E-14 20198.66 100 3.01E-14 5.05E-14 34858.85 100 2.90E-14 4.80E-14 22442.95 100

Dekkkers and

Aarts 5.49E-03 4.91E-01 1407.78 100 5.20E-03 4.90E-01 2181.96 100 4.56E-03 4.89E-01 1564.2 100

Hosaki Problem 6.47E-06 6.23E-06 23193.59 88 6.35E-06 5.83E-06 25821.46 88 6.54E-06 6.34E-06 36012.41 83

McCormick 6.80E-06 8.84E-05 937.332 100 6.82E-06 8.77E-05 1202.85 100 6.38E-06 8.80E-05 1041.48 100

Meyer and Roth

Problem 2.74E-06 1.95E-03 2437.272 100 2.86E-06 1.95E-03 3827.12 100 2.93E-06 1.94E-03 2708.08 100

Shubert 5.63E-06 5.07E-06 4921.884 100 5.78E-06 5.15E-06 10377.09 100 5.36E-06 4.68E-06 5468.76 100

Sinusoidal 2.53E-03 8.29E-03 127494.4 93 4.51E-03 8.77E-03 147882.4 86 7.35E-03 1.14E-02 144957.7 72

Moved axis

parallel hyper-

ellipsoid

1.32E-16 8.38E-16 39369.73 100 1.27E-16 9.68E-16 154819.9 95 9.10E-17 8.90E-16 43744.14 100

Pressure Vessel

Design problem 1.58E-01 1.21E-01 14212.9 84 1.13E-04 4.58E-05 163569.9 61 5.78E-02 1.93E-02 202147 13

Lennard-Jones

problem 2.55E-03 2.33E-03 192194 46 1.31E-03 1.55E-03 181965.2 48 1.49E-03 1.79E-03 180215.5 44

Parameter

Estimation for

FM Sound

Waves Problem

5.89E+00 5.76E+00 105576.9 83 4.23E+00 2.20E+00 161490.1 64 3.68E+00 1.51E+00 117307.7 74

Compression

Spring problem 1.01E-02 6.26E-03 26004.47 91 3.11E-04 5.26E-04 28893.86 100 2.43E-03 1.47E-03 87129.91 83

6. RESULTS AND DISCUSSION

The results obtained with basic SMO, MPU-SMO and

proposed SaSMO are reported in table 2. Table 2 discuss

results in terms of Success rate, average number of function

evaluations, mean error and standard deviation.

It can be observed from table 2 and figure 1 that proposed

SaSMO algorithm achieve better success rate in comparison

to considered other algorithms. The SaSMO algorithm is

also able to achieve optima in less number of function

evaluations. Additionally, the speed of convergence for the

considered algorithms also computed with the help of AFE.

It is considered that the less number of AFE indicate that the

algorithm has higher rate of convergence. The SMO

algorithm is stochastic in nature. So as to diminish the

significance of this stochastic nature the measured AFEs for

all algorithms averaged over 100 runs for considered

problem. Acceleration Rate (AR) computed in order to

check rate of convergence. Acceleration Rate (AR) is

defined as follows based on the AFEs for the two algorithms

ALGO and SaSMO:



ALGO

SaSMO

AFEAR

AFE (12)

Here { , }ALGO SMO MPU SMO . The AR>1 indicates

that SaSMO has higher rate of convergence. Comparison of

rate of convergence for SaSMO-SMO and SaSMO-

MPUSMO reported in table 3. It can be observed from table

3 that SaSMO algorithm always has higher rate of

convergence.

Table 3: Acceleration Rate (AR) of SaSMO compare to the Basic SMO and MPU-SMO for Test Problems

Test Function MPU-SMO SMO Test Function MPU-SMO SMO

Sphere 3.04 1.11 Braninss Function 1.70 1.11

De-Jong’s 2.39 1.12 Kowalik function 1.26 1.11

Griewank 3.10 1.10 2D Tripod function 1.11 1.85

Rosenbrock 1.14 1.15 Shifted Rosenbrock 1.11 1.59

Rastrigin 1.13 1.11 Shifted Sphere 2.06 1.11

Alpine 2.66 1.13 Shifted Rastrigin 1.11 1.09

Michalewicz 1.48 1.12 Shifted Schwefel 1.12 1.11

Cosine Mixture 1.98 1.11 Shifted Griewank 1.68 1.11

Exponential Problem 2.82 1.09 Shifted Ackley 2.22 1.11

Brown 3 3.70 1.11 Goldstein-Price 1.76 1.11

Schewel 2.57 1.11 Six-hump camel back 1.11 1.24

Salomon Problem 4.57 1.09 Easom’s function 1.73 1.11

Axis Parallel hyper-ellipsoid 3.16 1.11 Dekkkers and Aarts 1.55 1.12

Sum of different powers 1.44 1.12 Hosaki Problem 1.11 1.55

Step Function 2.74 1.11 McCormick 1.28 1.11

Inverted Cosine wave function 1.69 1.11 Meyer and Roth Problem 1.57 1.11

Neumaier 3 Problem (NF3) 1.11 2.08 Shubert 2.11 1.12

Rotated Hypere-ellipsoid 2.95 1.11 Sinusoidal 1.16 1.14

Levy montalvo -1 2.03 1.11 Moved axis parallel hyper-ellipsoid 3.93 1.11

Levy montalvo -2 2.61 1.13 Pressure Vessel Design problem 1.11 1.37

Ellipsoidal 3.74 1.11 Lennard-Jones problem 1.12 1.12

Beale function 1.52 1.16 Parameter Estimation for FM Sound Waves Problem 1.53 1.11

Colville function 1.11 1.27 Compression Spring problem 1.11 3.35

Figure 1: Boxplot graph for Average Number of Function Evaluation using SMO, MPU-SMO and SaSMO



7. CONCLUSION

Self-Adaptive Spider Monkey Optimization algorithm is an

efficient variant of SMO algorithm as it takes very less AFE

in comparison to basic SMO and MPU-SMO algorithms.

Comparison of AFE shown in figure 1 in form of box-plot

graph. This paper proposed a novel approach for

optimization problems by modifying basic Spider Monkey

Optimization algorithm. This approach add probability based

position update in global leader phase and dynamically

decrease size of search radius for local leader with

advancement in iterations and named as Self-Adaptive

Spider Monkey Optimization algorithm. The proposed

SaSMO algorithm easily solved the considered problems

with great success rate and with less number of function

evaluations i.e. with higher rate of convergence.

Convergence rate of SaSMO is four times of convergence

rate of basic SMO. SaSMO algorithm is significantly better

than their current variants in terms of Reliability (due to

success rate), Efficiency (due to average number of function

evaluations) and Accuracy (due to mean objective function

value).

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Self-Adaptive Spider Monkey Optimization Algorithm for Engineering Optimization Problems

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