research article smallest-small-world cellular harmony...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 635608 9 pageshttpdxdoiorg1011552013635608

Research ArticleSmallest-Small-World Cellular Harmony Search forOptimization of Unconstrained Benchmark Problems

Sung Soo Im1 Do Guen Yoo2 and Joong Hoon Kim3

1 Daelim Industrial Co Ltd Seoul 110-140 Republic of Korea2 Research Institute of Engineering and Technology Korea University Seoul 136-713 Republic of Korea3 School of Civil Environmental and Architectural Engineering Korea University Seoul 136-713 Republic of Korea

Correspondence should be addressed to Joong Hoon Kim jaykimkoreaackr

Received 28 June 2013 Revised 28 September 2013 Accepted 8 October 2013

Academic Editor Zong Woo Geem

Copyright copy 2013 Sung Soo Im 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

We presented a new hybrid method that combines cellular harmony search algorithms with the Smallest-Small-World theoryA harmony search (HS) algorithm is based on musical performance processes that occur when a musician searches for a betterstate of harmony Harmony search has successfully been applied to a wide variety of practical optimization problems Most ofthe previous researches have sought to improve the performance of the HS algorithm by changing the pitch adjusting rate andharmony memory considering rate However there has been a lack of studies to improve the performance of the algorithm by theformation of population structures Therefore we proposed an improved HS algorithm that uses the cellular automata formationand the topological structure of Smallest-Small-World network The improved HS algorithm has a high clustering coefficient anda short characteristic path length having good exploration and exploitation efficiencies Nine benchmark functions were appliedto evaluate the performance of the proposed algorithm Unlike the existing improved HS algorithm the proposed algorithm isexpected to have improved algorithmic efficiency from the formation of the population structure

1 Introduction

Studies on network maps provide in-depth understandingof the basic features and requirements of various systemsMany network connection topologies assumed to be eithercompletely regular or completely random have been studied[1 2] Networks which can be formally described by thetools of graph theory are critical for describing many sci-entific social and technological phenomena (Newman [1])Typical examples include the Internet World Wide Websocial acquaintances electric power networks and neuralnetworks In recent years new theoretical and applied resultshave motivated substantial researches in network scienceThe pioneering studies of Watts and Strogatz [2] have beenperformed and they have been followed by many others inthe subsequent years They constructed a simple computermodel of a regular network or lattice in which each nodeof the network was connected by a line or edge to eachof its four nearest neighbors This network structure ortopology is highly clustered or cliquish by design however

movement from one node to another node on the oppositeside of the lattice requires traversing a large number ofshort-range connections In other words although the pathlength (or number of mediating edges) between neighboringnodes is short the path length between distant nodes islong Therefore the minimum path length averaged overall possible pairs of nodes in the network which is quitelong Watts and Strogatz [2] investigated the change innetwork topology (measured in terms of local clustering andminimum path length) that resulted from randomly rewiringsome of the lattice edges to create long-range connectionsbetween distant nodes If many lattice edges were randomlyrewired the network would naturally acquire the topologicalcharacteristics of a random graph (short path length andlow clustering) It is worth mentioning that they foundthat even a few long-range connections greatly reduced theminimum path length of the network without affecting itslocal clustering Thus they defined algorithmically for thefirst time a class of networks having topological propertiessimilar to social networks and demonstrated both the high

2 Journal of Applied Mathematics

(a) Neumann (b) Moore

Figure 1 Neighboring cells

clustering of a lattice and the short path length of a randomgraphThese networkswere called Small-World networksOnthe other hand Nishikawa et al [3] suggested a Smallest-Small-World network (SSWN) and confirmed their theorywith examples They showed that the average path lengthof a Small-World network with fixed shortcuts became theminimum when there was a ldquocenterrdquo node from which allshortcuts are connected to uniformly distributed nodes in thenetwork [4] Research on efficient network structures can beapplied to fields of optimization Genetic Algorithms (GAs)Artificial Neural Networks (ANNs) and Particle SwarmOptimization (PSO) In GAs cellular Genetic Algorithms(cGAs) have been widely studied using cellular automata(CA) Kang [5] particularly proposed Smallest-Small-Worldcellular Genetic Algorithms (SSWcGAs) to apply the SSWNto the cGAs and evaluated the performance of the algorithm

The harmony search (HS) algorithm [6 7] is based onmusical performance processes that occur when a musiciansearches for a better state of harmonyTheHS has successfullybeen applied to many mathematical functions and a widevariety of practical optimization problems like pipe-networkdesign structural optimization vehicle routing combinedheat and power economic dispatching multiple-dam-systemscheduling and so forth In addition hybrid HS algorithmsthat combine with other optimization techniques such as thePSO and ANNs have been proposed

So far researches have been carried out to improve theperformance of theHS algorithmby changing the parameterssuch as pitch adjusting rate (PAR) and harmony memoryconsidering rate (HMCR) However there has been a lack ofstudies on the performance improvement of the HS throughthe formation of population structures to transform it intoa cGA In this study therefore we proposed the improvedHS algorithm which is used the CA and has the topologicalstructure of the SSWN The improved HS algorithm has ahigh clustering coefficient and a short characteristic path

length possessing good exploration and exploitation efficien-cies

2 Related Theory

21 Cellular Automata A cellular automaton is a discretemodel studied in computation theory mathematics physicscomplexity science theoretical biology and microstructuremodeling [8] Von Neumann [9] has used the concept ofcellular automaton space regularly arranged in a grid cellIndividual cells updated simultaneously in a discrete timestep Each cell is a finite state machine Each cell entered thestate of its own and neighboring and then printout the statein the next time step A collection of these cellular automatonsis calledCA space Other terms for theCA are ldquocellular spacerdquoldquotessellation automatardquo ldquohomogeneous structuresrdquo ldquocellularstructuresrdquo ldquotessellation structuresrdquo ldquoiterative arraysrdquo and soon [8]

The cellrsquos dimension in the CA can be defined as one-two- and three-dimensional cells Typically a 2D cell is themost widely used because it can deal with spatial phenomenaTalking about the shape of the cell a square cell is usuallyused because of its highest computer-processing efficiencycompared to the other types of cells Neighbors in theCA can be defined depending on the form of the adjacentsurrounding cells In the formof a 2D square cell CA it can bedefined as a Von Neumann neighborhood with neighbors infour directions or as aMoore neighborhoodhaving neighborsin eight directions as shown in Figure 1

The concept of the CA as described in this subsectionhas been applied to a variety of optimization techniquessuch as GAs [10 11] ANNs [12] and PSO [13 14] Forinstance cGA is a subclass of GA in which each individualis placed in a given topological distribution It is a latticegraph as one individual can only interact with its nearest

Journal of Applied Mathematics 3

Increasingly random connectivity

Regular high L high C low L high C low L low C

Small-World Random

Figure 2 Classification of networks according to connectivity

Table 1 Efficiency of average path length (119871) and clusteringcoefficient (119862)

Property DescriptionAverage path length(119871)

Exploitation (local search)Low LrArr good exploitation efficiency

Clustering coefficient(119862)

Exploration (Global Search)High C rArr good exploration efficiency

neighbors Therefore these relations can be applied on a setof each individual and the surrounding neighbors promotingneighborhood exploitation and exploration of the searchspace

22 Smallest-Small-World Theory The Small-World networkmodels have receivedmuch attention since their introductionby Watts and Strogatz [2] as models of real networks havingcharacteristics which lie between random and regular asshown in Figure 2 They are characterized by two factorsthe average path length (119871) which measures efficiency ofcommunication or passage time between nodes and theclustering coefficient (119862) which represents the degree oflocal order The average path length (119871) is defined as theaverage number of links in the shortest path between a pairof nodes in the network Accordingly clustering coefficient(119862) is defined as the probability that two nodes connectedto a common node are also connected to each other Ingeneral cGArsquos population space is a regular network thathas relatively high 119871 and 119862 from the point of view of thenetwork theory High 119871 makes the interactions of remotenodes difficult Otherwise a Small-World network representslow 119871 between any two arbitrary nodes and contains high 119862It can be characterized by lots of local connectivity betweennodes as well as the occasional longer links defined as ashortcut Only a few of these longer links are required in orderto obtain a Small-World network state

Nishikawa et al [3] suggested a SSWN and verified theirtheory with examples They showed that the 119871 of a Small-World networkwith fixed shortcuts is aminimumwhen there

Figure 3 Examples of shortcut configuration with a center node

is a ldquocenterrdquo node from which all shortcuts are connected touniformly distributed nodes in the network [4] An exampleof such a configuration is illustrated in Figure 3

They defined the SSWN theory as follows

(i) A SSWN is composed of two parts the underlyingnetwork (eg a regular lattice) and the subnetworkof shortcuts containing only the shortcuts and theirnodes

(ii) The nodes in the subnetwork of shortcuts must beuniformly distributed over the network

(iii) Finally among all possible configurations of con-nected subnetworks of shortcuts with uniformly dis-tributed nodes the ones with a single center involvethe largest number of nodes

These arguments indicate that given a fixed number ofshortcuts the networks connected with a sub-network ofshortcuts having uniformly distributed nodes have smaller 119871than a typical random configuration and among these theones with a single center minimize 119871 In other words theSmallest-Small-World networks are characterized by thesestructures


Do ReMi

Sol La Si

Mi Fa Sol

100 mm200 mm300 mm

500 mm600 mm700 mm

300 mm400 mm500 mm

= Do = Mi = Sol

= 100 mm = 300 mm = 500 mm

x1 x2 x

f(100 300 500)

3

Reservoir22 1

2 3 4

5 6 78 9

10 11 12

1415 16

13

17

18

19

20

2122

23 24

25

26

27

28

29

30

31

1 2 3 4

5 6 7

8 9 10 11

12

1314

15 16

17

18

19

20

21

119863 = 75

119863 = 75 119863 = 75

119863 = 75

119863=75

119863=75

119863=75

119863=75

119863=75

119863=75

119863=200

119863 = 500

119863 = 150

119863 = 150

119863=150

119863=150

119863=250

119863=150

119863=150

119863 = 100119863 = 100119863 = 100

119863 = 300 119863 = 300

119863 = 100 119863 = 100 119863 = 100

119863=100

119863=100119863=100

119863=100

Figure 4 Relation between music improvisation and engineering optimization [15 16]

Table 2 Definitions and specifications of two design variables problems

Functions Definition

Rosenbrockrsquos valleyMinimize 119891 (119909

1 1199092) = 100 (119909

2

1minus 1199092

2)2

+ (1 minus 1199091)2

minus2048 le 1199091le 2048 minus2048 le 119909

2le 2048

1199091= 1 119909

2= 1 119891 (1 1) = 0

Braninrsquos function

Minimize 119891 (1199091 1199092) = 119886(119909

2minus 1198871199092

1+ 1198881199091minus 119889)2

+ 119890 (1 minus 119891) cos (1199091) + 10

119886 = 1 119887 = 5141205872 119888 = 5120587 119889 = 6 119890 = 10 119891 = 18120587

minus5 le 1199091le 10 0 le 119909

2le 15

(1199091 1199092) = (minus120587 12275) (120587 2275) (942478 2475)

119891 (1199091 1199092) = 0397887

Easomrsquos functionMinimize 119891 (119909

1 1199092) = minus cos (119909

1) cos (119909

2) exp (minus(119909

1minus 120587)2

minus (1199092minus 120587)2

)


2le 100

(1199091 1199092) = (120587 120587) 119891 (119909

1 1199092) = minus1

Goldstein pricersquos function

Minimize 119891 (1199091 1199092) = [1 + (119909

1+ 1199092+ 1)2(19 minus 14119909

1+ 31199092

1minus 14119909

2+ 611990911199092+ 31199092

2)]

times [30 + (21199091minus 31199092)2

(18 minus 321199091+ 12119909

2

1+ 48119909

2minus 36119909

11199092+ 27119909

2

2)]


2le 2

(1199091 1199092) = (0 minus1) 119891 (119909

1 1199092) = 3

Six-hump camel back functionMinimize 119891 (119909

1 1199092) = (4 minus 21119909

2

1+ 1199094

13) 1199092

1+ 11990911199092+ (minus4 + 4119909

2

2) 1199092

2


2le 2

(1199091 1199092) = (minus00898 07126) (00898 minus07126) 119891 (119909

1 1199092) = minus10316

23 Harmony Search Algorithm The HS algorithm is basedon musical performance processes that occur when a musi-cian searches for a better state of harmony such as during jazzimprovisation Jazz improvisation seeks musically pleasingharmony (a perfect state) as determined by an aestheticstandard just as an optimization process seeks a global

solution (a perfect state) as determined by an objectivefunction The pitch of each musical instrument determinesthe aesthetic quality just as a set of values assigned toeach decision determines the value of the objective functionFigure 4 shows the details of the analogy between musicimprovisation and engineering optimization [15 16]


In musical improvisation each player sounds any pitchwithin the possible range together making one harmonyvector If all the pitches make a good harmony that experi-ence is stored in each playerrsquos memory and the possibility ofmaking a good harmony is increased next the time Similarlyin engineering optimization each decision variable initiallychooses any value within the possible range together makingone solution vector If all the values of the decision variablesresult in a good solution that experience is stored in eachvariablersquos memory and the possibility of making a goodsolution is also increased at the next iteration In brief thesteps of HS algorithm are given as follows [16 17]

Step 1 (initialize the problem and algorithm parameters) Theoptimization problem is defined as minimize 119891(119909) subjectto 119909119894119871

le 119909119894

le 119909119894119880

(119894 = 1 2 119873) and other existingconstraints 119909

119894119871and 119909

119894119880are the lower and upper bounds of

decision variables respectivelyTheHS algorithmparametersare also specified in this step They are the harmony memorysize (HMS) or the number of solution vectors in the harmonymemory HMCR bandwidth (119887

119908) PAR and the number of

improvisations (119870) or stopping criterion

Step 2 (initialize the harmonymemory) The initial harmonymemory (HM) is generated from a uniform distribution inthe ranges [119909

119894119871 119909119894119880] (119894 = 1 2 119873) as given in (1)

119867119872 =

[[[[[[[[[[[[[

[

1199091

11199091

2sdot sdot sdot 119909

1

119873

1199092

11199091


2

119873

119909HMSminus11

119909HMSminus12

sdot sdot sdot 119909HMSminus1119873

119909HMS1

119909HMS2

sdot sdot sdot 119909HMS119873

]]]]]]]]]]]]]

]

(1)

Step 3 (improvise a new harmony) Generating a new har-mony is called improvisation The new harmony vector1199091015840= (1199091015840

1 1199091015840

2 119909

1015840

119873) is determined by three rules memory

consideration pitch adjustment and random selection Thepseudocode for generating a new harmony search is given asshown in Algorithm 1 [17]

1199091015840

119894(119894 = 1 2 119873) is the 119894th component of1199091015840 and119909

119895

119894(119895 =

1 2 HMS) is the 119894th component of the 119895th candidatesolution vector in the HM rand is a uniformly generatedrandom number in the region of [0 1] and 119887

119908is an arbitrary

distance bandwidth

Step 4 (update harmony memory) If the fitness of theimprovised harmony vector 119909

1015840= (1199091015840

1 1199091015840

2 119909

1015840

119873) is better

than that of the worst harmony replace the worst harmonyin the HM with 119909

1015840

Step 5 (check the stopping criterion) If the stopping criterion(maximum number of iterations119870) is satisfied computation

Table 3 Obtained statistical results of 2D benchmark problem

SSWCHS CHS SHS1 Rosenbrock

Mean 174119864 minus 06 399119864 minus 08 237119864 minus 04

Best 360119864 minus 11 160119864 minus 11 407119864 minus 09

Worst 230119864 minus 05 429119864 minus 07 163119864 minus 02

SD 377119864 minus 06 716119864 minus 08 165119864 minus 03

Feasible solution 100 100 89Mean iteration 69868 110225 2014262Iteration SD 57489 103011 109192

2 Braninrsquos functionMean 398119864 minus 01 398119864 minus 01 398119864 minus 01





3 Easomrsquos functionMean minus670119864 minus 01 minus941119864 minus 01 minus490119864 minus 01


Worst 100119864 + 00 744119864 minus 01 100119864 + 00



4 Goldstein pricersquos functionMean 300119864 + 00 300119864 + 00 192119864 + 01

Best 000119864 + 00 000119864 + 00 000119864 + 00

Worst 121119864 minus 05 112119864 minus 05 270119864 + 01

SD 162119864 minus 06 193119864 minus 06 132119864 + 01


5 Six-hump camel back functionMean minus103119864 + 00 minus103119864 + 00 minus103119864 + 00




Feasible solution 100 100 100Mean iteration 8734 15280 85206Iteration SD 3313 6730 6730ldquoSDrdquo stands for standard deviation

is terminated Otherwise go to Step 3 The most importantstep of the HS algorithm is Step 3 which includes memoryconsideration pitch adjustment and random selection ThePAR and 119887

119908have a profound effect on the performance of the

HS


for each 119894 isin [1N] doif rand() leHMCR then1199091015840

119894= 119909119895

119894(119895 = 1 2 HMS)memory consideration

if rand() le PAR then1199091015840

119894= 1199091015840

119894plusmn rand() times 119887

119908 pitch adjustment

if 1199091015840

119894gt 119909119894119880

1199091015840

119894= 119909119894119880

elseif 1199091015840119894lt 119909119894119871

1199091015840

119894= 119909119894119871

endend

else1199091015840

119894= 119909119894119871+ rand() times (119909

119894119880minus 119909119894119871) random selection

endend

Algorithm 1

Build initial population

Calculation of initial objective function value of HM

Improvise a new harmony from HM by HMCR PAR

Update the HM (center node in subpopulations)

Stopping criteria

End

Divide the population to subpopulations

No

Ok

Figure 5 Flowchart of the CHS

3 Smallest-Small-World CellularHarmony Search

In the present paper we proposed the improvedHS algorithmthat uses the CA concept and has the topological structure ofthe SSWN A population is just a group of certain number ofarbitrary objects in the same generation as for the generationin the original HS Meanwhile these objects are not relatedto each other In the configuration of these populations as akind of random network the 119871 and 119862 are also very low Incontrast if the population only consists of a grid of cellularnetworks the 119862 is relatively high however the 119871 is alsohigh Because high 119871makes the interactions of remote nodesdifficult we need to reduce the 119871 Table 1 shows the efficiencyand comparison of clustering coefficient and path lengthThe Small-World network models have the advantages ofboth random and regular network They have low 119871 for fastinteraction between nodes and they have high 119862 ensuringsufficient redundancy for high fault tolerance [3] In thisstudy therefore the population of the original HS consistsof a form of cellular networks And then we can reduce

characteristic path length (119871) and increase the clusteringcoefficient (119862) using the shortcuts concepts of Small-Worldnetwork

31 Cellular Harmony Search (CHS) The operation processof the cellular harmony search (CHS) is shown in Figure 5The CHSrsquos operation process is the same as the existingoriginal HS However the CHSrsquos initial population has acellular form In this process new generations in the sub-network from the PAR and HMCR processes are comparedwith the existing population and then replace the lowestranked object After each small grid within the objects ofthe highest priority a node is located in the center of thesub-network Finally the centers of the sub-network nodesare compared and the object of the entire population of thehighest priority and the central node are replaced

32 Smallest Small World Cellular Harmony Search(SSWCHS) Smallest Small World cellular harmony search(SWCHS) performs its operation between the center nodes


Table 4 Definitions and specification of 30D benchmark problems


Step functionMinimize 119891 (119909) =

119899

sum

119894=1

([119909119894+ 05])

2

minus 30 le 119909119894le 30

min (119891) = 119891 (0 0) = 0

The Rastrigin function Minimize 119891 (119909) = 10119899 +

119899

sum

119894=1

[1199092

119894minus 10 cos(2120587119909

119894)] minus 512 le 119909

119894le 512 min (119891) =

119891 (0 0) = 0

The Griewank functionMinimize 119891 (119909) = (14000)

119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1

cos (119909119894radic119894) + 1 minus 600 le 119909

119894le 600

min (119891) = 119891 (0 0) = 0

The Ackley functionMinimize 119891 (119909) = minus119886 exp(minus119887radic(1119899)

119899

sum

119894=1

1199092

119894)) minus exp((1119899)

119899

sum

119894=1

cos (119888119909119894)) + 119886 + exp(1)

119886 = 20 119887 = 02 119888 = 2120587 minus 32768 le 119909119894le 32768

min (119891) = 119891 (0 0) = 0

Table 5 Results of 30ND Problems

SSWCHS CHS SHS1 Step function

Mean 000119864 + 00 109119864 + 00 560119864 + 00

Best 000119864 + 00 000119864 + 00 000119864 + 00

Worst 000119864 + 00 300119864 + 00 120119864 + 01

SD 000119864 + 00 981119864 minus 01 253119864 + 00

Feasible solution 100 36 1Mean iteration 3067372 14871739 9135300Iteration SD 4190201 4924037 mdash

2 Rastrigin functionMean 469119864 minus 02 299119864 + 00 609119864 + 00

Best 278119864 minus 03 112119864 minus 02 101119864 + 00

Worst 100119864 + 00 501119864 + 00 898119864 + 00

SD 195119864 minus 01 131119864 + 00 199119864 + 00

Feasible solution 92 0 0Mean iteration 11379016 mdash mdashIteration SD 4725760 mdash mdash

3 Griewank functionMean 103119864 minus 01 143119864 minus 01 694119864 minus 01




Feasible solution 10 0 3Mean iteration 7364200 mdash 20872600Iteration SD 4391252 mdash 2691884

4 Ackley functionMean 462119864 minus 02 131119864 + 00 158119864 + 00


Worst 150119864 + 00 165119864 + 00 346119864 + 00



by adding shortcuts in the CHSThe CHS operation betweenthe center nodes is added in the calculation process as shown

in Figure 6 In this process the SSWCHS is performed toobtain the final optimal solution through operations betweenthe small grids of optimum object

4 Applications and Results

In this study the proposed SSWCHS was applied for solvingunconstrained benchmark functions widely examined in theliterature The SSWCHS CHS and original simple harmonysearch (SHS) problems were performed in this study to havecomparisons among optimizers The optimization task wascarried out using 100 independent runs based on the resultsdepending on the type of problem Statistical values includ-ing best worst and mean values and mean iteration numberwere obtained to evaluate the performance of the reportedalgorithms Benchmark functions were utilized to evaluatethe performance of considered optimization techniquesAmong benchmark functions five benchmark functions have2 design variables and the rest have 30 design variables [18]The benchmark functions have difficulty in terms of numberof local optimum points and also the search space of thesefunctions is almost wide and can challenge the efficiencyof methods The SSWCHS CHS and SHS algorithms wereapplied and run separately The SSWCHS and CHS thatoccur at random initial solution were applied equally andthe initial solution in the SHS was applied differently becauseof the size of the HM The HMS in the SSWCHS and CHSwas composed by 225 cellular structures and the SHS wascomposed by 5 HMSTherefore the number of shortcuts canbe predetermined as shown in Figure 7 Two and 30 designvariable problems were initialized using 03 PAR and 095HMCR respectively The 2 and 30 design variable functionswere performedwith 50000 iterations and 250000 iterationsrespectively Feasible solution has a margin of error of 10minus4and 10minus2 in the 2 and 30 design variable functionsMean valueand standard deviation of iteration can indicate the degreeof convergence From the number of feasible solution theaccuracy percentage of each algorithm can be compared

41 Problems Having Two Design Variables In this study thefollowing problems were applied as shown in Table 2






Stopping criteria

End


Build second population (best node in subpopulations)

Improvise a new harmony from HM by HMCR PAR (second population)

Update the HM (global center node)

No

Ok

Considering smallest-Small-World theory

Figure 6 Flowchart of the SSWCHS

Table 3 shows the obtained results of the 2D benchmarkfunctions The SSWCHS and CHS relatively showed goodresults compared to the SHS The result of each benchmarkfunction indicated that Easomrsquos function could not findperfect optimal solution by the SSWCHS and CHS Theparameters of Easomrsquos function have a wide search range Forthis reason Easomrsquos function showed less search capabilityfor the optimal value than the other functions In such caseincreasing the number of iterations is considered to reachthe optimal solution so that the ratio can be improvedOne can see that the value of the entire functions which ismean iteration and iteration standard deviation value of theSSWCHS are smaller than those of the CHSThese results canconclude that the SSWCHS converged faster than the CHSand it can estimate a reliable optimal solution

42 Problems Having 30 Design Variables In this study thefollowing benchmark problems were applied as shown inTable 4

Table 5 shows the attained results of 30D benchmarkfunctionsThe SSWCHS relatively showed good results com-pared with the CHS and SHS For all benchmark problemsgiven in Table 4 except Griewankrsquos function the SSWCHSshowed its superiority over other reported methods In caseof Griewangkrsquos function we can see that with the ratio of 10the optimal solution is reached Because it had wide searchranges of parameters from minus600 to 600 By contrast to the 2D

problems 30D functions needmore time to reach the optimalsolutionHowever the SSWCHS showedmuchhigher perfor-mance than the CHS and SHS In particular the arithmeticmean and the standard deviation in the SSWCHS are lowerthan those in other algorithms indicating stability and fasterconvergence For the Ackley function and the Griewangkfunction however the CHS and SHS offered lower standarddeviation of iteration than the SWSCHS However in thiscase the number of feasible solutions was very small andmean iterationwas larger than that obtained by the SSWCHSIn case of the percentage of optimal solutions of the SHS thevalues of the Griewangk function and Ackley function werelarger than those of the CHS

5 Conclusions

In this study an improved HS algorithm which combines theCA and the topological structure of Smallest-Small-Worldnetwork is proposed Most of previous studies there havebeen a lack of studies on the performance improvement of theharmony search algorithm by use of population or memorystructures A new hybrid harmony search algorithm hav-ing high clustering coefficient and short characteristic pathlength was required The hybrid HS algorithm developed inthis paper has good exploration and exploitation efficienciesNine benchmark functions were applied to assess the perfor-mance of the proposed algorithm The applied benchmark


1 2 3 10 11 12 19 20 21 28 29 30 37 38 39

4 5 6 13 14 15 22 23 24 31 32 33 40 41 42

7 8 9 16 17 18 25 26 27 34 35 36 43 44 45

46 47 48 55 56 57 64 65 66 73 74 75 82 83 84

49 50 51 58 59 60 67 68 69 76 77 78 85 86 87

52 53 54 61 62 63 70 71 72 79 80 81 88 89 90

91 92 93 100 101 102 109 110 111 118 119 120 127 128 129

94 95 96 103 104 105 112 113 114 121 122 123 130 131 132

97 98 99 106 107 108 115 116 117 124 125 126 133 134 135

136 137 138 145 146 147 154 155 156 163 164 165 172 173 174

139 140 141 148 149 150 157 158 159 166 167 168 175 176 177

142 143 144 151 152 153 160 161 162 169 170 171 178 179 180

181 182 183 190 191 192 199 200 201 208 209 210 217 218 219

184 185 186 193 194 195 202 203 204 211 212 213 220 221 222

187 188 189 196 197 198 205 206 207 214 215 216 223 224 225

Center node (global)Shortcut

The number of shortcuts 24

Figure 7 Smallest Small World harmony search network

functions consist of five 2D functions and four 30D functionsThe evaluation indexes of the SSWCHSwere better than thoseof CHS and SHS in terms of solution quality The SSWCHSalgorithm showed generally faster convergence and morestability than the CHS or SHS It shows very competitivesolutionswith less number of iterations than other consideredalgorithms It is recommended that the optimization tech-niques as new algorithms became available be used in a widerange of engineering optimization problems However theSSWCHS has so many of the HS structures that it can affectcomputation time Therefore it remains a complementarypart As a further research parameter variations are expectedto develop using the proposed SSWCHS

Acknowledgment

This work was supported by the National Research Founda-tion of Korea (NRF)Grant funded by the Korean government(MSIP) (no 2013R1A2A1A01013886)

References

[1] M E J Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[2] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networkrdquo Nature vol 398 pp 440ndash442 1998

[3] T Nishikawa A E Motter Y C Lai and F C HoppensteadtldquoSmallest small-world networkrdquo Physical Review E - Statistical

Nonlinear and Soft Matter Physics vol 66 no 4 Article ID046139 2002

[4] V Latora and M Marchiori ldquoNotions of local and globalefficiency of a networkrdquo Physical Review Letters vol 87 ArticleID 198701 2001

[5] T W Kang ldquoSmallest-small-world cellular genetic algorithmrdquoJournal of Computing Science and Engineering vol 34 no 11pp 971ndash983 2007

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

[7] J H Kim Z W Geem and E S Kim ldquoParameter estimationof the nonlinear muskingum model using harmony searchrdquoJournal of the AmericanWater Resources Association vol 37 no5 pp 1131ndash1138 2001

[8] S Wolfram ldquoStatistical mechanics of cellular automatardquoReviews of Modern Physics vol 55 no 3 pp 601ndash644 1983

[9] J VonNeumannTheory of Self-Reproducing Automata Univer-sity of Illinois Press Urbana Ill USA 1966

[10] B Dorronsoro and E Alba ldquoA simple cellular genetic algo-rithm for continuous optimizationrdquo in Proceedings of the IEEECongress on Evolutionary Computation (CEC rsquo06) pp 2838ndash2844 Vancouver Canada July 2006

[11] G Folino C Pizzuti and G Spezzano ldquoCombining cellulargenetic algorithms and local search for solving satisfiabilityproblemsrdquo in Proceedings of the IEEE 10th International Confer-ence on Tools with Artificial Intelligence pp 192ndash198 November1998

[12] G Timar and D Balya ldquoRegular small-world cellular neuralnetworks key properties and experimentsrdquo in Proceedings of theIEEE International Symposium on Cirquits and System pp 69ndash72 British Columbia Canada May 2004

[13] A B Hashemi and M R Meybodi ldquoA multi-role cellular PSOfor dynamic environmentsrdquo in Proceedings of the 14th Inter-national CSI Computer Conference (CSICC rsquo09) pp 412ndash417October 2009

[14] V Noroozi A B Hashemi and M R Meybodi ldquoCellularDEa cellular based differential evolution for dynamic optimizationproblemsrdquo in Proceedings of the ICANNGA vol 6593 of LectureNotes in Computer Science pp 340ndash349 2011

[15] 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

[16] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoAppliedMathematics and Computation vol 188 no 2 pp 1567ndash1579 2007

[17] M G H Omran and M Mahdavi ldquoGlobal-best harmonysearchrdquo Applied Mathematics and Computation vol 198 no 2pp 643ndash656 2008

[18] M Molga and C Smutnicki ldquoTest functions for optimizationneedsrdquo 2005 httpwwwzsdictpwrwrocplfilesdocsfunc-tionspdf

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


(a) Neumann (b) Moore

Figure 1 Neighboring cells

clustering of a lattice and the short path length of a randomgraphThese networkswere called Small-World networksOnthe other hand Nishikawa et al [3] suggested a Smallest-Small-World network (SSWN) and confirmed their theorywith examples They showed that the average path lengthof a Small-World network with fixed shortcuts became theminimum when there was a ldquocenterrdquo node from which allshortcuts are connected to uniformly distributed nodes in thenetwork [4] Research on efficient network structures can beapplied to fields of optimization Genetic Algorithms (GAs)Artificial Neural Networks (ANNs) and Particle SwarmOptimization (PSO) In GAs cellular Genetic Algorithms(cGAs) have been widely studied using cellular automata(CA) Kang [5] particularly proposed Smallest-Small-Worldcellular Genetic Algorithms (SSWcGAs) to apply the SSWNto the cGAs and evaluated the performance of the algorithm

The harmony search (HS) algorithm [6 7] is based onmusical performance processes that occur when a musiciansearches for a better state of harmonyTheHS has successfullybeen applied to many mathematical functions and a widevariety of practical optimization problems like pipe-networkdesign structural optimization vehicle routing combinedheat and power economic dispatching multiple-dam-systemscheduling and so forth In addition hybrid HS algorithmsthat combine with other optimization techniques such as thePSO and ANNs have been proposed

So far researches have been carried out to improve theperformance of theHS algorithmby changing the parameterssuch as pitch adjusting rate (PAR) and harmony memoryconsidering rate (HMCR) However there has been a lack ofstudies on the performance improvement of the HS throughthe formation of population structures to transform it intoa cGA In this study therefore we proposed the improvedHS algorithm which is used the CA and has the topologicalstructure of the SSWN The improved HS algorithm has ahigh clustering coefficient and a short characteristic path

length possessing good exploration and exploitation efficien-cies

2 Related Theory

21 Cellular Automata A cellular automaton is a discretemodel studied in computation theory mathematics physicscomplexity science theoretical biology and microstructuremodeling [8] Von Neumann [9] has used the concept ofcellular automaton space regularly arranged in a grid cellIndividual cells updated simultaneously in a discrete timestep Each cell is a finite state machine Each cell entered thestate of its own and neighboring and then printout the statein the next time step A collection of these cellular automatonsis calledCA space Other terms for theCA are ldquocellular spacerdquoldquotessellation automatardquo ldquohomogeneous structuresrdquo ldquocellularstructuresrdquo ldquotessellation structuresrdquo ldquoiterative arraysrdquo and soon [8]

The cellrsquos dimension in the CA can be defined as one-two- and three-dimensional cells Typically a 2D cell is themost widely used because it can deal with spatial phenomenaTalking about the shape of the cell a square cell is usuallyused because of its highest computer-processing efficiencycompared to the other types of cells Neighbors in theCA can be defined depending on the form of the adjacentsurrounding cells In the formof a 2D square cell CA it can bedefined as a Von Neumann neighborhood with neighbors infour directions or as aMoore neighborhoodhaving neighborsin eight directions as shown in Figure 1

The concept of the CA as described in this subsectionhas been applied to a variety of optimization techniquessuch as GAs [10 11] ANNs [12] and PSO [13 14] Forinstance cGA is a subclass of GA in which each individualis placed in a given topological distribution It is a latticegraph as one individual can only interact with its nearest




Small-World Random


















Do ReMi

Sol La Si

Mi Fa Sol

100 mm200 mm300 mm

500 mm600 mm700 mm

300 mm400 mm500 mm

= Do = Mi = Sol

= 100 mm = 300 mm = 500 mm

x1 x2 x

f(100 300 500)

3

Reservoir22 1

2 3 4

5 6 78 9

10 11 12

1415 16

13

17

18

19

20

2122

23 24

25

26

27

28

29

30

31

1 2 3 4

5 6 7

8 9 10 11

12

1314

15 16

17

18

19

20

21

119863 = 75

119863 = 75 119863 = 75

119863 = 75

119863=75

119863=75

119863=75

119863=75

119863=75

119863=75

119863=200

119863 = 500

119863 = 150

119863 = 150

119863=150

119863=150

119863=250

119863=150

119863=150

119863 = 100119863 = 100119863 = 100

119863 = 300 119863 = 300

119863 = 100 119863 = 100 119863 = 100

119863=100

119863=100119863=100

119863=100





1 1199092) = 100 (119909

2

1minus 1199092

2)2

+ (1 minus 1199091)2


2le 2048

1199091= 1 119909

2= 1 119891 (1 1) = 0


Minimize 119891 (1199091 1199092) = 119886(119909

2minus 1198871199092

1+ 1198881199091minus 119889)2

+ 119890 (1 minus 119891) cos (1199091) + 10

119886 = 1 119887 = 5141205872 119888 = 5120587 119889 = 6 119890 = 10 119891 = 18120587

minus5 le 1199091le 10 0 le 119909

2le 15

(1199091 1199092) = (minus120587 12275) (120587 2275) (942478 2475)

119891 (1199091 1199092) = 0397887


1 1199092) = minus cos (119909

1) cos (119909


1minus 120587)2

minus (1199092minus 120587)2

)


2le 100

(1199091 1199092) = (120587 120587) 119891 (119909

1 1199092) = minus1


Minimize 119891 (1199091 1199092) = [1 + (119909

1+ 1199092+ 1)2(19 minus 14119909

1+ 31199092

1minus 14119909

2+ 611990911199092+ 31199092

2)]

times [30 + (21199091minus 31199092)2

(18 minus 321199091+ 12119909

2

1+ 48119909

2minus 36119909

11199092+ 27119909

2

2)]


2le 2

(1199091 1199092) = (0 minus1) 119891 (119909

1 1199092) = 3


1 1199092) = (4 minus 21119909

2

1+ 1199094

13) 1199092

1+ 11990911199092+ (minus4 + 4119909

2

2) 1199092

2


2le 2

(1199091 1199092) = (minus00898 07126) (00898 minus07126) 119891 (119909

1 1199092) = minus10316






le 119909119894

le 119909119894119880


119894119871and 119909






119894119871 119909119894119880] (119894 = 1 2 119873) as given in (1)

119867119872 =

[[[[[[[[[[[[[

[

1199091

11199091


1

119873

1199092

11199091


2

119873

119909HMSminus11

119909HMSminus12


119909HMS1

119909HMS2


]]]]]]]]]]]]]

]

(1)


1 1199091015840

2 119909

1015840



1199091015840


119895

119894(119895 =



distance bandwidth


1015840= (1199091015840

1 1199091015840

2 119909

1015840

119873) is better


1015840
















Worst 100119864 + 00 744119864 minus 01 100119864 + 00




Best 000119864 + 00 000119864 + 00 000119864 + 00


SD 162119864 minus 06 193119864 minus 06 132119864 + 01









HS



119894= 119909119895



119894= 1199091015840



if 1199091015840

119894gt 119909119894119880

1199091015840

119894= 119909119894119880

elseif 1199091015840119894lt 119909119894119871

1199091015840

119894= 119909119894119871

endend

else1199091015840

119894= 119909119894119871+ rand() times (119909


endend

Algorithm 1





Stopping criteria

End


No

Ok











119899

sum

119894=1

([119909119894+ 05])

2

minus 30 le 119909119894le 30

min (119891) = 119891 (0 0) = 0


119899

sum

119894=1

[1199092

119894minus 10 cos(2120587119909

119894)] minus 512 le 119909

119894le 512 min (119891) =

119891 (0 0) = 0


119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1


119894le 600

min (119891) = 119891 (0 0) = 0


119899

sum

119894=1

1199092

119894)) minus exp((1119899)

119899

sum

119894=1

cos (119888119909119894)) + 119886 + exp(1)

119886 = 20 119887 = 02 119888 = 2120587 minus 32768 le 119909119894le 32768

min (119891) = 119891 (0 0) = 0



Mean 000119864 + 00 109119864 + 00 560119864 + 00

Best 000119864 + 00 000119864 + 00 000119864 + 00

Worst 000119864 + 00 300119864 + 00 120119864 + 01

SD 000119864 + 00 981119864 minus 01 253119864 + 00



Best 278119864 minus 03 112119864 minus 02 101119864 + 00

Worst 100119864 + 00 501119864 + 00 898119864 + 00

SD 195119864 minus 01 131119864 + 00 199119864 + 00









Worst 150119864 + 00 165119864 + 00 346119864 + 00













Stopping criteria

End





No

Ok







5 Conclusions



1 2 3 10 11 12 19 20 21 28 29 30 37 38 39

4 5 6 13 14 15 22 23 24 31 32 33 40 41 42

7 8 9 16 17 18 25 26 27 34 35 36 43 44 45

46 47 48 55 56 57 64 65 66 73 74 75 82 83 84

49 50 51 58 59 60 67 68 69 76 77 78 85 86 87

52 53 54 61 62 63 70 71 72 79 80 81 88 89 90

91 92 93 100 101 102 109 110 111 118 119 120 127 128 129

94 95 96 103 104 105 112 113 114 121 122 123 130 131 132

97 98 99 106 107 108 115 116 117 124 125 126 133 134 135

136 137 138 145 146 147 154 155 156 163 164 165 172 173 174

139 140 141 148 149 150 157 158 159 166 167 168 175 176 177

142 143 144 151 152 153 160 161 162 169 170 171 178 179 180

181 182 183 190 191 192 199 200 201 208 209 210 217 218 219

184 185 186 193 194 195 202 203 204 211 212 213 220 221 222

187 188 189 196 197 198 205 206 207 214 215 216 223 224 225





Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Small-World Random


















Do ReMi

Sol La Si

Mi Fa Sol

100 mm200 mm300 mm

500 mm600 mm700 mm

300 mm400 mm500 mm

= Do = Mi = Sol

= 100 mm = 300 mm = 500 mm

x1 x2 x

f(100 300 500)

3

Reservoir22 1

2 3 4

5 6 78 9

10 11 12

1415 16

13

17

18

19

20

2122

23 24

25

26

27

28

29

30

31

1 2 3 4

5 6 7

8 9 10 11

12

1314

15 16

17

18

19

20

21

119863 = 75

119863 = 75 119863 = 75

119863 = 75

119863=75

119863=75

119863=75

119863=75

119863=75

119863=75

119863=200

119863 = 500

119863 = 150

119863 = 150

119863=150

119863=150

119863=250

119863=150

119863=150

119863 = 100119863 = 100119863 = 100

119863 = 300 119863 = 300

119863 = 100 119863 = 100 119863 = 100

119863=100

119863=100119863=100

119863=100





1 1199092) = 100 (119909

2

1minus 1199092

2)2

+ (1 minus 1199091)2


2le 2048

1199091= 1 119909

2= 1 119891 (1 1) = 0


Minimize 119891 (1199091 1199092) = 119886(119909

2minus 1198871199092

1+ 1198881199091minus 119889)2

+ 119890 (1 minus 119891) cos (1199091) + 10

119886 = 1 119887 = 5141205872 119888 = 5120587 119889 = 6 119890 = 10 119891 = 18120587

minus5 le 1199091le 10 0 le 119909

2le 15

(1199091 1199092) = (minus120587 12275) (120587 2275) (942478 2475)

119891 (1199091 1199092) = 0397887


1 1199092) = minus cos (119909

1) cos (119909


1minus 120587)2

minus (1199092minus 120587)2

)


2le 100

(1199091 1199092) = (120587 120587) 119891 (119909

1 1199092) = minus1


Minimize 119891 (1199091 1199092) = [1 + (119909

1+ 1199092+ 1)2(19 minus 14119909

1+ 31199092

1minus 14119909

2+ 611990911199092+ 31199092

2)]

times [30 + (21199091minus 31199092)2

(18 minus 321199091+ 12119909

2

1+ 48119909

2minus 36119909

11199092+ 27119909

2

2)]


2le 2

(1199091 1199092) = (0 minus1) 119891 (119909

1 1199092) = 3


1 1199092) = (4 minus 21119909

2

1+ 1199094

13) 1199092

1+ 11990911199092+ (minus4 + 4119909

2

2) 1199092

2


2le 2

(1199091 1199092) = (minus00898 07126) (00898 minus07126) 119891 (119909

1 1199092) = minus10316






le 119909119894

le 119909119894119880


119894119871and 119909






119894119871 119909119894119880] (119894 = 1 2 119873) as given in (1)

119867119872 =

[[[[[[[[[[[[[

[

1199091

11199091


1

119873

1199092

11199091


2

119873

119909HMSminus11

119909HMSminus12


119909HMS1

119909HMS2


]]]]]]]]]]]]]

]

(1)


1 1199091015840

2 119909

1015840



1199091015840


119895

119894(119895 =



distance bandwidth


1015840= (1199091015840

1 1199091015840

2 119909

1015840

119873) is better


1015840
















Worst 100119864 + 00 744119864 minus 01 100119864 + 00




Best 000119864 + 00 000119864 + 00 000119864 + 00


SD 162119864 minus 06 193119864 minus 06 132119864 + 01









HS



119894= 119909119895



119894= 1199091015840



if 1199091015840

119894gt 119909119894119880

1199091015840

119894= 119909119894119880

elseif 1199091015840119894lt 119909119894119871

1199091015840

119894= 119909119894119871

endend

else1199091015840

119894= 119909119894119871+ rand() times (119909


endend

Algorithm 1





Stopping criteria

End


No

Ok











119899

sum

119894=1

([119909119894+ 05])

2

minus 30 le 119909119894le 30

min (119891) = 119891 (0 0) = 0


119899

sum

119894=1

[1199092

119894minus 10 cos(2120587119909

119894)] minus 512 le 119909

119894le 512 min (119891) =

119891 (0 0) = 0


119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1


119894le 600

min (119891) = 119891 (0 0) = 0


119899

sum

119894=1

1199092

119894)) minus exp((1119899)

119899

sum

119894=1

cos (119888119909119894)) + 119886 + exp(1)

119886 = 20 119887 = 02 119888 = 2120587 minus 32768 le 119909119894le 32768

min (119891) = 119891 (0 0) = 0



Mean 000119864 + 00 109119864 + 00 560119864 + 00

Best 000119864 + 00 000119864 + 00 000119864 + 00

Worst 000119864 + 00 300119864 + 00 120119864 + 01

SD 000119864 + 00 981119864 minus 01 253119864 + 00



Best 278119864 minus 03 112119864 minus 02 101119864 + 00

Worst 100119864 + 00 501119864 + 00 898119864 + 00

SD 195119864 minus 01 131119864 + 00 199119864 + 00









Worst 150119864 + 00 165119864 + 00 346119864 + 00













Stopping criteria

End





No

Ok







5 Conclusions



1 2 3 10 11 12 19 20 21 28 29 30 37 38 39

4 5 6 13 14 15 22 23 24 31 32 33 40 41 42

7 8 9 16 17 18 25 26 27 34 35 36 43 44 45

46 47 48 55 56 57 64 65 66 73 74 75 82 83 84

49 50 51 58 59 60 67 68 69 76 77 78 85 86 87

52 53 54 61 62 63 70 71 72 79 80 81 88 89 90

91 92 93 100 101 102 109 110 111 118 119 120 127 128 129

94 95 96 103 104 105 112 113 114 121 122 123 130 131 132

97 98 99 106 107 108 115 116 117 124 125 126 133 134 135

136 137 138 145 146 147 154 155 156 163 164 165 172 173 174

139 140 141 148 149 150 157 158 159 166 167 168 175 176 177

142 143 144 151 152 153 160 161 162 169 170 171 178 179 180

181 182 183 190 191 192 199 200 201 208 209 210 217 218 219

184 185 186 193 194 195 202 203 204 211 212 213 220 221 222

187 188 189 196 197 198 205 206 207 214 215 216 223 224 225





Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Do ReMi

Sol La Si

Mi Fa Sol

100 mm200 mm300 mm

500 mm600 mm700 mm

300 mm400 mm500 mm

= Do = Mi = Sol

= 100 mm = 300 mm = 500 mm

x1 x2 x

f(100 300 500)

3

Reservoir22 1

2 3 4

5 6 78 9

10 11 12

1415 16

13

17

18

19

20

2122

23 24

25

26

27

28

29

30

31

1 2 3 4

5 6 7

8 9 10 11

12

1314

15 16

17

18

19

20

21

119863 = 75

119863 = 75 119863 = 75

119863 = 75

119863=75

119863=75

119863=75

119863=75

119863=75

119863=75

119863=200

119863 = 500

119863 = 150

119863 = 150

119863=150

119863=150

119863=250

119863=150

119863=150

119863 = 100119863 = 100119863 = 100

119863 = 300 119863 = 300

119863 = 100 119863 = 100 119863 = 100

119863=100

119863=100119863=100

119863=100





1 1199092) = 100 (119909

2

1minus 1199092

2)2

+ (1 minus 1199091)2


2le 2048

1199091= 1 119909

2= 1 119891 (1 1) = 0


Minimize 119891 (1199091 1199092) = 119886(119909

2minus 1198871199092

1+ 1198881199091minus 119889)2

+ 119890 (1 minus 119891) cos (1199091) + 10

119886 = 1 119887 = 5141205872 119888 = 5120587 119889 = 6 119890 = 10 119891 = 18120587

minus5 le 1199091le 10 0 le 119909

2le 15

(1199091 1199092) = (minus120587 12275) (120587 2275) (942478 2475)

119891 (1199091 1199092) = 0397887


1 1199092) = minus cos (119909

1) cos (119909


1minus 120587)2

minus (1199092minus 120587)2

)


2le 100

(1199091 1199092) = (120587 120587) 119891 (119909

1 1199092) = minus1


Minimize 119891 (1199091 1199092) = [1 + (119909

1+ 1199092+ 1)2(19 minus 14119909

1+ 31199092

1minus 14119909

2+ 611990911199092+ 31199092

2)]

times [30 + (21199091minus 31199092)2

(18 minus 321199091+ 12119909

2

1+ 48119909

2minus 36119909

11199092+ 27119909

2

2)]


2le 2

(1199091 1199092) = (0 minus1) 119891 (119909

1 1199092) = 3


1 1199092) = (4 minus 21119909

2

1+ 1199094

13) 1199092

1+ 11990911199092+ (minus4 + 4119909

2

2) 1199092

2


2le 2

(1199091 1199092) = (minus00898 07126) (00898 minus07126) 119891 (119909

1 1199092) = minus10316






le 119909119894

le 119909119894119880


119894119871and 119909






119894119871 119909119894119880] (119894 = 1 2 119873) as given in (1)

119867119872 =

[[[[[[[[[[[[[

[

1199091

11199091


1

119873

1199092

11199091


2

119873

119909HMSminus11

119909HMSminus12


119909HMS1

119909HMS2


]]]]]]]]]]]]]

]

(1)


1 1199091015840

2 119909

1015840



1199091015840


119895

119894(119895 =



distance bandwidth


1015840= (1199091015840

1 1199091015840

2 119909

1015840

119873) is better


1015840
















Worst 100119864 + 00 744119864 minus 01 100119864 + 00




Best 000119864 + 00 000119864 + 00 000119864 + 00


SD 162119864 minus 06 193119864 minus 06 132119864 + 01









HS



119894= 119909119895



119894= 1199091015840



if 1199091015840

119894gt 119909119894119880

1199091015840

119894= 119909119894119880

elseif 1199091015840119894lt 119909119894119871

1199091015840

119894= 119909119894119871

endend

else1199091015840

119894= 119909119894119871+ rand() times (119909


endend

Algorithm 1





Stopping criteria

End


No

Ok











119899

sum

119894=1

([119909119894+ 05])

2

minus 30 le 119909119894le 30

min (119891) = 119891 (0 0) = 0


119899

sum

119894=1

[1199092

119894minus 10 cos(2120587119909

119894)] minus 512 le 119909

119894le 512 min (119891) =

119891 (0 0) = 0


119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1


119894le 600

min (119891) = 119891 (0 0) = 0


119899

sum

119894=1

1199092

119894)) minus exp((1119899)

119899

sum

119894=1

cos (119888119909119894)) + 119886 + exp(1)

119886 = 20 119887 = 02 119888 = 2120587 minus 32768 le 119909119894le 32768

min (119891) = 119891 (0 0) = 0



Mean 000119864 + 00 109119864 + 00 560119864 + 00

Best 000119864 + 00 000119864 + 00 000119864 + 00

Worst 000119864 + 00 300119864 + 00 120119864 + 01

SD 000119864 + 00 981119864 minus 01 253119864 + 00



Best 278119864 minus 03 112119864 minus 02 101119864 + 00

Worst 100119864 + 00 501119864 + 00 898119864 + 00

SD 195119864 minus 01 131119864 + 00 199119864 + 00









Worst 150119864 + 00 165119864 + 00 346119864 + 00













Stopping criteria

End





No

Ok







5 Conclusions



1 2 3 10 11 12 19 20 21 28 29 30 37 38 39

4 5 6 13 14 15 22 23 24 31 32 33 40 41 42

7 8 9 16 17 18 25 26 27 34 35 36 43 44 45

46 47 48 55 56 57 64 65 66 73 74 75 82 83 84

49 50 51 58 59 60 67 68 69 76 77 78 85 86 87

52 53 54 61 62 63 70 71 72 79 80 81 88 89 90

91 92 93 100 101 102 109 110 111 118 119 120 127 128 129

94 95 96 103 104 105 112 113 114 121 122 123 130 131 132

97 98 99 106 107 108 115 116 117 124 125 126 133 134 135

136 137 138 145 146 147 154 155 156 163 164 165 172 173 174

139 140 141 148 149 150 157 158 159 166 167 168 175 176 177

142 143 144 151 152 153 160 161 162 169 170 171 178 179 180

181 182 183 190 191 192 199 200 201 208 209 210 217 218 219

184 185 186 193 194 195 202 203 204 211 212 213 220 221 222

187 188 189 196 197 198 205 206 207 214 215 216 223 224 225





Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












le 119909119894

le 119909119894119880


119894119871and 119909






119894119871 119909119894119880] (119894 = 1 2 119873) as given in (1)

119867119872 =

[[[[[[[[[[[[[

[

1199091

11199091


1

119873

1199092

11199091


2

119873

119909HMSminus11

119909HMSminus12


119909HMS1

119909HMS2


]]]]]]]]]]]]]

]

(1)


1 1199091015840

2 119909

1015840



1199091015840


119895

119894(119895 =



distance bandwidth


1015840= (1199091015840

1 1199091015840

2 119909

1015840

119873) is better


1015840
















Worst 100119864 + 00 744119864 minus 01 100119864 + 00




Best 000119864 + 00 000119864 + 00 000119864 + 00


SD 162119864 minus 06 193119864 minus 06 132119864 + 01









HS



119894= 119909119895



119894= 1199091015840



if 1199091015840

119894gt 119909119894119880

1199091015840

119894= 119909119894119880

elseif 1199091015840119894lt 119909119894119871

1199091015840

119894= 119909119894119871

endend

else1199091015840

119894= 119909119894119871+ rand() times (119909


endend

Algorithm 1





Stopping criteria

End


No

Ok











119899

sum

119894=1

([119909119894+ 05])

2

minus 30 le 119909119894le 30

min (119891) = 119891 (0 0) = 0


119899

sum

119894=1

[1199092

119894minus 10 cos(2120587119909

119894)] minus 512 le 119909

119894le 512 min (119891) =

119891 (0 0) = 0


119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1


119894le 600

min (119891) = 119891 (0 0) = 0


119899

sum

119894=1

1199092

119894)) minus exp((1119899)

119899

sum

119894=1

cos (119888119909119894)) + 119886 + exp(1)

119886 = 20 119887 = 02 119888 = 2120587 minus 32768 le 119909119894le 32768

min (119891) = 119891 (0 0) = 0



Mean 000119864 + 00 109119864 + 00 560119864 + 00

Best 000119864 + 00 000119864 + 00 000119864 + 00

Worst 000119864 + 00 300119864 + 00 120119864 + 01

SD 000119864 + 00 981119864 minus 01 253119864 + 00



Best 278119864 minus 03 112119864 minus 02 101119864 + 00

Worst 100119864 + 00 501119864 + 00 898119864 + 00

SD 195119864 minus 01 131119864 + 00 199119864 + 00









Worst 150119864 + 00 165119864 + 00 346119864 + 00













Stopping criteria

End





No

Ok







5 Conclusions



1 2 3 10 11 12 19 20 21 28 29 30 37 38 39

4 5 6 13 14 15 22 23 24 31 32 33 40 41 42

7 8 9 16 17 18 25 26 27 34 35 36 43 44 45

46 47 48 55 56 57 64 65 66 73 74 75 82 83 84

49 50 51 58 59 60 67 68 69 76 77 78 85 86 87

52 53 54 61 62 63 70 71 72 79 80 81 88 89 90

91 92 93 100 101 102 109 110 111 118 119 120 127 128 129

94 95 96 103 104 105 112 113 114 121 122 123 130 131 132

97 98 99 106 107 108 115 116 117 124 125 126 133 134 135

136 137 138 145 146 147 154 155 156 163 164 165 172 173 174

139 140 141 148 149 150 157 158 159 166 167 168 175 176 177

142 143 144 151 152 153 160 161 162 169 170 171 178 179 180

181 182 183 190 191 192 199 200 201 208 209 210 217 218 219

184 185 186 193 194 195 202 203 204 211 212 213 220 221 222

187 188 189 196 197 198 205 206 207 214 215 216 223 224 225





Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894= 119909119895



119894= 1199091015840



if 1199091015840

119894gt 119909119894119880

1199091015840

119894= 119909119894119880

elseif 1199091015840119894lt 119909119894119871

1199091015840

119894= 119909119894119871

endend

else1199091015840

119894= 119909119894119871+ rand() times (119909


endend

Algorithm 1





Stopping criteria

End


No

Ok











119899

sum

119894=1

([119909119894+ 05])

2

minus 30 le 119909119894le 30

min (119891) = 119891 (0 0) = 0


119899

sum

119894=1

[1199092

119894minus 10 cos(2120587119909

119894)] minus 512 le 119909

119894le 512 min (119891) =

119891 (0 0) = 0


119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1


119894le 600

min (119891) = 119891 (0 0) = 0


119899

sum

119894=1

1199092

119894)) minus exp((1119899)

119899

sum

119894=1

cos (119888119909119894)) + 119886 + exp(1)

119886 = 20 119887 = 02 119888 = 2120587 minus 32768 le 119909119894le 32768

min (119891) = 119891 (0 0) = 0



Mean 000119864 + 00 109119864 + 00 560119864 + 00

Best 000119864 + 00 000119864 + 00 000119864 + 00

Worst 000119864 + 00 300119864 + 00 120119864 + 01

SD 000119864 + 00 981119864 minus 01 253119864 + 00



Best 278119864 minus 03 112119864 minus 02 101119864 + 00

Worst 100119864 + 00 501119864 + 00 898119864 + 00

SD 195119864 minus 01 131119864 + 00 199119864 + 00









Worst 150119864 + 00 165119864 + 00 346119864 + 00













Stopping criteria

End





No

Ok







5 Conclusions



1 2 3 10 11 12 19 20 21 28 29 30 37 38 39

4 5 6 13 14 15 22 23 24 31 32 33 40 41 42

7 8 9 16 17 18 25 26 27 34 35 36 43 44 45

46 47 48 55 56 57 64 65 66 73 74 75 82 83 84

49 50 51 58 59 60 67 68 69 76 77 78 85 86 87

52 53 54 61 62 63 70 71 72 79 80 81 88 89 90

91 92 93 100 101 102 109 110 111 118 119 120 127 128 129

94 95 96 103 104 105 112 113 114 121 122 123 130 131 132

97 98 99 106 107 108 115 116 117 124 125 126 133 134 135

136 137 138 145 146 147 154 155 156 163 164 165 172 173 174

139 140 141 148 149 150 157 158 159 166 167 168 175 176 177

142 143 144 151 152 153 160 161 162 169 170 171 178 179 180

181 182 183 190 191 192 199 200 201 208 209 210 217 218 219

184 185 186 193 194 195 202 203 204 211 212 213 220 221 222

187 188 189 196 197 198 205 206 207 214 215 216 223 224 225





Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119899

sum

119894=1

([119909119894+ 05])

2

minus 30 le 119909119894le 30

min (119891) = 119891 (0 0) = 0


119899

sum

119894=1

[1199092

119894minus 10 cos(2120587119909

119894)] minus 512 le 119909

119894le 512 min (119891) =

119891 (0 0) = 0


119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1


119894le 600

min (119891) = 119891 (0 0) = 0


119899

sum

119894=1

1199092

119894)) minus exp((1119899)

119899

sum

119894=1

cos (119888119909119894)) + 119886 + exp(1)

119886 = 20 119887 = 02 119888 = 2120587 minus 32768 le 119909119894le 32768

min (119891) = 119891 (0 0) = 0



Mean 000119864 + 00 109119864 + 00 560119864 + 00

Best 000119864 + 00 000119864 + 00 000119864 + 00

Worst 000119864 + 00 300119864 + 00 120119864 + 01

SD 000119864 + 00 981119864 minus 01 253119864 + 00



Best 278119864 minus 03 112119864 minus 02 101119864 + 00

Worst 100119864 + 00 501119864 + 00 898119864 + 00

SD 195119864 minus 01 131119864 + 00 199119864 + 00









Worst 150119864 + 00 165119864 + 00 346119864 + 00













Stopping criteria

End





No

Ok







5 Conclusions



1 2 3 10 11 12 19 20 21 28 29 30 37 38 39

4 5 6 13 14 15 22 23 24 31 32 33 40 41 42

7 8 9 16 17 18 25 26 27 34 35 36 43 44 45

46 47 48 55 56 57 64 65 66 73 74 75 82 83 84

49 50 51 58 59 60 67 68 69 76 77 78 85 86 87

52 53 54 61 62 63 70 71 72 79 80 81 88 89 90

91 92 93 100 101 102 109 110 111 118 119 120 127 128 129

94 95 96 103 104 105 112 113 114 121 122 123 130 131 132

97 98 99 106 107 108 115 116 117 124 125 126 133 134 135

136 137 138 145 146 147 154 155 156 163 164 165 172 173 174

139 140 141 148 149 150 157 158 159 166 167 168 175 176 177

142 143 144 151 152 153 160 161 162 169 170 171 178 179 180

181 182 183 190 191 192 199 200 201 208 209 210 217 218 219

184 185 186 193 194 195 202 203 204 211 212 213 220 221 222

187 188 189 196 197 198 205 206 207 214 215 216 223 224 225





Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Stopping criteria

End





No

Ok







5 Conclusions



1 2 3 10 11 12 19 20 21 28 29 30 37 38 39

4 5 6 13 14 15 22 23 24 31 32 33 40 41 42

7 8 9 16 17 18 25 26 27 34 35 36 43 44 45

46 47 48 55 56 57 64 65 66 73 74 75 82 83 84

49 50 51 58 59 60 67 68 69 76 77 78 85 86 87

52 53 54 61 62 63 70 71 72 79 80 81 88 89 90

91 92 93 100 101 102 109 110 111 118 119 120 127 128 129

94 95 96 103 104 105 112 113 114 121 122 123 130 131 132

97 98 99 106 107 108 115 116 117 124 125 126 133 134 135

136 137 138 145 146 147 154 155 156 163 164 165 172 173 174

139 140 141 148 149 150 157 158 159 166 167 168 175 176 177

142 143 144 151 152 153 160 161 162 169 170 171 178 179 180

181 182 183 190 191 192 199 200 201 208 209 210 217 218 219

184 185 186 193 194 195 202 203 204 211 212 213 220 221 222

187 188 189 196 197 198 205 206 207 214 215 216 223 224 225





Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 10 11 12 19 20 21 28 29 30 37 38 39

4 5 6 13 14 15 22 23 24 31 32 33 40 41 42

7 8 9 16 17 18 25 26 27 34 35 36 43 44 45

46 47 48 55 56 57 64 65 66 73 74 75 82 83 84

49 50 51 58 59 60 67 68 69 76 77 78 85 86 87

52 53 54 61 62 63 70 71 72 79 80 81 88 89 90

91 92 93 100 101 102 109 110 111 118 119 120 127 128 129

94 95 96 103 104 105 112 113 114 121 122 123 130 131 132

97 98 99 106 107 108 115 116 117 124 125 126 133 134 135

136 137 138 145 146 147 154 155 156 163 164 165 172 173 174

139 140 141 148 149 150 157 158 159 166 167 168 175 176 177

142 143 144 151 152 153 160 161 162 169 170 171 178 179 180

181 182 183 190 191 192 199 200 201 208 209 210 217 218 219

184 185 186 193 194 195 202 203 204 211 212 213 220 221 222

187 188 189 196 197 198 205 206 207 214 215 216 223 224 225





Acknowledgment


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article smallest-small-world cellular harmony...

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