Novel global harmony search algorithm for unconstrained problems

Letters

Novel global harmony search algorithm for unconstrained problems

Dexuan Zou a,?,Liqun Gao a,Jianhua Wu a,Steven Li b

a School of Information Science and Engineering,Northeastern University,Shenyang,Liaoning110004,PR China

b Division of Business University of South Australia GPO Box2471,Adelaide,SA5001,Australia

a r t i c l e i n f o

Article history:

Received3January2010

Received in revised form

30July2010

Accepted31July2010

Communicated by Y.Liu

Available online27August2010

Keywords:

Novel global harmony search algorithm

Position updating

Genetic mutation

Convergence

Stability

Harmony search algorithm

a b s t r a c t

In this paper,we use a recently proposed algorithm—novel global harmony search(NGHS)algorithm to

solve unconstrained problems.The NGHS algorithm includes two important operations:position

updating and genetic mutation with a low probability.The former can enhance the convergence of the

NGHS,and the latter can effectively prevent the NGHS from being trapped into the local optimum.

Based on a large number of experiments,the NGHS has demonstrated stronger convergence and

stability than original harmony search(HS)algorithm and its two improved algorithms(IHS and SGHS).

&2010Elsevier B.V.All rights reserved.

1.Introduction

Due to the computational drawbacks of existing numerical

methods,researchers have to rely on metaheuristic algorithms

based on simulations to solve some complex optimization

problems.A common feature in meta-heuristic algorithms is that

they combine rules and randomness to imitate natural phenomena.

These phenomena include the biological evolutionary process(e.g.,

the genetic algorithm(GA)[1]and the differential evolution(DE)

[2]),animal behavior(e.g.,particle swarm optimization(PSO)[3]

and ant colony algorithm(ACA)[4]),and the physical annealing

process(e.g.,simulated annealing(SA)[5]).Over the last decades,

many meta-heuristic algorithms and their improved algorithms

have been successfully applied to various engineering optimization

problems[6–12].They have outperformed conventional numerical

methods on providing better solutions for some dif?cult and

complicated real-world optimization problems.

Geem et al.proposed a new metaheuristic algorithm—harmony

search(HS)[13]in2001,which is inspired by the phenomenon of

musician attuning.The HS algorithm is based on natural musical

performance processes that occur when a musician searches for a

better state of harmony,such as during jazz improvisation.Jazz

improvisation seeks to?nd musically pleasing harmony as

determined by an aesthetic standard,just as the optimization

process seeks to?nd a global solution as determined by an

objective function.Due to its own advantage and potential,the HS

has been successfully applied to many engineering optimization

problems[14–17]during the last decade.Recently,many improved

harmony search algorithms have been proposed,such as:improved

harmony search(IHS)algorithm[18],global-best harmony search

(GHS)algorithm[19],self-adaptive global-best harmony search

(SGHS)algorithm[20],and local-best harmony search algorithm

with dynamic subpopulations(DLHS)[21].In addition,Das et al.

presented a simple mathematical analysis of the explorative search

behavior of the HS[22],and proposed a simple but very useful

modi?cation to the classical HS in light of the mathematical

analysis.

On the other hand,the novel global harmony search(NGHS)

algorithm[23]is a recently proposed algorithm,and it has been

successfully applied into constrained reliability problems.The

NGHS is an improved version of the HS,and it has shown better

performance than the HS on solving constrained reliability

problems.In this paper,we use the NGHS to solve unconstrained

problems so as to profoundly study and analyze the performance

of the NGHS.We also compare the NGHS with the other three

harmony search algorithms on solving unconstrained problems.

Experimental results indicate that the NGHS has stronger

convergence and stability than the HS and its two improved

algorithms[18,20].

This paper is organized as follows.In Section2,a brief

overview of three harmony search algorithms is provided,and

they are the original harmony search(HS)algorithm,the

Contents lists available at ScienceDirect

journal homepage:https://www.360docs.net/doc/966346959.html,/locate/neucom

Neurocomputing

0925-2312/$-see front matter&2010Elsevier B.V.All rights reserved.

doi:10.1016/j.neucom.2010.07.010

?Corresponding author.

E-mail address:zoudexuan@https://www.360docs.net/doc/966346959.html,(D.Zou).

Neurocomputing73(2010)3308–3318

improved harmony search(IHS)algorithm,and the self-adaptive global-best harmony search(SGHS)algorithm,respectively.In Section3,a novel global harmony search(NGHS)is introduced, and the procedure of the NGHS is described adequately.In Section 4,a large number of experiments are carried out to test and compare the performance of the NGHS and the other three algorithms.We end this paper with some conclusions and comments for further research in Section5.

2.HS,IHS and SGHS algorithms

2.1.The HS algorithm

In this section,we will introduce three harmony search algorithms.The?rst is the original harmony search(HS) algorithm,and it works as follows:

Step1:Initialize the HS algorithm parameters.

The parameters are as follows:the harmony memory size (HMS),or the number of solution vectors in the harmony memory;harmony memory considering rate(HMCR);bandwidth (bw);pitch adjusting rate(PAR);and the number of improvisa-tions(K),or stopping criterion.

Step2:Initialize the harmony memory.

The initial harmony memory(HM)is generated from a uniform distribution in the ranges[x iL,x iU]ei?1,2,...,NT,as

HM?

x1

1

x1

2

áááx1

N

x2

1

x2

2

áááx2

N

^^&^

x HMS

1

x HMS

2

áááx HMS

N

B B

B B

@

1

C C

C C

A

e1T

Step3:Improvise a new harmony.

Generating a new harmony is called improvisation.The new harmony vector x u?ex u1,x u2,...,x u NTis determined by three rules: memory consideration,pitch adjustment and random selection. The procedure works as follows:

for each i A?1,N do

if randeTr HMCR then

x u i?x j

i

ej?1,2,...,HMST%memory consideration

if randeTr PAR then

x u i?x u i7r?bw%pitch adjustment

if x u i4x iU

x u i?x iU

elseif x u i o x iL

x u i?x iL

end

end

else

x u i?x iLtrandeT?ex iUàx iLT%random selection end

end

x u iei?1,2,...,NTis the i th component of x u,and x i j ej?1,2,...,HMSTis the i th component of the j th candidate solution vector in HM.Both r and rand()are uniformly generated random numbers in the region of[0,1],and bw is an arbitrary distance bandwidth.

Step4:Update harmony memory.

If the?tness of the improvised harmony vector x u?ex u1,x u2,...,x u NTis better than that of the worst harmony,replace the worst harmony in the HM with x u.

Step5:Check the stopping criterion.

If the maximal iteration number(K)is satis?ed,computation is terminated.Otherwise,Steps3and4are repeated.

The most important step of the HS algorithm is Step3,and it includes memory consideration,pitch adjustment and random selection.PAR and bw have profound effects on the performance of the HS.Specially,suitable PAR and bw values can not only diversify harmony memory(HM),but also improve the quality of harmonies in HM.

2.2.The IHS algorithm

Mahdavi et al.[18]proposed a new variant of the HS,called the improved harmony search(IHS).The IHS dynamically updates PAR and bw according to

PARekT?PAR mint

PAR maxàPAR min

K

ke2TbwekT?bw max exp

ln bw min

bw max

k

@

1

Ae3T

where K is the maximum number of iterations,and k is the current number of iterations;PAR min and PAR max are the minimum adjusting rate and the maximum adjusting rate,respectively; bw min and bw max are the minimum bandwidth and the maximum bandwidth,respectively.A large number of experiments and studies show that the IHS based on improved PAR and bw has better optimization performance than the HS in most cases.

2.3.The SGHS algorithm

Pan et al.[20]proposed a self-adaptive global-best harmony search(SGHS)algorithm,which is inspired by the GHS algorithm [19].The best harmony vector is utilized to produce a new vector, and a dynamically changing BW is introduced to well balance the exploration and exploitation of the SGHS algorithm.In addition, HCMR and PAR are dynamically adapted to a suitable range by recording their historic values corresponding to generated harmonies entering the HM.Specially,The HMCR(PAR)value is normally distributed in the range of[0.9,1.0]([0.0,1.0])with mean HMCRm(PARm)and standard deviation0.01(0.05). Initially,HMCRm(PARm)is set at0.98(0.9).And then SGHS starts with a HMCR(PAR)value generated according to the normal distribution.During the evolution,the HMCR(PAR)value associated with the generated harmony successfully replacing the worst member in the HM is recorded.After a speci?ed number of generations LP(i.e.,the learning period which is100in their experiment),HMCRm(PARm)is recalculated by averaging all the recorded HMCR(PAR)values during this period.With the new mean and the given standard deviation of0.01(0.05),new HMCR (PAR)value is produced and used in the subsequent iterations. The above procedure is repeated.As a result,an appropriate HMCR(PAR)value can be gradually learned to suit the particular problem and the particular phases of the search process.

3.The novel global harmony search(NGHS)algorithm

A feature of PSO is that the individual is inclined to mimic its successful companion.Inspired by the swarm intelligence of particle swarm,a new variation of HS is proposed in this paper. The new approach,called NGHS,modi?es the improvisation step of the HS such that the new harmony can mimic the global-best harmony in the HM.The NGHS and the HS are different in three aspects as follows:

(1)In Step1,harmony memory considering rate(HMCR)and pitch adjusting rate(PAR)are excluded from the NGHS,and genetic mutation probability(p m)is included in the NGHS;

D.Zou et al./Neurocomputing73(2010)3308–33183309

(2)In Step 3,the NGHS modi?es improvisation step of the HS,and it works as follows:

for each i A ?1,N do x R ?2?x best i àx worst i if x R 4x iU x R ?x iU

elseif x R o x iL x R ?x iL end

x u i ?x worst i tr ?ex R àx worst i T%position updating if rand eTr p m then

x u i ?x iL trand eT?ex iU àiL T%genetic mutation end end

Here,‘‘best’’and ‘‘worst’’are the indexes of the best harmony

and the worst harmony in HM,respectively.r and rand ()are all uniformly generated random number in [0,1].

Fig.1is used to illustrate the principle of position updating:step i ?j x best i àx worst i

j is de?ned as adaptive step of the i th decision variable.The region between P and R is de?ned as trust region for the i th decision variable.The trust region is actually a region near the global-best harmony.A reasonable explanation is as follows:In the early stage of optimization,all solution vectors are sporadic in solution space,so most adaptive steps are large and most trust regions are wide,which is bene?cial to the global search of the NGHS;while in the late stage of optimization,all non-best solution vectors are inclined to move to the global-best solution vector,so most solution vectors are close to each other.In this case,most adaptive steps are small and most trust regions are narrow,which is bene?cial to the local search of the NGHS;the reasonable design for step i can guarantee that the proposed algorithm has strong global search ability in the early stage of optimization,and has strong local search ability in the late stage of optimization.Dynamically adjusted step i keeps a balance between the global search and the local search.

Genetic mutation operation with a small probability is carried out for the worst harmony of harmony memory after updating position,for it can enhance the capacity of escaping from the local optimum for the proposed algorithm.

(3)In Step 4,the NGHS replaces the worst harmony x worst in HM with the new harmony x u even if x u is worse than x worst .

Now,we will illustrate that the modi?cations introduced in this paper are compatible with the HS framework.Detailed illustrations are as follows:

(1)The HS randomly selects harmony components from the harmony memory (HM)with the probability HMCR.On the other hand,the NGHS selects the best harmony components from the HM with probability 100%.Both algorithms aim at using existing good harmonies to improvise a new harmony.

(2)For the HS,pitch adjustment will be carried out if both HMCR and PAR are satis?ed,and the adjusting step bw is de?ned as distance bandwidth.For the NGHS,location updating is carried out with probability 100%,and adjusting step step i ei ?1,2,...,N Tis de?ned as adaptive step.Similarly,both bw and step i are

utilized to determine the promising search ranges of their corresponding algorithms.

(3)The HS carries out random selection with the probability 1-HMCR,and the NGHS carries out genetic mutation with the probability p m .In fact,both operations are exactly the same,and they are used to diversify the harmony memory,which can effectively prevent both algorithms from being trapped into the local optimum.

4.Experimental results and analysis

Sixteen test functions are considered to extensively investigate the performance of the NGHS algorithm,and they are presented as follows:

The ?rst is Sphere function,de?ned as min f 1?

X N i ?1

x 2i

e4T

where global optimum x *?(0,0,y ,0)and f (x *)?0for à100r x i r 100.

The second is Rosenbrock function,de?ned as min f 2?

X N à1i ?1

e100ex i t1àx 2i T2tex i à1T2

T

e5T

where global optimum x *?(1,1,y ,1)and f (x *)?0for à100r

x i r 100.

The third is generalized Rastrigrin function,de?ned as min f 3?

X N i ?1

ex 2i à10cos e2p x i Tt10T

e6T

where global optimum x *?(0,0,y ,0)and f (x *)?0for à10r

x i r 10.

The fourth is generalized Griewank function,de?ned as

min f 4?14000X N i ?1x 2

i àY N i ?1cos x i ?i p t1

e7Twhere global optimum x *?(0,0,y ,0)and f (x *)?0for à600r

x i r 600.

The ?fth is Ackley’s function,de?ned as

min f 5?20te à20exp à0:2

???????????????????P N i ?1x 2i N

s 0@1A àexp P N i ?1

cos e2p x i TN !e8T

where global optimum x *?(0,0,y ,0)and f (x *)?0for à32r

x i r 32.

The sixth is Schwefel’s problem 2.22,de?ned as min f 6?

X N i ?1

j x i jt

Y N i ?1

j x i j

e9T

where global optimum x *?(0,0,y ,0)and f (x *)?0for à100r

x i r 100.

The seventh is Schwefel’s problem 2.26,de?ned as min f 7?418:9829N à

X N i ?1

ex i sin e

???????

j x i j p TT

e10T

where global optimum x *?(420.9687,420.9687,y ,420.9687)and f (x *)?0for à500r x i r 500.

The eighth is rotated hyper-ellipsoid function,de?ned as

min f 8?

X N i ?1

X i j ?1

x j 0@1A 2e11

TFig.1.The schematic diagram of position updating.

D.Zou et al./Neurocomputing 73(2010)3308–3318

3310

where global optimum x*?(0,0,y,0)and f(x*)?0forà100r x i r100.

The ninth[24]is as follows:

min f9?

X N

i?1z2

i

à450e12T

where z?xào;o?[o1,o2,y,o n]is the shifted global optimum; global optimum x*?o and f(x*)?à450forà100r x i r100.

The10th[24]is as follows:

min f10?

X N

i?1

X i

j?1

z j

@

1

A

2

à450e13T

where z?xào;o?[o1,o2,y,o N]is the shifted global optimum; global optimum x*?o and f(x*)?à450forà100r x i r100.

The11th[24]is as follows:

min f11?

X N

i?1e106Teià1T=eNà1Tz2

i

à450e14T

where z?exàoT?M,and o?[o1,o2,y,o N]is the shifted global optimum,M is orthogonal matrix;global optimum x*?o and f(x*)?à450forà100r x i r100.

The12th[24]is as follows:

min f12?

X N

i?1

X i

j?1

z j

@

1

A

2

e1t0:4?j Ne0,1TjTà450e15T

where z?xào;o?[o1,o2,y,o N]is the shifted global optimum; global optimum x*?o and f(x*)?à450forà100r x i r100.

The13th[24]is as follows:

min f13?X

Nà1

i?1

e100ez it1àz2

i

T2tez ià1T2Tt390e16T

where z?xào+1;o?[o1,o2,y,o N]is the shifted global optimum; global optimum x*?o and f(x*)?390forà100r x i r100.

The14th[24]is as follows:

min f14?

X N

i?1z2

ià

Y N

i?1

cos

z i

?

i

p

t1à180e17T

where z?exàoT?M,and o?[o1,o2,y,o N]is the shifted global optimum,M is linear transformation matrix;global optimum x*?o and f(x*)?à180forà600r x i r600.

The15th[24]is as follows:

min f15?20teà20expà0:2??????????????????P N

i?1

z2

i

s 0

@1 A

àexp P N

i?1

cose2p z iT

!

à140e18T

where z?exàoT?M,and o?[o1,o2,y,o N]is the shifted global optimum,specially,o2jà1?à32,and o2j are randomly distributed in the search range,for j?1,2,...b N=2c.M is linear transformation matrix;global optimum x*?o and f(x*)?à140forà32r x i r32.

The16th[24]is as follows:

min f16?

X N

i?1ez2

i

à10cose2p z iTt10Tà330e19T

where z?xào;o?[o1,o2,y,o N]is the shifted global optimum; global optimum x*?o and f(x*)?à330forà5r x i https://www.360docs.net/doc/966346959.html,parison between NGHS,HS,IHS and SGHS algorithms

We experimentally compare the NGHS with the other three harmony search algorithms on a large number of problem instances(f1–f16).In the experiments,the settings of the compared HS algorithms are as follows:For the HS algorithm, HMS?5,HMCR?0.9,PAR?0.3and bw?0.01;For the IHS algorithm,HMS?5,HMCR?0.9,PAR min?0.01,PAR max?0.99, bw min?0.0001,and bw max?(x Uàx L)/20;For the SGHS algorithm, HMS?5,HMCRm?0.98,PARm?0.9,bw min?0.0005,bw max?(x Uàx L)/10,and LP?100;For the NGHS,HMS?5,p m?0.005.

According to Sections2.2and2.3,both the IHS and the SGHS dynamically adjust their algorithm parameters throughout the whole iterations,which adds additional computing time.On the other hand,the NGHS uses position updating to replace the harmony memory considering and pitch adjusting of the HS,and its mutation operation is the same as the random selection of the HS.It seems that the HS and the NGHS have similar computational complexity. To provide a more fair comparison,we increase the maximal iteration numbers of both the HS and the NGHS when the maximal iteration numbers of both the IHS and the SGHS are?xed.For each function,three different dimension sizes are tested,and they are N?10,30and50,respectively.Thirty independent experiments are carried out in each case,and the optimization results using four harmony search algorithms are reported in Tables1–3.

Here,‘‘K’’is the maximal iteration number,‘‘AOT’’represents average optimization time,and‘‘SD’’represents standard devia-tion.From Table1,we can see that the NGHS always uses the least average optimization time to solve any test function even if its maximal iteration number is larger than those of the IHS and the SGHS.The NGHS can?nd the best objective function values for all test functions except f14and f15.The IHS found the best objective function values for f14and f15,and the NGHS results are comparable to both best IHS results.

From Table2,it can be observed that the average optimization times using the NGHS are always less than those of the other three harmony search algorithms for all test functions except f12, and the AOT of the NGHS is slightly larger than that of the SGHS for f12.According to four criterions(Best,Worst,Mean and SD), the NGHS outperforms the other three harmony search algo-rithms for all test functions except f2and f15.More specially,the Mean and SD obtained by the NGHS are better than those of the other three approaches for most test functions,which indicates the strong convergence and stability of the NGHS.Based on the above simulation analysis,the NGHS can generate better solutions with smaller computational requirements compared to the other three approaches in most cases.

It can be seen from Table3,that NGHS performs better than the other three approaches in terms of solution accuracy on the utilized16test functions.Specially,the NGHS algorithm gen-erates15best results out of16functions,and the worst results using the NGHS are even better than the best results using three harmony search algorithms for f1,f5,f6,f7,f8,f10,f12,f14and f16.So, the HS,the IHS and the SGHS cannot show better results than NGHS even on one run for any of these test functions.

According to the above observations,we concluded that the performance of the NGHS is superior to the other three methods when used to solve most of these unconstrained optimization problems.More experiments,especially with larger problems may, however,be needed in order to con?rm such good performance.

4.2.Effects of p m and HMS

In this section,all the above16test functions are used to investigate the effects of p m and HMS on the performance of the

D.Zou et al./Neurocomputing73(2010)3308–33183311

Table 1

The optimization results of four harmony search algorithms for f 1àf 16(N ?10).Function Algorithm K AOT

Best

Worst

Mean

SD

f 1

HS 30000 5.7660e à001 2.9129e à008 6.1045e à007 2.0999e à007 1.2421e à007IHS 10000 6.3643e à001 6.1585e à009 3.3831e à008 2.1536e à008 6.4556e à009SGHS 100008.7347e à001 3.1385e à009 5.1117e à0028.6474e à003 1.1695e à002NGHS 20000 3.6770e à001 3.9296e à041 2.9471e à031 2.1333e à032 6.9650e à032f 2

HS 30000 5.8433e à001 1.6436e à002 4.9914e+003 2.3417e+0029.0439e+002IHS 10000 6.3910e à001 6.5349e+0009.4594e+003 1.5642e+003 2.8488e+003SGHS 100008.8440e à001 5.0470e à001 5.1306e+003 2.5233e+0029.2618e+002NGHS 20000 3.7760e à001 5.3463e à004 5.4967e+003 4.9368e+002 1.3524e+003f 3

HS 30000 6.3130e à001 1.1661e à005 1.3006e à004 4.5934e à005 2.5863e à005IHS 10000 6.6923e à001 2.4508e à006 2.9849e+0009.7833e à0018.8010e à001SGHS 100008.9847e à001 5.4866e à008 2.3348e+000 5.2420e à0017.9395e à001NGHS 20000 4.0100e à0010

9.9496e à001 3.3165e à002 1.8165e à001f 4

HS 30000 6.8443e à001 2.6485e à007 1.4754e à001 6.0457e à002 3.6097e à002IHS 10000 6.7507e à0019.0619e à003 2.0167e à0017.6168e à002 4.6718e à002SGHS 100009.1817e à001 5.4412e à002 2.6813e à001 1.4351e à001 5.7705e à002NGHS 20000 4.1453e à0010

1.2046e à001 6.8529e à002 3.3204e à002f 5

HS 30000 6.2083e à001 2.6202e à004 1.0237e à003 5.4888e à004 1.7181e à004IHS 10000 6.8070e à0019.7298e à005 1.8063e à001 6.1802e à003 3.2949e à002SGHS 100009.1300e à001 2.8494e à005 5.4666e à002 5.0686e à003 1.1268e à002NGHS 20000 4.1933e à0017.9936e à015 5.0626e à014 2.6468e à0149.4872e à015f 6

HS 30000 6.2397e à001 5.9582e à004 2.1651e à003 1.1033e à003 3.4149e à004IHS 10000 6.6090e à001 2.6277e à004 4.9911e à004 3.7745e à004 6.3608e à005SGHS 100009.1510e à001 2.0507e à002 4.7466e à001 1.4771e à0019.3862e à002NGHS 20000 3.9327e à001 5.4770e à021 2.6652e à017 2.5681e à018 6.4570e à018f 7

HS 300007.8127e à001 1.2731e à0049.8712e à003 6.5087e à004 2.0524e à003IHS 100008.1307e à001 1.2728e à004 1.2728e à004 1.2728e à004 1.0104e à009SGHS 10000 1.0687e+000 3.8367e à003 4.4140e+001 4.3980e+000 1.0872e+001NGHS 20000 5.0827e à001 1.2728e à004 1.1844e+002 3.9481e+000 2.1624e+001f 8

HS 300007.9223e à001 2.9975e+001 3.3575e+002 1.1424e+0027.7109e+001IHS 100008.1667e à001 1.2325e+001 6.8309e+002 2.4584e+002 1.9314e+002SGHS 10000 1.0832e+000 1.1472e+000 2.2166e+002 4.4718e+001 5.0450e+001NGHS 20000 5.2767e à001 1.0224e à0019.4449e+000 2.5160e+000 2.3538e+000f 9

HS 30000 6.4173e à001à4.5000e+002à4.5000e+002à4.5000e+002 1.2193e à007IHS 100007.1923e à001à4.5000e+002à4.5000e+002à4.5000e+002 5.7304e à009SGHS 100009.7077e à001à4.5000e+002à4.4993e+002à4.4999e+002 1.6373e à002NGHS 20000 4.5263e à001à450

à450

à450

f 10

HS 300008.6130e à001à4.4180e+002à1.5849e+002à3.3589e+0027.3644e+001IHS 100007.4760e à001à4.4870e+002 5.3709e+002à1.9303e+002 2.3832e+002SGHS 100009.9853e à001à4.4627e+002à2.6309e+002à4.0418e+002 5.1129e+001NGHS 20000 5.7227e à001à4.4983e+002à4.1824e+002à4.4334e+0028.0172e+000f 11

HS 30000 1.0529e+000 3.0837e+005 1.6868e+007 4.6601e+006 4.1172e+006IHS 100008.5843e à001 1.9761e+005 1.3515e+007 3.4793e+006 3.2989e+006SGHS 10000 1.1334e+000 2.1771e+005 1.0386e+007 2.8299e+006 2.2564e+006NGHS 200007.5890e à001 1.8886e+005 4.2554e+006 1.2129e+006 1.0627e+006f 12

HS 300009.2593e à001à4.3759e+002à8.4163e+000à2.9502e+002 1.1333e+002IHS 100008.0577e à001à4.4782e+002 1.0248e+003à4.6045e+001 4.3384e+002SGHS 10000 1.0329e+000à4.4131e+002à3.2378e+002à3.9487e+002 3.2845e+001NGHS 20000 5.7863e à001à4.4984e+002à4.1765e+002à4.4158e+0028.4856e+000f 13

HS 300007.3793e à001 3.9254e+002 1.0272e+004 1.1280e+003 2.2521e+003IHS 100007.0680e à001 3.9014e+0028.2056e+003 1.6553e+003 2.3782e+003SGHS 100009.3850e à001 3.9118e+002 1.1485e+004 3.1457e+003 4.2465e+003NGHS 20000 4.3240e à001 3.9002e+0029.6400e+003 2.0091e+003 3.3979e+003f 14

HS 30000 1.1510e+000à1.7988e+002à1.7716e+002à1.7915e+002 6.5803e à001IHS 100008.8337e à001à1.7994e+002à1.7131e+002à1.7800e+002 2.1163e+000SGHS 10000 1.1942e+000à1.7973e+002à1.7841e+002à1.7913e+002 4.2704e à001NGHS 200007.5427e à001à1.7987e+002à1.7147e+002à1.7639e+002 2.4661e+000f 15

HS 30000 1.1292e+000à1.1996e+002à1.1960e+002à1.1981e+0029.4518e à002IHS 100008.7710e à001à1.1997e+002à1.1936e+002à1.1969e+002 1.7267e à001SGHS 10000 1.1870e+000à1.1993e+002à1.1946e+002à1.1978e+002 1.1715e à001NGHS 200007.2650e à001à1.1996e+002à1.1954e+002à1.1978e+002 1.2150e à001f 16

HS 300007.4210e à001à3.3000e+002à3.3000e+002à3.3000e+002 1.4282e à005IHS 100007.0420e à001à3.3000e+002à3.2901e+002à3.2995e+002 1.8624e à001SGHS 100009.3697e à001à3.3000e+002à3.2501e+002à3.2800e+002 1.3135e+000NGHS

20000

4.7507e à001

à330

à3.2901e+002

à3.2997e+002

1.8165e à001

D.Zou et al./Neurocomputing 73(2010)3308–3318

3312

D.Zou et al./Neurocomputing73(2010)3308–33183313

Table2

The optimization results of four harmony search algorithms for f1–f16(N?30).

Function Algorithm K AOT Best Worst Mean SD

f1HS90000 4.5290e+000 5.9433eà002 3.1769e+000 6.9084eà0017.5714eà001 IHS30000 4.6819e+000 1.6633e+000 1.6300e+0017.1249e+000 3.4260e+000

SGHS30000 3.0687e+000 5.2824eà007 5.5217eà002 1.3529eà002 1.3744eà002

NGHS60000 2.2387e+000 5.7294eà018 3.9233eà015 5.9866eà016 1.0009eà015

f2HS90000 4.4417e+000 1.7814e+002 1.0026e+004 1.0228e+003 1.8587e+003 IHS30000 4.6833e+000 2.4381e+003 3.6220e+004 1.1604e+0047.7731e+003

SGHS30000 3.0953e+000 3.8906e+001 6.2642e+003 4.9478e+002 1.3856e+003

NGHS60000 2.2937e+000 2.1716eà0037.2347e+003 1.1163e+003 2.0391e+003

f3HS90000 4.7698e+000 2.5679eà002 3.0349e+0009.3737eà0019.3018eà001 IHS30000 4.8902e+0007.5656e+000 1.7846e+001 1.1807e+001 2.4043e+000

SGHS30000 3.1735e+000 2.3796eà0067.2831e+000 2.5508e+000 1.5670e+000

NGHS60000 2.5150e+0000 4.1353eà010 1.5694eà0117.5579eà011

f4HS90000 4.9225e+0007.2972eà001 1.0516e+0009.7764eà0017.4106eà002 IHS30000 4.9886e+000 1.0572e+000 1.3285e+000 1.1516e+000 6.4516eà002

SGHS30000 3.2848e+000 4.1045eà002 3.4484eà001 1.6738eà0017.8560eà002

NGHS60000 2.6151e+0000 1.8765eà001 5.5000eà002 4.9350eà002

f5HS90000 4.7823e+0008.4348eà003 6.3452eà001 6.1483eà002 1.3074eà001 IHS30000 4.9312e+000 1.0436eà001 2.6117e+000 1.8408e+000 4.5176eà001

SGHS30000 3.1797e+000 4.5161eà005 1.2398eà002 1.1591eà003 3.2218eà003

NGHS60000 2.4079e+0007.3700eà010 2.3065eà008 5.0506eà009 4.8921eà009

f6HS90000 4.7161e+000 3.1756eà001 3.2593e+000 1.5317e+0007.0225eà001 IHS30000 4.8510e+000 4.1647eà001 6.3598e+000 2.4069e+000 1.4358e+000

SGHS30000 3.1668e+000 1.2858eà0018.7032eà001 4.1507eà001 1.7614eà001

NGHS60000 2.5135e+000 1.3387eà0099.0607eà008 2.0414eà008 2.3302eà008

f7HS90000 5.3157e+000 3.7153e+000 1.9469e+0018.2052e+000 3.5276e+000 IHS30000 5.0390e+000 3.4680e+000 3.6983e+001 1.5504e+0019.4119e+000

SGHS30000 3.3552e+000 1.0536eà001 6.8470e+001 6.4187e+000 1.7870e+001

NGHS60000 2.8839e+000 3.8183eà004 3.8183eà004 3.8183eà004 1.0080eà012

f8HS90000 6.3901e+000 1.4499e+003 4.4411e+003 2.6404e+0038.9778e+002 IHS30000 5.3772e+000 3.1668e+0039.7564e+003 6.4600e+003 1.8053e+003

SGHS30000 3.7547e+000 3.8900e+002 2.0936e+0039.5742e+002 3.9842e+002

NGHS60000 3.6587e+000 1.4414e+001 2.4507e+0028.1775e+001 5.5860e+001

f9HS90000 4.8208e+000à4.4994e+002à4.4779e+002à4.4947e+002 4.6568eà001 IHS30000 5.0301e+000à4.4993e+002à4.3415e+002à4.4308e+002 4.1650e+000

SGHS30000 3.3276e+000à4.5000e+002à4.4993e+002à4.4999e+002 1.6583eà002

NGHS60000 2.4313e+000à450à4.5000e+002à450 5.0623eà014

f10HS900007.2657e+000 1.1187e+003 4.9553e+003 2.4254e+0039.1445e+002 IHS30000 5.9093e+000 4.1374e+003 1.7549e+0047.7504e+003 3.0011e+003

SGHS30000 4.1807e+000 1.7644e+002 2.8867e+0039.7098e+002 6.3131e+002

NGHS60000 4.1526e+000à3.9857e+002à2.9674e+000à3.0693e+0028.5679e+001

f11HS900007.6589e+000 1.3191e+007 5.0627e+007 3.3232e+007 1.0172e+007 IHS30000 6.1760e+000 1.6372e+0078.1516e+007 5.1501e+007 1.8883e+007

SGHS30000 4.7110e+000 5.8176e+006 2.1411e+007 1.2625e+007 4.1470e+006

NGHS60000 4.5360e+000 1.1506e+0068.1823e+006 3.4080e+006 1.5773e+006

f12HS900007.0880e+000 1.3297e+003 6.9905e+003 3.2367e+003 1.1122e+003 IHS30000 5.6652e+000 4.8642e+003 1.7237e+0048.5947e+003 2.8593e+003

SGHS30000 4.0302e+000 3.1185e+002 3.5944e+003 1.3484e+0038.0101e+002

NGHS60000 4.1833e+000à4.1304e+002à7.8426e+001à2.7939e+0028.7641e+001

f13HS90000 4.8709e+000 4.8810e+002 1.1203e+004 2.6713e+003 3.1923e+003 IHS30000 5.2942e+000 3.1273e+003 2.6193e+0049.8223e+003 5.1948e+003

SGHS30000 3.5121e+000 4.3287e+002 1.1863e+004 2.8462e+003 4.0781e+003

NGHS60000 2.7963e+000 3.9047e+002 1.1109e+004 1.6664e+003 2.8307e+003

f14HS900007.4739e+000à1.7867e+002à1.7686e+002à1.7819e+002 4.1038eà001 IHS30000 6.1693e+000à1.7775e+002à1.7065e+002à1.7517e+002 1.6656e+000

SGHS30000 4.6812e+000à1.7928e+002à1.7891e+002à1.7900e+0029.3573eà002

NGHS60000 4.3907e+000à1.8000e+002à1.7992e+002à1.7998e+002 1.7561eà002

f15HS900008.4621e+000à1.1911e+002à1.1893e+002à1.1900e+002 4.4332eà002 IHS300007.6494e+000à1.1907e+002à1.1888e+002à1.1896e+002 4.8804eà002

SGHS30000 5.6287e+000à1.1988e+002à1.1954e+002à1.1972e+0027.8421eà002

NGHS60000 3.6126e+000à1.1987e+002à1.1952e+002à1.1968e+0028.8529eà002

f16HS90000 5.2827e+000à3.2998e+002à3.2897e+002à3.2987e+002 3.0290eà001 IHS30000 5.0381e+000à3.2794e+002à3.2207e+002à3.2544e+002 1.5255e+000

SGHS30000 3.3412e+000à3.2891e+002à3.2478e+002à3.2695e+002 1.2779e+000

NGHS60000 2.8833e+000à330à3.3000e+002à3.3000e+0028.5981eà013

Table 3

The optimization results of four harmony search algorithms for f 1–f 16(N ?50).Function Algorithm K AOT

Best

Worst

Mean

SD

f 1

HS 100000 6.3780e+000 6.0076e+001 2.5205e+002 1.7603e+002 4.4359e+001IHS 50000 1.2900e+001 2.3228e+0027.5158e+002 4.9421e+002 1.1559e+002SGHS 50000 6.1854e+000 1.8998e à006 1.6033e à001 2.8402e à002 3.2096e à002NGHS 80000 4.7459e+000 2.2757e à0107.0950e à009 1.8091e à009 1.9488e à009f 2

HS 100000 6.5327e+0009.6629e+0048.9231e+005 3.3964e+005 1.9909e+005IHS 50000 1.2686e+0017.5693e+005 5.7741e+006 2.4476e+006 1.1118e+006SGHS 50000 6.2088e+000 1.5578e+0029.6556e+003 1.2443e+003 2.7350e+003NGHS 80000 4.7349e+000 2.4458e à0018.1098e+003 5.3162e+002 1.7208e+003f 3

HS 1000007.2161e+000 2.9511e+001 5.2810e+001 4.2476e+001 5.7278e+000IHS 50000 1.3192e+001 5.0142e+001 1.0121e+0027.6374e+001 1.0771e+001SGHS 50000 6.4016e+000 1.0903e à001 1.0875e+001 5.7834e+000 2.2214e+000NGHS 80000 4.9380e+000 3.7993e à0099.9496e à001 3.3166e à002 1.8165e à001f 4

HS 1000008.0198e+000 2.0566e+000 3.2913e+000 2.5946e+000 3.2134e à001IHS 50000 1.3743e+001 3.9497e+0009.6680e+000 5.8018e+000 1.2152e+000SGHS 50000 6.8679e+0007.6282e à002 3.1431e à001 1.8212e à001 6.0839e à002NGHS 80000 5.5714e+000 4.4744e à010 1.3045e à001 3.7959e à002 3.8783e à002f 5

HS 1000007.4250e+000 3.2228e+000 4.4313e+000 3.7857e+000 3.2904e à001IHS 50000 1.3500e+001 4.7196e+000 5.9813e+000 5.2778e+000 2.9940e à001SGHS 50000 6.5453e+000 6.8535e à0059.4342e à0038.1216e à004 2.0654e à003NGHS 80000 4.8640e+000 2.1342e à006 1.7391e à0057.7723e à006 3.4193e à006f 6

HS 1000007.0240e+000 3.1353e+001 5.0629e+001 4.2350e+001 4.9070e+000IHS 50000 1.3060e+001 3.5501e+0018.4063e+001 5.8769e+001 1.0974e+001SGHS 50000 6.4645e+000 4.7367e à001 1.3476e+0009.0272e à001 2.2061e à001NGHS 80000 5.0724e+000 3.6976e à005 2.2687e à0048.1858e à005 3.7660e à005f 7

HS 1000008.4332e+000 2.9427e+002 6.7011e+002 4.1774e+0029.5933e+001IHS 50000 1.3977e+001 2.8093e+002 1.1779e+0037.4093e+002 1.9622e+002SGHS 500007.2183e+000 2.0680e à001 1.5987e+001 1.4906e+000 2.9687e+000NGHS 80000 6.3162e+000 6.3638e à004 6.3681e à004 6.3644e à0048.7793e à008f 8

HS 100000 1.1377e+001 1.2951e+004 3.2445e+004 2.1537e+004 5.1131e+003IHS 50000 1.5177e+001 2.1066e+004 4.4783e+004 2.9112e+004 4.8624e+003SGHS 500008.6996e+000 2.3241e+003 6.2041e+003 3.9058e+0039.2316e+002NGHS 800008.7239e+000 3.0708e+002 1.2443e+003 6.9208e+002 2.1710e+002f 9

HS 100000 6.4484e+000à3.5043e+002à1.6889e+002à2.5642e+002 4.4840e+001IHS 50000 1.2916e+001à1.5674e+002 4.8751e+0029.7197e+001 1.4285e+002SGHS 50000 6.0876e+000à4.5000e+002à4.4989e+002à4.4998e+002 2.3295e à002NGHS 80000 4.5729e+000à4.5000e+002à4.5000e+002à4.5000e+002 1.2822e à009f 10

HS 100000 1.2535e+001 1.6278e+004 3.5761e+004 2.3503e+004 5.1672e+003IHS 50000 1.6052e+001 1.8107e+004 5.3281e+004 3.3209e+0047.0436e+003SGHS 500009.3468e+000 2.6940e+003 1.0618e+004 5.3262e+003 1.8789e+003NGHS 800009.7527e+000 3.9438e+002 2.3064e+0039.2796e+002 4.4507e+002f 11

HS 100000 1.1155e+001 6.0448e+007 1.8797e+008 1.2265e+008 3.2867e+007IHS 50000 1.5359e+0017.9658e+007 2.3900e+008 1.4841e+008 4.0123e+007SGHS 500009.7390e+0008.0843e+006 3.7497e+007 1.8978e+007 6.1005e+006NGHS 800008.6995e+000 3.8070e+006 1.4028e+007 6.6177e+006 1.9688e+006f 12

HS 100000 1.2926e+001 1.5686e+004 4.0686e+004 2.5813e+004 6.0171e+003IHS 50000 1.6085e+001 2.1726e+004 5.4777e+004 3.8369e+0047.7422e+003SGHS 500009.5588e+000 3.0714e+003 1.3240e+004 6.4690e+003 2.0085e+003NGHS 80000 1.0260e+001 3.1526e+002 2.6327e+003 1.0597e+003 6.0928e+002f 13

HS 100000 6.6739e+000 1.7211e+005 1.8291e+006 4.0277e+005 3.0845e+005IHS 50000 1.3041e+001 1.2263e+006 4.5379e+006 2.8450e+0068.9783e+005SGHS 50000 6.3203e+000 4.4911e+002 1.6233e+004 3.2408e+003 5.0484e+003NGHS 80000 4.8637e+000 3.9004e+002 1.1862e+004 2.0039e+003 3.3640e+003f 14

HS 1000009.8458e+000à1.6607e+002à1.4317e+002à1.5772e+002 6.2514e+000IHS 50000 1.4928e+001à1.5332e+002à8.2651e+001à1.2930e+002 1.6724e+001SGHS 500009.0807e+000à1.7896e+002à1.7880e+002à1.7891e+002 3.8661e à002NGHS 800007.3657e+000à1.7998e+002à1.7990e+002à1.7994e+002 2.1517e à002f 15

HS 1000009.1332e+000à1.1891e+002à1.1874e+002à1.1883e+002 3.7003e à002IHS 50000 1.4616e+001à1.1885e+002à1.1873e+002à1.1878e+002 2.8639e à002SGHS 500008.9495e+000à1.1984e+002à1.1966e+002à1.1975e+002 5.0678e à002NGHS 80000 6.9954e+000à1.1976e+002à1.1946e+002à1.1961e+0027.0579e à002f 16

HS 1000007.1546e+000à3.1671e+002à3.0395e+002à3.1038e+002 2.5907e+000IHS 50000 1.3261e+001à2.9968e+002à2.7562e+002à2.8991e+002 5.2346e+000SGHS 50000 6.4401e+000à3.2997e+002à3.2266e+002à3.2592e+002 1.8642e+000NGHS

80000

4.9802e+000

à3.3000e+002

à3.3000e+002

à3.3000e+002

4.0850e à005

D.Zou et al./Neurocomputing 73(2010)3308–3318

3314

NGHS.The dimension sizes of all functions are set to30,and the maximal iteration number is set to30000accordingly.Thirty independent runs are carried out in each case,and the mean and standard deviation(7SD)obtained using different p m and HMS values are reported in Tables4and5.

As can be seen from Table4,that a large p m(Such as0.05and 0.1)or a small p m(Such as0.0005and0.0001)is harmful to the convergence of the NGHS in most cases,and will deteriorate the performance of the NGHS.In contrast,using p m?0.005(or0.01) can improve the convergence and stability of the NGHS signi?cantly on solving most test functions.In short,there is no single choice for p m,and it should be adjusted according to practical optimization problems.

According to Table5,there is no indication that one setting of HMS is superior to the other settings.Generally speaking,small HMS is reasonable and logical,for it is similar to the short memory of musician.

4.3.Mann–Whitney U test

In this section,Mann–Whitney U test is employed to con?rm that a statistical signi?cant difference in performance is obtained by the NGHS algorithm when compared to the other three algorithms.The Mann–Whitney U test(also called as Mann–Whitney Wilcoxon test)[25,26]is a non-parametric rank-based test for identifying a difference between populations.The test statistic U is given by

U1?R1à

n1en1t1T

e20TU2?R2à

n2en2t1T

2

e21Twhere n1and n2are the sizes of sample1and sample2, respectively;R1and R2are the sums of the ranks in sample1and sample2,respectively.U1is the number of sample1observations beaten by sample2observations,and U2is the number of sample 2observations beaten by sample1observations.The smaller value of U1and U2is the one used when consulting signi?cance tables.The sum of the two values is given by

U1tU2?R1à

n1en1t1T

2

tR2à

n2en2t1T

2

e22TKnowing that R1+R2?n(n+1)/2and n?n1+n2,and doing some algebra,we?nd that the sum is

U1tU2?n1n2e23T

Before using Mann–Whitney U test,it is necessary to set a rule to determine who is the winner in a comparison.Suppose x i(i?1,y,n1)is a solution obtained by A algorithm for a problem, and y j(j?1,y,n2)is a solution obtained by B algorithm for the same problem.If x i(i?1,y,n1)is better than y j(j?1,y,n2),then A algorithm is the winner,otherwise,B algorithm is the winner.If

Table4

The effect of p m on the performance of the NGHS(HMS?5).

Function p m?0.0005p m?0.001p m?0.005p m?0.01p m?0.05p m?0.1

f1 1.1672eà012 4.9313eà021 3.4411eà0129.6784eà007 2.6428e+0017.9350e+002

(6.3851eà012)(1.9566eà020)(1.0646eà011)(9.0368eà007)(8.6339e+000)(1.5884e+002)

f2 1.6314e+003 1.3591e+0039.5434e+0029.2395e+002 1.4643e+004 6.1203e+006

(3.1221e+003)(2.9679e+003)(2.3201e+003)(2.2704e+003)(6.2798e+003)(2.9657e+006)

f3 1.1377e+001 3.6987e+000 1.9908eà001 2.3350eà001 5.0222e+001 1.5816e+002

(4.1109e+000)(2.1059e+000)(4.0474eà001)(4.2735eà001)(8.3951e+000)(2.0305e+001)

f4 1.7727eà001 1.2763eà001 6.4404eà002 3.6069eà002 1.2331e+0008.2507e+000

(1.7375eà001)(9.1172eà002)(4.1008eà002)(3.2762eà002)(7.2446eà002)(2.0658e+000)

f57.8514eà001 6.6771eà007 5.6981eà007 2.2327eà004 2.6356e+0007.4782e+000

(6.8145eà001)(3.1800eà006)(4.3733eà007)(9.2461eà005)(2.4915eà001)(4.8084eà001)

f6 6.9220eà0128.4738eà012 1.1684eà006 1.7768eà003 2.0990e+0017.0245e+003

(2.2300eà011)(3.0308eà011)(8.2811eà007)(8.3204eà004)(3.1491e+000)(2.2161e+004)

f78.5539e+002 2.8030e+002 1.1738eà003 3.9484e+000 4.1721e+002 2.6858e+003

(3.0424e+002)(1.4748e+002)(4.3166eà003)(2.1624e+001)(1.4368e+002)(3.4031e+002)

f89.7757e+003 5.4738e+003 5.9327e+002 3.4043e+002 2.5208e+0038.0722e+003

(2.2508e+003)(1.5381e+003)(2.4670e+002)(1.6069e+002)(7.7844e+002)(2.0146e+003)

f9à4.5000e+002à450à4.5000e+002à4.5000e+002à4.2585e+002 3.2059e+002

(1.2655eà008)(1.1942eà013)(1.4911eà011)(1.2235eà006)(4.5090e+000)(1.9590e+002)

f10 2.2198e+004 1.1990e+0048.9591e+002 1.0115e+002 2.2995e+0038.8052e+003

(5.5295e+003)(3.8401e+003)(5.7904e+002)(2.2891e+002)(1.0218e+003)(2.0989e+003)

f11 4.1851e+007 2.0056e+007 6.7207e+006 6.3989e+006 2.6575e+007 6.8280e+007

(1.7194e+007)(8.2315e+006)(2.6293e+006)(2.5094e+006)(8.3295e+006)(2.4975e+007)

f12 3.0884e+004 1.8353e+004 1.0991e+003 2.9116e+002 3.7482e+0038.9484e+003

(1.0873e+004)(7.4429e+003)(5.4102e+002)(3.5765e+002)(1.2450e+003)(3.2689e+003)

f13 4.7132e+003 2.9687e+003 1.4566e+003 3.0345e+003 1.3774e+004 4.3094e+006

(6.3945e+003)(5.1419e+003)(2.3845e+003)(4.7491e+003)(7.7463e+003)(1.8392e+006)

f14à1.7636e+002à1.7898e+002à1.7989e+002à1.7975e+002à1.7650e+002à1.3888e+002

(2.7628e+000)(3.3633eà001)(5.4303eà002)(9.7887eà002)(9.7624eà001)(1.0368e+001)

f15à1.1947e+002à1.1951e+002à1.1957e+002à1.1946e+002à1.1896e+002à1.1897e+002

(8.9905eà002)(1.1339eà001)(1.0229eà001)(1.1727eà001)(6.0986eà002)(5.8540eà002)

f16à3.1812e+002à3.2688e+002à3.2993e+002à3.2983e+002à3.2987e+002à2.2752e+002

(4.4830e+000)(1.9506e+000)(2.5243eà001)(3.7710eà001)(3.4405eà001)(1.3046e+001)

D.Zou et al./Neurocomputing73(2010)3308–33183315

x i (i ?1,y ,n 1)is the same as y j (j ?1,y ,n 2),then we should consider the optimization times of two algorithms.If the optimization time of A algorithm is less than that of B algorithm,then A algorithm is the winner,otherwise,B algorithm is the winner.

After learning the above rule,three groups of Mann–Whitney U tests are carried out,and they are (U NGHS ,U HS ),(U NGHS ,U IHS )and (U NGHS ,U SGHS ),respectively.All 16unconstrained problems (f 1àf 16,N ?50)are selected to carry out these tests.The parameters settings of the four harmony search algorithms are the same as those of Section 4.1,and 30independent experiments are carried out in each case.The three groups of Mann–Whitney U tests can be stated as Tables 6–8.

As can be seen from Table 6,the values of U NGHS are all equal to 0,in contrast,the values of U HS are all equal to 900which is much larger than 0.Statistically,the NGHS can beat the HS on solving all 16unconstrained functions,which indicates that the performance of the NGHS is better than that of the HS.

As can be seen from Table 7,the values of U NGHS are all smaller than those of U IHS for all functions.Specially,the values of U NGHS and U IHS are 0and 900,respectively,for any function.In other words,the numbers of NGHS solutions beaten by IHS solutions are 0for all 16unconstrained problems.Thus,we can safely conclude that the IHS has been badly defeated by the NGHS on solving all 16functions.

As can be seen from Table 8,the values of U NGHS are smaller than 450for all functions except f 15,in the mean time,the values of U SGHS are larger than 450for all functions except f 15.In short,the

NGHS has obtained a larger number of wins when compared to the SGHS,which indicates that the NGHS is superior to the SGHS.

5.Conclusions

A novel global harmony search (NGHS)algorithm is introduced in this paper.Unlike the HS algorithm,the NGHS designs a novel

Table 5

The effect of HMS on the performance of the NGHS (p m ?0.005).Function HMS ?5

HMS ?10HMS ?20HMS ?40HMS ?80f 1 3.7455e à012 6.6008e à013 6.4703e à012 6.7822e à011 1.0727e à009(1.1753e à011)(8.1519e à013)(1.0961e à011)(1.3287e à010)(1.1924e à009)f 2 5.5693e+002 1.8485e+002 6.6884e+001 2.8060e+001 4.3644e+001(1.6224e+003)(5.6518e+002)(7.5718e+001)(2.7791e+001)(4.3756e+001)f 3 6.6345e à002 6.6406e à002 1.6608e à001 2.3356e à001 6.8403e à001(2.5247e à001)(2.5241e à001)(4.5872e à001)(5.0089e à001)(8.0939e à001)f 48.8581e à002 4.9286e à002 3.7184e à002 3.9671e à002 2.3203e à002(5.9176e à002)(4.7725e à002)(3.2936e à002)(4.2763e à002)(3.5180e à002)f 5 5.3823e à007 2.7800e à007 6.7582e à007 1.6858e à0069.2713e à006(6.2054e à007)(1.7985e à007)(3.8313e à007)(7.5629e à007)(3.5933e à006)f 6 1.9709e à006 1.2651e à0067.0816e à006 3.5219e à005 2.2310e à004(3.3050e à006)(8.0173e à007)(6.7054e à006)(2.3731e à005)(1.0196e à004)f 7 3.8189e à004 3.8193e à0047.8963e+0007.8963e+000 1.1847e+001(2.6756e à007)(4.6899e à007)(3.0049e+001)(3.0049e+001)(3.6138e+001)f 8 5.4900e+002 5.2437e+001 1.5879e+0009.8972e à001 1.2598e+000(2.3962e+002)(2.7859e+001)(7.1301e à001)(6.2141e à001)(7.4038e à001)f 9à4.5000e+002à4.5000e+002à4.5000e+002à4.5000e+002à4.5000e+002(4.6250e à012)(1.9141e à012)(1.6327e à012)(3.9962e à011)(1.0251e à009)f 10 6.8696e+002à3.8233e+002à4.4801e+002à4.4887e+002à4.4805e+002(4.9857e+002)(4.0714e+001)(1.0446e+000)(6.7255e à001)(1.1716e+000)f 117.0622e+006 3.5850e+006 3.1335e+006 2.4065e+006 2.7199e+006(2.7765e+006)(1.7565e+006)(1.6418e+006)(1.1816e+006)(1.4497e+006)f 128.1370e+002à3.7498e+002à4.4778e+002à4.4827e+002à4.4767e+002(5.3745e+002)(3.8696e+001)(8.7792e à001)(1.3109e+000)(1.6420e+000)f 13 2.1212e+003 1.1022e+003 5.1917e+002 4.9355e+002 4.6619e+002(3.0182e+003)(1.7786e+003)(2.2059e+002)(1.3083e+002)(9.5065e+001)f 14à1.7988e+002à1.7995e+002à1.7996e+002à1.7998e+002à1.7998e+002(5.3773e à002)(2.6152e à002)(1.9606e à002)(1.8976e à002)(1.8882e à002)f 15à1.1956e+002à1.1963e+002à1.1971e+002à1.1972e+002à1.1974e+002(1.0329e à001)(7.5334e à002)(9.8542e à002)(9.4335e à002)(7.2315e à002)f 16

à3.2997e+002à3.2993e+002à3.2997e+002à3.2977e+002à3.2978e+002(1.8165e à001)

(2.5243e à001)

(1.8165e à001)

(5.0145e à001)

(4.1944e à001)

Table 6

Mann–Whitney U test result obtained using the NGHS and the HS.Function U NGHS U HS f 10900f 20900f 30900f 40900f 50900f 60900f 70900f 80900f 90900f 100900f 110900f 120900f 130900f 140900f 150900f 16

900

D.Zou et al./Neurocomputing 73(2010)3308–3318

3316

location updating strategy,which makes it easier to converge.In addition,the genetic mutation operation of the NGHS is the same as the random selection of the HS,and it can effectively prevent the NGHS from being trapped into local optima.Based on a large number of experiments,the NGHS has demonstrated better performance on solving most unconstrained problems when compared to the other three harmony search algorithms. Acknowledgement

This work was supported by National Science Foundation of P.R.China under Grants60674021.

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Dexuan Zou received the B.Sc.degree in electronic

information science and technology from Liaoning

University in2005,and he obtained his M.S.degree

in signal and information processing from North-

eastern University,Shenyang,China,in2008.He is

currently working towards the Ph.D.degree in the?eld

of control theory&control Engineering at North-

eastern University.His current research interests are

evolutionary algorithms,arti?cial intelligence,and

optimal

control.

Liqun Gao received the M.S.and Ph.D.degrees in auto-

matic control from Northeastern University,Shenyang,

China,in1985and1991,respectively.Currently,he is a

professor in the control theory and navigation technology

department,Northeastern University.His current research

interests are arti?cial intelligence,control theory and

control methods,and pattern recognition.

Table7

Mann–Whitney U test result obtained using the NGHS and the IHS.

Function U NGHS U IHS

f10900

f20900

f30900

f40900

f50900

f60900

f70900

f80900

f90900

f100900

f110900

f120900

f130900

f140900

f150900

f160900

Table8

Mann–Whitney U test result obtained using the NGHS and the SGHS.

Function U NGHS U SGHS

f10900

f2182718

f30900

f422878

f50900

f60900

f70900

f80900

f90900

f100900

f1120880

f120900

f13172728

f140900

f1584555

f160900

D.Zou et al./Neurocomputing73(2010)3308–33183317

Jianhua Wu received the B.Sc.and M.S.degrees in automatic control from Northeastern University, Shenyang,China,in1978and1986,respectively. Currently,she is a professor in the electric power system and automation department,Northeastern University.Her current research interests are digital image processing,research and development of power converter,and pattern

recognition.Steven Li received the Ph.D degree from Delft University of Technology,The Netherlands.Currently, he is an associate professor in the division of business, University of South Australia.His current research interests are quantitative?nance,corporate?nance, asset pricing,ef?ciency of?nancial markets,and investments.

D.Zou et al./Neurocomputing73(2010)3308–3318 3318