Research ArticleModified Grey Wolf Optimizer forGlobal Engineering Optimization

Nitin Mittal1 Urvinder Singh2 and Balwinder Singh Sohi1

1Department of Electronics and Communication Engineering Chandigarh University Mohali Punjab 140413 India2Department of Electronics and Communication Engineering Thapar University Patiala Punjab 147004 India

Correspondence should be addressed to Nitin Mittal mittalnitin84gmailcom

Received 30 November 2015 Accepted 3 April 2016

Academic Editor Samuel Huang

Copyright copy 2016 Nitin Mittal et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Nature-inspired algorithms are becoming popular among researchers due to their simplicity and flexibility The nature-inspiredmetaheuristic algorithms are analysed in terms of their key features like their diversity and adaptation exploration and exploitationand attractions and diffusion mechanisms The success and challenges concerning these algorithms are based on their parametertuning and parameter control A comparatively new algorithm motivated by the social hierarchy and hunting behavior of greywolves is Grey Wolf Optimizer (GWO) which is a very successful algorithm for solving real mechanical and optical engineeringproblems In the original GWO half of the iterations are devoted to exploration and the other half are dedicated to exploitationoverlooking the impact of right balance between these two to guarantee an accurate approximation of global optimum To overcomethis shortcoming a modified GWO (mGWO) is proposed which focuses on proper balance between exploration and exploitationthat leads to an optimal performance of the algorithm Simulations based on benchmark problems and WSN clustering problemdemonstrate the effectiveness efficiency and stability of mGWO compared with the basic GWO and some well-known algorithms

1 Introduction

Metaheuristic algorithms are powerful methods for solvingmany real-world engineering problemsThemajority of thesealgorithms have been derived from the survival of fittesttheory of evolutionary algorithms collective intelligence ofswarm particles behavior of biological inspired algorithmsandor logical behavior of physical algorithms in nature

Evolutionary algorithms are those who mimic the evolu-tionary processes in nature The evolutionary algorithms arebased on survival of fittest candidate for a given environmentThese algorithms begin with a population (set of solutions)which tries to survive in an environment (definedwith fitnessevaluation) The parent population shares its properties ofadaptation to the environment to the children with variousmechanisms of evolution such as genetic crossover andmutation The process continues over a number of gener-ations (iterative process) till the solutions are found to bemost suitable for the environment Some of the evolutionaryalgorithms are Genetic Algorithm (GA) [1] Evolution Strate-gies (ES) [2] Genetic Programming (GP) [3] Differential

Evolution (DE) [4] and Biogeography-Based Optimization(BBO) [5ndash9]

Thephysical algorithms are inspired by physical processessuch as heating and cooling of materials (Simulated Anneal-ing [10]) discrete cultural information which is treated as inbetween genetic and culture evolution (Memetic Algorithm[11]) harmony of music played by musicians (HarmonySearch [12 13]) cultural behavior of frogs (Shuffled Frog-LeapingAlgorithm [14]) Gravitational Search algorithm [15]Multiverse Optimizer (MVO) [16] and Chemical ReactionOptimization (CRO) [17]

Swarm intelligence is the group of natural metaheuristicsinspired by the ldquocollective intelligencerdquo of swarms The col-lective intelligence is built up through a population of homo-geneous agents interacting with each other and with theirenvironment Example of such intelligence is found amongcolonies of ants flocks of birds schools of fish and so forthParticle Swarm Optimization [18] is developed based on theswarm behavior of birds The firefly algorithm [19] is formu-lated based on the flashing behavior of fireflies Bat Algorithm(BA) [20] is based on the echolocation behavior of bats

Hindawi Publishing CorporationApplied Computational Intelligence and So ComputingVolume 2016 Article ID 7950348 16 pageshttpdxdoiorg10115520167950348

2 Applied Computational Intelligence and Soft Computing

Ant Colony Optimization (ACO) [21 22] is inspired by thepheromone trail laying behavior of real ant colonies A newevolutionary optimization algorithm Cuckoo Search (CS)Algorithm [23] is inspired by lifestyle of cuckoo birds Themajor algorithms include Ant Colony Optimization (ACO)[21 22] Particle Swarm Optimization (PSO) [18] ArtificialBee Colony (ABC) Algorithm [24] Fish Swarm Algorithm(FSA) [25] Glowworm Swarm Optimization (GSO) [26]Grey Wolf Optimizer (GWO) [27] Fruit Fly OptimizationAlgorithm (FFOA) [28] Bat Algorithm (BA) [20] NovelBat Algorithm (NBA) [29] Dragonfly Algorithm (DA) [30]Cat Swarm Optimization (CSO) [31] Cuckoo Search (CS)Algorithm [23] Cuckoo Optimization Algorithm (COA)[32] and Spider Monkey Optimization (SMO) Algorithm[33]

The biologically inspired algorithms comprise naturalmetaheuristics derived from living phenomena and behav-ior of biological organisms The intelligence derived withbioinspired algorithms is decentralized distributed self-organizing and adaptive in nature under uncertain environ-ments The major algorithms in this field include ArtificialImmune Systems (AIS) [34] Bacterial ForagingOptimization(BFO) [35] and Krill Herd Algorithm [36]

Because of their inherent advantages such algorithmscan be applied to various applications including powersystems operations and control job scheduling problemsclustering and routing problems batch process schedulingimage processing and pattern recognition problems

GWO is recently developed heuristics inspired from theleadership hierarchy and hunting mechanism of grey wolvesin nature and has been successfully applied for solving eco-nomic dispatch problems [37] feature subset selection [38]optimal design of double later grids [39] time forecasting[40] flow shop scheduling problem [41] optimal power flowproblem [42] and optimizing key values in the cryptographyalgorithms [43] A number of variants are also proposedto improve the performance of basic GWO that includebinary GWO [44] a hybrid version of GWO with PSO [45]integration of DE with GWO [46] and parallelized GWO[47 48]

Every optimization algorithm stated above needs toaddress the exploration and exploitation of a search spaceIn order to be successful an optimization algorithm needs toestablish a good ratio between exploration and exploitationIn this paper a modified GWO (mGWO) is proposed tobalance the exploration and exploitation trade-off in originalGWO algorithm Different functions with diverse slopes areemployed to tune the parameters of GWO algorithm forvarying exploration and exploitation combinations over thecourse of iterations Increasing the exploration in comparisonto exploitation increases the convergence speed and avoidsthe local minima trapping effect

The rest of the paper is organized as follows Section 2gives the overview of original GWO The proposed mGWOalgorithm is explained in Section 3 The experimental resultsare demonstrated in Section 4 Section 5 solves the clusteringproblem in WSN for cluster head selection to demonstratethe applicability of the proposed algorithm Finally Section 6concludes the paper

2 Overview of Grey Wolf Optimizer Algorithm

Grey Wolf Optimizer (GWO) is a typical swarm-intelligencealgorithm which is inspired from the leadership hierarchyandhuntingmechanismof greywolves in natureGreywolvesare considered as apex predators they have average group sizeof 5ndash12 In the hierarchy of GWO alpha (120572) is consideredthe most dominating member among the group The rest ofthe subordinates to 120572 are beta (120573) and delta (120575) which helpto control the majority of wolves in the hierarchy that areconsidered as omega (120596) The 120596 wolves are of lowest rankingin the hierarchy

The mathematical model of hunting mechanism of greywolves consists of the following

(i) Tracking chasing and approaching the prey

(ii) Pursuing encircling and harassing the prey until itstops moving

(iii) Attacking the prey

21 Encircling Prey Grey wolves encircle the prey during thehunt which can be mathematically written as [27]

=

100381610038161003816100381610038161003816

sdot

997888997888rarr

119883119901 (119905) minus (119905)

100381610038161003816100381610038161003816

(119905 + 1) =

997888997888rarr

119883119901 (119905) minus sdot

(1)

where 119905 indicates the current iteration and are coefficientvectors 997888997888rarr119883119901 is the position vector of the prey and indicatesthe position vector of a grey wolf

The vectors and are calculated as follows

= 2 sdot997888rarr1199031 minus

= 2 sdot997888rarr1199032

(2)

where components of 119886 are linearly decreased from 2 to 0 overthe course of iterations and 1199031 and 1199032 are random vectors in[0 1]

22 Hunting Hunting of prey is usually guided by 120572 and120573 and 120575 will participate occasionally The best candidatesolutions that is 120572 120573 and 120575 have better knowledge aboutthe potential location of prey The other search agents (120596)update their positions according to the position of three bestsearch agents The following formulas are proposed in thisregard

997888rarr

119863120572 =

100381610038161003816100381610038161003816

997888rarr

1198621 sdot997888rarr

119883120572 minus

100381610038161003816100381610038161003816

997888rarr

119863120573 =

100381610038161003816100381610038161003816

997888rarr

1198622 sdot997888rarr

119883120573 minus

100381610038161003816100381610038161003816

997888rarr

119863120575 =

100381610038161003816100381610038161003816

997888rarr

1198623 sdot997888rarr

119883120575 minus

100381610038161003816100381610038161003816

(3)

Applied Computational Intelligence and Soft Computing 3

997888rarr

1198831 =997888rarr

119883120572 minus997888rarr

1198601 sdot (997888rarr

119863120572)

997888rarr

1198832 =997888rarr

119883120573 minus997888rarr

1198602 sdot (997888rarr

119863120573)

997888rarr

1198833 =997888rarr

119883120575 minus997888rarr

1198603 sdot (997888rarr

119863120575)

(4)

(119905 + 1) =

997888rarr

1198831 (119905) +997888rarr

1198832 (119905) +997888rarr

1198833 (119905)

3

(5)

23 Attacking Prey In order to mathematically model forapproaching the prey we decrease the value of Thefluctuation range of is also decreased by is a randomvalue in the interval [minus119886 119886]where 119886 is decreased linearly from2 to 0 over the course of iterationsWhen random values of are in [minus1 1] the next position of a search agent can be in anyposition between its current position and the position of theprey The value |119860| lt 1 forces the wolves to attack the prey

After the attack again they search for the prey in the nextiteration wherein they again find the next best solution 120572among all wolves This process repeats till the terminationcriterion is fulfilled

3 Modified GWO Algorithm

Finding the global minimum is a common challenging taskamong all minimization methods In population-based opti-mization methods generally the desirable way to convergetowards the global minimum can be divided into two basicphases In the early stages of the optimization the individualsshould be encouraged to scatter throughout the entire searchspace In other words they should try to explore the wholesearch space instead of clustering around local minima Inthe latter stages the individuals have to exploit informationgathered to converge on the global minimum In GWO withfine-adjusting of the parameters 119886 and 119860 we can balancethese two phases in order to find global minimum with fastconvergence speed

Although different improvements of individual-basedalgorithms promote local optima avoidance the litera-ture shows that population-based algorithms are better inhandling this issue Regardless of the differences betweenpopulation-based algorithms the common approach is thedivision of optimization process to two conflicting mile-stones exploration versus exploitation The explorationencourages candidate solutions to change abruptly andstochastically This mechanism improves the diversity of thesolutions and causes high exploration of the search space Incontrast the exploitation aims for improving the quality ofsolutions by searching locally around the obtained promisingsolutions in the exploration In this milestone candidatesolutions are obliged to change less suddenly and searchlocally

Exploration and exploitation are two conflicting mile-stones where promoting one results in degrading the otherA right balance between these two milestones can guaran-tee a very accurate approximation of the global optimumusing population-based algorithms On the one hand mere

exploration of the search space prevents an algorithm fromfinding an accurate approximation of the global optimumOnthe other handmere exploitation results in local optima stag-nation and again low quality of the approximated optimum

In GWO the transition between exploration andexploitation is generated by the adaptive values of 119886 and119860 Inthis half of the iterations are devoted to exploration (|119860| ge 1)and the other half are used for exploitation (|119860| lt 1) asshown in Figure 1(a) Generally higher exploration of searchspace results in lower probability of local optima stagnationThere are various possibilities to enhance the explorationrate as shown in Figure 1(b) in which exponential functionsare used instead of linear function to decrease the value of 119886over the course of iterations Too much exploration is similarto too much randomness and will probably not give goodoptimization results But too much exploitation is relatedto too little randomness Therefore there must be a balancebetween exploration and exploitation

In GWO the value of 119886 decreases linearly from 2 to 0using the update equation as follows

119886 = 2 (1 minus

119905

119879

) (6)

where 119879 indicates the maximum number of iterations and119905 is the current iteration Our mGWO employs exponentialfunction for the decay of 119886 over the course of iterationsConsider

119886 = 2(1 minus

1199052

1198792) (7)

as shown in Figure 1(c) Using this exponential decay func-tion the numbers of iterations used for exploration andexploitation are 70 and 30 respectively

The pseudocode of mGWO is given in Algorithm 1

4 Results and Discussion

This section investigates the effectiveness of mGWO inpractice It is common in this field to benchmark theperformance of algorithms on a set ofmathematical functionswith known global optima We also follow the same processand employ 27 benchmark functions for comparisonThe testfunctions are divided to four groups unimodal multimodalfixed-dimension multimodal and composite benchmarkfunctions The unimodal functions (1198651ndash1198657) are suitable forbenchmarking the exploitation of algorithms since they haveone global optimum and no local optima On the contrarymultimodal functions (1198658ndash11986513) have a large number of localoptima and are helpful to examine exploration and localoptima avoidance of algorithms

Themathematical formulation of the employed test func-tions is presented in Tables 1ndash4 We consider 30 variablesfor unimodal and multimodal test function for furtherimproving their difficulties

Since heuristic algorithms are stochastic optimizationtechniques they have to be run at least more than 10times to generate meaningful statistical results It is againa common strategy that an algorithm is run on a problem

4 Applied Computational Intelligence and Soft Computing

Valu

e of a

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

2

Iteration

ExplorationExploitation

(a)

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

2

Iteration

Valu

e of a

ExplorationExploitation

(b)

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

2

Iteration

Valu

e of a

ExplorationExploitation

(c)

Figure 1 (a) Updating the value of 119886 for GWO (b) some samples of possible functions for updating 119886 over the course of iterations and (c)updating the value of 119886 over the course of iterations for mGWO

119898 times and averagestandard deviationmedian of the bestobtained solution in the last iteration are calculated as themetrics of performance We follow the same method togenerate and report the results over 30 independent runsIn order to verify the performance of mGWO algorithmPSO BA CS and GWO algorithms are chosen Note that we

utilized 30 search agents and 3000 iterations for each of thealgorithms

The convergence curves of unimodal multimodal fixed-dimensionmultimodal and composite benchmark functionsfor the competitive optimization algorithms are given inFigures 2 3 4 and 5 respectively As Table 5 shows

Applied Computational Intelligence and Soft Computing 5

Initialize the search agent (grey wolf) population119883119894 (119894 = 1 2 119899)Initialize 119886 119860 and 119862Calculate the fitness of each search agent119883120572 = the best (or dominating) search agent119883120573 = the second best search agent119883120575 = the third best search agentwhile (119905 ltMaximum number of iterations)

for each search agentupdate the position of the current search agent by (5)

end forupdate 119886 by (7)update 119860 and 119862 by (2)calculate the fitness of all search agentsupdate119883120572 119883120573 and119883120575119905 = 119905 + 1

end whileReturn119883120572

Algorithm 1 Pseudocode of mGWO algorithm

Table 1 Unimodal benchmark functions

Function Dim Range 119891min

1198651 (119909) =

119899

sum

119894=1

1199092

11989430 [minus100 100] 0

1198652 (119909) =

119899

sum

119894=1

|119909119894| +

119899

prod

119894=1

|119909119894| 30 [minus10 10] 0

1198653 (119909) =

119899

sum

119894=1

(

119894

sum

119895minus1

119909119895)

2

30 [minus100 100] 0

1198654 (119909) = max1198941003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119899 30 [minus100 100] 0

1198655 (119909) =

119899minus1

sum

119894=1

[100 (119909119894+1 minus 1199092

119894)

2

+ (119909119894 minus 1)2] 30 [minus30 30] 0

1198656 (119909) =

119899

sum

119894=1

([119909119894 + 05])2 30 [minus100 100] 0

1198657 (119909) =

119899

sum

119894=1

1198941199094

119894+ random (0 1) 30 [minus128 128] 0

mGWO algorithm provides the best results in 5 out of 7unimodal benchmark test functions The mGWO algorithmalso provides very competitive results compared to CS on 1198655and 1198656 As discussed above unimodal functions are suitablefor benchmarking exploitation of the algorithms Thereforethese results evidence high exploitation capability of themGWO algorithm

The statistical results of the algorithms on multimodaltest function are presented in Table 6 It may be seen thatmGWO algorithm highly outperforms other algorithms on11986591198651011986511 and11986512 It should be noted thatmGWOalgorithmoutperforms other algorithms on these multimodal testfunctions except PSO for 11986513 The results of multimodal testfunction strongly prove that high exploration of mGWOalgorithm is a suitable mechanism for avoiding local solu-tions Since the multimodal functions have an exponentialnumber of local solutions the results show that mGWOalgorithm is able to explore the search space extensively and

find promising regions of the search space In addition highlocal optima avoidance of this algorithm is another findingthat can be inferred from these results

The rest of the results which belong to 11986514ndash11986523 and11986524ndash11986527 are provided in Tables 7 and 8 respectively Theresults are consistent with those of other test functions inwhich mGWO shows very competitive results compared toother algorithms

5 Cluster Head Selection inWSN Using mGWO

Cluster head (CH) selection problem is a well-known prob-lem in the field of wireless sensor networks (WSNs) inwhich the energy consumption cost of the network shouldbe minimized [49ndash53] In this paper this problem is solvedusing mGWO algorithm and compared with GA PSO BACS and GWO

The main challenges in designing and planning theoperations ofWSNs are to optimize energy consumption andprolong network lifetime Cluster-based routing techniquessuch as the well-known low-energy adaptive clustering hier-archy (LEACH) [50] are used to achieve scalable solutionsand extend the network lifetime until the last node dies(LND) In order to achieve prolonged network lifetime incluster-based routing techniques the lifetime of the CHsplays an important role Improper cluster formation maycause some CHs to be overloaded Such overload maycause high energy consumption of the CH and degradethe overall performance of the WSN Therefore proper CHselection is the most important issue for clustering sensornodes Designing an energy efficient clustering algorithmis not an easy task Therefore nature-inspired optimizationalgorithms may be applied to tackle cluster-based routingproblem in WSN Evolutionary algorithms (EAs) have beenused in recent years as metaheuristics to address energy-aware routing challenges by designing intelligent models

6 Applied Computational Intelligence and Soft Computing

Table2Multim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891min

1198658(119909)=

119899

sum 119894=1

minus119909119894sin(radic1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816)

30[minus500500]

minus4189829times5

1198659(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

30[minus512512]

0

11986510(119909)=minus20exp(minus02radic

1 119899

119899

sum 119894=1

1199092 119894)minusexp(

1 119899

119899

sum 119894=1

cos(2120587119909119894))+20+119890

30[minus3232]

0

11986511(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos(119909119894

radic119894

)+1

30[minus600600]

0

11986512(119909)=

120587 119899

10sin(1205871199101)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+

119909119894+1

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)119898

119909119894gt119886

0minus119886lt119909119894lt119886

119896(minus119909119894minus119886)119898119909119894ltminus119886

30[minus5050]

0

11986513(119909)=01sin2(31205871199091)+

119899

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

30[minus5050]

0

Applied Computational Intelligence and Soft Computing 7

Table 3 Fixed-dimension multimodal benchmark functions

Function Dim Range 119891min

11986514 (119909) = (

1

500

+

25

sum

119895=1

1

119895 + sum2

119894=1(119909119894 minus 119886119894119895)

6)

minus1

2 [minus65 65] 1

11986515 (119909) =

11

sum

119894=1

[119886119894 minus

1199091 (1198872

119894+ 1198871198941199092)

1198872119894+ 1198871198941199093 + 1199094

]

2

4 [minus5 5] 000030

11986516 (119909) = 41199092

1minus 21119909

4

1+

1

3

1199096

1+ 11990911199092 minus 4119909

2

2+ 41199094

2 2 [minus5 5] minus10316

11986517 (119909) = (1199092 minus

51

412058721199092

1+

5

120587

1199091 minus 6)

2

+ 10 (1 minus

1

8120587

) cos1199091 + 10 2 [minus5 5] 0398

11986518 (119909) = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909

2

1minus 141199092 + 611990911199092 + 3119909

2

2)]

sdot [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909

2

1+ 481199092 minus 3611990911199092 + 27119909

2

2)]

2 [minus2 2] 3

11986519 (119909) = minus

4

sum

119894=1

119888119894 exp(minus3

sum

119895=1

119886119894119895 (119909119895 minus 119901119894119895)

2

) 3 [1 3] minus386

11986520 (119909) = minus

4

sum

119894=1

119888119894 exp(minus6

sum

119895=1

119886119894119895 (119909119895 minus 119901119894119895)

2

) 6 [0 1] minus332

11986521 (119909) = minus

5

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus101532

11986522 (119909) = minus

7

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus104028

11986523 (119909) = minus

10

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus105363

that collaborate together to optimize an appropriate energy-aware objective function [52] GWO is one of the powerfulheuristics that can be applied for efficient load balancedclustering In this paper mGWO based clustering algorithmis used to solve the abovementioned load balancing problemThe algorithm forms clusters in such a way that the overallenergy consumption of the network is minimized Totalenergy consumption in the network is the sum of the totalenergy dissipated from the non-CHs to send information totheir respective CHs and the total energy consumed by CHnodes to aggregate the information and send it to the basestation (BS)

Consider a WSN of 119899 sensor nodes randomly deployedin the sensing field and organized into 119870 clusters1198621 1198622 119862119870 The fitness function for the energy consump-tion may be defined as

119891 = (

119870

sum

119894=1

sum

119904isin119862119894

119864TX119904CH119894+ 119864RX + 119864DA) +

119870

sum

119894=1

119864TXCH119894 BS (8)

where 119870 is the total number of CHs 119904 isin 119862119894 is a non-CHassociated with the 119894th CH and 119864TXnode1node2

is the energydissipated for transmitting data from node1 to node2

In order to calculate radio energy transmission andreception costs a 119896-bit message and also the transmitter-receiver separation distance 119889 are given by

119864TX =

119896119864elec + 119896120576friss amp1198892 if 119889 lt 1198890

119896119864elec + 119896120576two ray amp1198894 if 119889 ge 1198890

(9)

The term 119864elec denotes the per-bit energy dissipation duringtransmission The per-bit amplification energy is propor-tional to 1198894 when the transmission distance exceeds thethreshold 1198890 (called crossover distance) and otherwise isproportional to 1198892 The parameters 120576friss amp and 120576two ray ampdenote transmitter amplification parameters for free-spaceand multipath fading models respectively The value of 1198890 isgiven by

1198890 = radic

120576friss amp

120576two ray amp (10)

The reception energy of the 119896-bit data message can beexpressed by

119864RX = 119896119864elec (11)

where 119864elec denotes the per-bit energy dissipation duringreception119864DA is the data aggregation energy expenditure and is set

as 119864DA = 5 njbit The values of other parameters are set to119864elec = 50 njbit 120576friss amp = 100 pjbitm

2 and 120576two ray amp =

00013 pjbitm4 respectively [51]For the simulation setup 100 nodes are randomly

deployed in a 100m times 100m area of the sensing field BSis placed at the center of the field The initial energy of allhomogeneous nodes is set to 1198640 = 1 J During this analysisthree parameters namely first node dead (FND) half nodesdead (HND) and last node dead (LND) are employed tooutline the network lifetime

8 Applied Computational Intelligence and Soft Computing

Table 4 Composite benchmark functions

Function Dim Range 119891min

11986524 (CF1)

10 [minus5 5] 0

1198911 1198912 = Rastriginrsquos function1198913 1198914 =Weierstrassrsquo function1198915 1198916 = Griewankrsquos function1198917 1198918 = Ackleyrsquos function1198919 11989110 = sphere function[1205901 1205902 1205903 12059010] = [1 1 1 1]

[1205821 1205822 1205823 12058210] = [15 15 505 505 5100 5100 532 532 5100 5100]

11986525 (CF2)

10 [minus5 5] 0

1198911 1198912 = Ackleyrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = sphere function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 2 15 15 1 1 15 15 2 2]

[1205821 1205822 1205823 12058210] = [2 lowast 532 532 2 lowast 1 1 2 lowast 5100 5100 2 lowast 10 10 2 lowast 560 560]

11986526 (CF3)

10 [minus5 5] 0

1198911 1198912 = expanded Schaffer Rosenbrockrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = expanded Griewankrsquos and Rosenbrockrsquos function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 1 1 1 1 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [5 lowast 5100 5100 5 lowast 1 1 5 lowast 1 1 5 lowast 10 10 5 lowast 5200 5200]

11986527 (CF4)

10 [minus5 5] 0

1198911 =Weierstrassrsquos function1198912 = expanded Schaffer Rosenbrockrsquos function1198913 = expanded Griewankrsquos and Rosenbrockrsquos function1198914 = Ackleyrsquos function1198915 = Rastriginrsquos function1198916 = Griewankrsquos function1198917 = expanded Schaffer Rosenbrockrsquos noncont function1198918 = Rastriginrsquos noncont function1198919 = elliptic function11989110 = sphere noise function[1205901 1205902 1205903 12059010] = [2 2 2 2 2 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [10 520 1 532 1 5100 550 1 5100 5100]

Table 5 Results of unimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198651 531119864 minus 11 976119864 minus 11 22546833 66683781 289119864 minus 16 333119864 minus 16 265119864 minus 155 118119864 minus 154 644E minus 205 000E + 001198652 567119864 minus 07 126119864 minus 06 28648141 94412388 592119864 minus 09 543119864 minus 09 723119864 minus 92 841119864 minus 92 334E minus 119 495E minus 1191198653 32288636 19324945 61977294 26054622 10408052 09448328 178119864 minus 42 414119864 minus 42 274E minus 52 119E minus 511198654 40759779 16786821 56421446 10918884 10803036 30246624 586119864 minus 37 257119864 minus 36 120E minus 51 425E minus 511198655 62716929 49490404 18305153 12467683 22524975 15539303 26799858 05767914 26900044 08521161198656 289119864 minus 11 427119864 minus 11 19775038 73616353 499E minus 16 130E minus 15 07070627 02983174 07862954 024492871198657 00542038 00175609 50503673 16183878 00466985 00214709 00003805 00002072 00002609 0000176

Applied Computational Intelligence and Soft Computing 9

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

10minus150

10minus200

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

108

106

104

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

10minus10

10minus15

100

F1 F2

F3 F4

F5 F6

Figure 2 Continued

10 Applied Computational Intelligence and Soft Computing

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus2

100

102

F7

Figure 2 Convergence graph of unimodal benchmark functions

Table 6 Results of multimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198658 mdash mdash mdash mdash minus99587573 38210737 minus56976258 10363742 minus5712462 924473821198659 65433009 14197664 96065525 32511237 33785055 69716023 0 0 0 011986510 10769295 91708499 17231924 11118681 09966469 04914999 888119864 minus 15 227119864 minus 15 782E minus 15 794E minus 1611986511 00173265 00205398 23738699 81345844 00008613 00038519 00025395 00066692 0 011986512 03185579 05824145 18644754 18385611 01635701 04113459 00478581 00208711 00469178 0021678911986513 0020326 00488385 76481054 53670684 03009574 11566959 06303231 01787968 06051939 01740609

Table 7 Results of fixed-dimension multimodal functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986514 09980038 0 97544907 70552637 09980038 0 41298947 42619933 48165637 4309756211986515 00011042 00003066 00032289 0005698 00003075 111E minus 19 00044241 00081813 0007327 0009814511986516 minus10316285 228119864 minus 16 minus09908202 01825 minus10316285 228E minus 16 minus10316285 175119864 minus 09 minus10316284 486119864 minus 09

11986517 03978874 0 03978874 134E minus 10 03978874 0 03978978 463119864 minus 05 03978876 508119864 minus 07

11986518 3 815E minus 16 1245 20119315 3 139119864 minus 15 70500022 1811215 30000017 287119864 minus 06

11986519 minus38627821 224E minus 15 minus38627821 172119864 minus 08 minus38627821 228119864 minus 15 minus3862402 00016777 minus38611515 00031911986520 minus32625486 00609909 minus32566017 00606853 minus33219952 519E minus 16 minus31770991 026433 minus32734635 0060991711986521 minus65146454 31931088 minus57607751 31257537 minus101532 365E minus 15 minus96433311 1569109 minus92742029 2193450111986522 minus79202597 34833672 minus52362723 27999111 minus10402941 195E minus 15 minus10402879 466119864 minus 05 minus10136952 1188478111986523 minus78374538 34395208 minus59516938 38940879 minus1053641 195E minus 15 minus9724888 24976263 minus10536237 00001251

Table 9 shows the best results obtained for theCH selection problem in WSN The results of Table 9show that mGWO algorithm is able to find the bestresults compared to other algorithms The results ofmGWO are closely followed by the CS and GWOalgorithms

6 Conclusion

This paper proposed a modification to the Grey Wolf Opti-mizer named mGWO inspired by the hunting behavior ofgrey wolves in nature An exponential decay function is usedto balance the exploration and exploitation in the search

Applied Computational Intelligence and Soft Computing 11

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

Best

scor

e obt

aine

d so

far

0 1000 2000 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus10100

minus10200

minus10300

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

10minus15

102

104

106

108

100

108

106

104

102

100

F8 F9

F10 F11

F12 F13

Figure 3 Convergence graph of multimodal benchmark functions

12 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

053

minus10055

minus10057

minus10051

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus1002

minus1003

minus1004

minus1005

minus1001

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus1

10minus2

10minus3

F14 F15

F17 F18

F19 F20

Figure 4 Continued

Applied Computational Intelligence and Soft Computing 13

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

0

minus101

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus101

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

minus100

minus101

F21 F22

F23

Figure 4 Convergence graph of fixed-dimension multimodal functions

Table 8 Results of composite functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986524 3751142 1491547 774141 2693536 4136706 17765048 3379168 1733873 3718084 194095411986525 1670097 3186513 4907166 1803477 132158 245817 1339568 2317741 1323655 258643111986526 9556119 2808871 1476118 1179418 9150283 2623901 9437718 2779337 9119168 319887611986527 4376602 1831154 1328464 1770158 395 26708204 5593846 301082 390904 279606

14 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

1028

1026

1024

1022

F24 F25

F26 F27

Figure 5 Convergence graph of composite functions

Table 9 Comparison results of CH selection problem in WSN

Algorithm Optimalcost of FF FND HND LND

GA 39798mJ 20471 26782 33375PSO 39854mJ 20348 26652 33246BA 40324mJ 19967 26548 32989CS 38412mJ 20792 27025 33724GWO 38560mJ 20739 26963 33698mGWO 38209mJ 20846 27144 33822

space over the course of iterations The results proved thatthe proposed algorithm benefits from high exploration incomparison to the standard GWO

The paper also considered the clustering problem inWSNin which the CH selection is performed using the proposedmGWO algorithm which is a challenging and NP hardproblemThe results show that the proposedmethod is foundto be very effective for real-world applications due to fastconvergence and fewer chances to get stuck at local minimaIt can be concluded that the proposed algorithm is able tooutperform the current well-known and powerful algorithmsin the literature The results prove the competence andsuperiority of mGWO to existing metaheuristic algorithmsand it has an ability to become an effective tool for solvingreal word optimization problems

Competing Interests

The authors declare that they have no competing interests

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 Applied Computational Intelligence and Soft Computing

Ant Colony Optimization (ACO) [21 22] is inspired by thepheromone trail laying behavior of real ant colonies A newevolutionary optimization algorithm Cuckoo Search (CS)Algorithm [23] is inspired by lifestyle of cuckoo birds Themajor algorithms include Ant Colony Optimization (ACO)[21 22] Particle Swarm Optimization (PSO) [18] ArtificialBee Colony (ABC) Algorithm [24] Fish Swarm Algorithm(FSA) [25] Glowworm Swarm Optimization (GSO) [26]Grey Wolf Optimizer (GWO) [27] Fruit Fly OptimizationAlgorithm (FFOA) [28] Bat Algorithm (BA) [20] NovelBat Algorithm (NBA) [29] Dragonfly Algorithm (DA) [30]Cat Swarm Optimization (CSO) [31] Cuckoo Search (CS)Algorithm [23] Cuckoo Optimization Algorithm (COA)[32] and Spider Monkey Optimization (SMO) Algorithm[33]

The biologically inspired algorithms comprise naturalmetaheuristics derived from living phenomena and behav-ior of biological organisms The intelligence derived withbioinspired algorithms is decentralized distributed self-organizing and adaptive in nature under uncertain environ-ments The major algorithms in this field include ArtificialImmune Systems (AIS) [34] Bacterial ForagingOptimization(BFO) [35] and Krill Herd Algorithm [36]

Because of their inherent advantages such algorithmscan be applied to various applications including powersystems operations and control job scheduling problemsclustering and routing problems batch process schedulingimage processing and pattern recognition problems

GWO is recently developed heuristics inspired from theleadership hierarchy and hunting mechanism of grey wolvesin nature and has been successfully applied for solving eco-nomic dispatch problems [37] feature subset selection [38]optimal design of double later grids [39] time forecasting[40] flow shop scheduling problem [41] optimal power flowproblem [42] and optimizing key values in the cryptographyalgorithms [43] A number of variants are also proposedto improve the performance of basic GWO that includebinary GWO [44] a hybrid version of GWO with PSO [45]integration of DE with GWO [46] and parallelized GWO[47 48]

Every optimization algorithm stated above needs toaddress the exploration and exploitation of a search spaceIn order to be successful an optimization algorithm needs toestablish a good ratio between exploration and exploitationIn this paper a modified GWO (mGWO) is proposed tobalance the exploration and exploitation trade-off in originalGWO algorithm Different functions with diverse slopes areemployed to tune the parameters of GWO algorithm forvarying exploration and exploitation combinations over thecourse of iterations Increasing the exploration in comparisonto exploitation increases the convergence speed and avoidsthe local minima trapping effect

The rest of the paper is organized as follows Section 2gives the overview of original GWO The proposed mGWOalgorithm is explained in Section 3 The experimental resultsare demonstrated in Section 4 Section 5 solves the clusteringproblem in WSN for cluster head selection to demonstratethe applicability of the proposed algorithm Finally Section 6concludes the paper

2 Overview of Grey Wolf Optimizer Algorithm

Grey Wolf Optimizer (GWO) is a typical swarm-intelligencealgorithm which is inspired from the leadership hierarchyandhuntingmechanismof greywolves in natureGreywolvesare considered as apex predators they have average group sizeof 5ndash12 In the hierarchy of GWO alpha (120572) is consideredthe most dominating member among the group The rest ofthe subordinates to 120572 are beta (120573) and delta (120575) which helpto control the majority of wolves in the hierarchy that areconsidered as omega (120596) The 120596 wolves are of lowest rankingin the hierarchy

The mathematical model of hunting mechanism of greywolves consists of the following

(i) Tracking chasing and approaching the prey

(ii) Pursuing encircling and harassing the prey until itstops moving

(iii) Attacking the prey

21 Encircling Prey Grey wolves encircle the prey during thehunt which can be mathematically written as [27]

=

100381610038161003816100381610038161003816

sdot

997888997888rarr

119883119901 (119905) minus (119905)

100381610038161003816100381610038161003816

(119905 + 1) =

997888997888rarr

119883119901 (119905) minus sdot

(1)

where 119905 indicates the current iteration and are coefficientvectors 997888997888rarr119883119901 is the position vector of the prey and indicatesthe position vector of a grey wolf

The vectors and are calculated as follows

= 2 sdot997888rarr1199031 minus

= 2 sdot997888rarr1199032

(2)

where components of 119886 are linearly decreased from 2 to 0 overthe course of iterations and 1199031 and 1199032 are random vectors in[0 1]

22 Hunting Hunting of prey is usually guided by 120572 and120573 and 120575 will participate occasionally The best candidatesolutions that is 120572 120573 and 120575 have better knowledge aboutthe potential location of prey The other search agents (120596)update their positions according to the position of three bestsearch agents The following formulas are proposed in thisregard

997888rarr

119863120572 =

100381610038161003816100381610038161003816

997888rarr

1198621 sdot997888rarr

119883120572 minus

100381610038161003816100381610038161003816

997888rarr

119863120573 =

100381610038161003816100381610038161003816

997888rarr

1198622 sdot997888rarr

119883120573 minus

100381610038161003816100381610038161003816

997888rarr

119863120575 =

100381610038161003816100381610038161003816

997888rarr

1198623 sdot997888rarr

119883120575 minus

100381610038161003816100381610038161003816

(3)

Applied Computational Intelligence and Soft Computing 3

997888rarr

1198831 =997888rarr

119883120572 minus997888rarr

1198601 sdot (997888rarr

119863120572)

997888rarr

1198832 =997888rarr

119883120573 minus997888rarr

1198602 sdot (997888rarr

119863120573)

997888rarr

1198833 =997888rarr

119883120575 minus997888rarr

1198603 sdot (997888rarr

119863120575)

(4)

(119905 + 1) =

997888rarr

1198831 (119905) +997888rarr

1198832 (119905) +997888rarr

1198833 (119905)

3

(5)

23 Attacking Prey In order to mathematically model forapproaching the prey we decrease the value of Thefluctuation range of is also decreased by is a randomvalue in the interval [minus119886 119886]where 119886 is decreased linearly from2 to 0 over the course of iterationsWhen random values of are in [minus1 1] the next position of a search agent can be in anyposition between its current position and the position of theprey The value |119860| lt 1 forces the wolves to attack the prey

After the attack again they search for the prey in the nextiteration wherein they again find the next best solution 120572among all wolves This process repeats till the terminationcriterion is fulfilled

3 Modified GWO Algorithm

Finding the global minimum is a common challenging taskamong all minimization methods In population-based opti-mization methods generally the desirable way to convergetowards the global minimum can be divided into two basicphases In the early stages of the optimization the individualsshould be encouraged to scatter throughout the entire searchspace In other words they should try to explore the wholesearch space instead of clustering around local minima Inthe latter stages the individuals have to exploit informationgathered to converge on the global minimum In GWO withfine-adjusting of the parameters 119886 and 119860 we can balancethese two phases in order to find global minimum with fastconvergence speed

Although different improvements of individual-basedalgorithms promote local optima avoidance the litera-ture shows that population-based algorithms are better inhandling this issue Regardless of the differences betweenpopulation-based algorithms the common approach is thedivision of optimization process to two conflicting mile-stones exploration versus exploitation The explorationencourages candidate solutions to change abruptly andstochastically This mechanism improves the diversity of thesolutions and causes high exploration of the search space Incontrast the exploitation aims for improving the quality ofsolutions by searching locally around the obtained promisingsolutions in the exploration In this milestone candidatesolutions are obliged to change less suddenly and searchlocally

Exploration and exploitation are two conflicting mile-stones where promoting one results in degrading the otherA right balance between these two milestones can guaran-tee a very accurate approximation of the global optimumusing population-based algorithms On the one hand mere

exploration of the search space prevents an algorithm fromfinding an accurate approximation of the global optimumOnthe other handmere exploitation results in local optima stag-nation and again low quality of the approximated optimum

In GWO the transition between exploration andexploitation is generated by the adaptive values of 119886 and119860 Inthis half of the iterations are devoted to exploration (|119860| ge 1)and the other half are used for exploitation (|119860| lt 1) asshown in Figure 1(a) Generally higher exploration of searchspace results in lower probability of local optima stagnationThere are various possibilities to enhance the explorationrate as shown in Figure 1(b) in which exponential functionsare used instead of linear function to decrease the value of 119886over the course of iterations Too much exploration is similarto too much randomness and will probably not give goodoptimization results But too much exploitation is relatedto too little randomness Therefore there must be a balancebetween exploration and exploitation

In GWO the value of 119886 decreases linearly from 2 to 0using the update equation as follows

119886 = 2 (1 minus

119905

119879

) (6)

where 119879 indicates the maximum number of iterations and119905 is the current iteration Our mGWO employs exponentialfunction for the decay of 119886 over the course of iterationsConsider

119886 = 2(1 minus

1199052

1198792) (7)

as shown in Figure 1(c) Using this exponential decay func-tion the numbers of iterations used for exploration andexploitation are 70 and 30 respectively

The pseudocode of mGWO is given in Algorithm 1

4 Results and Discussion

This section investigates the effectiveness of mGWO inpractice It is common in this field to benchmark theperformance of algorithms on a set ofmathematical functionswith known global optima We also follow the same processand employ 27 benchmark functions for comparisonThe testfunctions are divided to four groups unimodal multimodalfixed-dimension multimodal and composite benchmarkfunctions The unimodal functions (1198651ndash1198657) are suitable forbenchmarking the exploitation of algorithms since they haveone global optimum and no local optima On the contrarymultimodal functions (1198658ndash11986513) have a large number of localoptima and are helpful to examine exploration and localoptima avoidance of algorithms

Themathematical formulation of the employed test func-tions is presented in Tables 1ndash4 We consider 30 variablesfor unimodal and multimodal test function for furtherimproving their difficulties

Since heuristic algorithms are stochastic optimizationtechniques they have to be run at least more than 10times to generate meaningful statistical results It is againa common strategy that an algorithm is run on a problem

4 Applied Computational Intelligence and Soft Computing

Valu

e of a

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

2

Iteration

ExplorationExploitation

(a)

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

2

Iteration

Valu

e of a

ExplorationExploitation

(b)

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

2

Iteration

Valu

e of a

ExplorationExploitation

(c)

Figure 1 (a) Updating the value of 119886 for GWO (b) some samples of possible functions for updating 119886 over the course of iterations and (c)updating the value of 119886 over the course of iterations for mGWO

119898 times and averagestandard deviationmedian of the bestobtained solution in the last iteration are calculated as themetrics of performance We follow the same method togenerate and report the results over 30 independent runsIn order to verify the performance of mGWO algorithmPSO BA CS and GWO algorithms are chosen Note that we

utilized 30 search agents and 3000 iterations for each of thealgorithms

The convergence curves of unimodal multimodal fixed-dimensionmultimodal and composite benchmark functionsfor the competitive optimization algorithms are given inFigures 2 3 4 and 5 respectively As Table 5 shows

Applied Computational Intelligence and Soft Computing 5

Initialize the search agent (grey wolf) population119883119894 (119894 = 1 2 119899)Initialize 119886 119860 and 119862Calculate the fitness of each search agent119883120572 = the best (or dominating) search agent119883120573 = the second best search agent119883120575 = the third best search agentwhile (119905 ltMaximum number of iterations)

for each search agentupdate the position of the current search agent by (5)

end forupdate 119886 by (7)update 119860 and 119862 by (2)calculate the fitness of all search agentsupdate119883120572 119883120573 and119883120575119905 = 119905 + 1

end whileReturn119883120572

Algorithm 1 Pseudocode of mGWO algorithm

Table 1 Unimodal benchmark functions

Function Dim Range 119891min

1198651 (119909) =

119899

sum

119894=1

1199092

11989430 [minus100 100] 0

1198652 (119909) =

119899

sum

119894=1

|119909119894| +

119899

prod

119894=1

|119909119894| 30 [minus10 10] 0

1198653 (119909) =

119899

sum

119894=1

(

119894

sum

119895minus1

119909119895)

2

30 [minus100 100] 0

1198654 (119909) = max1198941003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119899 30 [minus100 100] 0

1198655 (119909) =

119899minus1

sum

119894=1

[100 (119909119894+1 minus 1199092

119894)

2

+ (119909119894 minus 1)2] 30 [minus30 30] 0

1198656 (119909) =

119899

sum

119894=1

([119909119894 + 05])2 30 [minus100 100] 0

1198657 (119909) =

119899

sum

119894=1

1198941199094

119894+ random (0 1) 30 [minus128 128] 0

mGWO algorithm provides the best results in 5 out of 7unimodal benchmark test functions The mGWO algorithmalso provides very competitive results compared to CS on 1198655and 1198656 As discussed above unimodal functions are suitablefor benchmarking exploitation of the algorithms Thereforethese results evidence high exploitation capability of themGWO algorithm

The statistical results of the algorithms on multimodaltest function are presented in Table 6 It may be seen thatmGWO algorithm highly outperforms other algorithms on11986591198651011986511 and11986512 It should be noted thatmGWOalgorithmoutperforms other algorithms on these multimodal testfunctions except PSO for 11986513 The results of multimodal testfunction strongly prove that high exploration of mGWOalgorithm is a suitable mechanism for avoiding local solu-tions Since the multimodal functions have an exponentialnumber of local solutions the results show that mGWOalgorithm is able to explore the search space extensively and

find promising regions of the search space In addition highlocal optima avoidance of this algorithm is another findingthat can be inferred from these results

The rest of the results which belong to 11986514ndash11986523 and11986524ndash11986527 are provided in Tables 7 and 8 respectively Theresults are consistent with those of other test functions inwhich mGWO shows very competitive results compared toother algorithms

5 Cluster Head Selection inWSN Using mGWO

Cluster head (CH) selection problem is a well-known prob-lem in the field of wireless sensor networks (WSNs) inwhich the energy consumption cost of the network shouldbe minimized [49ndash53] In this paper this problem is solvedusing mGWO algorithm and compared with GA PSO BACS and GWO

The main challenges in designing and planning theoperations ofWSNs are to optimize energy consumption andprolong network lifetime Cluster-based routing techniquessuch as the well-known low-energy adaptive clustering hier-archy (LEACH) [50] are used to achieve scalable solutionsand extend the network lifetime until the last node dies(LND) In order to achieve prolonged network lifetime incluster-based routing techniques the lifetime of the CHsplays an important role Improper cluster formation maycause some CHs to be overloaded Such overload maycause high energy consumption of the CH and degradethe overall performance of the WSN Therefore proper CHselection is the most important issue for clustering sensornodes Designing an energy efficient clustering algorithmis not an easy task Therefore nature-inspired optimizationalgorithms may be applied to tackle cluster-based routingproblem in WSN Evolutionary algorithms (EAs) have beenused in recent years as metaheuristics to address energy-aware routing challenges by designing intelligent models

6 Applied Computational Intelligence and Soft Computing

Table2Multim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891min

1198658(119909)=

119899

sum 119894=1

minus119909119894sin(radic1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816)

30[minus500500]

minus4189829times5

1198659(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

30[minus512512]

0

11986510(119909)=minus20exp(minus02radic

1 119899

119899

sum 119894=1

1199092 119894)minusexp(

1 119899

119899

sum 119894=1

cos(2120587119909119894))+20+119890

30[minus3232]

0

11986511(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos(119909119894

radic119894

)+1

30[minus600600]

0

11986512(119909)=

120587 119899

10sin(1205871199101)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+

119909119894+1

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)119898

119909119894gt119886

0minus119886lt119909119894lt119886

119896(minus119909119894minus119886)119898119909119894ltminus119886

30[minus5050]

0

11986513(119909)=01sin2(31205871199091)+

119899

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

30[minus5050]

0

Applied Computational Intelligence and Soft Computing 7

Table 3 Fixed-dimension multimodal benchmark functions

Function Dim Range 119891min

11986514 (119909) = (

1

500

+

25

sum

119895=1

1

119895 + sum2

119894=1(119909119894 minus 119886119894119895)

6)

minus1

2 [minus65 65] 1

11986515 (119909) =

11

sum

119894=1

[119886119894 minus

1199091 (1198872

119894+ 1198871198941199092)

1198872119894+ 1198871198941199093 + 1199094

]

2

4 [minus5 5] 000030

11986516 (119909) = 41199092

1minus 21119909

4

1+

1

3

1199096

1+ 11990911199092 minus 4119909

2

2+ 41199094

2 2 [minus5 5] minus10316

11986517 (119909) = (1199092 minus

51

412058721199092

1+

5

120587

1199091 minus 6)

2

+ 10 (1 minus

1

8120587

) cos1199091 + 10 2 [minus5 5] 0398

11986518 (119909) = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909

2

1minus 141199092 + 611990911199092 + 3119909

2

2)]

sdot [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909

2

1+ 481199092 minus 3611990911199092 + 27119909

2

2)]

2 [minus2 2] 3

11986519 (119909) = minus

4

sum

119894=1

119888119894 exp(minus3

sum

119895=1

119886119894119895 (119909119895 minus 119901119894119895)

2

) 3 [1 3] minus386

11986520 (119909) = minus

4

sum

119894=1

119888119894 exp(minus6

sum

119895=1

119886119894119895 (119909119895 minus 119901119894119895)

2

) 6 [0 1] minus332

11986521 (119909) = minus

5

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus101532

11986522 (119909) = minus

7

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus104028

11986523 (119909) = minus

10

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus105363

that collaborate together to optimize an appropriate energy-aware objective function [52] GWO is one of the powerfulheuristics that can be applied for efficient load balancedclustering In this paper mGWO based clustering algorithmis used to solve the abovementioned load balancing problemThe algorithm forms clusters in such a way that the overallenergy consumption of the network is minimized Totalenergy consumption in the network is the sum of the totalenergy dissipated from the non-CHs to send information totheir respective CHs and the total energy consumed by CHnodes to aggregate the information and send it to the basestation (BS)

Consider a WSN of 119899 sensor nodes randomly deployedin the sensing field and organized into 119870 clusters1198621 1198622 119862119870 The fitness function for the energy consump-tion may be defined as

119891 = (

119870

sum

119894=1

sum

119904isin119862119894

119864TX119904CH119894+ 119864RX + 119864DA) +

119870

sum

119894=1

119864TXCH119894 BS (8)

where 119870 is the total number of CHs 119904 isin 119862119894 is a non-CHassociated with the 119894th CH and 119864TXnode1node2

is the energydissipated for transmitting data from node1 to node2

In order to calculate radio energy transmission andreception costs a 119896-bit message and also the transmitter-receiver separation distance 119889 are given by

119864TX =

119896119864elec + 119896120576friss amp1198892 if 119889 lt 1198890

119896119864elec + 119896120576two ray amp1198894 if 119889 ge 1198890

(9)

The term 119864elec denotes the per-bit energy dissipation duringtransmission The per-bit amplification energy is propor-tional to 1198894 when the transmission distance exceeds thethreshold 1198890 (called crossover distance) and otherwise isproportional to 1198892 The parameters 120576friss amp and 120576two ray ampdenote transmitter amplification parameters for free-spaceand multipath fading models respectively The value of 1198890 isgiven by

1198890 = radic

120576friss amp

120576two ray amp (10)

The reception energy of the 119896-bit data message can beexpressed by

119864RX = 119896119864elec (11)

where 119864elec denotes the per-bit energy dissipation duringreception119864DA is the data aggregation energy expenditure and is set

as 119864DA = 5 njbit The values of other parameters are set to119864elec = 50 njbit 120576friss amp = 100 pjbitm

2 and 120576two ray amp =

00013 pjbitm4 respectively [51]For the simulation setup 100 nodes are randomly

deployed in a 100m times 100m area of the sensing field BSis placed at the center of the field The initial energy of allhomogeneous nodes is set to 1198640 = 1 J During this analysisthree parameters namely first node dead (FND) half nodesdead (HND) and last node dead (LND) are employed tooutline the network lifetime

8 Applied Computational Intelligence and Soft Computing

Table 4 Composite benchmark functions

Function Dim Range 119891min

11986524 (CF1)

10 [minus5 5] 0

1198911 1198912 = Rastriginrsquos function1198913 1198914 =Weierstrassrsquo function1198915 1198916 = Griewankrsquos function1198917 1198918 = Ackleyrsquos function1198919 11989110 = sphere function[1205901 1205902 1205903 12059010] = [1 1 1 1]

[1205821 1205822 1205823 12058210] = [15 15 505 505 5100 5100 532 532 5100 5100]

11986525 (CF2)

10 [minus5 5] 0

1198911 1198912 = Ackleyrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = sphere function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 2 15 15 1 1 15 15 2 2]

[1205821 1205822 1205823 12058210] = [2 lowast 532 532 2 lowast 1 1 2 lowast 5100 5100 2 lowast 10 10 2 lowast 560 560]

11986526 (CF3)

10 [minus5 5] 0

1198911 1198912 = expanded Schaffer Rosenbrockrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = expanded Griewankrsquos and Rosenbrockrsquos function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 1 1 1 1 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [5 lowast 5100 5100 5 lowast 1 1 5 lowast 1 1 5 lowast 10 10 5 lowast 5200 5200]

11986527 (CF4)

10 [minus5 5] 0

1198911 =Weierstrassrsquos function1198912 = expanded Schaffer Rosenbrockrsquos function1198913 = expanded Griewankrsquos and Rosenbrockrsquos function1198914 = Ackleyrsquos function1198915 = Rastriginrsquos function1198916 = Griewankrsquos function1198917 = expanded Schaffer Rosenbrockrsquos noncont function1198918 = Rastriginrsquos noncont function1198919 = elliptic function11989110 = sphere noise function[1205901 1205902 1205903 12059010] = [2 2 2 2 2 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [10 520 1 532 1 5100 550 1 5100 5100]

Table 5 Results of unimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198651 531119864 minus 11 976119864 minus 11 22546833 66683781 289119864 minus 16 333119864 minus 16 265119864 minus 155 118119864 minus 154 644E minus 205 000E + 001198652 567119864 minus 07 126119864 minus 06 28648141 94412388 592119864 minus 09 543119864 minus 09 723119864 minus 92 841119864 minus 92 334E minus 119 495E minus 1191198653 32288636 19324945 61977294 26054622 10408052 09448328 178119864 minus 42 414119864 minus 42 274E minus 52 119E minus 511198654 40759779 16786821 56421446 10918884 10803036 30246624 586119864 minus 37 257119864 minus 36 120E minus 51 425E minus 511198655 62716929 49490404 18305153 12467683 22524975 15539303 26799858 05767914 26900044 08521161198656 289119864 minus 11 427119864 minus 11 19775038 73616353 499E minus 16 130E minus 15 07070627 02983174 07862954 024492871198657 00542038 00175609 50503673 16183878 00466985 00214709 00003805 00002072 00002609 0000176

Applied Computational Intelligence and Soft Computing 9

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

10minus150

10minus200

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

108

106

104

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

10minus10

10minus15

100

F1 F2

F3 F4

F5 F6

Figure 2 Continued

10 Applied Computational Intelligence and Soft Computing

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus2

100

102

F7

Figure 2 Convergence graph of unimodal benchmark functions

Table 6 Results of multimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198658 mdash mdash mdash mdash minus99587573 38210737 minus56976258 10363742 minus5712462 924473821198659 65433009 14197664 96065525 32511237 33785055 69716023 0 0 0 011986510 10769295 91708499 17231924 11118681 09966469 04914999 888119864 minus 15 227119864 minus 15 782E minus 15 794E minus 1611986511 00173265 00205398 23738699 81345844 00008613 00038519 00025395 00066692 0 011986512 03185579 05824145 18644754 18385611 01635701 04113459 00478581 00208711 00469178 0021678911986513 0020326 00488385 76481054 53670684 03009574 11566959 06303231 01787968 06051939 01740609

Table 7 Results of fixed-dimension multimodal functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986514 09980038 0 97544907 70552637 09980038 0 41298947 42619933 48165637 4309756211986515 00011042 00003066 00032289 0005698 00003075 111E minus 19 00044241 00081813 0007327 0009814511986516 minus10316285 228119864 minus 16 minus09908202 01825 minus10316285 228E minus 16 minus10316285 175119864 minus 09 minus10316284 486119864 minus 09

11986517 03978874 0 03978874 134E minus 10 03978874 0 03978978 463119864 minus 05 03978876 508119864 minus 07

11986518 3 815E minus 16 1245 20119315 3 139119864 minus 15 70500022 1811215 30000017 287119864 minus 06

11986519 minus38627821 224E minus 15 minus38627821 172119864 minus 08 minus38627821 228119864 minus 15 minus3862402 00016777 minus38611515 00031911986520 minus32625486 00609909 minus32566017 00606853 minus33219952 519E minus 16 minus31770991 026433 minus32734635 0060991711986521 minus65146454 31931088 minus57607751 31257537 minus101532 365E minus 15 minus96433311 1569109 minus92742029 2193450111986522 minus79202597 34833672 minus52362723 27999111 minus10402941 195E minus 15 minus10402879 466119864 minus 05 minus10136952 1188478111986523 minus78374538 34395208 minus59516938 38940879 minus1053641 195E minus 15 minus9724888 24976263 minus10536237 00001251

Table 9 shows the best results obtained for theCH selection problem in WSN The results of Table 9show that mGWO algorithm is able to find the bestresults compared to other algorithms The results ofmGWO are closely followed by the CS and GWOalgorithms

6 Conclusion

This paper proposed a modification to the Grey Wolf Opti-mizer named mGWO inspired by the hunting behavior ofgrey wolves in nature An exponential decay function is usedto balance the exploration and exploitation in the search

Applied Computational Intelligence and Soft Computing 11

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

Best

scor

e obt

aine

d so

far

0 1000 2000 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus10100

minus10200

minus10300

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

10minus15

102

104

106

108

100

108

106

104

102

100

F8 F9

F10 F11

F12 F13

Figure 3 Convergence graph of multimodal benchmark functions

12 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

053

minus10055

minus10057

minus10051

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus1002

minus1003

minus1004

minus1005

minus1001

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus1

10minus2

10minus3

F14 F15

F17 F18

F19 F20

Figure 4 Continued

Applied Computational Intelligence and Soft Computing 13

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

0

minus101

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus101

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

minus100

minus101

F21 F22

F23

Figure 4 Convergence graph of fixed-dimension multimodal functions

Table 8 Results of composite functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986524 3751142 1491547 774141 2693536 4136706 17765048 3379168 1733873 3718084 194095411986525 1670097 3186513 4907166 1803477 132158 245817 1339568 2317741 1323655 258643111986526 9556119 2808871 1476118 1179418 9150283 2623901 9437718 2779337 9119168 319887611986527 4376602 1831154 1328464 1770158 395 26708204 5593846 301082 390904 279606

14 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

1028

1026

1024

1022

F24 F25

F26 F27

Figure 5 Convergence graph of composite functions

Table 9 Comparison results of CH selection problem in WSN

Algorithm Optimalcost of FF FND HND LND

GA 39798mJ 20471 26782 33375PSO 39854mJ 20348 26652 33246BA 40324mJ 19967 26548 32989CS 38412mJ 20792 27025 33724GWO 38560mJ 20739 26963 33698mGWO 38209mJ 20846 27144 33822

space over the course of iterations The results proved thatthe proposed algorithm benefits from high exploration incomparison to the standard GWO

The paper also considered the clustering problem inWSNin which the CH selection is performed using the proposedmGWO algorithm which is a challenging and NP hardproblemThe results show that the proposedmethod is foundto be very effective for real-world applications due to fastconvergence and fewer chances to get stuck at local minimaIt can be concluded that the proposed algorithm is able tooutperform the current well-known and powerful algorithmsin the literature The results prove the competence andsuperiority of mGWO to existing metaheuristic algorithmsand it has an ability to become an effective tool for solvingreal word optimization problems

Competing Interests

The authors declare that they have no competing interests

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

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Applied Computational Intelligence and Soft Computing 3

997888rarr

1198831 =997888rarr

119883120572 minus997888rarr

1198601 sdot (997888rarr

119863120572)

997888rarr

1198832 =997888rarr

119883120573 minus997888rarr

1198602 sdot (997888rarr

119863120573)

997888rarr

1198833 =997888rarr

119883120575 minus997888rarr

1198603 sdot (997888rarr

119863120575)

(4)

(119905 + 1) =

997888rarr

1198831 (119905) +997888rarr

1198832 (119905) +997888rarr

1198833 (119905)

3

(5)

23 Attacking Prey In order to mathematically model forapproaching the prey we decrease the value of Thefluctuation range of is also decreased by is a randomvalue in the interval [minus119886 119886]where 119886 is decreased linearly from2 to 0 over the course of iterationsWhen random values of are in [minus1 1] the next position of a search agent can be in anyposition between its current position and the position of theprey The value |119860| lt 1 forces the wolves to attack the prey

After the attack again they search for the prey in the nextiteration wherein they again find the next best solution 120572among all wolves This process repeats till the terminationcriterion is fulfilled

3 Modified GWO Algorithm

Finding the global minimum is a common challenging taskamong all minimization methods In population-based opti-mization methods generally the desirable way to convergetowards the global minimum can be divided into two basicphases In the early stages of the optimization the individualsshould be encouraged to scatter throughout the entire searchspace In other words they should try to explore the wholesearch space instead of clustering around local minima Inthe latter stages the individuals have to exploit informationgathered to converge on the global minimum In GWO withfine-adjusting of the parameters 119886 and 119860 we can balancethese two phases in order to find global minimum with fastconvergence speed

Although different improvements of individual-basedalgorithms promote local optima avoidance the litera-ture shows that population-based algorithms are better inhandling this issue Regardless of the differences betweenpopulation-based algorithms the common approach is thedivision of optimization process to two conflicting mile-stones exploration versus exploitation The explorationencourages candidate solutions to change abruptly andstochastically This mechanism improves the diversity of thesolutions and causes high exploration of the search space Incontrast the exploitation aims for improving the quality ofsolutions by searching locally around the obtained promisingsolutions in the exploration In this milestone candidatesolutions are obliged to change less suddenly and searchlocally

Exploration and exploitation are two conflicting mile-stones where promoting one results in degrading the otherA right balance between these two milestones can guaran-tee a very accurate approximation of the global optimumusing population-based algorithms On the one hand mere

exploration of the search space prevents an algorithm fromfinding an accurate approximation of the global optimumOnthe other handmere exploitation results in local optima stag-nation and again low quality of the approximated optimum

In GWO the transition between exploration andexploitation is generated by the adaptive values of 119886 and119860 Inthis half of the iterations are devoted to exploration (|119860| ge 1)and the other half are used for exploitation (|119860| lt 1) asshown in Figure 1(a) Generally higher exploration of searchspace results in lower probability of local optima stagnationThere are various possibilities to enhance the explorationrate as shown in Figure 1(b) in which exponential functionsare used instead of linear function to decrease the value of 119886over the course of iterations Too much exploration is similarto too much randomness and will probably not give goodoptimization results But too much exploitation is relatedto too little randomness Therefore there must be a balancebetween exploration and exploitation

In GWO the value of 119886 decreases linearly from 2 to 0using the update equation as follows

119886 = 2 (1 minus

119905

119879

) (6)

where 119879 indicates the maximum number of iterations and119905 is the current iteration Our mGWO employs exponentialfunction for the decay of 119886 over the course of iterationsConsider

119886 = 2(1 minus

1199052

1198792) (7)

as shown in Figure 1(c) Using this exponential decay func-tion the numbers of iterations used for exploration andexploitation are 70 and 30 respectively

The pseudocode of mGWO is given in Algorithm 1

4 Results and Discussion

This section investigates the effectiveness of mGWO inpractice It is common in this field to benchmark theperformance of algorithms on a set ofmathematical functionswith known global optima We also follow the same processand employ 27 benchmark functions for comparisonThe testfunctions are divided to four groups unimodal multimodalfixed-dimension multimodal and composite benchmarkfunctions The unimodal functions (1198651ndash1198657) are suitable forbenchmarking the exploitation of algorithms since they haveone global optimum and no local optima On the contrarymultimodal functions (1198658ndash11986513) have a large number of localoptima and are helpful to examine exploration and localoptima avoidance of algorithms

Themathematical formulation of the employed test func-tions is presented in Tables 1ndash4 We consider 30 variablesfor unimodal and multimodal test function for furtherimproving their difficulties

Since heuristic algorithms are stochastic optimizationtechniques they have to be run at least more than 10times to generate meaningful statistical results It is againa common strategy that an algorithm is run on a problem

4 Applied Computational Intelligence and Soft Computing

Valu

e of a

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

2

Iteration

ExplorationExploitation

(a)

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

2

Iteration

Valu

e of a

ExplorationExploitation

(b)

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

2

Iteration

Valu

e of a

ExplorationExploitation

(c)

Figure 1 (a) Updating the value of 119886 for GWO (b) some samples of possible functions for updating 119886 over the course of iterations and (c)updating the value of 119886 over the course of iterations for mGWO

119898 times and averagestandard deviationmedian of the bestobtained solution in the last iteration are calculated as themetrics of performance We follow the same method togenerate and report the results over 30 independent runsIn order to verify the performance of mGWO algorithmPSO BA CS and GWO algorithms are chosen Note that we

utilized 30 search agents and 3000 iterations for each of thealgorithms

The convergence curves of unimodal multimodal fixed-dimensionmultimodal and composite benchmark functionsfor the competitive optimization algorithms are given inFigures 2 3 4 and 5 respectively As Table 5 shows

Applied Computational Intelligence and Soft Computing 5

Initialize the search agent (grey wolf) population119883119894 (119894 = 1 2 119899)Initialize 119886 119860 and 119862Calculate the fitness of each search agent119883120572 = the best (or dominating) search agent119883120573 = the second best search agent119883120575 = the third best search agentwhile (119905 ltMaximum number of iterations)

for each search agentupdate the position of the current search agent by (5)

end forupdate 119886 by (7)update 119860 and 119862 by (2)calculate the fitness of all search agentsupdate119883120572 119883120573 and119883120575119905 = 119905 + 1

end whileReturn119883120572

Algorithm 1 Pseudocode of mGWO algorithm

Table 1 Unimodal benchmark functions

Function Dim Range 119891min

1198651 (119909) =

119899

sum

119894=1

1199092

11989430 [minus100 100] 0

1198652 (119909) =

119899

sum

119894=1

|119909119894| +

119899

prod

119894=1

|119909119894| 30 [minus10 10] 0

1198653 (119909) =

119899

sum

119894=1

(

119894

sum

119895minus1

119909119895)

2

30 [minus100 100] 0

1198654 (119909) = max1198941003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119899 30 [minus100 100] 0

1198655 (119909) =

119899minus1

sum

119894=1

[100 (119909119894+1 minus 1199092

119894)

2

+ (119909119894 minus 1)2] 30 [minus30 30] 0

1198656 (119909) =

119899

sum

119894=1

([119909119894 + 05])2 30 [minus100 100] 0

1198657 (119909) =

119899

sum

119894=1

1198941199094

119894+ random (0 1) 30 [minus128 128] 0

mGWO algorithm provides the best results in 5 out of 7unimodal benchmark test functions The mGWO algorithmalso provides very competitive results compared to CS on 1198655and 1198656 As discussed above unimodal functions are suitablefor benchmarking exploitation of the algorithms Thereforethese results evidence high exploitation capability of themGWO algorithm

The statistical results of the algorithms on multimodaltest function are presented in Table 6 It may be seen thatmGWO algorithm highly outperforms other algorithms on11986591198651011986511 and11986512 It should be noted thatmGWOalgorithmoutperforms other algorithms on these multimodal testfunctions except PSO for 11986513 The results of multimodal testfunction strongly prove that high exploration of mGWOalgorithm is a suitable mechanism for avoiding local solu-tions Since the multimodal functions have an exponentialnumber of local solutions the results show that mGWOalgorithm is able to explore the search space extensively and

find promising regions of the search space In addition highlocal optima avoidance of this algorithm is another findingthat can be inferred from these results

The rest of the results which belong to 11986514ndash11986523 and11986524ndash11986527 are provided in Tables 7 and 8 respectively Theresults are consistent with those of other test functions inwhich mGWO shows very competitive results compared toother algorithms

5 Cluster Head Selection inWSN Using mGWO

Cluster head (CH) selection problem is a well-known prob-lem in the field of wireless sensor networks (WSNs) inwhich the energy consumption cost of the network shouldbe minimized [49ndash53] In this paper this problem is solvedusing mGWO algorithm and compared with GA PSO BACS and GWO

The main challenges in designing and planning theoperations ofWSNs are to optimize energy consumption andprolong network lifetime Cluster-based routing techniquessuch as the well-known low-energy adaptive clustering hier-archy (LEACH) [50] are used to achieve scalable solutionsand extend the network lifetime until the last node dies(LND) In order to achieve prolonged network lifetime incluster-based routing techniques the lifetime of the CHsplays an important role Improper cluster formation maycause some CHs to be overloaded Such overload maycause high energy consumption of the CH and degradethe overall performance of the WSN Therefore proper CHselection is the most important issue for clustering sensornodes Designing an energy efficient clustering algorithmis not an easy task Therefore nature-inspired optimizationalgorithms may be applied to tackle cluster-based routingproblem in WSN Evolutionary algorithms (EAs) have beenused in recent years as metaheuristics to address energy-aware routing challenges by designing intelligent models

6 Applied Computational Intelligence and Soft Computing

Table2Multim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891min

1198658(119909)=

119899

sum 119894=1

minus119909119894sin(radic1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816)

30[minus500500]

minus4189829times5

1198659(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

30[minus512512]

0

11986510(119909)=minus20exp(minus02radic

1 119899

119899

sum 119894=1

1199092 119894)minusexp(

1 119899

119899

sum 119894=1

cos(2120587119909119894))+20+119890

30[minus3232]

0

11986511(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos(119909119894

radic119894

)+1

30[minus600600]

0

11986512(119909)=

120587 119899

10sin(1205871199101)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+

119909119894+1

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)119898

119909119894gt119886

0minus119886lt119909119894lt119886

119896(minus119909119894minus119886)119898119909119894ltminus119886

30[minus5050]

0

11986513(119909)=01sin2(31205871199091)+

119899

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

30[minus5050]

0

Applied Computational Intelligence and Soft Computing 7

Table 3 Fixed-dimension multimodal benchmark functions

Function Dim Range 119891min

11986514 (119909) = (

1

500

+

25

sum

119895=1

1

119895 + sum2

119894=1(119909119894 minus 119886119894119895)

6)

minus1

2 [minus65 65] 1

11986515 (119909) =

11

sum

119894=1

[119886119894 minus

1199091 (1198872

119894+ 1198871198941199092)

1198872119894+ 1198871198941199093 + 1199094

]

2

4 [minus5 5] 000030

11986516 (119909) = 41199092

1minus 21119909

4

1+

1

3

1199096

1+ 11990911199092 minus 4119909

2

2+ 41199094

2 2 [minus5 5] minus10316

11986517 (119909) = (1199092 minus

51

412058721199092

1+

5

120587

1199091 minus 6)

2

+ 10 (1 minus

1

8120587

) cos1199091 + 10 2 [minus5 5] 0398

11986518 (119909) = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909

2

1minus 141199092 + 611990911199092 + 3119909

2

2)]

sdot [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909

2

1+ 481199092 minus 3611990911199092 + 27119909

2

2)]

2 [minus2 2] 3

11986519 (119909) = minus

4

sum

119894=1

119888119894 exp(minus3

sum

119895=1

119886119894119895 (119909119895 minus 119901119894119895)

2

) 3 [1 3] minus386

11986520 (119909) = minus

4

sum

119894=1

119888119894 exp(minus6

sum

119895=1

119886119894119895 (119909119895 minus 119901119894119895)

2

) 6 [0 1] minus332

11986521 (119909) = minus

5

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus101532

11986522 (119909) = minus

7

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus104028

11986523 (119909) = minus

10

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus105363

that collaborate together to optimize an appropriate energy-aware objective function [52] GWO is one of the powerfulheuristics that can be applied for efficient load balancedclustering In this paper mGWO based clustering algorithmis used to solve the abovementioned load balancing problemThe algorithm forms clusters in such a way that the overallenergy consumption of the network is minimized Totalenergy consumption in the network is the sum of the totalenergy dissipated from the non-CHs to send information totheir respective CHs and the total energy consumed by CHnodes to aggregate the information and send it to the basestation (BS)

Consider a WSN of 119899 sensor nodes randomly deployedin the sensing field and organized into 119870 clusters1198621 1198622 119862119870 The fitness function for the energy consump-tion may be defined as

119891 = (

119870

sum

119894=1

sum

119904isin119862119894

119864TX119904CH119894+ 119864RX + 119864DA) +

119870

sum

119894=1

119864TXCH119894 BS (8)

where 119870 is the total number of CHs 119904 isin 119862119894 is a non-CHassociated with the 119894th CH and 119864TXnode1node2

is the energydissipated for transmitting data from node1 to node2

In order to calculate radio energy transmission andreception costs a 119896-bit message and also the transmitter-receiver separation distance 119889 are given by

119864TX =

119896119864elec + 119896120576friss amp1198892 if 119889 lt 1198890

119896119864elec + 119896120576two ray amp1198894 if 119889 ge 1198890

(9)

The term 119864elec denotes the per-bit energy dissipation duringtransmission The per-bit amplification energy is propor-tional to 1198894 when the transmission distance exceeds thethreshold 1198890 (called crossover distance) and otherwise isproportional to 1198892 The parameters 120576friss amp and 120576two ray ampdenote transmitter amplification parameters for free-spaceand multipath fading models respectively The value of 1198890 isgiven by

1198890 = radic

120576friss amp

120576two ray amp (10)

The reception energy of the 119896-bit data message can beexpressed by

119864RX = 119896119864elec (11)

where 119864elec denotes the per-bit energy dissipation duringreception119864DA is the data aggregation energy expenditure and is set

as 119864DA = 5 njbit The values of other parameters are set to119864elec = 50 njbit 120576friss amp = 100 pjbitm

2 and 120576two ray amp =

00013 pjbitm4 respectively [51]For the simulation setup 100 nodes are randomly

deployed in a 100m times 100m area of the sensing field BSis placed at the center of the field The initial energy of allhomogeneous nodes is set to 1198640 = 1 J During this analysisthree parameters namely first node dead (FND) half nodesdead (HND) and last node dead (LND) are employed tooutline the network lifetime

8 Applied Computational Intelligence and Soft Computing

Table 4 Composite benchmark functions

Function Dim Range 119891min

11986524 (CF1)

10 [minus5 5] 0

1198911 1198912 = Rastriginrsquos function1198913 1198914 =Weierstrassrsquo function1198915 1198916 = Griewankrsquos function1198917 1198918 = Ackleyrsquos function1198919 11989110 = sphere function[1205901 1205902 1205903 12059010] = [1 1 1 1]

[1205821 1205822 1205823 12058210] = [15 15 505 505 5100 5100 532 532 5100 5100]

11986525 (CF2)

10 [minus5 5] 0

1198911 1198912 = Ackleyrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = sphere function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 2 15 15 1 1 15 15 2 2]

[1205821 1205822 1205823 12058210] = [2 lowast 532 532 2 lowast 1 1 2 lowast 5100 5100 2 lowast 10 10 2 lowast 560 560]

11986526 (CF3)

10 [minus5 5] 0

1198911 1198912 = expanded Schaffer Rosenbrockrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = expanded Griewankrsquos and Rosenbrockrsquos function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 1 1 1 1 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [5 lowast 5100 5100 5 lowast 1 1 5 lowast 1 1 5 lowast 10 10 5 lowast 5200 5200]

11986527 (CF4)

10 [minus5 5] 0

1198911 =Weierstrassrsquos function1198912 = expanded Schaffer Rosenbrockrsquos function1198913 = expanded Griewankrsquos and Rosenbrockrsquos function1198914 = Ackleyrsquos function1198915 = Rastriginrsquos function1198916 = Griewankrsquos function1198917 = expanded Schaffer Rosenbrockrsquos noncont function1198918 = Rastriginrsquos noncont function1198919 = elliptic function11989110 = sphere noise function[1205901 1205902 1205903 12059010] = [2 2 2 2 2 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [10 520 1 532 1 5100 550 1 5100 5100]

Table 5 Results of unimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198651 531119864 minus 11 976119864 minus 11 22546833 66683781 289119864 minus 16 333119864 minus 16 265119864 minus 155 118119864 minus 154 644E minus 205 000E + 001198652 567119864 minus 07 126119864 minus 06 28648141 94412388 592119864 minus 09 543119864 minus 09 723119864 minus 92 841119864 minus 92 334E minus 119 495E minus 1191198653 32288636 19324945 61977294 26054622 10408052 09448328 178119864 minus 42 414119864 minus 42 274E minus 52 119E minus 511198654 40759779 16786821 56421446 10918884 10803036 30246624 586119864 minus 37 257119864 minus 36 120E minus 51 425E minus 511198655 62716929 49490404 18305153 12467683 22524975 15539303 26799858 05767914 26900044 08521161198656 289119864 minus 11 427119864 minus 11 19775038 73616353 499E minus 16 130E minus 15 07070627 02983174 07862954 024492871198657 00542038 00175609 50503673 16183878 00466985 00214709 00003805 00002072 00002609 0000176

Applied Computational Intelligence and Soft Computing 9

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

10minus150

10minus200

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

108

106

104

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

10minus10

10minus15

100

F1 F2

F3 F4

F5 F6

Figure 2 Continued

10 Applied Computational Intelligence and Soft Computing

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus2

100

102

F7

Figure 2 Convergence graph of unimodal benchmark functions

Table 6 Results of multimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198658 mdash mdash mdash mdash minus99587573 38210737 minus56976258 10363742 minus5712462 924473821198659 65433009 14197664 96065525 32511237 33785055 69716023 0 0 0 011986510 10769295 91708499 17231924 11118681 09966469 04914999 888119864 minus 15 227119864 minus 15 782E minus 15 794E minus 1611986511 00173265 00205398 23738699 81345844 00008613 00038519 00025395 00066692 0 011986512 03185579 05824145 18644754 18385611 01635701 04113459 00478581 00208711 00469178 0021678911986513 0020326 00488385 76481054 53670684 03009574 11566959 06303231 01787968 06051939 01740609

Table 7 Results of fixed-dimension multimodal functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986514 09980038 0 97544907 70552637 09980038 0 41298947 42619933 48165637 4309756211986515 00011042 00003066 00032289 0005698 00003075 111E minus 19 00044241 00081813 0007327 0009814511986516 minus10316285 228119864 minus 16 minus09908202 01825 minus10316285 228E minus 16 minus10316285 175119864 minus 09 minus10316284 486119864 minus 09

11986517 03978874 0 03978874 134E minus 10 03978874 0 03978978 463119864 minus 05 03978876 508119864 minus 07

11986518 3 815E minus 16 1245 20119315 3 139119864 minus 15 70500022 1811215 30000017 287119864 minus 06

11986519 minus38627821 224E minus 15 minus38627821 172119864 minus 08 minus38627821 228119864 minus 15 minus3862402 00016777 minus38611515 00031911986520 minus32625486 00609909 minus32566017 00606853 minus33219952 519E minus 16 minus31770991 026433 minus32734635 0060991711986521 minus65146454 31931088 minus57607751 31257537 minus101532 365E minus 15 minus96433311 1569109 minus92742029 2193450111986522 minus79202597 34833672 minus52362723 27999111 minus10402941 195E minus 15 minus10402879 466119864 minus 05 minus10136952 1188478111986523 minus78374538 34395208 minus59516938 38940879 minus1053641 195E minus 15 minus9724888 24976263 minus10536237 00001251

Table 9 shows the best results obtained for theCH selection problem in WSN The results of Table 9show that mGWO algorithm is able to find the bestresults compared to other algorithms The results ofmGWO are closely followed by the CS and GWOalgorithms

6 Conclusion

This paper proposed a modification to the Grey Wolf Opti-mizer named mGWO inspired by the hunting behavior ofgrey wolves in nature An exponential decay function is usedto balance the exploration and exploitation in the search

Applied Computational Intelligence and Soft Computing 11

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

Best

scor

e obt

aine

d so

far

0 1000 2000 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus10100

minus10200

minus10300

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

10minus15

102

104

106

108

100

108

106

104

102

100

F8 F9

F10 F11

F12 F13

Figure 3 Convergence graph of multimodal benchmark functions

12 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

053

minus10055

minus10057

minus10051

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus1002

minus1003

minus1004

minus1005

minus1001

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus1

10minus2

10minus3

F14 F15

F17 F18

F19 F20

Figure 4 Continued

Applied Computational Intelligence and Soft Computing 13

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

0

minus101

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus101

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

minus100

minus101

F21 F22

F23

Figure 4 Convergence graph of fixed-dimension multimodal functions

Table 8 Results of composite functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986524 3751142 1491547 774141 2693536 4136706 17765048 3379168 1733873 3718084 194095411986525 1670097 3186513 4907166 1803477 132158 245817 1339568 2317741 1323655 258643111986526 9556119 2808871 1476118 1179418 9150283 2623901 9437718 2779337 9119168 319887611986527 4376602 1831154 1328464 1770158 395 26708204 5593846 301082 390904 279606

14 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

1028

1026

1024

1022

F24 F25

F26 F27

Figure 5 Convergence graph of composite functions

Table 9 Comparison results of CH selection problem in WSN

Algorithm Optimalcost of FF FND HND LND

GA 39798mJ 20471 26782 33375PSO 39854mJ 20348 26652 33246BA 40324mJ 19967 26548 32989CS 38412mJ 20792 27025 33724GWO 38560mJ 20739 26963 33698mGWO 38209mJ 20846 27144 33822

space over the course of iterations The results proved thatthe proposed algorithm benefits from high exploration incomparison to the standard GWO

The paper also considered the clustering problem inWSNin which the CH selection is performed using the proposedmGWO algorithm which is a challenging and NP hardproblemThe results show that the proposedmethod is foundto be very effective for real-world applications due to fastconvergence and fewer chances to get stuck at local minimaIt can be concluded that the proposed algorithm is able tooutperform the current well-known and powerful algorithmsin the literature The results prove the competence andsuperiority of mGWO to existing metaheuristic algorithmsand it has an ability to become an effective tool for solvingreal word optimization problems

Competing Interests

The authors declare that they have no competing interests

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 Applied Computational Intelligence and Soft Computing

Valu

e of a

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

2

Iteration

ExplorationExploitation

(a)

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

2

Iteration

Valu

e of a

ExplorationExploitation

(b)

0 20 40 60 80 1000

02

04

06

08

1

12

14

16

18

2

Iteration

Valu

e of a

ExplorationExploitation

(c)

Figure 1 (a) Updating the value of 119886 for GWO (b) some samples of possible functions for updating 119886 over the course of iterations and (c)updating the value of 119886 over the course of iterations for mGWO

119898 times and averagestandard deviationmedian of the bestobtained solution in the last iteration are calculated as themetrics of performance We follow the same method togenerate and report the results over 30 independent runsIn order to verify the performance of mGWO algorithmPSO BA CS and GWO algorithms are chosen Note that we

utilized 30 search agents and 3000 iterations for each of thealgorithms

The convergence curves of unimodal multimodal fixed-dimensionmultimodal and composite benchmark functionsfor the competitive optimization algorithms are given inFigures 2 3 4 and 5 respectively As Table 5 shows

Applied Computational Intelligence and Soft Computing 5

Initialize the search agent (grey wolf) population119883119894 (119894 = 1 2 119899)Initialize 119886 119860 and 119862Calculate the fitness of each search agent119883120572 = the best (or dominating) search agent119883120573 = the second best search agent119883120575 = the third best search agentwhile (119905 ltMaximum number of iterations)

for each search agentupdate the position of the current search agent by (5)

end forupdate 119886 by (7)update 119860 and 119862 by (2)calculate the fitness of all search agentsupdate119883120572 119883120573 and119883120575119905 = 119905 + 1

end whileReturn119883120572

Algorithm 1 Pseudocode of mGWO algorithm

Table 1 Unimodal benchmark functions

Function Dim Range 119891min

1198651 (119909) =

119899

sum

119894=1

1199092

11989430 [minus100 100] 0

1198652 (119909) =

119899

sum

119894=1

|119909119894| +

119899

prod

119894=1

|119909119894| 30 [minus10 10] 0

1198653 (119909) =

119899

sum

119894=1

(

119894

sum

119895minus1

119909119895)

2

30 [minus100 100] 0

1198654 (119909) = max1198941003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119899 30 [minus100 100] 0

1198655 (119909) =

119899minus1

sum

119894=1

[100 (119909119894+1 minus 1199092

119894)

2

+ (119909119894 minus 1)2] 30 [minus30 30] 0

1198656 (119909) =

119899

sum

119894=1

([119909119894 + 05])2 30 [minus100 100] 0

1198657 (119909) =

119899

sum

119894=1

1198941199094

119894+ random (0 1) 30 [minus128 128] 0

mGWO algorithm provides the best results in 5 out of 7unimodal benchmark test functions The mGWO algorithmalso provides very competitive results compared to CS on 1198655and 1198656 As discussed above unimodal functions are suitablefor benchmarking exploitation of the algorithms Thereforethese results evidence high exploitation capability of themGWO algorithm

The statistical results of the algorithms on multimodaltest function are presented in Table 6 It may be seen thatmGWO algorithm highly outperforms other algorithms on11986591198651011986511 and11986512 It should be noted thatmGWOalgorithmoutperforms other algorithms on these multimodal testfunctions except PSO for 11986513 The results of multimodal testfunction strongly prove that high exploration of mGWOalgorithm is a suitable mechanism for avoiding local solu-tions Since the multimodal functions have an exponentialnumber of local solutions the results show that mGWOalgorithm is able to explore the search space extensively and

find promising regions of the search space In addition highlocal optima avoidance of this algorithm is another findingthat can be inferred from these results

The rest of the results which belong to 11986514ndash11986523 and11986524ndash11986527 are provided in Tables 7 and 8 respectively Theresults are consistent with those of other test functions inwhich mGWO shows very competitive results compared toother algorithms

5 Cluster Head Selection inWSN Using mGWO

Cluster head (CH) selection problem is a well-known prob-lem in the field of wireless sensor networks (WSNs) inwhich the energy consumption cost of the network shouldbe minimized [49ndash53] In this paper this problem is solvedusing mGWO algorithm and compared with GA PSO BACS and GWO

The main challenges in designing and planning theoperations ofWSNs are to optimize energy consumption andprolong network lifetime Cluster-based routing techniquessuch as the well-known low-energy adaptive clustering hier-archy (LEACH) [50] are used to achieve scalable solutionsand extend the network lifetime until the last node dies(LND) In order to achieve prolonged network lifetime incluster-based routing techniques the lifetime of the CHsplays an important role Improper cluster formation maycause some CHs to be overloaded Such overload maycause high energy consumption of the CH and degradethe overall performance of the WSN Therefore proper CHselection is the most important issue for clustering sensornodes Designing an energy efficient clustering algorithmis not an easy task Therefore nature-inspired optimizationalgorithms may be applied to tackle cluster-based routingproblem in WSN Evolutionary algorithms (EAs) have beenused in recent years as metaheuristics to address energy-aware routing challenges by designing intelligent models

6 Applied Computational Intelligence and Soft Computing

Table2Multim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891min

1198658(119909)=

119899

sum 119894=1

minus119909119894sin(radic1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816)

30[minus500500]

minus4189829times5

1198659(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

30[minus512512]

0

11986510(119909)=minus20exp(minus02radic

1 119899

119899

sum 119894=1

1199092 119894)minusexp(

1 119899

119899

sum 119894=1

cos(2120587119909119894))+20+119890

30[minus3232]

0

11986511(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos(119909119894

radic119894

)+1

30[minus600600]

0

11986512(119909)=

120587 119899

10sin(1205871199101)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+

119909119894+1

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)119898

119909119894gt119886

0minus119886lt119909119894lt119886

119896(minus119909119894minus119886)119898119909119894ltminus119886

30[minus5050]

0

11986513(119909)=01sin2(31205871199091)+

119899

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

30[minus5050]

0

Applied Computational Intelligence and Soft Computing 7

Table 3 Fixed-dimension multimodal benchmark functions

Function Dim Range 119891min

11986514 (119909) = (

1

500

+

25

sum

119895=1

1

119895 + sum2

119894=1(119909119894 minus 119886119894119895)

6)

minus1

2 [minus65 65] 1

11986515 (119909) =

11

sum

119894=1

[119886119894 minus

1199091 (1198872

119894+ 1198871198941199092)

1198872119894+ 1198871198941199093 + 1199094

]

2

4 [minus5 5] 000030

11986516 (119909) = 41199092

1minus 21119909

4

1+

1

3

1199096

1+ 11990911199092 minus 4119909

2

2+ 41199094

2 2 [minus5 5] minus10316

11986517 (119909) = (1199092 minus

51

412058721199092

1+

5

120587

1199091 minus 6)

2

+ 10 (1 minus

1

8120587

) cos1199091 + 10 2 [minus5 5] 0398

11986518 (119909) = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909

2

1minus 141199092 + 611990911199092 + 3119909

2

2)]

sdot [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909

2

1+ 481199092 minus 3611990911199092 + 27119909

2

2)]

2 [minus2 2] 3

11986519 (119909) = minus

4

sum

119894=1

119888119894 exp(minus3

sum

119895=1

119886119894119895 (119909119895 minus 119901119894119895)

2

) 3 [1 3] minus386

11986520 (119909) = minus

4

sum

119894=1

119888119894 exp(minus6

sum

119895=1

119886119894119895 (119909119895 minus 119901119894119895)

2

) 6 [0 1] minus332

11986521 (119909) = minus

5

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus101532

11986522 (119909) = minus

7

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus104028

11986523 (119909) = minus

10

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus105363

that collaborate together to optimize an appropriate energy-aware objective function [52] GWO is one of the powerfulheuristics that can be applied for efficient load balancedclustering In this paper mGWO based clustering algorithmis used to solve the abovementioned load balancing problemThe algorithm forms clusters in such a way that the overallenergy consumption of the network is minimized Totalenergy consumption in the network is the sum of the totalenergy dissipated from the non-CHs to send information totheir respective CHs and the total energy consumed by CHnodes to aggregate the information and send it to the basestation (BS)

Consider a WSN of 119899 sensor nodes randomly deployedin the sensing field and organized into 119870 clusters1198621 1198622 119862119870 The fitness function for the energy consump-tion may be defined as

119891 = (

119870

sum

119894=1

sum

119904isin119862119894

119864TX119904CH119894+ 119864RX + 119864DA) +

119870

sum

119894=1

119864TXCH119894 BS (8)

where 119870 is the total number of CHs 119904 isin 119862119894 is a non-CHassociated with the 119894th CH and 119864TXnode1node2

is the energydissipated for transmitting data from node1 to node2

In order to calculate radio energy transmission andreception costs a 119896-bit message and also the transmitter-receiver separation distance 119889 are given by

119864TX =

119896119864elec + 119896120576friss amp1198892 if 119889 lt 1198890

119896119864elec + 119896120576two ray amp1198894 if 119889 ge 1198890

(9)

The term 119864elec denotes the per-bit energy dissipation duringtransmission The per-bit amplification energy is propor-tional to 1198894 when the transmission distance exceeds thethreshold 1198890 (called crossover distance) and otherwise isproportional to 1198892 The parameters 120576friss amp and 120576two ray ampdenote transmitter amplification parameters for free-spaceand multipath fading models respectively The value of 1198890 isgiven by

1198890 = radic

120576friss amp

120576two ray amp (10)

The reception energy of the 119896-bit data message can beexpressed by

119864RX = 119896119864elec (11)

where 119864elec denotes the per-bit energy dissipation duringreception119864DA is the data aggregation energy expenditure and is set

as 119864DA = 5 njbit The values of other parameters are set to119864elec = 50 njbit 120576friss amp = 100 pjbitm

2 and 120576two ray amp =

00013 pjbitm4 respectively [51]For the simulation setup 100 nodes are randomly

deployed in a 100m times 100m area of the sensing field BSis placed at the center of the field The initial energy of allhomogeneous nodes is set to 1198640 = 1 J During this analysisthree parameters namely first node dead (FND) half nodesdead (HND) and last node dead (LND) are employed tooutline the network lifetime

8 Applied Computational Intelligence and Soft Computing

Table 4 Composite benchmark functions

Function Dim Range 119891min

11986524 (CF1)

10 [minus5 5] 0

1198911 1198912 = Rastriginrsquos function1198913 1198914 =Weierstrassrsquo function1198915 1198916 = Griewankrsquos function1198917 1198918 = Ackleyrsquos function1198919 11989110 = sphere function[1205901 1205902 1205903 12059010] = [1 1 1 1]

[1205821 1205822 1205823 12058210] = [15 15 505 505 5100 5100 532 532 5100 5100]

11986525 (CF2)

10 [minus5 5] 0

1198911 1198912 = Ackleyrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = sphere function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 2 15 15 1 1 15 15 2 2]

[1205821 1205822 1205823 12058210] = [2 lowast 532 532 2 lowast 1 1 2 lowast 5100 5100 2 lowast 10 10 2 lowast 560 560]

11986526 (CF3)

10 [minus5 5] 0

1198911 1198912 = expanded Schaffer Rosenbrockrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = expanded Griewankrsquos and Rosenbrockrsquos function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 1 1 1 1 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [5 lowast 5100 5100 5 lowast 1 1 5 lowast 1 1 5 lowast 10 10 5 lowast 5200 5200]

11986527 (CF4)

10 [minus5 5] 0

1198911 =Weierstrassrsquos function1198912 = expanded Schaffer Rosenbrockrsquos function1198913 = expanded Griewankrsquos and Rosenbrockrsquos function1198914 = Ackleyrsquos function1198915 = Rastriginrsquos function1198916 = Griewankrsquos function1198917 = expanded Schaffer Rosenbrockrsquos noncont function1198918 = Rastriginrsquos noncont function1198919 = elliptic function11989110 = sphere noise function[1205901 1205902 1205903 12059010] = [2 2 2 2 2 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [10 520 1 532 1 5100 550 1 5100 5100]

Table 5 Results of unimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198651 531119864 minus 11 976119864 minus 11 22546833 66683781 289119864 minus 16 333119864 minus 16 265119864 minus 155 118119864 minus 154 644E minus 205 000E + 001198652 567119864 minus 07 126119864 minus 06 28648141 94412388 592119864 minus 09 543119864 minus 09 723119864 minus 92 841119864 minus 92 334E minus 119 495E minus 1191198653 32288636 19324945 61977294 26054622 10408052 09448328 178119864 minus 42 414119864 minus 42 274E minus 52 119E minus 511198654 40759779 16786821 56421446 10918884 10803036 30246624 586119864 minus 37 257119864 minus 36 120E minus 51 425E minus 511198655 62716929 49490404 18305153 12467683 22524975 15539303 26799858 05767914 26900044 08521161198656 289119864 minus 11 427119864 minus 11 19775038 73616353 499E minus 16 130E minus 15 07070627 02983174 07862954 024492871198657 00542038 00175609 50503673 16183878 00466985 00214709 00003805 00002072 00002609 0000176

Applied Computational Intelligence and Soft Computing 9

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

10minus150

10minus200

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

108

106

104

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

10minus10

10minus15

100

F1 F2

F3 F4

F5 F6

Figure 2 Continued

10 Applied Computational Intelligence and Soft Computing

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus2

100

102

F7

Figure 2 Convergence graph of unimodal benchmark functions

Table 6 Results of multimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198658 mdash mdash mdash mdash minus99587573 38210737 minus56976258 10363742 minus5712462 924473821198659 65433009 14197664 96065525 32511237 33785055 69716023 0 0 0 011986510 10769295 91708499 17231924 11118681 09966469 04914999 888119864 minus 15 227119864 minus 15 782E minus 15 794E minus 1611986511 00173265 00205398 23738699 81345844 00008613 00038519 00025395 00066692 0 011986512 03185579 05824145 18644754 18385611 01635701 04113459 00478581 00208711 00469178 0021678911986513 0020326 00488385 76481054 53670684 03009574 11566959 06303231 01787968 06051939 01740609

Table 7 Results of fixed-dimension multimodal functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986514 09980038 0 97544907 70552637 09980038 0 41298947 42619933 48165637 4309756211986515 00011042 00003066 00032289 0005698 00003075 111E minus 19 00044241 00081813 0007327 0009814511986516 minus10316285 228119864 minus 16 minus09908202 01825 minus10316285 228E minus 16 minus10316285 175119864 minus 09 minus10316284 486119864 minus 09

11986517 03978874 0 03978874 134E minus 10 03978874 0 03978978 463119864 minus 05 03978876 508119864 minus 07

11986518 3 815E minus 16 1245 20119315 3 139119864 minus 15 70500022 1811215 30000017 287119864 minus 06

11986519 minus38627821 224E minus 15 minus38627821 172119864 minus 08 minus38627821 228119864 minus 15 minus3862402 00016777 minus38611515 00031911986520 minus32625486 00609909 minus32566017 00606853 minus33219952 519E minus 16 minus31770991 026433 minus32734635 0060991711986521 minus65146454 31931088 minus57607751 31257537 minus101532 365E minus 15 minus96433311 1569109 minus92742029 2193450111986522 minus79202597 34833672 minus52362723 27999111 minus10402941 195E minus 15 minus10402879 466119864 minus 05 minus10136952 1188478111986523 minus78374538 34395208 minus59516938 38940879 minus1053641 195E minus 15 minus9724888 24976263 minus10536237 00001251

Table 9 shows the best results obtained for theCH selection problem in WSN The results of Table 9show that mGWO algorithm is able to find the bestresults compared to other algorithms The results ofmGWO are closely followed by the CS and GWOalgorithms

6 Conclusion

This paper proposed a modification to the Grey Wolf Opti-mizer named mGWO inspired by the hunting behavior ofgrey wolves in nature An exponential decay function is usedto balance the exploration and exploitation in the search

Applied Computational Intelligence and Soft Computing 11

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

Best

scor

e obt

aine

d so

far

0 1000 2000 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus10100

minus10200

minus10300

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

10minus15

102

104

106

108

100

108

106

104

102

100

F8 F9

F10 F11

F12 F13

Figure 3 Convergence graph of multimodal benchmark functions

12 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

053

minus10055

minus10057

minus10051

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus1002

minus1003

minus1004

minus1005

minus1001

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus1

10minus2

10minus3

F14 F15

F17 F18

F19 F20

Figure 4 Continued

Applied Computational Intelligence and Soft Computing 13

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

0

minus101

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus101

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

minus100

minus101

F21 F22

F23

Figure 4 Convergence graph of fixed-dimension multimodal functions

Table 8 Results of composite functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986524 3751142 1491547 774141 2693536 4136706 17765048 3379168 1733873 3718084 194095411986525 1670097 3186513 4907166 1803477 132158 245817 1339568 2317741 1323655 258643111986526 9556119 2808871 1476118 1179418 9150283 2623901 9437718 2779337 9119168 319887611986527 4376602 1831154 1328464 1770158 395 26708204 5593846 301082 390904 279606

14 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

1028

1026

1024

1022

F24 F25

F26 F27

Figure 5 Convergence graph of composite functions

Table 9 Comparison results of CH selection problem in WSN

Algorithm Optimalcost of FF FND HND LND

GA 39798mJ 20471 26782 33375PSO 39854mJ 20348 26652 33246BA 40324mJ 19967 26548 32989CS 38412mJ 20792 27025 33724GWO 38560mJ 20739 26963 33698mGWO 38209mJ 20846 27144 33822

space over the course of iterations The results proved thatthe proposed algorithm benefits from high exploration incomparison to the standard GWO

The paper also considered the clustering problem inWSNin which the CH selection is performed using the proposedmGWO algorithm which is a challenging and NP hardproblemThe results show that the proposedmethod is foundto be very effective for real-world applications due to fastconvergence and fewer chances to get stuck at local minimaIt can be concluded that the proposed algorithm is able tooutperform the current well-known and powerful algorithmsin the literature The results prove the competence andsuperiority of mGWO to existing metaheuristic algorithmsand it has an ability to become an effective tool for solvingreal word optimization problems

Competing Interests

The authors declare that they have no competing interests

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

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Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Computational Intelligence and Neuroscience

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing 5

Initialize the search agent (grey wolf) population119883119894 (119894 = 1 2 119899)Initialize 119886 119860 and 119862Calculate the fitness of each search agent119883120572 = the best (or dominating) search agent119883120573 = the second best search agent119883120575 = the third best search agentwhile (119905 ltMaximum number of iterations)

for each search agentupdate the position of the current search agent by (5)

end forupdate 119886 by (7)update 119860 and 119862 by (2)calculate the fitness of all search agentsupdate119883120572 119883120573 and119883120575119905 = 119905 + 1

end whileReturn119883120572

Algorithm 1 Pseudocode of mGWO algorithm

Table 1 Unimodal benchmark functions

Function Dim Range 119891min

1198651 (119909) =

119899

sum

119894=1

1199092

11989430 [minus100 100] 0

1198652 (119909) =

119899

sum

119894=1

|119909119894| +

119899

prod

119894=1

|119909119894| 30 [minus10 10] 0

1198653 (119909) =

119899

sum

119894=1

(

119894

sum

119895minus1

119909119895)

2

30 [minus100 100] 0

1198654 (119909) = max1198941003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119899 30 [minus100 100] 0

1198655 (119909) =

119899minus1

sum

119894=1

[100 (119909119894+1 minus 1199092

119894)

2

+ (119909119894 minus 1)2] 30 [minus30 30] 0

1198656 (119909) =

119899

sum

119894=1

([119909119894 + 05])2 30 [minus100 100] 0

1198657 (119909) =

119899

sum

119894=1

1198941199094

119894+ random (0 1) 30 [minus128 128] 0

mGWO algorithm provides the best results in 5 out of 7unimodal benchmark test functions The mGWO algorithmalso provides very competitive results compared to CS on 1198655and 1198656 As discussed above unimodal functions are suitablefor benchmarking exploitation of the algorithms Thereforethese results evidence high exploitation capability of themGWO algorithm

The statistical results of the algorithms on multimodaltest function are presented in Table 6 It may be seen thatmGWO algorithm highly outperforms other algorithms on11986591198651011986511 and11986512 It should be noted thatmGWOalgorithmoutperforms other algorithms on these multimodal testfunctions except PSO for 11986513 The results of multimodal testfunction strongly prove that high exploration of mGWOalgorithm is a suitable mechanism for avoiding local solu-tions Since the multimodal functions have an exponentialnumber of local solutions the results show that mGWOalgorithm is able to explore the search space extensively and

find promising regions of the search space In addition highlocal optima avoidance of this algorithm is another findingthat can be inferred from these results

The rest of the results which belong to 11986514ndash11986523 and11986524ndash11986527 are provided in Tables 7 and 8 respectively Theresults are consistent with those of other test functions inwhich mGWO shows very competitive results compared toother algorithms

5 Cluster Head Selection inWSN Using mGWO

Cluster head (CH) selection problem is a well-known prob-lem in the field of wireless sensor networks (WSNs) inwhich the energy consumption cost of the network shouldbe minimized [49ndash53] In this paper this problem is solvedusing mGWO algorithm and compared with GA PSO BACS and GWO

The main challenges in designing and planning theoperations ofWSNs are to optimize energy consumption andprolong network lifetime Cluster-based routing techniquessuch as the well-known low-energy adaptive clustering hier-archy (LEACH) [50] are used to achieve scalable solutionsand extend the network lifetime until the last node dies(LND) In order to achieve prolonged network lifetime incluster-based routing techniques the lifetime of the CHsplays an important role Improper cluster formation maycause some CHs to be overloaded Such overload maycause high energy consumption of the CH and degradethe overall performance of the WSN Therefore proper CHselection is the most important issue for clustering sensornodes Designing an energy efficient clustering algorithmis not an easy task Therefore nature-inspired optimizationalgorithms may be applied to tackle cluster-based routingproblem in WSN Evolutionary algorithms (EAs) have beenused in recent years as metaheuristics to address energy-aware routing challenges by designing intelligent models

6 Applied Computational Intelligence and Soft Computing

Table2Multim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891min

1198658(119909)=

119899

sum 119894=1

minus119909119894sin(radic1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816)

30[minus500500]

minus4189829times5

1198659(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

30[minus512512]

0

11986510(119909)=minus20exp(minus02radic

1 119899

119899

sum 119894=1

1199092 119894)minusexp(

1 119899

119899

sum 119894=1

cos(2120587119909119894))+20+119890

30[minus3232]

0

11986511(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos(119909119894

radic119894

)+1

30[minus600600]

0

11986512(119909)=

120587 119899

10sin(1205871199101)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+

119909119894+1

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)119898

119909119894gt119886

0minus119886lt119909119894lt119886

119896(minus119909119894minus119886)119898119909119894ltminus119886

30[minus5050]

0

11986513(119909)=01sin2(31205871199091)+

119899

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

30[minus5050]

0

Applied Computational Intelligence and Soft Computing 7

Table 3 Fixed-dimension multimodal benchmark functions

Function Dim Range 119891min

11986514 (119909) = (

1

500

+

25

sum

119895=1

1

119895 + sum2

119894=1(119909119894 minus 119886119894119895)

6)

minus1

2 [minus65 65] 1

11986515 (119909) =

11

sum

119894=1

[119886119894 minus

1199091 (1198872

119894+ 1198871198941199092)

1198872119894+ 1198871198941199093 + 1199094

]

2

4 [minus5 5] 000030

11986516 (119909) = 41199092

1minus 21119909

4

1+

1

3

1199096

1+ 11990911199092 minus 4119909

2

2+ 41199094

2 2 [minus5 5] minus10316

11986517 (119909) = (1199092 minus

51

412058721199092

1+

5

120587

1199091 minus 6)

2

+ 10 (1 minus

1

8120587

) cos1199091 + 10 2 [minus5 5] 0398

11986518 (119909) = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909

2

1minus 141199092 + 611990911199092 + 3119909

2

2)]

sdot [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909

2

1+ 481199092 minus 3611990911199092 + 27119909

2

2)]

2 [minus2 2] 3

11986519 (119909) = minus

4

sum

119894=1

119888119894 exp(minus3

sum

119895=1

119886119894119895 (119909119895 minus 119901119894119895)

2

) 3 [1 3] minus386

11986520 (119909) = minus

4

sum

119894=1

119888119894 exp(minus6

sum

119895=1

119886119894119895 (119909119895 minus 119901119894119895)

2

) 6 [0 1] minus332

11986521 (119909) = minus

5

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus101532

11986522 (119909) = minus

7

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus104028

11986523 (119909) = minus

10

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus105363

that collaborate together to optimize an appropriate energy-aware objective function [52] GWO is one of the powerfulheuristics that can be applied for efficient load balancedclustering In this paper mGWO based clustering algorithmis used to solve the abovementioned load balancing problemThe algorithm forms clusters in such a way that the overallenergy consumption of the network is minimized Totalenergy consumption in the network is the sum of the totalenergy dissipated from the non-CHs to send information totheir respective CHs and the total energy consumed by CHnodes to aggregate the information and send it to the basestation (BS)

Consider a WSN of 119899 sensor nodes randomly deployedin the sensing field and organized into 119870 clusters1198621 1198622 119862119870 The fitness function for the energy consump-tion may be defined as

119891 = (

119870

sum

119894=1

sum

119904isin119862119894

119864TX119904CH119894+ 119864RX + 119864DA) +

119870

sum

119894=1

119864TXCH119894 BS (8)

where 119870 is the total number of CHs 119904 isin 119862119894 is a non-CHassociated with the 119894th CH and 119864TXnode1node2

is the energydissipated for transmitting data from node1 to node2

In order to calculate radio energy transmission andreception costs a 119896-bit message and also the transmitter-receiver separation distance 119889 are given by

119864TX =

119896119864elec + 119896120576friss amp1198892 if 119889 lt 1198890

119896119864elec + 119896120576two ray amp1198894 if 119889 ge 1198890

(9)

The term 119864elec denotes the per-bit energy dissipation duringtransmission The per-bit amplification energy is propor-tional to 1198894 when the transmission distance exceeds thethreshold 1198890 (called crossover distance) and otherwise isproportional to 1198892 The parameters 120576friss amp and 120576two ray ampdenote transmitter amplification parameters for free-spaceand multipath fading models respectively The value of 1198890 isgiven by

1198890 = radic

120576friss amp

120576two ray amp (10)

The reception energy of the 119896-bit data message can beexpressed by

119864RX = 119896119864elec (11)

where 119864elec denotes the per-bit energy dissipation duringreception119864DA is the data aggregation energy expenditure and is set

as 119864DA = 5 njbit The values of other parameters are set to119864elec = 50 njbit 120576friss amp = 100 pjbitm

2 and 120576two ray amp =

00013 pjbitm4 respectively [51]For the simulation setup 100 nodes are randomly

deployed in a 100m times 100m area of the sensing field BSis placed at the center of the field The initial energy of allhomogeneous nodes is set to 1198640 = 1 J During this analysisthree parameters namely first node dead (FND) half nodesdead (HND) and last node dead (LND) are employed tooutline the network lifetime

8 Applied Computational Intelligence and Soft Computing

Table 4 Composite benchmark functions

Function Dim Range 119891min

11986524 (CF1)

10 [minus5 5] 0

1198911 1198912 = Rastriginrsquos function1198913 1198914 =Weierstrassrsquo function1198915 1198916 = Griewankrsquos function1198917 1198918 = Ackleyrsquos function1198919 11989110 = sphere function[1205901 1205902 1205903 12059010] = [1 1 1 1]

[1205821 1205822 1205823 12058210] = [15 15 505 505 5100 5100 532 532 5100 5100]

11986525 (CF2)

10 [minus5 5] 0

1198911 1198912 = Ackleyrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = sphere function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 2 15 15 1 1 15 15 2 2]

[1205821 1205822 1205823 12058210] = [2 lowast 532 532 2 lowast 1 1 2 lowast 5100 5100 2 lowast 10 10 2 lowast 560 560]

11986526 (CF3)

10 [minus5 5] 0

1198911 1198912 = expanded Schaffer Rosenbrockrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = expanded Griewankrsquos and Rosenbrockrsquos function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 1 1 1 1 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [5 lowast 5100 5100 5 lowast 1 1 5 lowast 1 1 5 lowast 10 10 5 lowast 5200 5200]

11986527 (CF4)

10 [minus5 5] 0

1198911 =Weierstrassrsquos function1198912 = expanded Schaffer Rosenbrockrsquos function1198913 = expanded Griewankrsquos and Rosenbrockrsquos function1198914 = Ackleyrsquos function1198915 = Rastriginrsquos function1198916 = Griewankrsquos function1198917 = expanded Schaffer Rosenbrockrsquos noncont function1198918 = Rastriginrsquos noncont function1198919 = elliptic function11989110 = sphere noise function[1205901 1205902 1205903 12059010] = [2 2 2 2 2 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [10 520 1 532 1 5100 550 1 5100 5100]

Table 5 Results of unimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198651 531119864 minus 11 976119864 minus 11 22546833 66683781 289119864 minus 16 333119864 minus 16 265119864 minus 155 118119864 minus 154 644E minus 205 000E + 001198652 567119864 minus 07 126119864 minus 06 28648141 94412388 592119864 minus 09 543119864 minus 09 723119864 minus 92 841119864 minus 92 334E minus 119 495E minus 1191198653 32288636 19324945 61977294 26054622 10408052 09448328 178119864 minus 42 414119864 minus 42 274E minus 52 119E minus 511198654 40759779 16786821 56421446 10918884 10803036 30246624 586119864 minus 37 257119864 minus 36 120E minus 51 425E minus 511198655 62716929 49490404 18305153 12467683 22524975 15539303 26799858 05767914 26900044 08521161198656 289119864 minus 11 427119864 minus 11 19775038 73616353 499E minus 16 130E minus 15 07070627 02983174 07862954 024492871198657 00542038 00175609 50503673 16183878 00466985 00214709 00003805 00002072 00002609 0000176

Applied Computational Intelligence and Soft Computing 9

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

10minus150

10minus200

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

108

106

104

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

10minus10

10minus15

100

F1 F2

F3 F4

F5 F6

Figure 2 Continued

10 Applied Computational Intelligence and Soft Computing

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus2

100

102

F7

Figure 2 Convergence graph of unimodal benchmark functions

Table 6 Results of multimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198658 mdash mdash mdash mdash minus99587573 38210737 minus56976258 10363742 minus5712462 924473821198659 65433009 14197664 96065525 32511237 33785055 69716023 0 0 0 011986510 10769295 91708499 17231924 11118681 09966469 04914999 888119864 minus 15 227119864 minus 15 782E minus 15 794E minus 1611986511 00173265 00205398 23738699 81345844 00008613 00038519 00025395 00066692 0 011986512 03185579 05824145 18644754 18385611 01635701 04113459 00478581 00208711 00469178 0021678911986513 0020326 00488385 76481054 53670684 03009574 11566959 06303231 01787968 06051939 01740609

Table 7 Results of fixed-dimension multimodal functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986514 09980038 0 97544907 70552637 09980038 0 41298947 42619933 48165637 4309756211986515 00011042 00003066 00032289 0005698 00003075 111E minus 19 00044241 00081813 0007327 0009814511986516 minus10316285 228119864 minus 16 minus09908202 01825 minus10316285 228E minus 16 minus10316285 175119864 minus 09 minus10316284 486119864 minus 09

11986517 03978874 0 03978874 134E minus 10 03978874 0 03978978 463119864 minus 05 03978876 508119864 minus 07

11986518 3 815E minus 16 1245 20119315 3 139119864 minus 15 70500022 1811215 30000017 287119864 minus 06

11986519 minus38627821 224E minus 15 minus38627821 172119864 minus 08 minus38627821 228119864 minus 15 minus3862402 00016777 minus38611515 00031911986520 minus32625486 00609909 minus32566017 00606853 minus33219952 519E minus 16 minus31770991 026433 minus32734635 0060991711986521 minus65146454 31931088 minus57607751 31257537 minus101532 365E minus 15 minus96433311 1569109 minus92742029 2193450111986522 minus79202597 34833672 minus52362723 27999111 minus10402941 195E minus 15 minus10402879 466119864 minus 05 minus10136952 1188478111986523 minus78374538 34395208 minus59516938 38940879 minus1053641 195E minus 15 minus9724888 24976263 minus10536237 00001251

Table 9 shows the best results obtained for theCH selection problem in WSN The results of Table 9show that mGWO algorithm is able to find the bestresults compared to other algorithms The results ofmGWO are closely followed by the CS and GWOalgorithms

6 Conclusion

This paper proposed a modification to the Grey Wolf Opti-mizer named mGWO inspired by the hunting behavior ofgrey wolves in nature An exponential decay function is usedto balance the exploration and exploitation in the search

Applied Computational Intelligence and Soft Computing 11

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

Best

scor

e obt

aine

d so

far

0 1000 2000 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus10100

minus10200

minus10300

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

10minus15

102

104

106

108

100

108

106

104

102

100

F8 F9

F10 F11

F12 F13

Figure 3 Convergence graph of multimodal benchmark functions

12 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

053

minus10055

minus10057

minus10051

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus1002

minus1003

minus1004

minus1005

minus1001

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus1

10minus2

10minus3

F14 F15

F17 F18

F19 F20

Figure 4 Continued

Applied Computational Intelligence and Soft Computing 13

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

0

minus101

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus101

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

minus100

minus101

F21 F22

F23

Figure 4 Convergence graph of fixed-dimension multimodal functions

Table 8 Results of composite functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986524 3751142 1491547 774141 2693536 4136706 17765048 3379168 1733873 3718084 194095411986525 1670097 3186513 4907166 1803477 132158 245817 1339568 2317741 1323655 258643111986526 9556119 2808871 1476118 1179418 9150283 2623901 9437718 2779337 9119168 319887611986527 4376602 1831154 1328464 1770158 395 26708204 5593846 301082 390904 279606

14 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

1028

1026

1024

1022

F24 F25

F26 F27

Figure 5 Convergence graph of composite functions

Table 9 Comparison results of CH selection problem in WSN

Algorithm Optimalcost of FF FND HND LND

GA 39798mJ 20471 26782 33375PSO 39854mJ 20348 26652 33246BA 40324mJ 19967 26548 32989CS 38412mJ 20792 27025 33724GWO 38560mJ 20739 26963 33698mGWO 38209mJ 20846 27144 33822

space over the course of iterations The results proved thatthe proposed algorithm benefits from high exploration incomparison to the standard GWO

The paper also considered the clustering problem inWSNin which the CH selection is performed using the proposedmGWO algorithm which is a challenging and NP hardproblemThe results show that the proposedmethod is foundto be very effective for real-world applications due to fastconvergence and fewer chances to get stuck at local minimaIt can be concluded that the proposed algorithm is able tooutperform the current well-known and powerful algorithmsin the literature The results prove the competence andsuperiority of mGWO to existing metaheuristic algorithmsand it has an ability to become an effective tool for solvingreal word optimization problems

Competing Interests

The authors declare that they have no competing interests

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Applied Computational Intelligence and Soft Computing

Table2Multim

odalbenchm

arkfunctio

ns

Functio

nDim

Range

119891min

1198658(119909)=

119899

sum 119894=1

minus119909119894sin(radic1003816 1003816 1003816 10038161199091198941003816 1003816 1003816 1003816)

30[minus500500]

minus4189829times5

1198659(119909)=

119899

sum 119894=1

[1199092 119894minus10cos(2120587119909119894)+10]

30[minus512512]

0

11986510(119909)=minus20exp(minus02radic

1 119899

119899

sum 119894=1

1199092 119894)minusexp(

1 119899

119899

sum 119894=1

cos(2120587119909119894))+20+119890

30[minus3232]

0

11986511(119909)=

1

4000

119899

sum 119894=1

1199092 119894minus

119899

prod 119894=1

cos(119909119894

radic119894

)+1

30[minus600600]

0

11986512(119909)=

120587 119899

10sin(1205871199101)+

119899minus1

sum 119894=1

(119910119894minus1)2[1+10sin2(120587119910119894+1)]+(119910119899minus1)2+

119899

sum 119894=1

119906(119909119894101004)

119910119894=1+

119909119894+1

4

119906(119909119894119886119896119898)=

119896(119909119894minus119886)119898

119909119894gt119886

0minus119886lt119909119894lt119886

119896(minus119909119894minus119886)119898119909119894ltminus119886

30[minus5050]

0

11986513(119909)=01sin2(31205871199091)+

119899

sum 119894=1

(119909119894minus1)2[1+sin2(3120587119909119894+1)]+(119909119899minus1)2[1+sin2(2120587119909119899)]+

119899

sum 119894=1

119906(11990911989451004)

30[minus5050]

0

Applied Computational Intelligence and Soft Computing 7

Table 3 Fixed-dimension multimodal benchmark functions

Function Dim Range 119891min

11986514 (119909) = (

1

500

+

25

sum

119895=1

1

119895 + sum2

119894=1(119909119894 minus 119886119894119895)

6)

minus1

2 [minus65 65] 1

11986515 (119909) =

11

sum

119894=1

[119886119894 minus

1199091 (1198872

119894+ 1198871198941199092)

1198872119894+ 1198871198941199093 + 1199094

]

2

4 [minus5 5] 000030

11986516 (119909) = 41199092

1minus 21119909

4

1+

1

3

1199096

1+ 11990911199092 minus 4119909

2

2+ 41199094

2 2 [minus5 5] minus10316

11986517 (119909) = (1199092 minus

51

412058721199092

1+

5

120587

1199091 minus 6)

2

+ 10 (1 minus

1

8120587

) cos1199091 + 10 2 [minus5 5] 0398

11986518 (119909) = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909

2

1minus 141199092 + 611990911199092 + 3119909

2

2)]

sdot [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909

2

1+ 481199092 minus 3611990911199092 + 27119909

2

2)]

2 [minus2 2] 3

11986519 (119909) = minus

4

sum

119894=1

119888119894 exp(minus3

sum

119895=1

119886119894119895 (119909119895 minus 119901119894119895)

2

) 3 [1 3] minus386

11986520 (119909) = minus

4

sum

119894=1

119888119894 exp(minus6

sum

119895=1

119886119894119895 (119909119895 minus 119901119894119895)

2

) 6 [0 1] minus332

11986521 (119909) = minus

5

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus101532

11986522 (119909) = minus

7

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus104028

11986523 (119909) = minus

10

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus105363

that collaborate together to optimize an appropriate energy-aware objective function [52] GWO is one of the powerfulheuristics that can be applied for efficient load balancedclustering In this paper mGWO based clustering algorithmis used to solve the abovementioned load balancing problemThe algorithm forms clusters in such a way that the overallenergy consumption of the network is minimized Totalenergy consumption in the network is the sum of the totalenergy dissipated from the non-CHs to send information totheir respective CHs and the total energy consumed by CHnodes to aggregate the information and send it to the basestation (BS)

Consider a WSN of 119899 sensor nodes randomly deployedin the sensing field and organized into 119870 clusters1198621 1198622 119862119870 The fitness function for the energy consump-tion may be defined as

119891 = (

119870

sum

119894=1

sum

119904isin119862119894

119864TX119904CH119894+ 119864RX + 119864DA) +

119870

sum

119894=1

119864TXCH119894 BS (8)

where 119870 is the total number of CHs 119904 isin 119862119894 is a non-CHassociated with the 119894th CH and 119864TXnode1node2

is the energydissipated for transmitting data from node1 to node2

In order to calculate radio energy transmission andreception costs a 119896-bit message and also the transmitter-receiver separation distance 119889 are given by

119864TX =

119896119864elec + 119896120576friss amp1198892 if 119889 lt 1198890

119896119864elec + 119896120576two ray amp1198894 if 119889 ge 1198890

(9)

The term 119864elec denotes the per-bit energy dissipation duringtransmission The per-bit amplification energy is propor-tional to 1198894 when the transmission distance exceeds thethreshold 1198890 (called crossover distance) and otherwise isproportional to 1198892 The parameters 120576friss amp and 120576two ray ampdenote transmitter amplification parameters for free-spaceand multipath fading models respectively The value of 1198890 isgiven by

1198890 = radic

120576friss amp

120576two ray amp (10)

The reception energy of the 119896-bit data message can beexpressed by

119864RX = 119896119864elec (11)

where 119864elec denotes the per-bit energy dissipation duringreception119864DA is the data aggregation energy expenditure and is set

as 119864DA = 5 njbit The values of other parameters are set to119864elec = 50 njbit 120576friss amp = 100 pjbitm

2 and 120576two ray amp =

00013 pjbitm4 respectively [51]For the simulation setup 100 nodes are randomly

deployed in a 100m times 100m area of the sensing field BSis placed at the center of the field The initial energy of allhomogeneous nodes is set to 1198640 = 1 J During this analysisthree parameters namely first node dead (FND) half nodesdead (HND) and last node dead (LND) are employed tooutline the network lifetime

8 Applied Computational Intelligence and Soft Computing

Table 4 Composite benchmark functions

Function Dim Range 119891min

11986524 (CF1)

10 [minus5 5] 0

1198911 1198912 = Rastriginrsquos function1198913 1198914 =Weierstrassrsquo function1198915 1198916 = Griewankrsquos function1198917 1198918 = Ackleyrsquos function1198919 11989110 = sphere function[1205901 1205902 1205903 12059010] = [1 1 1 1]

[1205821 1205822 1205823 12058210] = [15 15 505 505 5100 5100 532 532 5100 5100]

11986525 (CF2)

10 [minus5 5] 0

1198911 1198912 = Ackleyrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = sphere function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 2 15 15 1 1 15 15 2 2]

[1205821 1205822 1205823 12058210] = [2 lowast 532 532 2 lowast 1 1 2 lowast 5100 5100 2 lowast 10 10 2 lowast 560 560]

11986526 (CF3)

10 [minus5 5] 0

1198911 1198912 = expanded Schaffer Rosenbrockrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = expanded Griewankrsquos and Rosenbrockrsquos function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 1 1 1 1 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [5 lowast 5100 5100 5 lowast 1 1 5 lowast 1 1 5 lowast 10 10 5 lowast 5200 5200]

11986527 (CF4)

10 [minus5 5] 0

1198911 =Weierstrassrsquos function1198912 = expanded Schaffer Rosenbrockrsquos function1198913 = expanded Griewankrsquos and Rosenbrockrsquos function1198914 = Ackleyrsquos function1198915 = Rastriginrsquos function1198916 = Griewankrsquos function1198917 = expanded Schaffer Rosenbrockrsquos noncont function1198918 = Rastriginrsquos noncont function1198919 = elliptic function11989110 = sphere noise function[1205901 1205902 1205903 12059010] = [2 2 2 2 2 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [10 520 1 532 1 5100 550 1 5100 5100]

Table 5 Results of unimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198651 531119864 minus 11 976119864 minus 11 22546833 66683781 289119864 minus 16 333119864 minus 16 265119864 minus 155 118119864 minus 154 644E minus 205 000E + 001198652 567119864 minus 07 126119864 minus 06 28648141 94412388 592119864 minus 09 543119864 minus 09 723119864 minus 92 841119864 minus 92 334E minus 119 495E minus 1191198653 32288636 19324945 61977294 26054622 10408052 09448328 178119864 minus 42 414119864 minus 42 274E minus 52 119E minus 511198654 40759779 16786821 56421446 10918884 10803036 30246624 586119864 minus 37 257119864 minus 36 120E minus 51 425E minus 511198655 62716929 49490404 18305153 12467683 22524975 15539303 26799858 05767914 26900044 08521161198656 289119864 minus 11 427119864 minus 11 19775038 73616353 499E minus 16 130E minus 15 07070627 02983174 07862954 024492871198657 00542038 00175609 50503673 16183878 00466985 00214709 00003805 00002072 00002609 0000176

Applied Computational Intelligence and Soft Computing 9

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

10minus150

10minus200

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

108

106

104

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

10minus10

10minus15

100

F1 F2

F3 F4

F5 F6

Figure 2 Continued

10 Applied Computational Intelligence and Soft Computing

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus2

100

102

F7

Figure 2 Convergence graph of unimodal benchmark functions

Table 6 Results of multimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198658 mdash mdash mdash mdash minus99587573 38210737 minus56976258 10363742 minus5712462 924473821198659 65433009 14197664 96065525 32511237 33785055 69716023 0 0 0 011986510 10769295 91708499 17231924 11118681 09966469 04914999 888119864 minus 15 227119864 minus 15 782E minus 15 794E minus 1611986511 00173265 00205398 23738699 81345844 00008613 00038519 00025395 00066692 0 011986512 03185579 05824145 18644754 18385611 01635701 04113459 00478581 00208711 00469178 0021678911986513 0020326 00488385 76481054 53670684 03009574 11566959 06303231 01787968 06051939 01740609

Table 7 Results of fixed-dimension multimodal functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986514 09980038 0 97544907 70552637 09980038 0 41298947 42619933 48165637 4309756211986515 00011042 00003066 00032289 0005698 00003075 111E minus 19 00044241 00081813 0007327 0009814511986516 minus10316285 228119864 minus 16 minus09908202 01825 minus10316285 228E minus 16 minus10316285 175119864 minus 09 minus10316284 486119864 minus 09

11986517 03978874 0 03978874 134E minus 10 03978874 0 03978978 463119864 minus 05 03978876 508119864 minus 07

11986518 3 815E minus 16 1245 20119315 3 139119864 minus 15 70500022 1811215 30000017 287119864 minus 06

11986519 minus38627821 224E minus 15 minus38627821 172119864 minus 08 minus38627821 228119864 minus 15 minus3862402 00016777 minus38611515 00031911986520 minus32625486 00609909 minus32566017 00606853 minus33219952 519E minus 16 minus31770991 026433 minus32734635 0060991711986521 minus65146454 31931088 minus57607751 31257537 minus101532 365E minus 15 minus96433311 1569109 minus92742029 2193450111986522 minus79202597 34833672 minus52362723 27999111 minus10402941 195E minus 15 minus10402879 466119864 minus 05 minus10136952 1188478111986523 minus78374538 34395208 minus59516938 38940879 minus1053641 195E minus 15 minus9724888 24976263 minus10536237 00001251

Table 9 shows the best results obtained for theCH selection problem in WSN The results of Table 9show that mGWO algorithm is able to find the bestresults compared to other algorithms The results ofmGWO are closely followed by the CS and GWOalgorithms

6 Conclusion

This paper proposed a modification to the Grey Wolf Opti-mizer named mGWO inspired by the hunting behavior ofgrey wolves in nature An exponential decay function is usedto balance the exploration and exploitation in the search

Applied Computational Intelligence and Soft Computing 11

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

Best

scor

e obt

aine

d so

far

0 1000 2000 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus10100

minus10200

minus10300

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

10minus15

102

104

106

108

100

108

106

104

102

100

F8 F9

F10 F11

F12 F13

Figure 3 Convergence graph of multimodal benchmark functions

12 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

053

minus10055

minus10057

minus10051

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus1002

minus1003

minus1004

minus1005

minus1001

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus1

10minus2

10minus3

F14 F15

F17 F18

F19 F20

Figure 4 Continued

Applied Computational Intelligence and Soft Computing 13

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

0

minus101

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus101

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

minus100

minus101

F21 F22

F23

Figure 4 Convergence graph of fixed-dimension multimodal functions

Table 8 Results of composite functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986524 3751142 1491547 774141 2693536 4136706 17765048 3379168 1733873 3718084 194095411986525 1670097 3186513 4907166 1803477 132158 245817 1339568 2317741 1323655 258643111986526 9556119 2808871 1476118 1179418 9150283 2623901 9437718 2779337 9119168 319887611986527 4376602 1831154 1328464 1770158 395 26708204 5593846 301082 390904 279606

14 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

1028

1026

1024

1022

F24 F25

F26 F27

Figure 5 Convergence graph of composite functions

Table 9 Comparison results of CH selection problem in WSN

Algorithm Optimalcost of FF FND HND LND

GA 39798mJ 20471 26782 33375PSO 39854mJ 20348 26652 33246BA 40324mJ 19967 26548 32989CS 38412mJ 20792 27025 33724GWO 38560mJ 20739 26963 33698mGWO 38209mJ 20846 27144 33822

space over the course of iterations The results proved thatthe proposed algorithm benefits from high exploration incomparison to the standard GWO

The paper also considered the clustering problem inWSNin which the CH selection is performed using the proposedmGWO algorithm which is a challenging and NP hardproblemThe results show that the proposedmethod is foundto be very effective for real-world applications due to fastconvergence and fewer chances to get stuck at local minimaIt can be concluded that the proposed algorithm is able tooutperform the current well-known and powerful algorithmsin the literature The results prove the competence andsuperiority of mGWO to existing metaheuristic algorithmsand it has an ability to become an effective tool for solvingreal word optimization problems

Competing Interests

The authors declare that they have no competing interests

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing 7

Table 3 Fixed-dimension multimodal benchmark functions

Function Dim Range 119891min

11986514 (119909) = (

1

500

+

25

sum

119895=1

1

119895 + sum2

119894=1(119909119894 minus 119886119894119895)

6)

minus1

2 [minus65 65] 1

11986515 (119909) =

11

sum

119894=1

[119886119894 minus

1199091 (1198872

119894+ 1198871198941199092)

1198872119894+ 1198871198941199093 + 1199094

]

2

4 [minus5 5] 000030

11986516 (119909) = 41199092

1minus 21119909

4

1+

1

3

1199096

1+ 11990911199092 minus 4119909

2

2+ 41199094

2 2 [minus5 5] minus10316

11986517 (119909) = (1199092 minus

51

412058721199092

1+

5

120587

1199091 minus 6)

2

+ 10 (1 minus

1

8120587

) cos1199091 + 10 2 [minus5 5] 0398

11986518 (119909) = [1 + (1199091 + 1199092 + 1)2(19 minus 141199091 + 3119909

2

1minus 141199092 + 611990911199092 + 3119909

2

2)]

sdot [30 + (21199091 minus 31199092)2(18 minus 321199091 + 12119909

2

1+ 481199092 minus 3611990911199092 + 27119909

2

2)]

2 [minus2 2] 3

11986519 (119909) = minus

4

sum

119894=1

119888119894 exp(minus3

sum

119895=1

119886119894119895 (119909119895 minus 119901119894119895)

2

) 3 [1 3] minus386

11986520 (119909) = minus

4

sum

119894=1

119888119894 exp(minus6

sum

119895=1

119886119894119895 (119909119895 minus 119901119894119895)

2

) 6 [0 1] minus332

11986521 (119909) = minus

5

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus101532

11986522 (119909) = minus

7

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus104028

11986523 (119909) = minus

10

sum

119894=1

[(119883 minus 119886119894) (119883 minus 119886119894)119879+ 119888119894]

minus1

4 [0 10] minus105363

that collaborate together to optimize an appropriate energy-aware objective function [52] GWO is one of the powerfulheuristics that can be applied for efficient load balancedclustering In this paper mGWO based clustering algorithmis used to solve the abovementioned load balancing problemThe algorithm forms clusters in such a way that the overallenergy consumption of the network is minimized Totalenergy consumption in the network is the sum of the totalenergy dissipated from the non-CHs to send information totheir respective CHs and the total energy consumed by CHnodes to aggregate the information and send it to the basestation (BS)

Consider a WSN of 119899 sensor nodes randomly deployedin the sensing field and organized into 119870 clusters1198621 1198622 119862119870 The fitness function for the energy consump-tion may be defined as

119891 = (

119870

sum

119894=1

sum

119904isin119862119894

119864TX119904CH119894+ 119864RX + 119864DA) +

119870

sum

119894=1

119864TXCH119894 BS (8)

where 119870 is the total number of CHs 119904 isin 119862119894 is a non-CHassociated with the 119894th CH and 119864TXnode1node2

is the energydissipated for transmitting data from node1 to node2

In order to calculate radio energy transmission andreception costs a 119896-bit message and also the transmitter-receiver separation distance 119889 are given by

119864TX =

119896119864elec + 119896120576friss amp1198892 if 119889 lt 1198890

119896119864elec + 119896120576two ray amp1198894 if 119889 ge 1198890

(9)

The term 119864elec denotes the per-bit energy dissipation duringtransmission The per-bit amplification energy is propor-tional to 1198894 when the transmission distance exceeds thethreshold 1198890 (called crossover distance) and otherwise isproportional to 1198892 The parameters 120576friss amp and 120576two ray ampdenote transmitter amplification parameters for free-spaceand multipath fading models respectively The value of 1198890 isgiven by

1198890 = radic

120576friss amp

120576two ray amp (10)

The reception energy of the 119896-bit data message can beexpressed by

119864RX = 119896119864elec (11)

where 119864elec denotes the per-bit energy dissipation duringreception119864DA is the data aggregation energy expenditure and is set

as 119864DA = 5 njbit The values of other parameters are set to119864elec = 50 njbit 120576friss amp = 100 pjbitm

2 and 120576two ray amp =

00013 pjbitm4 respectively [51]For the simulation setup 100 nodes are randomly

deployed in a 100m times 100m area of the sensing field BSis placed at the center of the field The initial energy of allhomogeneous nodes is set to 1198640 = 1 J During this analysisthree parameters namely first node dead (FND) half nodesdead (HND) and last node dead (LND) are employed tooutline the network lifetime

8 Applied Computational Intelligence and Soft Computing

Table 4 Composite benchmark functions

Function Dim Range 119891min

11986524 (CF1)

10 [minus5 5] 0

1198911 1198912 = Rastriginrsquos function1198913 1198914 =Weierstrassrsquo function1198915 1198916 = Griewankrsquos function1198917 1198918 = Ackleyrsquos function1198919 11989110 = sphere function[1205901 1205902 1205903 12059010] = [1 1 1 1]

[1205821 1205822 1205823 12058210] = [15 15 505 505 5100 5100 532 532 5100 5100]

11986525 (CF2)

10 [minus5 5] 0

1198911 1198912 = Ackleyrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = sphere function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 2 15 15 1 1 15 15 2 2]

[1205821 1205822 1205823 12058210] = [2 lowast 532 532 2 lowast 1 1 2 lowast 5100 5100 2 lowast 10 10 2 lowast 560 560]

11986526 (CF3)

10 [minus5 5] 0

1198911 1198912 = expanded Schaffer Rosenbrockrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = expanded Griewankrsquos and Rosenbrockrsquos function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 1 1 1 1 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [5 lowast 5100 5100 5 lowast 1 1 5 lowast 1 1 5 lowast 10 10 5 lowast 5200 5200]

11986527 (CF4)

10 [minus5 5] 0

1198911 =Weierstrassrsquos function1198912 = expanded Schaffer Rosenbrockrsquos function1198913 = expanded Griewankrsquos and Rosenbrockrsquos function1198914 = Ackleyrsquos function1198915 = Rastriginrsquos function1198916 = Griewankrsquos function1198917 = expanded Schaffer Rosenbrockrsquos noncont function1198918 = Rastriginrsquos noncont function1198919 = elliptic function11989110 = sphere noise function[1205901 1205902 1205903 12059010] = [2 2 2 2 2 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [10 520 1 532 1 5100 550 1 5100 5100]

Table 5 Results of unimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198651 531119864 minus 11 976119864 minus 11 22546833 66683781 289119864 minus 16 333119864 minus 16 265119864 minus 155 118119864 minus 154 644E minus 205 000E + 001198652 567119864 minus 07 126119864 minus 06 28648141 94412388 592119864 minus 09 543119864 minus 09 723119864 minus 92 841119864 minus 92 334E minus 119 495E minus 1191198653 32288636 19324945 61977294 26054622 10408052 09448328 178119864 minus 42 414119864 minus 42 274E minus 52 119E minus 511198654 40759779 16786821 56421446 10918884 10803036 30246624 586119864 minus 37 257119864 minus 36 120E minus 51 425E minus 511198655 62716929 49490404 18305153 12467683 22524975 15539303 26799858 05767914 26900044 08521161198656 289119864 minus 11 427119864 minus 11 19775038 73616353 499E minus 16 130E minus 15 07070627 02983174 07862954 024492871198657 00542038 00175609 50503673 16183878 00466985 00214709 00003805 00002072 00002609 0000176

Applied Computational Intelligence and Soft Computing 9

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

10minus150

10minus200

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

108

106

104

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

10minus10

10minus15

100

F1 F2

F3 F4

F5 F6

Figure 2 Continued

10 Applied Computational Intelligence and Soft Computing

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus2

100

102

F7

Figure 2 Convergence graph of unimodal benchmark functions

Table 6 Results of multimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198658 mdash mdash mdash mdash minus99587573 38210737 minus56976258 10363742 minus5712462 924473821198659 65433009 14197664 96065525 32511237 33785055 69716023 0 0 0 011986510 10769295 91708499 17231924 11118681 09966469 04914999 888119864 minus 15 227119864 minus 15 782E minus 15 794E minus 1611986511 00173265 00205398 23738699 81345844 00008613 00038519 00025395 00066692 0 011986512 03185579 05824145 18644754 18385611 01635701 04113459 00478581 00208711 00469178 0021678911986513 0020326 00488385 76481054 53670684 03009574 11566959 06303231 01787968 06051939 01740609

Table 7 Results of fixed-dimension multimodal functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986514 09980038 0 97544907 70552637 09980038 0 41298947 42619933 48165637 4309756211986515 00011042 00003066 00032289 0005698 00003075 111E minus 19 00044241 00081813 0007327 0009814511986516 minus10316285 228119864 minus 16 minus09908202 01825 minus10316285 228E minus 16 minus10316285 175119864 minus 09 minus10316284 486119864 minus 09

11986517 03978874 0 03978874 134E minus 10 03978874 0 03978978 463119864 minus 05 03978876 508119864 minus 07

11986518 3 815E minus 16 1245 20119315 3 139119864 minus 15 70500022 1811215 30000017 287119864 minus 06

11986519 minus38627821 224E minus 15 minus38627821 172119864 minus 08 minus38627821 228119864 minus 15 minus3862402 00016777 minus38611515 00031911986520 minus32625486 00609909 minus32566017 00606853 minus33219952 519E minus 16 minus31770991 026433 minus32734635 0060991711986521 minus65146454 31931088 minus57607751 31257537 minus101532 365E minus 15 minus96433311 1569109 minus92742029 2193450111986522 minus79202597 34833672 minus52362723 27999111 minus10402941 195E minus 15 minus10402879 466119864 minus 05 minus10136952 1188478111986523 minus78374538 34395208 minus59516938 38940879 minus1053641 195E minus 15 minus9724888 24976263 minus10536237 00001251

Table 9 shows the best results obtained for theCH selection problem in WSN The results of Table 9show that mGWO algorithm is able to find the bestresults compared to other algorithms The results ofmGWO are closely followed by the CS and GWOalgorithms

6 Conclusion

This paper proposed a modification to the Grey Wolf Opti-mizer named mGWO inspired by the hunting behavior ofgrey wolves in nature An exponential decay function is usedto balance the exploration and exploitation in the search

Applied Computational Intelligence and Soft Computing 11

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

Best

scor

e obt

aine

d so

far

0 1000 2000 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus10100

minus10200

minus10300

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

10minus15

102

104

106

108

100

108

106

104

102

100

F8 F9

F10 F11

F12 F13

Figure 3 Convergence graph of multimodal benchmark functions

12 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

053

minus10055

minus10057

minus10051

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus1002

minus1003

minus1004

minus1005

minus1001

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus1

10minus2

10minus3

F14 F15

F17 F18

F19 F20

Figure 4 Continued

Applied Computational Intelligence and Soft Computing 13

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

0

minus101

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus101

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

minus100

minus101

F21 F22

F23

Figure 4 Convergence graph of fixed-dimension multimodal functions

Table 8 Results of composite functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986524 3751142 1491547 774141 2693536 4136706 17765048 3379168 1733873 3718084 194095411986525 1670097 3186513 4907166 1803477 132158 245817 1339568 2317741 1323655 258643111986526 9556119 2808871 1476118 1179418 9150283 2623901 9437718 2779337 9119168 319887611986527 4376602 1831154 1328464 1770158 395 26708204 5593846 301082 390904 279606

14 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

1028

1026

1024

1022

F24 F25

F26 F27

Figure 5 Convergence graph of composite functions

Table 9 Comparison results of CH selection problem in WSN

Algorithm Optimalcost of FF FND HND LND

GA 39798mJ 20471 26782 33375PSO 39854mJ 20348 26652 33246BA 40324mJ 19967 26548 32989CS 38412mJ 20792 27025 33724GWO 38560mJ 20739 26963 33698mGWO 38209mJ 20846 27144 33822

space over the course of iterations The results proved thatthe proposed algorithm benefits from high exploration incomparison to the standard GWO

The paper also considered the clustering problem inWSNin which the CH selection is performed using the proposedmGWO algorithm which is a challenging and NP hardproblemThe results show that the proposedmethod is foundto be very effective for real-world applications due to fastconvergence and fewer chances to get stuck at local minimaIt can be concluded that the proposed algorithm is able tooutperform the current well-known and powerful algorithmsin the literature The results prove the competence andsuperiority of mGWO to existing metaheuristic algorithmsand it has an ability to become an effective tool for solvingreal word optimization problems

Competing Interests

The authors declare that they have no competing interests

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 Applied Computational Intelligence and Soft Computing

Table 4 Composite benchmark functions

Function Dim Range 119891min

11986524 (CF1)

10 [minus5 5] 0

1198911 1198912 = Rastriginrsquos function1198913 1198914 =Weierstrassrsquo function1198915 1198916 = Griewankrsquos function1198917 1198918 = Ackleyrsquos function1198919 11989110 = sphere function[1205901 1205902 1205903 12059010] = [1 1 1 1]

[1205821 1205822 1205823 12058210] = [15 15 505 505 5100 5100 532 532 5100 5100]

11986525 (CF2)

10 [minus5 5] 0

1198911 1198912 = Ackleyrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = sphere function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 2 15 15 1 1 15 15 2 2]

[1205821 1205822 1205823 12058210] = [2 lowast 532 532 2 lowast 1 1 2 lowast 5100 5100 2 lowast 10 10 2 lowast 560 560]

11986526 (CF3)

10 [minus5 5] 0

1198911 1198912 = expanded Schaffer Rosenbrockrsquos function1198913 1198914 = Rastriginrsquos function1198915 1198916 = expanded Griewankrsquos and Rosenbrockrsquos function1198917 1198918 =Weierstrassrsquos function1198919 11989110 = Griewankrsquos function[1205901 1205902 1205903 12059010] = [1 1 1 1 1 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [5 lowast 5100 5100 5 lowast 1 1 5 lowast 1 1 5 lowast 10 10 5 lowast 5200 5200]

11986527 (CF4)

10 [minus5 5] 0

1198911 =Weierstrassrsquos function1198912 = expanded Schaffer Rosenbrockrsquos function1198913 = expanded Griewankrsquos and Rosenbrockrsquos function1198914 = Ackleyrsquos function1198915 = Rastriginrsquos function1198916 = Griewankrsquos function1198917 = expanded Schaffer Rosenbrockrsquos noncont function1198918 = Rastriginrsquos noncont function1198919 = elliptic function11989110 = sphere noise function[1205901 1205902 1205903 12059010] = [2 2 2 2 2 2 2 2 2 2]

[1205821 1205822 1205823 12058210] = [10 520 1 532 1 5100 550 1 5100 5100]

Table 5 Results of unimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198651 531119864 minus 11 976119864 minus 11 22546833 66683781 289119864 minus 16 333119864 minus 16 265119864 minus 155 118119864 minus 154 644E minus 205 000E + 001198652 567119864 minus 07 126119864 minus 06 28648141 94412388 592119864 minus 09 543119864 minus 09 723119864 minus 92 841119864 minus 92 334E minus 119 495E minus 1191198653 32288636 19324945 61977294 26054622 10408052 09448328 178119864 minus 42 414119864 minus 42 274E minus 52 119E minus 511198654 40759779 16786821 56421446 10918884 10803036 30246624 586119864 minus 37 257119864 minus 36 120E minus 51 425E minus 511198655 62716929 49490404 18305153 12467683 22524975 15539303 26799858 05767914 26900044 08521161198656 289119864 minus 11 427119864 minus 11 19775038 73616353 499E minus 16 130E minus 15 07070627 02983174 07862954 024492871198657 00542038 00175609 50503673 16183878 00466985 00214709 00003805 00002072 00002609 0000176

Applied Computational Intelligence and Soft Computing 9

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

10minus150

10minus200

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

108

106

104

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

10minus10

10minus15

100

F1 F2

F3 F4

F5 F6

Figure 2 Continued

10 Applied Computational Intelligence and Soft Computing

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus2

100

102

F7

Figure 2 Convergence graph of unimodal benchmark functions

Table 6 Results of multimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198658 mdash mdash mdash mdash minus99587573 38210737 minus56976258 10363742 minus5712462 924473821198659 65433009 14197664 96065525 32511237 33785055 69716023 0 0 0 011986510 10769295 91708499 17231924 11118681 09966469 04914999 888119864 minus 15 227119864 minus 15 782E minus 15 794E minus 1611986511 00173265 00205398 23738699 81345844 00008613 00038519 00025395 00066692 0 011986512 03185579 05824145 18644754 18385611 01635701 04113459 00478581 00208711 00469178 0021678911986513 0020326 00488385 76481054 53670684 03009574 11566959 06303231 01787968 06051939 01740609

Table 7 Results of fixed-dimension multimodal functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986514 09980038 0 97544907 70552637 09980038 0 41298947 42619933 48165637 4309756211986515 00011042 00003066 00032289 0005698 00003075 111E minus 19 00044241 00081813 0007327 0009814511986516 minus10316285 228119864 minus 16 minus09908202 01825 minus10316285 228E minus 16 minus10316285 175119864 minus 09 minus10316284 486119864 minus 09

11986517 03978874 0 03978874 134E minus 10 03978874 0 03978978 463119864 minus 05 03978876 508119864 minus 07

11986518 3 815E minus 16 1245 20119315 3 139119864 minus 15 70500022 1811215 30000017 287119864 minus 06

11986519 minus38627821 224E minus 15 minus38627821 172119864 minus 08 minus38627821 228119864 minus 15 minus3862402 00016777 minus38611515 00031911986520 minus32625486 00609909 minus32566017 00606853 minus33219952 519E minus 16 minus31770991 026433 minus32734635 0060991711986521 minus65146454 31931088 minus57607751 31257537 minus101532 365E minus 15 minus96433311 1569109 minus92742029 2193450111986522 minus79202597 34833672 minus52362723 27999111 minus10402941 195E minus 15 minus10402879 466119864 minus 05 minus10136952 1188478111986523 minus78374538 34395208 minus59516938 38940879 minus1053641 195E minus 15 minus9724888 24976263 minus10536237 00001251

Table 9 shows the best results obtained for theCH selection problem in WSN The results of Table 9show that mGWO algorithm is able to find the bestresults compared to other algorithms The results ofmGWO are closely followed by the CS and GWOalgorithms

6 Conclusion

This paper proposed a modification to the Grey Wolf Opti-mizer named mGWO inspired by the hunting behavior ofgrey wolves in nature An exponential decay function is usedto balance the exploration and exploitation in the search

Applied Computational Intelligence and Soft Computing 11

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

Best

scor

e obt

aine

d so

far

0 1000 2000 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus10100

minus10200

minus10300

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

10minus15

102

104

106

108

100

108

106

104

102

100

F8 F9

F10 F11

F12 F13

Figure 3 Convergence graph of multimodal benchmark functions

12 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

053

minus10055

minus10057

minus10051

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus1002

minus1003

minus1004

minus1005

minus1001

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus1

10minus2

10minus3

F14 F15

F17 F18

F19 F20

Figure 4 Continued

Applied Computational Intelligence and Soft Computing 13

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

0

minus101

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus101

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

minus100

minus101

F21 F22

F23

Figure 4 Convergence graph of fixed-dimension multimodal functions

Table 8 Results of composite functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986524 3751142 1491547 774141 2693536 4136706 17765048 3379168 1733873 3718084 194095411986525 1670097 3186513 4907166 1803477 132158 245817 1339568 2317741 1323655 258643111986526 9556119 2808871 1476118 1179418 9150283 2623901 9437718 2779337 9119168 319887611986527 4376602 1831154 1328464 1770158 395 26708204 5593846 301082 390904 279606

14 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

1028

1026

1024

1022

F24 F25

F26 F27

Figure 5 Convergence graph of composite functions

Table 9 Comparison results of CH selection problem in WSN

Algorithm Optimalcost of FF FND HND LND

GA 39798mJ 20471 26782 33375PSO 39854mJ 20348 26652 33246BA 40324mJ 19967 26548 32989CS 38412mJ 20792 27025 33724GWO 38560mJ 20739 26963 33698mGWO 38209mJ 20846 27144 33822

space over the course of iterations The results proved thatthe proposed algorithm benefits from high exploration incomparison to the standard GWO

The paper also considered the clustering problem inWSNin which the CH selection is performed using the proposedmGWO algorithm which is a challenging and NP hardproblemThe results show that the proposedmethod is foundto be very effective for real-world applications due to fastconvergence and fewer chances to get stuck at local minimaIt can be concluded that the proposed algorithm is able tooutperform the current well-known and powerful algorithmsin the literature The results prove the competence andsuperiority of mGWO to existing metaheuristic algorithmsand it has an ability to become an effective tool for solvingreal word optimization problems

Competing Interests

The authors declare that they have no competing interests

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing 9

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

10minus150

10minus200

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

10minus50

10minus100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

10minus20

10minus40

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

108

106

104

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

10minus10

10minus15

100

F1 F2

F3 F4

F5 F6

Figure 2 Continued

10 Applied Computational Intelligence and Soft Computing

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus2

100

102

F7

Figure 2 Convergence graph of unimodal benchmark functions

Table 6 Results of multimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198658 mdash mdash mdash mdash minus99587573 38210737 minus56976258 10363742 minus5712462 924473821198659 65433009 14197664 96065525 32511237 33785055 69716023 0 0 0 011986510 10769295 91708499 17231924 11118681 09966469 04914999 888119864 minus 15 227119864 minus 15 782E minus 15 794E minus 1611986511 00173265 00205398 23738699 81345844 00008613 00038519 00025395 00066692 0 011986512 03185579 05824145 18644754 18385611 01635701 04113459 00478581 00208711 00469178 0021678911986513 0020326 00488385 76481054 53670684 03009574 11566959 06303231 01787968 06051939 01740609

Table 7 Results of fixed-dimension multimodal functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986514 09980038 0 97544907 70552637 09980038 0 41298947 42619933 48165637 4309756211986515 00011042 00003066 00032289 0005698 00003075 111E minus 19 00044241 00081813 0007327 0009814511986516 minus10316285 228119864 minus 16 minus09908202 01825 minus10316285 228E minus 16 minus10316285 175119864 minus 09 minus10316284 486119864 minus 09

11986517 03978874 0 03978874 134E minus 10 03978874 0 03978978 463119864 minus 05 03978876 508119864 minus 07

11986518 3 815E minus 16 1245 20119315 3 139119864 minus 15 70500022 1811215 30000017 287119864 minus 06

11986519 minus38627821 224E minus 15 minus38627821 172119864 minus 08 minus38627821 228119864 minus 15 minus3862402 00016777 minus38611515 00031911986520 minus32625486 00609909 minus32566017 00606853 minus33219952 519E minus 16 minus31770991 026433 minus32734635 0060991711986521 minus65146454 31931088 minus57607751 31257537 minus101532 365E minus 15 minus96433311 1569109 minus92742029 2193450111986522 minus79202597 34833672 minus52362723 27999111 minus10402941 195E minus 15 minus10402879 466119864 minus 05 minus10136952 1188478111986523 minus78374538 34395208 minus59516938 38940879 minus1053641 195E minus 15 minus9724888 24976263 minus10536237 00001251

Table 9 shows the best results obtained for theCH selection problem in WSN The results of Table 9show that mGWO algorithm is able to find the bestresults compared to other algorithms The results ofmGWO are closely followed by the CS and GWOalgorithms

6 Conclusion

This paper proposed a modification to the Grey Wolf Opti-mizer named mGWO inspired by the hunting behavior ofgrey wolves in nature An exponential decay function is usedto balance the exploration and exploitation in the search

Applied Computational Intelligence and Soft Computing 11

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

Best

scor

e obt

aine

d so

far

0 1000 2000 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus10100

minus10200

minus10300

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

10minus15

102

104

106

108

100

108

106

104

102

100

F8 F9

F10 F11

F12 F13

Figure 3 Convergence graph of multimodal benchmark functions

12 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

053

minus10055

minus10057

minus10051

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus1002

minus1003

minus1004

minus1005

minus1001

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus1

10minus2

10minus3

F14 F15

F17 F18

F19 F20

Figure 4 Continued

Applied Computational Intelligence and Soft Computing 13

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

0

minus101

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus101

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

minus100

minus101

F21 F22

F23

Figure 4 Convergence graph of fixed-dimension multimodal functions

Table 8 Results of composite functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986524 3751142 1491547 774141 2693536 4136706 17765048 3379168 1733873 3718084 194095411986525 1670097 3186513 4907166 1803477 132158 245817 1339568 2317741 1323655 258643111986526 9556119 2808871 1476118 1179418 9150283 2623901 9437718 2779337 9119168 319887611986527 4376602 1831154 1328464 1770158 395 26708204 5593846 301082 390904 279606

14 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

1028

1026

1024

1022

F24 F25

F26 F27

Figure 5 Convergence graph of composite functions

Table 9 Comparison results of CH selection problem in WSN

Algorithm Optimalcost of FF FND HND LND

GA 39798mJ 20471 26782 33375PSO 39854mJ 20348 26652 33246BA 40324mJ 19967 26548 32989CS 38412mJ 20792 27025 33724GWO 38560mJ 20739 26963 33698mGWO 38209mJ 20846 27144 33822

space over the course of iterations The results proved thatthe proposed algorithm benefits from high exploration incomparison to the standard GWO

The paper also considered the clustering problem inWSNin which the CH selection is performed using the proposedmGWO algorithm which is a challenging and NP hardproblemThe results show that the proposedmethod is foundto be very effective for real-world applications due to fastconvergence and fewer chances to get stuck at local minimaIt can be concluded that the proposed algorithm is able tooutperform the current well-known and powerful algorithmsin the literature The results prove the competence andsuperiority of mGWO to existing metaheuristic algorithmsand it has an ability to become an effective tool for solvingreal word optimization problems

Competing Interests

The authors declare that they have no competing interests

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 Applied Computational Intelligence and Soft Computing

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus2

100

102

F7

Figure 2 Convergence graph of unimodal benchmark functions

Table 6 Results of multimodal benchmark functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

1198658 mdash mdash mdash mdash minus99587573 38210737 minus56976258 10363742 minus5712462 924473821198659 65433009 14197664 96065525 32511237 33785055 69716023 0 0 0 011986510 10769295 91708499 17231924 11118681 09966469 04914999 888119864 minus 15 227119864 minus 15 782E minus 15 794E minus 1611986511 00173265 00205398 23738699 81345844 00008613 00038519 00025395 00066692 0 011986512 03185579 05824145 18644754 18385611 01635701 04113459 00478581 00208711 00469178 0021678911986513 0020326 00488385 76481054 53670684 03009574 11566959 06303231 01787968 06051939 01740609

Table 7 Results of fixed-dimension multimodal functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986514 09980038 0 97544907 70552637 09980038 0 41298947 42619933 48165637 4309756211986515 00011042 00003066 00032289 0005698 00003075 111E minus 19 00044241 00081813 0007327 0009814511986516 minus10316285 228119864 minus 16 minus09908202 01825 minus10316285 228E minus 16 minus10316285 175119864 minus 09 minus10316284 486119864 minus 09

11986517 03978874 0 03978874 134E minus 10 03978874 0 03978978 463119864 minus 05 03978876 508119864 minus 07

11986518 3 815E minus 16 1245 20119315 3 139119864 minus 15 70500022 1811215 30000017 287119864 minus 06

11986519 minus38627821 224E minus 15 minus38627821 172119864 minus 08 minus38627821 228119864 minus 15 minus3862402 00016777 minus38611515 00031911986520 minus32625486 00609909 minus32566017 00606853 minus33219952 519E minus 16 minus31770991 026433 minus32734635 0060991711986521 minus65146454 31931088 minus57607751 31257537 minus101532 365E minus 15 minus96433311 1569109 minus92742029 2193450111986522 minus79202597 34833672 minus52362723 27999111 minus10402941 195E minus 15 minus10402879 466119864 minus 05 minus10136952 1188478111986523 minus78374538 34395208 minus59516938 38940879 minus1053641 195E minus 15 minus9724888 24976263 minus10536237 00001251

Table 9 shows the best results obtained for theCH selection problem in WSN The results of Table 9show that mGWO algorithm is able to find the bestresults compared to other algorithms The results ofmGWO are closely followed by the CS and GWOalgorithms

6 Conclusion

This paper proposed a modification to the Grey Wolf Opti-mizer named mGWO inspired by the hunting behavior ofgrey wolves in nature An exponential decay function is usedto balance the exploration and exploitation in the search

Applied Computational Intelligence and Soft Computing 11

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

Best

scor

e obt

aine

d so

far

0 1000 2000 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus10100

minus10200

minus10300

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

10minus15

102

104

106

108

100

108

106

104

102

100

F8 F9

F10 F11

F12 F13

Figure 3 Convergence graph of multimodal benchmark functions

12 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

053

minus10055

minus10057

minus10051

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus1002

minus1003

minus1004

minus1005

minus1001

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus1

10minus2

10minus3

F14 F15

F17 F18

F19 F20

Figure 4 Continued

Applied Computational Intelligence and Soft Computing 13

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

0

minus101

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus101

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

minus100

minus101

F21 F22

F23

Figure 4 Convergence graph of fixed-dimension multimodal functions

Table 8 Results of composite functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986524 3751142 1491547 774141 2693536 4136706 17765048 3379168 1733873 3718084 194095411986525 1670097 3186513 4907166 1803477 132158 245817 1339568 2317741 1323655 258643111986526 9556119 2808871 1476118 1179418 9150283 2623901 9437718 2779337 9119168 319887611986527 4376602 1831154 1328464 1770158 395 26708204 5593846 301082 390904 279606

14 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

1028

1026

1024

1022

F24 F25

F26 F27

Figure 5 Convergence graph of composite functions

Table 9 Comparison results of CH selection problem in WSN

Algorithm Optimalcost of FF FND HND LND

GA 39798mJ 20471 26782 33375PSO 39854mJ 20348 26652 33246BA 40324mJ 19967 26548 32989CS 38412mJ 20792 27025 33724GWO 38560mJ 20739 26963 33698mGWO 38209mJ 20846 27144 33822

space over the course of iterations The results proved thatthe proposed algorithm benefits from high exploration incomparison to the standard GWO

The paper also considered the clustering problem inWSNin which the CH selection is performed using the proposedmGWO algorithm which is a challenging and NP hardproblemThe results show that the proposedmethod is foundto be very effective for real-world applications due to fastconvergence and fewer chances to get stuck at local minimaIt can be concluded that the proposed algorithm is able tooutperform the current well-known and powerful algorithmsin the literature The results prove the competence andsuperiority of mGWO to existing metaheuristic algorithmsand it has an ability to become an effective tool for solvingreal word optimization problems

Competing Interests

The authors declare that they have no competing interests

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing 11

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

Best

scor

e obt

aine

d so

far

0 1000 2000 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus10100

minus10200

minus10300

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus5

100

10minus10

10minus15

102

104

106

108

100

108

106

104

102

100

F8 F9

F10 F11

F12 F13

Figure 3 Convergence graph of multimodal benchmark functions

12 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

053

minus10055

minus10057

minus10051

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus1002

minus1003

minus1004

minus1005

minus1001

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus1

10minus2

10minus3

F14 F15

F17 F18

F19 F20

Figure 4 Continued

Applied Computational Intelligence and Soft Computing 13

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

0

minus101

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus101

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

minus100

minus101

F21 F22

F23

Figure 4 Convergence graph of fixed-dimension multimodal functions

Table 8 Results of composite functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986524 3751142 1491547 774141 2693536 4136706 17765048 3379168 1733873 3718084 194095411986525 1670097 3186513 4907166 1803477 132158 245817 1339568 2317741 1323655 258643111986526 9556119 2808871 1476118 1179418 9150283 2623901 9437718 2779337 9119168 319887611986527 4376602 1831154 1328464 1770158 395 26708204 5593846 301082 390904 279606

14 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

1028

1026

1024

1022

F24 F25

F26 F27

Figure 5 Convergence graph of composite functions

Table 9 Comparison results of CH selection problem in WSN

Algorithm Optimalcost of FF FND HND LND

GA 39798mJ 20471 26782 33375PSO 39854mJ 20348 26652 33246BA 40324mJ 19967 26548 32989CS 38412mJ 20792 27025 33724GWO 38560mJ 20739 26963 33698mGWO 38209mJ 20846 27144 33822

space over the course of iterations The results proved thatthe proposed algorithm benefits from high exploration incomparison to the standard GWO

The paper also considered the clustering problem inWSNin which the CH selection is performed using the proposedmGWO algorithm which is a challenging and NP hardproblemThe results show that the proposedmethod is foundto be very effective for real-world applications due to fastconvergence and fewer chances to get stuck at local minimaIt can be concluded that the proposed algorithm is able tooutperform the current well-known and powerful algorithmsin the literature The results prove the competence andsuperiority of mGWO to existing metaheuristic algorithmsand it has an ability to become an effective tool for solvingreal word optimization problems

Competing Interests

The authors declare that they have no competing interests

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

12 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

100

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

101

102

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

053

minus10055

minus10057

minus10051

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus1002

minus1003

minus1004

minus1005

minus1001

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

10minus1

10minus2

10minus3

F14 F15

F17 F18

F19 F20

Figure 4 Continued

Applied Computational Intelligence and Soft Computing 13

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

0

minus101

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus101

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

minus100

minus101

F21 F22

F23

Figure 4 Convergence graph of fixed-dimension multimodal functions

Table 8 Results of composite functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986524 3751142 1491547 774141 2693536 4136706 17765048 3379168 1733873 3718084 194095411986525 1670097 3186513 4907166 1803477 132158 245817 1339568 2317741 1323655 258643111986526 9556119 2808871 1476118 1179418 9150283 2623901 9437718 2779337 9119168 319887611986527 4376602 1831154 1328464 1770158 395 26708204 5593846 301082 390904 279606

14 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

1028

1026

1024

1022

F24 F25

F26 F27

Figure 5 Convergence graph of composite functions

Table 9 Comparison results of CH selection problem in WSN

Algorithm Optimalcost of FF FND HND LND

GA 39798mJ 20471 26782 33375PSO 39854mJ 20348 26652 33246BA 40324mJ 19967 26548 32989CS 38412mJ 20792 27025 33724GWO 38560mJ 20739 26963 33698mGWO 38209mJ 20846 27144 33822

space over the course of iterations The results proved thatthe proposed algorithm benefits from high exploration incomparison to the standard GWO

The paper also considered the clustering problem inWSNin which the CH selection is performed using the proposedmGWO algorithm which is a challenging and NP hardproblemThe results show that the proposedmethod is foundto be very effective for real-world applications due to fastconvergence and fewer chances to get stuck at local minimaIt can be concluded that the proposed algorithm is able tooutperform the current well-known and powerful algorithmsin the literature The results prove the competence andsuperiority of mGWO to existing metaheuristic algorithmsand it has an ability to become an effective tool for solvingreal word optimization problems

Competing Interests

The authors declare that they have no competing interests

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing 13

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far minus10

0

minus101

500 1000 1500 2000 2500 3000Iteration

Best

scor

e obt

aine

d so

far

minus100

minus101

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

minus100

minus101

F21 F22

F23

Figure 4 Convergence graph of fixed-dimension multimodal functions

Table 8 Results of composite functions

119865

PSO BA CS GWO mGWOAvg Std Avg Std Avg Std Avg Std Avg Std

11986524 3751142 1491547 774141 2693536 4136706 17765048 3379168 1733873 3718084 194095411986525 1670097 3186513 4907166 1803477 132158 245817 1339568 2317741 1323655 258643111986526 9556119 2808871 1476118 1179418 9150283 2623901 9437718 2779337 9119168 319887611986527 4376602 1831154 1328464 1770158 395 26708204 5593846 301082 390904 279606

14 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

1028

1026

1024

1022

F24 F25

F26 F27

Figure 5 Convergence graph of composite functions

Table 9 Comparison results of CH selection problem in WSN

Algorithm Optimalcost of FF FND HND LND

GA 39798mJ 20471 26782 33375PSO 39854mJ 20348 26652 33246BA 40324mJ 19967 26548 32989CS 38412mJ 20792 27025 33724GWO 38560mJ 20739 26963 33698mGWO 38209mJ 20846 27144 33822

space over the course of iterations The results proved thatthe proposed algorithm benefits from high exploration incomparison to the standard GWO

The paper also considered the clustering problem inWSNin which the CH selection is performed using the proposedmGWO algorithm which is a challenging and NP hardproblemThe results show that the proposedmethod is foundto be very effective for real-world applications due to fastconvergence and fewer chances to get stuck at local minimaIt can be concluded that the proposed algorithm is able tooutperform the current well-known and powerful algorithmsin the literature The results prove the competence andsuperiority of mGWO to existing metaheuristic algorithmsand it has an ability to become an effective tool for solvingreal word optimization problems

Competing Interests

The authors declare that they have no competing interests

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

14 Applied Computational Intelligence and Soft Computing

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

500 1000 1500 2000 2500 3000Iteration

PSOBACS

GWOmGWO

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

103

Best

scor

e obt

aine

d so

far

1028

1026

1024

1022

F24 F25

F26 F27

Figure 5 Convergence graph of composite functions

Table 9 Comparison results of CH selection problem in WSN

Algorithm Optimalcost of FF FND HND LND

GA 39798mJ 20471 26782 33375PSO 39854mJ 20348 26652 33246BA 40324mJ 19967 26548 32989CS 38412mJ 20792 27025 33724GWO 38560mJ 20739 26963 33698mGWO 38209mJ 20846 27144 33822

space over the course of iterations The results proved thatthe proposed algorithm benefits from high exploration incomparison to the standard GWO

The paper also considered the clustering problem inWSNin which the CH selection is performed using the proposedmGWO algorithm which is a challenging and NP hardproblemThe results show that the proposedmethod is foundto be very effective for real-world applications due to fastconvergence and fewer chances to get stuck at local minimaIt can be concluded that the proposed algorithm is able tooutperform the current well-known and powerful algorithmsin the literature The results prove the competence andsuperiority of mGWO to existing metaheuristic algorithmsand it has an ability to become an effective tool for solvingreal word optimization problems

Competing Interests

The authors declare that they have no competing interests

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing 15

References

[1] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[2] T Back F Hoffmeister and H P Schwefel ldquoA survey ofevolution strategiesrdquo in Proceedings of the 4th InternationalConference on Genetic Algorithms San Diego Calif USA July1991

[3] J R Koza Genetic Programming On the Programming ofComputers by Natural Selection MIT Press Cambridge UK1992

[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[5] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[6] W Gong Z Cai C X Ling and H Li ldquoA real-codedbiogeography-based optimization with mutationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2749ndash27582010

[7] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid differentialevolution with biogeography-based optimization for globalnumerical optimizationrdquo Soft Computing vol 15 no 4 pp 645ndash665 2011

[8] H Ma and D Simon ldquoBlended biogeography-based optimiza-tion for constrained optimizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 3 pp 517ndash525 2011

[9] U Singh and T S Kamal ldquoDesign of non-uniform circularantenna arrays using biogeography-based optimisationrdquo IETMicrowaves Antennas and Propagation vol 5 no 11 pp 1365ndash1370 2011

[10] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[11] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts towards Memetic Algorithmsrdquo CaltechConcurrent Computation Program Report 826 1989

[12] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[13] K S Lee and Z W Geem ldquoA new meta-heuristic algorithm forcontinuous engineering optimization harmony search theoryand practicerdquo Computer Methods in Applied Mechanics andEngineering vol 194 no 36ndash38 pp 3902ndash3933 2005

[14] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leapingalgorithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

[15] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoGSA agravitational search algorithmrdquo Information Sciences vol 179no 13 pp 2232ndash2248 2009

[16] S Mirjalili S M Mirjalili and A Hatamlou ldquoMulti-verseoptimizer a nature-inspired algorithm for global optimizationrdquoNeural Computing amp Applications vol 27 no 2 pp 495ndash5132016

[17] A Y S Lam and V O K Li ldquoChemical-reaction-inspiredmeta-heuristic for optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 14 no 3 pp 381ndash399 2010

[18] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetwork vol 4 pp 1942ndash1948 Perth Australia December 1995

[19] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[20] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[21] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[22] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[23] X-S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[24] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[25] X Li Z Shao and J Qian ldquoAn optimizing method baseon autonomous animates fish swarm algorithmrdquo SystemsEngineeringmdashTheory amp Practice vol 22 pp 32ndash38 2002

[26] K Krishnanand and D Ghose ldquoGlowworm swarm optimi-sation a new method for optimising multi-modal functionsrdquoInternational Journal of Computational Intelligence Studies vol1 no 1 pp 93ndash119 2009

[27] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014

[28] W-T Pan ldquoA new fruit fly optimization algorithm takingthe financial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015

[30] S Mirjalili ldquoDragonfly algorithm a new meta-heuristic opti-mization technique for solving single-objective discrete andmulti-objective problemsrdquo Neural Computing amp Applications2015

[31] S C Chu and P W Tsai ldquoComputational intelligence basedon the behaviour of catsrdquo International Journal of InnovativeComputing Information and Control vol 3 pp 163ndash173 2007

[32] R Rajabioun ldquoCuckoo optimization algorithmrdquo Applied SoftComputing Journal vol 11 no 8 pp 5508ndash5518 2011

[33] J C Bansal H Sharma S S Jadon and M Clerc ldquoSpiderMonkey Optimization algorithm for numerical optimizationrdquoMemetic Computing vol 6 no 1 pp 31ndash47 2014

[34] D Dasgupta Artificial Immune Systems and Their ApplicationsSpringer 1999

[35] S Das A Biswas S Dasgupta and A Abraham ldquoBacterial for-aging optimization algorithm theoretical foundations analysisand applicationsrdquo Studies inComputational Intelligence vol 203pp 23ndash55 2009

[36] A H Gandomi and A H Alavi ldquoKrill herd a new bio-inspiredoptimization algorithmrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 12 pp 4831ndash4845 2012

[37] V K Kamboj S K Bath and J S Dhillon ldquoSolution ofnon-convex economic load dispatch problem using Grey WolfOptimizerrdquo Neural Computing and Applications 2015

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

16 Applied Computational Intelligence and Soft Computing

[38] E Emary H M Zawbaa C Grosan and A E HassenianldquoFeature subset selection approach by gray-wolf optimizationrdquoin Afro-European Conference for Industrial Advancement vol334 of Advances in Intelligent Systems and Computing Springer2015

[39] S Gholizadeh ldquoOptimal design of double layer grids consid-ering nonlinear behaviour by sequential grey wolf algorithmrdquoJournal of Optimization in Civil Engineering vol 5 no 4 pp511ndash523 2015

[40] Y Yusof and Z Mustaffa ldquoTime series forecasting of energycommodity using grey wolf optimizerrdquo in Proceedings of theInternational Multi Conference of Engineers and ComputerScientists (IMECS rsquo15) vol 1 Hong Kong March 2015

[41] G M Komaki and V Kayvanfar ldquoGrey wolf optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015

[42] A A El-Fergany and H M Hasanien ldquoSingle and multi-objective optimal power flow using grey wolf optimizer anddifferential evolution algorithmsrdquo Electric Power Componentsand Systems vol 43 no 13 pp 1548ndash1559 2015

[43] K Shankar and P Eswaran ldquoA secure visual secret share (VSS)creation scheme in visual cryptography using elliptic curvecryptography with optimization techniquerdquo Australian Journalof Basic amp Applied Science vol 9 no 36 pp 150ndash163 2015

[44] E Emary H M Zawbaa and A E Hassanien ldquoBinary greywolf optimization approaches for feature selectionrdquo Neurocom-puting vol 172 pp 371ndash381 2016

[45] V K Kamboj ldquoA novel hybrid PSOGWOapproach for unitcommitment problemrdquo Neural Computing and Applications2015

[46] A Zhu C Xu Z Li J Wu and Z Liu ldquoHybridizing grey Wolfoptimization with differential evolution for global optimizationand test scheduling for 3D stacked SoCrdquo Journal of SystemsEngineering and Electronics vol 26 no 2 pp 317ndash328 2015

[47] T-S Pan T-K Dao T-T Nguyen and S-C Chu ldquoA communi-cation strategy for paralleling grey wolf optimizerrdquo Advances inIntelligent Systems and Computing vol 388 pp 253ndash262 2015

[48] J Jayapriya and M Arock ldquoA parallel GWO technique foraligning multiple molecular sequencesrdquo in Proceedings of theInternational Conference on Advances in Computing Communi-cations and Informatics (ICACCI rsquo15) pp 210ndash215 IEEE KochiIndia August 2015

[49] M M Afsar and M-H Tayarani-N ldquoClustering in sensornetworks a literature surveyrdquo Journal of Network and ComputerApplications vol 46 pp 198ndash226 2014

[50] W B Heinzelman A Chandrakasan and H Balakrish-nan ldquoEnergy-efficient communication protocol for wirelessmicrosensor networksrdquo in Proceedings of the 33rd AnnualHawaii International Conference on System Siences (HICSS rsquo00)p 223 IEEE January 2000

[51] N Mittal and U Singh ldquoDistance-based residual energy-efficient stable election protocol for WSNsrdquo Arabian Journal forScience and Engineering vol 40 no 6 pp 1637ndash1646 2015

[52] E A Khalil and B A Attea ldquoEnergy-aware evolutionaryrouting protocol for dynamic clustering of wireless sensornetworksrdquo Swarm and Evolutionary Computation vol 1 no 4pp 195ndash203 2011

[53] B A Attea and E A Khalil ldquoA new evolutionary based routingprotocol for clustered heterogeneous wireless sensor networksrdquoApplied Soft Computing vol 12 no 7 pp 1950ndash1957 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014