Research ArticleA Novel Artificial Bee Colony Algorithm forFunction Optimization

Song Zhang and Sanyang Liu

School of Mathematics and Statistics Xidian University Xirsquoan 710071 China

Correspondence should be addressed to Song Zhang yxszhang163com

Received 6 October 2014 Revised 1 March 2015 Accepted 11 March 2015

Academic Editor Sishaj P Simon

Copyright copy 2015 S Zhang and S LiuThis 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

It is known that both exploration and exploitation are important in the search equations of ABC algorithms How to well balancethe two abilities in the search process is still a challenging problem in ABC algorithms In this paper we propose a novel artificialbee algorithm named as ldquoNABCrdquo by incorporating the information of the global best solution into the solution search equation ofthe onlookers stage to improve the exploitation At the same time we improve the search equation of the employed bees to keep theexploration The experimental results of NABC tested on a set of 11 numerical benchmark functions show good performance andfast convergence in solving function optimization problems compared with variant ABC DE and PSO algorithmsThe applicationof NABC on solving five standard knapsack problems shows its effectiveness and practicability

1 Introduction

Intelligent algorithms such as genetic algorithm (GA) [1]particle swarm optimization (PSO) [2] ant colony opti-mization (ACO) [3] and Biogeography-Based Optimization(BBO) [4] have shown great success in solving optimizationproblems which are nonconvex discontinuous nondiffer-entiable and so on At the same time they have manyadvantages such as simplicity ease of implementation andoutstanding performance Inspired by the intelligent behav-iors of honey bee swarms Karaboga in 2005 [5] proposedthe artificial bee colony (ABC) algorithm for function opti-mizations A set of experimental results [6ndash9] show that ABCalgorithm is competitive to some other intelligent algorithmsSo the ABC algorithm has captured many attentions and hasbeen applied to solve neural networks flowshop schedulingproblems constrained optimization problems [10ndash12] and soon

We know that both exploration and exploitation are nec-essary for an intelligent algorithm In order to achieve goodperformances on problem optimizations in practice the twoabilities should be well balanced while the two abilities con-tradict each other and the solution search equations of ABCalgorithm are good at exploration but poor at exploitationSomany researchersmended the search solutions to improve

the exploitation and the convergence during the past decadesInspired by PSO Zhu and Kwong [13] took advantage ofthe information of the global best solution to improve theperformance of the ABC algorithm named GABC Basturkand Karaboga [14] proposed a modified ABC algorithm bycontrolling the frequency of perturbation and introducingthe ratio of the variance operator Gao et al proposed anovel ABC method called EABC [15] to further improve theperformance of ABC In EABC in order to balance the explo-ration and the exploitation they presented two new searchequations to generate candidate solutions in the employed beephase and the onlookers phase respectively Karaboga andGorkemli presented a new version of ABC algorithm namedquick artificial bee colony (qABC) [16] which modeled thebehavior of onlooker bees more accurately and improved theperformance of ABC in terms of local search ability And theperformance of the qABC depended on the neighborhoodradius

In this paper we propose two new solution search equa-tions in the stage of the employed bees and the onlookersrespectively in order to improve the exploitation and keep theexploration We name the novel ABC algorithm as ldquoNABCrdquoThe experiment results tested on 11 numerical functionoptimizations show that the novel ABC algorithm is superiorto the standard ABC algorithm and other improved ABC

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 129271 10 pageshttpdxdoiorg1011552015129271

2 Mathematical Problems in Engineering

algorithms inmost cases And it is better than variant DE andPSO algorithms At the same time five knapsack problemsare solved by ABC and NABC which shows that NABCis superior to ABC on solving engineering optimizationproblems

The rest of this paper is organized as follows Section 2summarizes the standard ABC algorithm The novel ABCalgorithm named NABC is presented and analyzed inSection 3 Section 4 presents and discusses the experimen-tal results of 11 benchmark functions Section 5 shows theapplication of NABC on knapsack problems Finally theconclusion is drawn in Section 6

2 Standard ABC Algorithm

Initialization is the first step in the standard ABC algorithm[5] The position of a food source represents a possiblesolution to the optimization problem Each food sourceis exploited by only one employed bee so the number ofemployed bees named SN is equal to the number of foodsources And a random initial population of SN food sourcesis generated as follows

119909119894119895

= 119909min119895 + rand (0 1) lowast (119909max119895 minus 119909min119895) (1)

where 119894 isin (1 2 SN) 119895 isin (1 2 119863) and 119863 is thedimension of the numerical function119909max119895 and 119909min119895 are thelower and upper bounds of 119909

119894119895

Second is the search processes of the employed beeswhere the onlookers and the scouts are started The standardABC algorithm uses the following equation to produce acandidate food source position from the old one

119881119894119895

= 119909119894119895

+ 120601119894119895

lowast (119909119894119895

minus 119909119896119895) (2)

where 119896 isin (1 2 SN) and 119895 isin (1 2 119863) are randomindexes 119896 is different from 119894 120601

119894119895is a random number in the

range [minus1 1] After each candidate source position is pro-duced and evaluated by the employed bees its performanceis compared with that of the old food source position If thenew food source position has an equal or better quality thanthe old source position the old one is replaced by the newone Otherwise the old one is retained The greedy selectionis used

Third the employed bees transfer their food informationto the onlookers The onlookers tend to further search thefood around the selected food source The onlooker choosesa food source depending on the probability value associatedwith that food source

119901119894=

fitness119894

sum119899

119894=1fitness

119894

(3)

where fitness119894is the fitness value of the solution which is

obtained by the following equation

fitness119894=

1

1 + 119891119894

119891119894ge 0

1 +1003816100381610038161003816119891119894

1003816100381610038161003816 119891119894lt 0

(4)

where119891119894is the objective function value If the new food source

position has an equal or better quality than the old source

position the old one is replaced by the new one Otherwisethe old one is retainedwhich is the same as the employed beesstage

At last if a position cannot be improved further through anumber of cycles which is called Limit for abandonment thatfood source is abandoned accordinglyThe scout will discovera new food source position randomly using (1) to replace it

3 A Novel Artificial BeeColony Algorithm (NABC)

31 Modified Search Solutions The exploration is the abilityof investigating the various unknown regions in the solutionspace So it is also very important inABC algorithm To retainthe exploration we present a modified search equation of theemployed bees as follows

119881119894119895

= 119909119903119895

+ 120601119894119895

lowast (119909119903119895

minus 119909119896119895) (5)

where the index 119903 119896 is randomly chosen from (1 2 SN)and different from the index 119894 119895 isin (1 2 119863) is a randomlychosen index 120601

119894119895is a random number in the range [minus1 1]

Differential evolution (DE) [17] is a simple and effi-cient evolutionary algorithm for many optimization prob-lems in the real-world applications [18ndash22] Inspired by itmany researchers improved variant intelligent algorithms Soinspired by DE and various modified search equations of theABC algorithms and taking advantage of the information ofthe global best solution to guide the search of the candidatesolutions a new equation is proposed to serve as the searchequation of onlookers as follows

119881119894119895

= 119909best119895 + 120601119894119895

lowast (119909best119895 minus 119909119896119895) (6)

where the index 119896 is a mutually exclusive integer randomlychosen from (1 2 SN) and different from the index 119894119909best119895 is the 119895th element of the global best solution and119895 isin (1 2 119863) is a randomly chosen index 120601

119894119895is a

randomnumber in the range [minus1 1]We name the novel ABCalgorithmusing (5) to replace (2) on the stage of the employedbees and using (6) to replace (2) on the stage of the onlookersas NABC which can drive the new candidate solution onlyaround the best solution of the previous iteration Thereforethe modified solution search equation described by (6) canincrease the exploitation of ABC algorithm greatly The bestsolution in the current population is a very useful sourcewhich can be used to improve the convergence performanceTherefore we modify the solution search equation by apply-ing the global best solution to guide the search of newcandidate solutions in order to improve the exploitation Atthe same time the exploration is improved by (5) from theexperiments in Section 4

The new candidate solution is generated around 119909119894119895

inGABC [13] It is different from (6) in which the one isgenerated around 119909best119895 EABC [15] adopts two differentsearch equations based on different the emphases in thedifferent search stages NABC uses (5) and (6) to generate thenew solution in the stages of employed bees and onlookersrespectively In the qABC [16] 119909

best119873119898

represents the best

Mathematical Problems in Engineering 3

Table 1 Benchmark functions used in experiments

Fun Name Search range Min

1198911=

119899

sum

119894=1

1199092

119894Sphere [minus100 100] 0

1198912=

119899

sum

119894=1

1198941199092

119894Sumsquare [minus10 10] 0

1198913=

119899

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119899

prod

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816

Schwefel222 [minus10 10] 0

1198914= max 100381610038161003816

1003816119909119894

1003816100381610038161003816 1 le 119894 le 119899 Schwefel221 [minus100 100] 0

1198915=

119899

sum

119894=1

(lfloor119909119894+ 05rfloor)

2 Step [minus100 100] 0

1198916=

119899

sum

119894=1

[100 (1199092

119894minus 119909119894+1

)

2

+ (119909119894minus 1)2

] Rosenbrock [minus5 10] 0

1198917=

119899

sum

119894=1

[1199092

119894minus 10 cos (2120587119909

119894) + 10] Rastrigin [minus512 512] 0

1198918=

1

4000

119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 Griewank [minus600 600] 0

1198919= 41898288727243369 lowast 119899 minus

119899

sum

119894=1

119909119894sin(radic

1003816100381610038161003816119909119894

1003816100381610038161003816) Schwefel226 [minus500 500] 0

11989110

= minus20 exp(minus02radic

1

119899

119899

sum

119894=1

1199092

119894) minus exp(

1

119899

119899

sum

119894=1

cos (2120587119909119894)) + 20 + 119890 Ackley [minus32 32] 0

11989111

=

1

119899

119899

sum

119894=1

(1199094

119894minus 16119909

2

119894+ 5119909119894) Himmelblau [minus5 5] minus7833236

solution among the neighbors of 119909119898and itself (119873

119898) But in

the NABC 119909best119895 represents the global best solution At thesame time in the stages of employed bees and onlookers thenew search equations used in qABC are different from theseof the NABC adopted

32 The Novel ABC Algorithm (NABC) The complete com-putational procedure of the novel ABC algorithm (NABC) isoutlined as follows

Algorithm 1 (NABC) Consider the following steps

Step 1 (initialization) Preset SN Limit and Maxcycle

Step 2 Randomly generate SN solutions from the search spaceto create an initial population 119875

Step 3 Calculate the function values 119891 of the initial popula-tion 119875

Step 4 The employed bees stage Generate a new solution V119894119895

by (5) if 119891(V119894119895) lt 119891(119909

119894119895) 119909119894119895

= V119894119895 trail119894= 1 else trail

119894=

trail119894+ 1

Step 5 Calculate the probability values 119901119894for the solutions 119909

119894119895

by (3)

Step 6 (the onlookers stage) If rand lt 119901119894 generate a new

solution V119894119895by (6) if 119891(V

119894119895) lt 119891(119909

119894119895) 119909119894119895

= V119894119895 trail119894= 1

else trail119894= trail

119894+ 1

Step 7 (the scouts stage) If trail gt Limit replace 119909119894119895with a

new produced solution by (1)

Step 8 If cycle number gt Maxcycle stop and memorize thebest solution achieved so far otherwise go to Step 2

4 Numerical Functionsand Experimental Results

41 Numerical Functions It is applied to minimize a set of 11scalable benchmark functions in Table 1 to test the efficiencyof the novel ABC algorithm The smaller the final result thebetter it is

42 Parameter Settings A set of experiments tested on 11numerical benchmark functions are performed in this sectionto compare the performance of the novel ABC algorithmwiththat of other improved ABC algorithms Functions 119891

1ndash1198914are

unimodal Function 1198915is a discontinuous step function The

Rosenbrock function of 1198916which is unimodal when 119863 =

2 and 3 may have multiple minima in the high dimensioncase [18] Functions 119891

7ndash11989111are multimodal To compare with

other variance algorithms the dimension119863 of every functionis 30 60 or 100 respectively the same as [15]The dimension119863 of 119891

11is 150 300 or 500 The maximum number of gen-

erations is 1500 3000 or 5000 respectively The populationsize is 100 namely SN = 50 and Limit is 200 which is thesame as [15] Each of the experiments is repeated 30 timesindependently

43 Experimental Results Figure 1 graphically presents theconvergence curves of 119891

1ndash11989110functions respectively We can

see from Figure 1 that in the later stage of evolution thestandard ABC algorithm enters a long period of stagnation

4 Mathematical Problems in Engineering

1010

100

10minus10

10minus20

10minus30

10minus40

10minus50

10minus60

0 500 1000 1500

Mea

n

Sphere Sumsquare10

10

100

10minus10

10minus20

10minus30

10minus40

10minus50

10minus60

0 500 1000 1500

Mea

n

Schwefel222

0 500 1000 1500

Mea

n

1020

1010

100

10minus10

10minus20

10minus30

0 500 1000 1500

Mea

n

Schwefel22110

2

101

100

10minus1

Mea

n

105

104

103

102

101

100

0 20 40 60 80 100 120 140 160 180

Step

0 500 1000 1500

Mea

n

Rosenbrock10

8

106

104

102

100

10minus2

10minus4

ABCNABC

Number of generations0 500 1000 1500

Mea

n

Rastrigin10

5

100

10minus5

10minus10

10minus15

0 500 1000 1500

Mea

n

Griewank10

5

100

10minus5

10minus10

10minus15

10minus20

Number of generations

ABCNABC

Figure 1 Continued

Mathematical Problems in Engineering 5

Number of generations0 500 1000 1500

Mea

n

Schwefel22610

5

100

10minus5

10minus10

10minus15

Number of generations0 500 1000 1500

Mea

n

Ackley10

2

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

ABCNABC

ABCNABC

Figure 1 Convergence curves of ABC and NABC algorithms for 1198911ndash11989110functions (119863 = 30)

However NABC still keeps fast convergence and best perfor-mance

The results of NABC compared with the standard ABCalgorithm are shown in Table 2 in terms of the best worstmedian mean and standard deviation of the solutionsTheseresults show that the convergence rate of NABC is betterthan the standard ABC algorithm for most test functionsThe rate of convergence obviously increases on 119891

1ndash1198913 1198915

1198917ndash11989111

functions from Figure 1 and Table 2 NABC can findthe optimal solutions on functions119891

511989171198918The convergence

rate of NABC is the same order of magnitude as the standardABC algorithm on 119891

6 So the superiority of standard ABC

algorithm is not very obvious From the results in Table 2NABC can obtain better and closer-to-optimal solutions thanthe standard ABC algorithm for most of the functions In aword NABC increases the exploitation greatly NABC alsoretains good exploration Here ldquo+rdquo indicates that NABCis statistically significantly better than its correspondingcompetitor algorithm ldquoasymprdquo stands for that the result of thecorresponding algorithm is statistically similar with that ofNABC

NABC is further compared with GABC [13] and EABC[15] in Tables 3 and 4 The results of GABC and EABCare gained from [15] directly It is clear that NABC worksbetter in almost all the cases and achieves better performancethan GABC The rate of convergence of NABC increasesobviously on all functions in Tables 3 and 4The convergencerate of NABC is a little worse than that of EABC fromTable 3 In particular the convergence rate of Schwefel221and Rosenbrock is better than EABC in Table 4

At the same time NABC is compared with four variantsof DE (DE [17] jDE [18] JADE [19] and SaDE [20]) and fivevariants of PSO (FIPS [21] HPSO-TVAC [22] CLPSO [23]FPSO [24] OLPSO-G [25]) The results of DEs and PSOs aregained from [15] directly The maximum number of functionevaluations (FEs) is shown in Tables 5 and 6 Here FEs isMaxgenlowastSN It can be seen that NABC is better on almostall the test functions except Ackley in JADE and OLPSO-G

In Table 7 we can find the execution time of ABC issimilar with that of NABC So NABC is largely faster thanABC and does not add the execution time

5 Application

51 Knapsack Problems We know that Knapsack problem[26] is an engineering optimization problem It can be definedan unconstrained optimization problem and described asfollows

min119891 (119909) = minus

119899

sum

119894=1

119901119894119909119894+ 120572 lowastmax(0

119899

sum

119894=1

119908119894119909119894minus 119888) (7)

where 119908119894is each itemrsquos weight and 119901

119894is each itemrsquos profit

respectively The knapsack has a limited weight capabilityof 119888 The objective function of this problem is to pack theknapsack so that the items in it have the maximal totalprofit The decision variable 119909

119894is the value one if the item

119894 is packed otherwise it is the value zero And we assumethat all profits and weights are positive and all weights aresmaller than the capacity of 119888 120572 is a penalty factor which isset to 10119890 minus 20 in this paper Tables 8 and 9 are parametersof five standard knapsack problems The experiments testedon five benchmark 0-1 knapsack problems are performed tocompare the performance of NABC with that of standardABC The maximum number of generations is set to 2000The population size is 100 namely SN = 50 and Limit is 10Each of the experiments is repeated 30 times independently

52 Data Analysis In Table 10 we can see that NABC caneasily solve the optimum of Knapsack problems And itcan find the optimum in most cases except 119891

5 ABC has

the worst performance because we can see that it can notfind the optimums for 119891

2and 119891

5 In short NABC has a

better performance than ABC on solving the five knapsackproblems

6 Mathematical Problems in Engineering

Table 2 Best worst medianmean and standard deviation values obtained byABC andNABC through 30 independent runs on 11 functions

Function Dim Best Worst Median Mean SD Significant

1198911

30 ABC 504119890 minus 16 725119890 minus 16 504119890 minus 16 606119890 minus 16 934119890 minus 17 +NABC 113119890 minus 55 966119890 minus 54 666119890 minus 55 110119890 minus 54 171119890 minus 54

60 ABC 140119890 minus 15 187119890 minus 15 152119890 minus 15 162119890 minus 15 202119890 minus 16 +NABC 103119890 minus 53 466119890 minus 52 906119890 minus 53 148119890 minus 52 137119890 minus 52

100 ABC 220119890 minus 15 932119890 minus 15 521119890 minus 15 336119890 minus 15 201119890 minus 15 +NABC 161119890 minus 52 121119890 minus 51 369119890 minus 51 145119890 minus 51 891119890 minus 52

1198912

30 ABC 430119890 minus 16 634119890 minus 16 450119890 minus 16 503119890 minus 16 711119890 minus 17 +NABC 164119890 minus 57 782119890 minus 56 116119890 minus 56 270119890 minus 56 248119890 minus 56

60 ABC 143119890 minus 15 200119890 minus 15 162119890 minus 15 167119890 minus 15 184119890 minus 16 +NABC 443119890 minus 55 782119890 minus 53 801119890 minus 54 208119890 minus 53 233119890 minus 53

100 ABC 247119890 minus 15 366119890 minus 15 343119890 minus 05 319119890 minus 15 421119890 minus 16 +NABC 351119890 minus 53 575119890 minus 52 266119890 minus 52 301119890 minus 52 156119890 minus 52

1198913

30 ABC 238119890 minus 11 134119890 minus 10 642119890 minus 11 705119890 minus 11 519119890 minus 11 +NABC 506119890 minus 30 411119890 minus 29 173119890 minus 29 202119890 minus 29 110119890 minus 29

60 ABC 902119890 minus 11 823119890 minus 10 512119890 minus 10 408119890 minus 10 157119890 minus 10 +NABC 712119890 minus 29 572119890 minus 28 257119890 minus 28 292119890 minus 28 154119890 minus 28

100 ABC 205119890 minus 09 536119890 minus 08 484119890 minus 09 367119890 minus 09 480119890 minus 09 +NABC 285119890 minus 28 281119890 minus 27 735119890 minus 28 104119890 minus 27 688119890 minus 28

1198914

30 ABC 868119890 + 00 120119890 + 01 184119890 + 01 110119890 + 01 129119890 minus 00 +NABC 177119890 minus 01 318119890 minus 01 268119890 minus 01 267119890 minus 01 355119890 minus 02

60 ABC 453119890 + 01 508119890 + 01 502119890 + 01 486119890 + 01 190119890 minus 00 +NABC 190119890 + 00 316119890 + 00 274119890 + 00 271119890 + 00 314119890 minus 01

100 ABC 692119890 + 01 743119890 + 01 726119890 + 01 725119890 + 01 186119890 minus 00 +NABC 832119890 + 00 110119890 + 01 979119890 + 00 992119890 + 00 704119890 minus 01

1198915

30 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

60 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

100 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

1198916

30 ABC 812119890 minus 03 168119890 minus 01 133119890 minus 02 486119890 minus 02 484119890 minus 02

asymp

NABC 221119890 minus 03 231119890 minus 01 332119890 minus 02 599119890 minus 02 564119890 minus 02

60 ABC 108119890 minus 02 588119890 minus 01 156119890 minus 01 141119890 minus 01 165119890 minus 01

NABC 288119890 minus 04 388119890 minus 01 473119890 minus 02 102119890 minus 01 995119890 minus 02 asymp

100 ABC 357119890 minus 02 994119890 minus 01 145119890 minus 01 297119890 minus 01 364119890 minus 01

NABC 292119890 minus 04 103119890 + 00 216119890 minus 01 350119890 minus 01 328119890 minus 01 asymp

1198917

30 ABC 536119890 minus 15 146119890 minus 01 456119890 minus 09 958119890 minus 04 105119890 minus 03 +NABC 0 0 0 0 0

60 ABC 587119890 minus 12 199119890 minus 00 125119890 minus 09 268119890 minus 02 632119890 minus 02 +NABC 0 0 0 0 0

100 ABC 865119890 minus 13 199119890 minus 00 115119890 minus 06 497119890 minus 02 216119890 minus 01 +NABC 0 0 0 0 0

1198918

30 ABC 801119890 minus 15 130119890 minus 12 814119890 minus 13 237119890 minus 13 418119890 minus 13 +NABC 0 0 0 0 0

60 ABC 832119890 minus 14 134119890 minus 11 720119890 minus 13 345119890 minus 12 289119890 minus 12 +NABC 0 0 0 0 0

100 ABC 536119890 minus 14 158119890 minus 09 427119890 minus 10 180119890 minus 10 203119890 minus 10 +NABC 0 0 0 0 0

Mathematical Problems in Engineering 7

Table 2 Continued

Function Dim Best Worst Median Mean SD Significant

1198919

30 ABC 154119890 minus 06 237119890 + 02 376119890 minus 01 886119890 + 01 862119890 + 01 +NABC minus364119890 minus 12 minus182119890 minus 12 minus182119890 minus 12 minus200119890 minus 12 555119890 minus 13

60 ABC 355119890 + 02 769119890 + 02 769119890 + 02 540119890 + 02 141119890 + 02 +NABC 291119890 minus 11 364119890 minus 11 364119890 minus 11 354119890 minus 11 252119890 minus 12

100 ABC 781119890 + 02 155119890 + 03 151119890 + 03 129119890 + 03 223119890 + 02 +NABC 102119890 minus 10 116119890 minus 10 109119890 minus 10 110119890 minus 10 458119890 minus 12

11989110

30 ABC 536119890 minus 10 123119890 minus 08 367119890 minus 09 145119890 minus 09 237119890 minus 09 +NABC 222119890 minus 14 293119890 minus 14 293119890 minus 14 266119890 minus 14 332119890 minus 15

60 ABC 814119890 minus 09 754119890 minus 08 538119890 minus 08 460119890 minus 08 204119890 minus 08 +NABC 506119890 minus 14 684119890 minus 14 648119890 minus 14 644119890 minus 14 526119890 minus 15

100 ABC 535119890 minus 08 428119890 minus 07 104119890 minus 07 283119890 minus 07 157119890 minus 07 +NABC 104119890 minus 13 122119890 minus 13 115119890 minus 13 116119890 minus 13 475119890 minus 15

11989111

150 ABC minus774221 minus769602 minus770460 minus771658 211119890 minus 01 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

300 ABC minus770391 minus768430 minus770391 minus769258 676119890 minus 02 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

500 ABC minus768562 minus765849 minus768562 minus767220 952119890 minus 02 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

Table 3 Comparison among GABC EABC and NABC on optimizing 8 benchmark functions

Function MaxFEs Dim GABC EABC NABCMean SD Mean SD Mean SD

Sphere150000 30 137119890 minus 25 270119890 minus 25 442119890 minus 67 271119890 minus 67 110119890 minus 54 171119890 minus 54

300000 60 486119890 minus 23 514119890 minus 23 230119890 minus 64 103119890 minus 64 148119890 minus 52 137119890 minus 52

500000 100 905119890 minus 22 428119890 minus 22 637119890 minus 63 255119890 minus 63 145119890 minus 51 891119890 minus 52

Schwefel222150000 30 556119890 minus 15 979119890 minus 15 551119890 minus 35 702119890 minus 35 202119890 minus 29 110119890 minus 29

300000 60 561119890 minus 14 993119890 minus 15 508119890 minus 33 195119890 minus 34 292119890 minus 28 154119890 minus 28

500000 100 229119890 minus 13 397119890 minus 13 468119890 minus 32 142119890 minus 32 104119890 minus 27 688119890 minus 28

Schwefel221150000 30 307119890 minus 00 472119890 minus 01 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

300000 60 385119890 + 01 342119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

500000 100 704119890 + 01 120119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock150000 30 130119890 minus 00 164119890 minus 00 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

300000 60 166119890 + 01 328119890 + 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

500000 100 239119890 + 01 334119890 + 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Griewank100000 30 140119890 minus 08 116119890 minus 08 0 0 0 0150000 60 158119890 minus 06 100119890 minus 06 0 0 0 0250000 100 244119890 minus 06 250119890 minus 06 0 0 0 0

Rastrigin100000 30 708119890 minus 03 119119890 minus 03 0 0 0 0150000 60 347119890 minus 00 878119890 minus 01 0 0 0 0250000 100 978119890 minus 00 291119890 minus 00 0 0 0 0

Schwefel22650000 30 638119890 minus 00 238119890 + 01 0 0 171119890 minus 09 301119890 minus 10

100000 60 330119890 + 01 494119890 + 01 392119890 minus 11 356119890 minus 12 124119890 minus 09 502119890 minus 09

200000 100 605119890 + 01 805119890 + 01 112119890 minus 10 356119890 minus 12 122119890 minus 10 297119890 minus 11

Ackley50000 30 645119890 minus 03 268119890 minus 03 339119890 minus 10 695119890 minus 11 204119890 minus 08 472119890 minus 09

100000 60 296119890 minus 03 214119890 minus 03 181119890 minus 09 270119890 minus 10 445119890 minus 08 150119890 minus 08

150000 100 601119890 minus 02 960119890 minus 02 453119890 minus 08 123119890 minus 08 601119890 minus 07 323119890 minus 07

8 Mathematical Problems in Engineering

Table 4 Comparison among ABC EABC and NABC on optimizing 4 benchmark functions with119863 = 30 60 100

Function Dim ABC EABC NABCMean SD Mean SD Mean SD

Schwefel22130 110119890 + 01 129119890 minus 00 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

60 486119890 + 01 190119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

100 725119890 + 01 186119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock30 486119890 minus 02 484119890 minus 02 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

60 141119890 minus 01 165119890 minus 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

100 297119890 minus 01 364119890 minus 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Schwefel22630 886119890 + 01 862119890 + 01 minus123119890 minus 13 109119890 minus 13 minus200119890 minus 12 555119890 minus 13

60 540119890 + 02 141119890 + 02 291119890 minus 11 0 354119890 minus 11 252119890 minus 12

100 129119890 + 03 223119890 + 02 945119890 minus 11 0 110119890 minus 10 458119890 minus 12

Ackley30 145119890 minus 09 237119890 minus 09 136119890 minus 14 174119890 minus 15 266119890 minus 14 332119890 minus 15

60 460119890 minus 08 204119890 minus 08 449119890 minus 14 284119890 minus 15 644119890 minus 14 526119890 minus 15

100 283119890 minus 07 157119890 minus 07 953119890 minus 14 284119890 minus 15 116119890 minus 13 475119890 minus 15

Table 5 Comparison among NABC DE jDE JADE and SaDE on optimizing 8 benchmark functions with119863 = 30

Function MaxFEs DE jDE JADE SaDE NABC

Sphere 150000 Mean 98119890 minus 14 146119890 minus 28 132119890 minus 54 328119890 minus 20 110119890 minus 54

SD 84119890 minus 14 178119890 minus 28 922119890 minus 54 362119890 minus 20 171119890 minus 54

Schwefel222 200000 Mean 16119890 minus 09 902119890 minus 24 318119890 minus 25 351119890 minus 25 720119890 minus 40

SD 11119890 minus 09 601119890 minus 24 205119890 minus 25 274119890 minus 25 333119890 minus 40

Step 10000 Mean 47119890 + 03 613119890 + 02 562119890 + 00 507119890 + 01 0SD 11119890 + 03 172119890 + 02 187119890 + 00 134119890 + 01 0

Rosenbrock 300000 Mean 21119890 + 00 13119890 + 01 32119890 minus 01 21119890 + 01 422119890 minus 02

SD 15119890 + 00 14119890 + 01 11119890 + 00 78119890 + 00 493119890 minus 02

Griewank 50000 Mean 20119890 minus 01 729119890 minus 06 157119890 minus 08 252119890 minus 09 109119890 minus 08

SD 11119890 minus 01 105119890 minus 05 109119890 minus 07 124119890 minus 08 286119890 minus 08

Rastrigin 100000 Mean 18119890 + 02 332119890 minus 04 133119890 minus 01 243119890 + 00 0

SD 13119890 + 01 639119890 minus 04 974119890 minus 02 160119890 + 00 0

Ackley 50000 Mean 11119890 minus 01 237119890 minus 04 335119890 minus 09 381119890 minus 06 204119890 minus 08

SD 39119890 minus 02 710119890 minus 05 284119890 minus 09 826119890 minus 07 472119890 minus 09

Schwefel226 100000 Mean 59119890 + 03 170119890 minus 10 262119890 minus 04 113119890 minus 08 minus194119890 minus 12

SD 11119890 + 03 171119890 minus 10 359119890 minus 04 108119890 minus 08 461119890 minus 13

Table 6 Comparison among NABC FIPS HPSO-TVAC CLPSO FPSO and OLPSO-G on optimizing 8 benchmark functions with 200000FEs

Function FIPS HPSO-TVAC CLPSO FPSO OLPSO-G NABC

Sphere Mean 242119890 minus 13 283119890 minus 33 158119890 minus 12 240119890 minus 16 412119890 minus 54 545119890 minus 75

SD 173119890 minus 13 319119890 minus 33 770119890 minus 13 200119890 minus 31 634119890 minus 54 355119890 minus 75

Schwefel222 Mean 276119890 minus 08 903119890 minus 20 251119890 minus 08 158119890 minus 11 985119890 minus 30 720119890 minus 40

SD 904119890 minus 09 958119890 minus 20 584119890 minus 09 103119890 minus 22 101119890 minus 29 333119890 minus 40

Step Mean 0 0 0 0 0 0SD 0 0 0 0 0 0

Rosenbrock Mean 251119890 + 01 239119890 + 01 113119890 + 01 281119890 + 01 215119890 + 01 633119890 minus 02

SD 510119890 minus 01 265119890 + 01 985119890 minus 00 231119890 + 02 299119890 + 01 634119890 minus 02

Griewank Mean 901119890 minus 12 975119890 minus 03 902119890 minus 09 147119890 minus 03 483119890 minus 03 0

SD 184119890 minus 11 833119890 minus 03 857119890 minus 09 128119890 minus 05 863119890 minus 03 0

Rastrigin Mean 651119890 + 01 943119890 minus 00 909119890 minus 05 738119890 + 01 107119890 minus 00 0

SD 133119890 + 01 348119890 minus 00 125119890 minus 04 370119890 + 02 992119890 minus 01 0

Ackley Mean 233119890 minus 07 729119890 minus 14 366119890 minus 07 217119890 minus 09 798119890 minus 15 278119890 minus 14

SD 719119890 minus 08 300119890 minus 14 757119890 minus 08 171119890 minus 18 203119890 minus 15 259119890 minus 15

Schwefel226 Mean 993119890 + 02 159119890 + 03 382119890 minus 04 134119890 + 03 384119890 + 02 minus243119890 minus 12

SD 509119890 + 02 326119890 + 02 128119890 minus 05 277119890 + 02 217119890 + 02 872119890 minus 13

Mathematical Problems in Engineering 9

Table 7 Execution time of ABC and NABC algorithm with119863 = 30 60 100 (second)

Dim Algorithm 1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918

1198919

11989110

30 ABC 19850 25535 33363 18768 3969 32338 40942 78680 94332 47626NABC 17661 25697 36352 20056 2768 30612 18229 31331 77497 42660

60 ABC 43236 72503 104475 47454 12304 81891 185687 259944 322573 205303NABC 44900 73449 108912 47991 8202 80281 95126 129497 265051 186870

100 ABC 89402 163012 255115 92929 37074 180789 371414 574304 841482 410198NABC 88507 168964 255134 91418 21081 175828 334624 308107 687463 365939

Table 8 The five benchmark knapsack problems

119891 Dim Parameter (119908 119901 119888)1198911

10 119908 = (95 4 60 32 23 72 80 62 65 46) 119901 = (55 10 47 5 4 50 8 61 85 87) 119888 = 269

1198912

20 119908 = (92 4 43 83 84 68 92 82 6 44 32 18 56 83 25 96 70 48 14 58) 119901 = (44 46 90 72 91 40 75 35 8 54 78 4077 15 61 17 75 29 75 63) 119888 = 878

1198913

4 119908 = (6 5 9 7) 119901 = (9 11 13 15) 119888 = 201198914

4 119908 = (2 4 6 7) 119901 = (6 10 12 13) 119888 = 11

1198915

15119908 = (56358531 80874050 47987304 89596240 74660482 85894345 51353496 1498459 36445204 1658986244569231 0466933 37788018 57118442 60716575) 119901 = (0125126 19330424 58500931 35029145 8228400517410810 71050142 30399487 9140294 14731285 98852504 11908322 0891140 53166295 60176397) 119888 = 375

Table 9 The optimal solutions of the five benchmark knapsack problems

119891 Optimal solution Optimal value Value of constraint1198911

(0 1 1 1 0 0 0 1 1 1) 295 01198912

(1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1) 1024 minus71198913

(1 1 0 1) 35 minus21198914

(0 1 0 1) 23 01198915

(0 0 1 0 1 0 1 1 0 1 1 1 0 1 1) 4810694 minus200392

Table 10 Best worst median mean and standard deviation values obtained by ABC and NABC

119891 Algorithm Best Worst Median Mean SD

1198911

ABC 295 295 295 295 0NDABC 295 295 295 295 0

1198912

ABC 1024 1013 1024 1021 452NDABC 1024 1024 1024 1024 0

1198913

ABC 35 35 35 35 0NDABC 35 35 35 35 0

1198914

ABC 23 23 23 23 0NDABC 23 23 23 23 0

1198915

ABC 481069 437935 475478 460729 1847NDABC 481069 435786 481069 470771 879

6 Conclusion

In this paper we develop a novel artificial bee colonyalgorithm named NABC We add the global best solutioninto the search equation to drive the new candidate solutiononly around the global best solution in order to improvethe exploitation And the search equation of the employedbees is improved to keep the exploration of algorithm Theexperimental results tested on 11 benchmark functions showthat the convergence of NABC is much faster than that of

other algorithms and the computing is more effective At thesame timewe applyNABCon solving five standardKnapsackproblems and get good optimums So it is fitted to solvemanyengineering practical problems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors are grateful to the editor and the anonymousreviewers for their valuable comments and suggestions

References

[1] K S Tang K FMan S Kwong andQ He ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal ProcessingMagazine vol 13no 6 pp 22ndash37 1996

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

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

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

[5] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 Erciyes University KayseriTurkey 2005

[6] B Basturk and D Karaboga ldquoAn artificial bee colony (ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium May 2006

[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

[10] D Karaboga B Akay and C Ozturk ldquoArtificial bee colony(ABC) optimization algorithm for training feed-forward neuralnetworksrdquo in Modeling Decisions for Artificial Intelligence vol4617 of LectureNotes in Computer Science pp 318ndash329 SpringerBerlin Germany 2007

[11] Y-F Liu and S-Y Liu ldquoA hybrid discrete artificial bee colonyalgorithm for permutation flowshop scheduling problemrdquoApplied Soft Computing Journal vol 13 no 3 pp 1459ndash14632013

[12] D Karaboga and B Basturk ldquoArtificial bee colony (ABC)optimization algorithm for solving constrained optimizationproblemsrdquo in Foundations of Fuzzy Logic and Soft Computingvol 4529 of Lecture Notes in Computer Science pp 789ndash798Springer Berlin Germany 2007

[13] G P Zhu and S Kwong ldquoGbest-guided artificial bee colonyalgorithm for numerical function optimizationrdquo Applied Math-ematics and Computation vol 217 no 7 pp 3166ndash3173 2010

[14] B Basturk and D Karaboga ldquoA modified artificial bee colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[15] W-F Gao S-Y Liu and L-L Huang ldquoEnhancing artificialbee colony algorithm using more information-based searchequationsrdquo Information Sciences vol 270 pp 112ndash133 2014

[16] D Karaboga and B Gorkemli ldquoA quick artificial bee colony(qABC) algorithm and its performance on optimization prob-lemsrdquo Applied Soft Computing vol 23 pp 227ndash238 2014

[17] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 23 pp 689ndash6942010

[18] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[19] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009

[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2008

[21] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[22] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[23] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[24] M A M de Oca T Stutzle M Birattari and M DorigoldquoFrankensteinrsquos PSO a composite particle swarm optimizationalgorithmrdquo IEEE Transactions on Evolutionary Computationvol 13 no 5 pp 1120ndash1132 2009

[25] Z-H Zhan J Zhang Y Li and Y-H Shi ldquoOrthogonal learningparticle swarm optimizationrdquo IEEE Transactions on Evolution-ary Computation vol 15 no 6 pp 832ndash847 2011

[26] D Zou L Gao S Li and J Wu ldquoSolving 0-1 knapsack problemby a novel global harmony search algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 1556ndash1564 2011

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

algorithms inmost cases And it is better than variant DE andPSO algorithms At the same time five knapsack problemsare solved by ABC and NABC which shows that NABCis superior to ABC on solving engineering optimizationproblems

The rest of this paper is organized as follows Section 2summarizes the standard ABC algorithm The novel ABCalgorithm named NABC is presented and analyzed inSection 3 Section 4 presents and discusses the experimen-tal results of 11 benchmark functions Section 5 shows theapplication of NABC on knapsack problems Finally theconclusion is drawn in Section 6

2 Standard ABC Algorithm

Initialization is the first step in the standard ABC algorithm[5] The position of a food source represents a possiblesolution to the optimization problem Each food sourceis exploited by only one employed bee so the number ofemployed bees named SN is equal to the number of foodsources And a random initial population of SN food sourcesis generated as follows

119909119894119895

= 119909min119895 + rand (0 1) lowast (119909max119895 minus 119909min119895) (1)

where 119894 isin (1 2 SN) 119895 isin (1 2 119863) and 119863 is thedimension of the numerical function119909max119895 and 119909min119895 are thelower and upper bounds of 119909

119894119895

Second is the search processes of the employed beeswhere the onlookers and the scouts are started The standardABC algorithm uses the following equation to produce acandidate food source position from the old one

119881119894119895

= 119909119894119895

+ 120601119894119895

lowast (119909119894119895

minus 119909119896119895) (2)

where 119896 isin (1 2 SN) and 119895 isin (1 2 119863) are randomindexes 119896 is different from 119894 120601

119894119895is a random number in the

range [minus1 1] After each candidate source position is pro-duced and evaluated by the employed bees its performanceis compared with that of the old food source position If thenew food source position has an equal or better quality thanthe old source position the old one is replaced by the newone Otherwise the old one is retained The greedy selectionis used

Third the employed bees transfer their food informationto the onlookers The onlookers tend to further search thefood around the selected food source The onlooker choosesa food source depending on the probability value associatedwith that food source

119901119894=

fitness119894

sum119899

119894=1fitness

119894

(3)

where fitness119894is the fitness value of the solution which is

obtained by the following equation

fitness119894=

1

1 + 119891119894

119891119894ge 0

1 +1003816100381610038161003816119891119894

1003816100381610038161003816 119891119894lt 0

(4)

where119891119894is the objective function value If the new food source

position has an equal or better quality than the old source

position the old one is replaced by the new one Otherwisethe old one is retainedwhich is the same as the employed beesstage

At last if a position cannot be improved further through anumber of cycles which is called Limit for abandonment thatfood source is abandoned accordinglyThe scout will discovera new food source position randomly using (1) to replace it

3 A Novel Artificial BeeColony Algorithm (NABC)

31 Modified Search Solutions The exploration is the abilityof investigating the various unknown regions in the solutionspace So it is also very important inABC algorithm To retainthe exploration we present a modified search equation of theemployed bees as follows

119881119894119895

= 119909119903119895

+ 120601119894119895

lowast (119909119903119895

minus 119909119896119895) (5)

where the index 119903 119896 is randomly chosen from (1 2 SN)and different from the index 119894 119895 isin (1 2 119863) is a randomlychosen index 120601

119894119895is a random number in the range [minus1 1]

Differential evolution (DE) [17] is a simple and effi-cient evolutionary algorithm for many optimization prob-lems in the real-world applications [18ndash22] Inspired by itmany researchers improved variant intelligent algorithms Soinspired by DE and various modified search equations of theABC algorithms and taking advantage of the information ofthe global best solution to guide the search of the candidatesolutions a new equation is proposed to serve as the searchequation of onlookers as follows

119881119894119895

= 119909best119895 + 120601119894119895

lowast (119909best119895 minus 119909119896119895) (6)

where the index 119896 is a mutually exclusive integer randomlychosen from (1 2 SN) and different from the index 119894119909best119895 is the 119895th element of the global best solution and119895 isin (1 2 119863) is a randomly chosen index 120601

119894119895is a

randomnumber in the range [minus1 1]We name the novel ABCalgorithmusing (5) to replace (2) on the stage of the employedbees and using (6) to replace (2) on the stage of the onlookersas NABC which can drive the new candidate solution onlyaround the best solution of the previous iteration Thereforethe modified solution search equation described by (6) canincrease the exploitation of ABC algorithm greatly The bestsolution in the current population is a very useful sourcewhich can be used to improve the convergence performanceTherefore we modify the solution search equation by apply-ing the global best solution to guide the search of newcandidate solutions in order to improve the exploitation Atthe same time the exploration is improved by (5) from theexperiments in Section 4

The new candidate solution is generated around 119909119894119895

inGABC [13] It is different from (6) in which the one isgenerated around 119909best119895 EABC [15] adopts two differentsearch equations based on different the emphases in thedifferent search stages NABC uses (5) and (6) to generate thenew solution in the stages of employed bees and onlookersrespectively In the qABC [16] 119909

best119873119898

represents the best

Mathematical Problems in Engineering 3

Table 1 Benchmark functions used in experiments

Fun Name Search range Min

1198911=

119899

sum

119894=1

1199092

119894Sphere [minus100 100] 0

1198912=

119899

sum

119894=1

1198941199092

119894Sumsquare [minus10 10] 0

1198913=

119899

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119899

prod

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816

Schwefel222 [minus10 10] 0

1198914= max 100381610038161003816

1003816119909119894

1003816100381610038161003816 1 le 119894 le 119899 Schwefel221 [minus100 100] 0

1198915=

119899

sum

119894=1

(lfloor119909119894+ 05rfloor)

2 Step [minus100 100] 0

1198916=

119899

sum

119894=1

[100 (1199092

119894minus 119909119894+1

)

2

+ (119909119894minus 1)2

] Rosenbrock [minus5 10] 0

1198917=

119899

sum

119894=1

[1199092

119894minus 10 cos (2120587119909

119894) + 10] Rastrigin [minus512 512] 0

1198918=

1

4000

119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 Griewank [minus600 600] 0

1198919= 41898288727243369 lowast 119899 minus

119899

sum

119894=1

119909119894sin(radic

1003816100381610038161003816119909119894

1003816100381610038161003816) Schwefel226 [minus500 500] 0

11989110

= minus20 exp(minus02radic

1

119899

119899

sum

119894=1

1199092

119894) minus exp(

1

119899

119899

sum

119894=1

cos (2120587119909119894)) + 20 + 119890 Ackley [minus32 32] 0

11989111

=

1

119899

119899

sum

119894=1

(1199094

119894minus 16119909

2

119894+ 5119909119894) Himmelblau [minus5 5] minus7833236

solution among the neighbors of 119909119898and itself (119873

119898) But in

the NABC 119909best119895 represents the global best solution At thesame time in the stages of employed bees and onlookers thenew search equations used in qABC are different from theseof the NABC adopted

32 The Novel ABC Algorithm (NABC) The complete com-putational procedure of the novel ABC algorithm (NABC) isoutlined as follows

Algorithm 1 (NABC) Consider the following steps

Step 1 (initialization) Preset SN Limit and Maxcycle

Step 2 Randomly generate SN solutions from the search spaceto create an initial population 119875

Step 3 Calculate the function values 119891 of the initial popula-tion 119875

Step 4 The employed bees stage Generate a new solution V119894119895

by (5) if 119891(V119894119895) lt 119891(119909

119894119895) 119909119894119895

= V119894119895 trail119894= 1 else trail

119894=

trail119894+ 1

Step 5 Calculate the probability values 119901119894for the solutions 119909

119894119895

by (3)

Step 6 (the onlookers stage) If rand lt 119901119894 generate a new

solution V119894119895by (6) if 119891(V

119894119895) lt 119891(119909

119894119895) 119909119894119895

= V119894119895 trail119894= 1

else trail119894= trail

119894+ 1

Step 7 (the scouts stage) If trail gt Limit replace 119909119894119895with a

new produced solution by (1)

Step 8 If cycle number gt Maxcycle stop and memorize thebest solution achieved so far otherwise go to Step 2

4 Numerical Functionsand Experimental Results

41 Numerical Functions It is applied to minimize a set of 11scalable benchmark functions in Table 1 to test the efficiencyof the novel ABC algorithm The smaller the final result thebetter it is

42 Parameter Settings A set of experiments tested on 11numerical benchmark functions are performed in this sectionto compare the performance of the novel ABC algorithmwiththat of other improved ABC algorithms Functions 119891

1ndash1198914are

unimodal Function 1198915is a discontinuous step function The

Rosenbrock function of 1198916which is unimodal when 119863 =

2 and 3 may have multiple minima in the high dimensioncase [18] Functions 119891

7ndash11989111are multimodal To compare with

other variance algorithms the dimension119863 of every functionis 30 60 or 100 respectively the same as [15]The dimension119863 of 119891

11is 150 300 or 500 The maximum number of gen-

erations is 1500 3000 or 5000 respectively The populationsize is 100 namely SN = 50 and Limit is 200 which is thesame as [15] Each of the experiments is repeated 30 timesindependently

43 Experimental Results Figure 1 graphically presents theconvergence curves of 119891

1ndash11989110functions respectively We can

see from Figure 1 that in the later stage of evolution thestandard ABC algorithm enters a long period of stagnation

4 Mathematical Problems in Engineering

1010

100

10minus10

10minus20

10minus30

10minus40

10minus50

10minus60

0 500 1000 1500

Mea

n

Sphere Sumsquare10

10

100

10minus10

10minus20

10minus30

10minus40

10minus50

10minus60

0 500 1000 1500

Mea

n

Schwefel222

0 500 1000 1500

Mea

n

1020

1010

100

10minus10

10minus20

10minus30

0 500 1000 1500

Mea

n

Schwefel22110

2

101

100

10minus1

Mea

n

105

104

103

102

101

100

0 20 40 60 80 100 120 140 160 180

Step

0 500 1000 1500

Mea

n

Rosenbrock10

8

106

104

102

100

10minus2

10minus4

ABCNABC

Number of generations0 500 1000 1500

Mea

n

Rastrigin10

5

100

10minus5

10minus10

10minus15

0 500 1000 1500

Mea

n

Griewank10

5

100

10minus5

10minus10

10minus15

10minus20

Number of generations

ABCNABC

Figure 1 Continued

Mathematical Problems in Engineering 5

Number of generations0 500 1000 1500

Mea

n

Schwefel22610

5

100

10minus5

10minus10

10minus15

Number of generations0 500 1000 1500

Mea

n

Ackley10

2

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

ABCNABC

ABCNABC

Figure 1 Convergence curves of ABC and NABC algorithms for 1198911ndash11989110functions (119863 = 30)

However NABC still keeps fast convergence and best perfor-mance

The results of NABC compared with the standard ABCalgorithm are shown in Table 2 in terms of the best worstmedian mean and standard deviation of the solutionsTheseresults show that the convergence rate of NABC is betterthan the standard ABC algorithm for most test functionsThe rate of convergence obviously increases on 119891

1ndash1198913 1198915

1198917ndash11989111

functions from Figure 1 and Table 2 NABC can findthe optimal solutions on functions119891

511989171198918The convergence

rate of NABC is the same order of magnitude as the standardABC algorithm on 119891

6 So the superiority of standard ABC

algorithm is not very obvious From the results in Table 2NABC can obtain better and closer-to-optimal solutions thanthe standard ABC algorithm for most of the functions In aword NABC increases the exploitation greatly NABC alsoretains good exploration Here ldquo+rdquo indicates that NABCis statistically significantly better than its correspondingcompetitor algorithm ldquoasymprdquo stands for that the result of thecorresponding algorithm is statistically similar with that ofNABC

NABC is further compared with GABC [13] and EABC[15] in Tables 3 and 4 The results of GABC and EABCare gained from [15] directly It is clear that NABC worksbetter in almost all the cases and achieves better performancethan GABC The rate of convergence of NABC increasesobviously on all functions in Tables 3 and 4The convergencerate of NABC is a little worse than that of EABC fromTable 3 In particular the convergence rate of Schwefel221and Rosenbrock is better than EABC in Table 4

At the same time NABC is compared with four variantsof DE (DE [17] jDE [18] JADE [19] and SaDE [20]) and fivevariants of PSO (FIPS [21] HPSO-TVAC [22] CLPSO [23]FPSO [24] OLPSO-G [25]) The results of DEs and PSOs aregained from [15] directly The maximum number of functionevaluations (FEs) is shown in Tables 5 and 6 Here FEs isMaxgenlowastSN It can be seen that NABC is better on almostall the test functions except Ackley in JADE and OLPSO-G

In Table 7 we can find the execution time of ABC issimilar with that of NABC So NABC is largely faster thanABC and does not add the execution time

5 Application

51 Knapsack Problems We know that Knapsack problem[26] is an engineering optimization problem It can be definedan unconstrained optimization problem and described asfollows

min119891 (119909) = minus

119899

sum

119894=1

119901119894119909119894+ 120572 lowastmax(0

119899

sum

119894=1

119908119894119909119894minus 119888) (7)

where 119908119894is each itemrsquos weight and 119901

119894is each itemrsquos profit

respectively The knapsack has a limited weight capabilityof 119888 The objective function of this problem is to pack theknapsack so that the items in it have the maximal totalprofit The decision variable 119909

119894is the value one if the item

119894 is packed otherwise it is the value zero And we assumethat all profits and weights are positive and all weights aresmaller than the capacity of 119888 120572 is a penalty factor which isset to 10119890 minus 20 in this paper Tables 8 and 9 are parametersof five standard knapsack problems The experiments testedon five benchmark 0-1 knapsack problems are performed tocompare the performance of NABC with that of standardABC The maximum number of generations is set to 2000The population size is 100 namely SN = 50 and Limit is 10Each of the experiments is repeated 30 times independently

52 Data Analysis In Table 10 we can see that NABC caneasily solve the optimum of Knapsack problems And itcan find the optimum in most cases except 119891

5 ABC has

the worst performance because we can see that it can notfind the optimums for 119891

2and 119891

5 In short NABC has a

better performance than ABC on solving the five knapsackproblems

6 Mathematical Problems in Engineering

Table 2 Best worst medianmean and standard deviation values obtained byABC andNABC through 30 independent runs on 11 functions

Function Dim Best Worst Median Mean SD Significant

1198911

30 ABC 504119890 minus 16 725119890 minus 16 504119890 minus 16 606119890 minus 16 934119890 minus 17 +NABC 113119890 minus 55 966119890 minus 54 666119890 minus 55 110119890 minus 54 171119890 minus 54

60 ABC 140119890 minus 15 187119890 minus 15 152119890 minus 15 162119890 minus 15 202119890 minus 16 +NABC 103119890 minus 53 466119890 minus 52 906119890 minus 53 148119890 minus 52 137119890 minus 52

100 ABC 220119890 minus 15 932119890 minus 15 521119890 minus 15 336119890 minus 15 201119890 minus 15 +NABC 161119890 minus 52 121119890 minus 51 369119890 minus 51 145119890 minus 51 891119890 minus 52

1198912

30 ABC 430119890 minus 16 634119890 minus 16 450119890 minus 16 503119890 minus 16 711119890 minus 17 +NABC 164119890 minus 57 782119890 minus 56 116119890 minus 56 270119890 minus 56 248119890 minus 56

60 ABC 143119890 minus 15 200119890 minus 15 162119890 minus 15 167119890 minus 15 184119890 minus 16 +NABC 443119890 minus 55 782119890 minus 53 801119890 minus 54 208119890 minus 53 233119890 minus 53

100 ABC 247119890 minus 15 366119890 minus 15 343119890 minus 05 319119890 minus 15 421119890 minus 16 +NABC 351119890 minus 53 575119890 minus 52 266119890 minus 52 301119890 minus 52 156119890 minus 52

1198913

30 ABC 238119890 minus 11 134119890 minus 10 642119890 minus 11 705119890 minus 11 519119890 minus 11 +NABC 506119890 minus 30 411119890 minus 29 173119890 minus 29 202119890 minus 29 110119890 minus 29

60 ABC 902119890 minus 11 823119890 minus 10 512119890 minus 10 408119890 minus 10 157119890 minus 10 +NABC 712119890 minus 29 572119890 minus 28 257119890 minus 28 292119890 minus 28 154119890 minus 28

100 ABC 205119890 minus 09 536119890 minus 08 484119890 minus 09 367119890 minus 09 480119890 minus 09 +NABC 285119890 minus 28 281119890 minus 27 735119890 minus 28 104119890 minus 27 688119890 minus 28

1198914

30 ABC 868119890 + 00 120119890 + 01 184119890 + 01 110119890 + 01 129119890 minus 00 +NABC 177119890 minus 01 318119890 minus 01 268119890 minus 01 267119890 minus 01 355119890 minus 02

60 ABC 453119890 + 01 508119890 + 01 502119890 + 01 486119890 + 01 190119890 minus 00 +NABC 190119890 + 00 316119890 + 00 274119890 + 00 271119890 + 00 314119890 minus 01

100 ABC 692119890 + 01 743119890 + 01 726119890 + 01 725119890 + 01 186119890 minus 00 +NABC 832119890 + 00 110119890 + 01 979119890 + 00 992119890 + 00 704119890 minus 01

1198915

30 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

60 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

100 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

1198916

30 ABC 812119890 minus 03 168119890 minus 01 133119890 minus 02 486119890 minus 02 484119890 minus 02

asymp

NABC 221119890 minus 03 231119890 minus 01 332119890 minus 02 599119890 minus 02 564119890 minus 02

60 ABC 108119890 minus 02 588119890 minus 01 156119890 minus 01 141119890 minus 01 165119890 minus 01

NABC 288119890 minus 04 388119890 minus 01 473119890 minus 02 102119890 minus 01 995119890 minus 02 asymp

100 ABC 357119890 minus 02 994119890 minus 01 145119890 minus 01 297119890 minus 01 364119890 minus 01

NABC 292119890 minus 04 103119890 + 00 216119890 minus 01 350119890 minus 01 328119890 minus 01 asymp

1198917

30 ABC 536119890 minus 15 146119890 minus 01 456119890 minus 09 958119890 minus 04 105119890 minus 03 +NABC 0 0 0 0 0

60 ABC 587119890 minus 12 199119890 minus 00 125119890 minus 09 268119890 minus 02 632119890 minus 02 +NABC 0 0 0 0 0

100 ABC 865119890 minus 13 199119890 minus 00 115119890 minus 06 497119890 minus 02 216119890 minus 01 +NABC 0 0 0 0 0

1198918

30 ABC 801119890 minus 15 130119890 minus 12 814119890 minus 13 237119890 minus 13 418119890 minus 13 +NABC 0 0 0 0 0

60 ABC 832119890 minus 14 134119890 minus 11 720119890 minus 13 345119890 minus 12 289119890 minus 12 +NABC 0 0 0 0 0

100 ABC 536119890 minus 14 158119890 minus 09 427119890 minus 10 180119890 minus 10 203119890 minus 10 +NABC 0 0 0 0 0

Mathematical Problems in Engineering 7

Table 2 Continued

Function Dim Best Worst Median Mean SD Significant

1198919

30 ABC 154119890 minus 06 237119890 + 02 376119890 minus 01 886119890 + 01 862119890 + 01 +NABC minus364119890 minus 12 minus182119890 minus 12 minus182119890 minus 12 minus200119890 minus 12 555119890 minus 13

60 ABC 355119890 + 02 769119890 + 02 769119890 + 02 540119890 + 02 141119890 + 02 +NABC 291119890 minus 11 364119890 minus 11 364119890 minus 11 354119890 minus 11 252119890 minus 12

100 ABC 781119890 + 02 155119890 + 03 151119890 + 03 129119890 + 03 223119890 + 02 +NABC 102119890 minus 10 116119890 minus 10 109119890 minus 10 110119890 minus 10 458119890 minus 12

11989110

30 ABC 536119890 minus 10 123119890 minus 08 367119890 minus 09 145119890 minus 09 237119890 minus 09 +NABC 222119890 minus 14 293119890 minus 14 293119890 minus 14 266119890 minus 14 332119890 minus 15

60 ABC 814119890 minus 09 754119890 minus 08 538119890 minus 08 460119890 minus 08 204119890 minus 08 +NABC 506119890 minus 14 684119890 minus 14 648119890 minus 14 644119890 minus 14 526119890 minus 15

100 ABC 535119890 minus 08 428119890 minus 07 104119890 minus 07 283119890 minus 07 157119890 minus 07 +NABC 104119890 minus 13 122119890 minus 13 115119890 minus 13 116119890 minus 13 475119890 minus 15

11989111

150 ABC minus774221 minus769602 minus770460 minus771658 211119890 minus 01 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

300 ABC minus770391 minus768430 minus770391 minus769258 676119890 minus 02 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

500 ABC minus768562 minus765849 minus768562 minus767220 952119890 minus 02 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

Table 3 Comparison among GABC EABC and NABC on optimizing 8 benchmark functions

Function MaxFEs Dim GABC EABC NABCMean SD Mean SD Mean SD

Sphere150000 30 137119890 minus 25 270119890 minus 25 442119890 minus 67 271119890 minus 67 110119890 minus 54 171119890 minus 54

300000 60 486119890 minus 23 514119890 minus 23 230119890 minus 64 103119890 minus 64 148119890 minus 52 137119890 minus 52

500000 100 905119890 minus 22 428119890 minus 22 637119890 minus 63 255119890 minus 63 145119890 minus 51 891119890 minus 52

Schwefel222150000 30 556119890 minus 15 979119890 minus 15 551119890 minus 35 702119890 minus 35 202119890 minus 29 110119890 minus 29

300000 60 561119890 minus 14 993119890 minus 15 508119890 minus 33 195119890 minus 34 292119890 minus 28 154119890 minus 28

500000 100 229119890 minus 13 397119890 minus 13 468119890 minus 32 142119890 minus 32 104119890 minus 27 688119890 minus 28

Schwefel221150000 30 307119890 minus 00 472119890 minus 01 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

300000 60 385119890 + 01 342119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

500000 100 704119890 + 01 120119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock150000 30 130119890 minus 00 164119890 minus 00 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

300000 60 166119890 + 01 328119890 + 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

500000 100 239119890 + 01 334119890 + 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Griewank100000 30 140119890 minus 08 116119890 minus 08 0 0 0 0150000 60 158119890 minus 06 100119890 minus 06 0 0 0 0250000 100 244119890 minus 06 250119890 minus 06 0 0 0 0

Rastrigin100000 30 708119890 minus 03 119119890 minus 03 0 0 0 0150000 60 347119890 minus 00 878119890 minus 01 0 0 0 0250000 100 978119890 minus 00 291119890 minus 00 0 0 0 0

Schwefel22650000 30 638119890 minus 00 238119890 + 01 0 0 171119890 minus 09 301119890 minus 10

100000 60 330119890 + 01 494119890 + 01 392119890 minus 11 356119890 minus 12 124119890 minus 09 502119890 minus 09

200000 100 605119890 + 01 805119890 + 01 112119890 minus 10 356119890 minus 12 122119890 minus 10 297119890 minus 11

Ackley50000 30 645119890 minus 03 268119890 minus 03 339119890 minus 10 695119890 minus 11 204119890 minus 08 472119890 minus 09

100000 60 296119890 minus 03 214119890 minus 03 181119890 minus 09 270119890 minus 10 445119890 minus 08 150119890 minus 08

150000 100 601119890 minus 02 960119890 minus 02 453119890 minus 08 123119890 minus 08 601119890 minus 07 323119890 minus 07

8 Mathematical Problems in Engineering

Table 4 Comparison among ABC EABC and NABC on optimizing 4 benchmark functions with119863 = 30 60 100

Function Dim ABC EABC NABCMean SD Mean SD Mean SD

Schwefel22130 110119890 + 01 129119890 minus 00 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

60 486119890 + 01 190119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

100 725119890 + 01 186119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock30 486119890 minus 02 484119890 minus 02 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

60 141119890 minus 01 165119890 minus 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

100 297119890 minus 01 364119890 minus 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Schwefel22630 886119890 + 01 862119890 + 01 minus123119890 minus 13 109119890 minus 13 minus200119890 minus 12 555119890 minus 13

60 540119890 + 02 141119890 + 02 291119890 minus 11 0 354119890 minus 11 252119890 minus 12

100 129119890 + 03 223119890 + 02 945119890 minus 11 0 110119890 minus 10 458119890 minus 12

Ackley30 145119890 minus 09 237119890 minus 09 136119890 minus 14 174119890 minus 15 266119890 minus 14 332119890 minus 15

60 460119890 minus 08 204119890 minus 08 449119890 minus 14 284119890 minus 15 644119890 minus 14 526119890 minus 15

100 283119890 minus 07 157119890 minus 07 953119890 minus 14 284119890 minus 15 116119890 minus 13 475119890 minus 15

Table 5 Comparison among NABC DE jDE JADE and SaDE on optimizing 8 benchmark functions with119863 = 30

Function MaxFEs DE jDE JADE SaDE NABC

Sphere 150000 Mean 98119890 minus 14 146119890 minus 28 132119890 minus 54 328119890 minus 20 110119890 minus 54

SD 84119890 minus 14 178119890 minus 28 922119890 minus 54 362119890 minus 20 171119890 minus 54

Schwefel222 200000 Mean 16119890 minus 09 902119890 minus 24 318119890 minus 25 351119890 minus 25 720119890 minus 40

SD 11119890 minus 09 601119890 minus 24 205119890 minus 25 274119890 minus 25 333119890 minus 40

Step 10000 Mean 47119890 + 03 613119890 + 02 562119890 + 00 507119890 + 01 0SD 11119890 + 03 172119890 + 02 187119890 + 00 134119890 + 01 0

Rosenbrock 300000 Mean 21119890 + 00 13119890 + 01 32119890 minus 01 21119890 + 01 422119890 minus 02

SD 15119890 + 00 14119890 + 01 11119890 + 00 78119890 + 00 493119890 minus 02

Griewank 50000 Mean 20119890 minus 01 729119890 minus 06 157119890 minus 08 252119890 minus 09 109119890 minus 08

SD 11119890 minus 01 105119890 minus 05 109119890 minus 07 124119890 minus 08 286119890 minus 08

Rastrigin 100000 Mean 18119890 + 02 332119890 minus 04 133119890 minus 01 243119890 + 00 0

SD 13119890 + 01 639119890 minus 04 974119890 minus 02 160119890 + 00 0

Ackley 50000 Mean 11119890 minus 01 237119890 minus 04 335119890 minus 09 381119890 minus 06 204119890 minus 08

SD 39119890 minus 02 710119890 minus 05 284119890 minus 09 826119890 minus 07 472119890 minus 09

Schwefel226 100000 Mean 59119890 + 03 170119890 minus 10 262119890 minus 04 113119890 minus 08 minus194119890 minus 12

SD 11119890 + 03 171119890 minus 10 359119890 minus 04 108119890 minus 08 461119890 minus 13

Table 6 Comparison among NABC FIPS HPSO-TVAC CLPSO FPSO and OLPSO-G on optimizing 8 benchmark functions with 200000FEs

Function FIPS HPSO-TVAC CLPSO FPSO OLPSO-G NABC

Sphere Mean 242119890 minus 13 283119890 minus 33 158119890 minus 12 240119890 minus 16 412119890 minus 54 545119890 minus 75

SD 173119890 minus 13 319119890 minus 33 770119890 minus 13 200119890 minus 31 634119890 minus 54 355119890 minus 75

Schwefel222 Mean 276119890 minus 08 903119890 minus 20 251119890 minus 08 158119890 minus 11 985119890 minus 30 720119890 minus 40

SD 904119890 minus 09 958119890 minus 20 584119890 minus 09 103119890 minus 22 101119890 minus 29 333119890 minus 40

Step Mean 0 0 0 0 0 0SD 0 0 0 0 0 0

Rosenbrock Mean 251119890 + 01 239119890 + 01 113119890 + 01 281119890 + 01 215119890 + 01 633119890 minus 02

SD 510119890 minus 01 265119890 + 01 985119890 minus 00 231119890 + 02 299119890 + 01 634119890 minus 02

Griewank Mean 901119890 minus 12 975119890 minus 03 902119890 minus 09 147119890 minus 03 483119890 minus 03 0

SD 184119890 minus 11 833119890 minus 03 857119890 minus 09 128119890 minus 05 863119890 minus 03 0

Rastrigin Mean 651119890 + 01 943119890 minus 00 909119890 minus 05 738119890 + 01 107119890 minus 00 0

SD 133119890 + 01 348119890 minus 00 125119890 minus 04 370119890 + 02 992119890 minus 01 0

Ackley Mean 233119890 minus 07 729119890 minus 14 366119890 minus 07 217119890 minus 09 798119890 minus 15 278119890 minus 14

SD 719119890 minus 08 300119890 minus 14 757119890 minus 08 171119890 minus 18 203119890 minus 15 259119890 minus 15

Schwefel226 Mean 993119890 + 02 159119890 + 03 382119890 minus 04 134119890 + 03 384119890 + 02 minus243119890 minus 12

SD 509119890 + 02 326119890 + 02 128119890 minus 05 277119890 + 02 217119890 + 02 872119890 minus 13

Mathematical Problems in Engineering 9

Table 7 Execution time of ABC and NABC algorithm with119863 = 30 60 100 (second)

Dim Algorithm 1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918

1198919

11989110

30 ABC 19850 25535 33363 18768 3969 32338 40942 78680 94332 47626NABC 17661 25697 36352 20056 2768 30612 18229 31331 77497 42660

60 ABC 43236 72503 104475 47454 12304 81891 185687 259944 322573 205303NABC 44900 73449 108912 47991 8202 80281 95126 129497 265051 186870

100 ABC 89402 163012 255115 92929 37074 180789 371414 574304 841482 410198NABC 88507 168964 255134 91418 21081 175828 334624 308107 687463 365939

Table 8 The five benchmark knapsack problems

119891 Dim Parameter (119908 119901 119888)1198911

10 119908 = (95 4 60 32 23 72 80 62 65 46) 119901 = (55 10 47 5 4 50 8 61 85 87) 119888 = 269

1198912

20 119908 = (92 4 43 83 84 68 92 82 6 44 32 18 56 83 25 96 70 48 14 58) 119901 = (44 46 90 72 91 40 75 35 8 54 78 4077 15 61 17 75 29 75 63) 119888 = 878

1198913

4 119908 = (6 5 9 7) 119901 = (9 11 13 15) 119888 = 201198914

4 119908 = (2 4 6 7) 119901 = (6 10 12 13) 119888 = 11

1198915

15119908 = (56358531 80874050 47987304 89596240 74660482 85894345 51353496 1498459 36445204 1658986244569231 0466933 37788018 57118442 60716575) 119901 = (0125126 19330424 58500931 35029145 8228400517410810 71050142 30399487 9140294 14731285 98852504 11908322 0891140 53166295 60176397) 119888 = 375

Table 9 The optimal solutions of the five benchmark knapsack problems

119891 Optimal solution Optimal value Value of constraint1198911

(0 1 1 1 0 0 0 1 1 1) 295 01198912

(1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1) 1024 minus71198913

(1 1 0 1) 35 minus21198914

(0 1 0 1) 23 01198915

(0 0 1 0 1 0 1 1 0 1 1 1 0 1 1) 4810694 minus200392

Table 10 Best worst median mean and standard deviation values obtained by ABC and NABC

119891 Algorithm Best Worst Median Mean SD

1198911

ABC 295 295 295 295 0NDABC 295 295 295 295 0

1198912

ABC 1024 1013 1024 1021 452NDABC 1024 1024 1024 1024 0

1198913

ABC 35 35 35 35 0NDABC 35 35 35 35 0

1198914

ABC 23 23 23 23 0NDABC 23 23 23 23 0

1198915

ABC 481069 437935 475478 460729 1847NDABC 481069 435786 481069 470771 879

6 Conclusion

In this paper we develop a novel artificial bee colonyalgorithm named NABC We add the global best solutioninto the search equation to drive the new candidate solutiononly around the global best solution in order to improvethe exploitation And the search equation of the employedbees is improved to keep the exploration of algorithm Theexperimental results tested on 11 benchmark functions showthat the convergence of NABC is much faster than that of

other algorithms and the computing is more effective At thesame timewe applyNABCon solving five standardKnapsackproblems and get good optimums So it is fitted to solvemanyengineering practical problems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors are grateful to the editor and the anonymousreviewers for their valuable comments and suggestions

References

[1] K S Tang K FMan S Kwong andQ He ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal ProcessingMagazine vol 13no 6 pp 22ndash37 1996

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

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

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

[5] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 Erciyes University KayseriTurkey 2005

[6] B Basturk and D Karaboga ldquoAn artificial bee colony (ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium May 2006

[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

[10] D Karaboga B Akay and C Ozturk ldquoArtificial bee colony(ABC) optimization algorithm for training feed-forward neuralnetworksrdquo in Modeling Decisions for Artificial Intelligence vol4617 of LectureNotes in Computer Science pp 318ndash329 SpringerBerlin Germany 2007

[11] Y-F Liu and S-Y Liu ldquoA hybrid discrete artificial bee colonyalgorithm for permutation flowshop scheduling problemrdquoApplied Soft Computing Journal vol 13 no 3 pp 1459ndash14632013

[12] D Karaboga and B Basturk ldquoArtificial bee colony (ABC)optimization algorithm for solving constrained optimizationproblemsrdquo in Foundations of Fuzzy Logic and Soft Computingvol 4529 of Lecture Notes in Computer Science pp 789ndash798Springer Berlin Germany 2007

[13] G P Zhu and S Kwong ldquoGbest-guided artificial bee colonyalgorithm for numerical function optimizationrdquo Applied Math-ematics and Computation vol 217 no 7 pp 3166ndash3173 2010

[14] B Basturk and D Karaboga ldquoA modified artificial bee colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[15] W-F Gao S-Y Liu and L-L Huang ldquoEnhancing artificialbee colony algorithm using more information-based searchequationsrdquo Information Sciences vol 270 pp 112ndash133 2014

[16] D Karaboga and B Gorkemli ldquoA quick artificial bee colony(qABC) algorithm and its performance on optimization prob-lemsrdquo Applied Soft Computing vol 23 pp 227ndash238 2014

[17] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 23 pp 689ndash6942010

[18] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[19] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009

[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2008

[21] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[22] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[23] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[24] M A M de Oca T Stutzle M Birattari and M DorigoldquoFrankensteinrsquos PSO a composite particle swarm optimizationalgorithmrdquo IEEE Transactions on Evolutionary Computationvol 13 no 5 pp 1120ndash1132 2009

[25] Z-H Zhan J Zhang Y Li and Y-H Shi ldquoOrthogonal learningparticle swarm optimizationrdquo IEEE Transactions on Evolution-ary Computation vol 15 no 6 pp 832ndash847 2011

[26] D Zou L Gao S Li and J Wu ldquoSolving 0-1 knapsack problemby a novel global harmony search algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 1556ndash1564 2011

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of

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Mathematical Problems in Engineering 3

Table 1 Benchmark functions used in experiments

Fun Name Search range Min

1198911=

119899

sum

119894=1

1199092

119894Sphere [minus100 100] 0

1198912=

119899

sum

119894=1

1198941199092

119894Sumsquare [minus10 10] 0

1198913=

119899

sum

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816+

119899

prod

119894=1

1003816100381610038161003816119909119894

1003816100381610038161003816

Schwefel222 [minus10 10] 0

1198914= max 100381610038161003816

1003816119909119894

1003816100381610038161003816 1 le 119894 le 119899 Schwefel221 [minus100 100] 0

1198915=

119899

sum

119894=1

(lfloor119909119894+ 05rfloor)

2 Step [minus100 100] 0

1198916=

119899

sum

119894=1

[100 (1199092

119894minus 119909119894+1

)

2

+ (119909119894minus 1)2

] Rosenbrock [minus5 10] 0

1198917=

119899

sum

119894=1

[1199092

119894minus 10 cos (2120587119909

119894) + 10] Rastrigin [minus512 512] 0

1198918=

1

4000

119899

sum

119894=1

1199092

119894minus

119899

prod

119894=1

cos(119909119894

radic119894

) + 1 Griewank [minus600 600] 0

1198919= 41898288727243369 lowast 119899 minus

119899

sum

119894=1

119909119894sin(radic

1003816100381610038161003816119909119894

1003816100381610038161003816) Schwefel226 [minus500 500] 0

11989110

= minus20 exp(minus02radic

1

119899

119899

sum

119894=1

1199092

119894) minus exp(

1

119899

119899

sum

119894=1

cos (2120587119909119894)) + 20 + 119890 Ackley [minus32 32] 0

11989111

=

1

119899

119899

sum

119894=1

(1199094

119894minus 16119909

2

119894+ 5119909119894) Himmelblau [minus5 5] minus7833236

solution among the neighbors of 119909119898and itself (119873

119898) But in

the NABC 119909best119895 represents the global best solution At thesame time in the stages of employed bees and onlookers thenew search equations used in qABC are different from theseof the NABC adopted

32 The Novel ABC Algorithm (NABC) The complete com-putational procedure of the novel ABC algorithm (NABC) isoutlined as follows

Algorithm 1 (NABC) Consider the following steps

Step 1 (initialization) Preset SN Limit and Maxcycle

Step 2 Randomly generate SN solutions from the search spaceto create an initial population 119875

Step 3 Calculate the function values 119891 of the initial popula-tion 119875

Step 4 The employed bees stage Generate a new solution V119894119895

by (5) if 119891(V119894119895) lt 119891(119909

119894119895) 119909119894119895

= V119894119895 trail119894= 1 else trail

119894=

trail119894+ 1

Step 5 Calculate the probability values 119901119894for the solutions 119909

119894119895

by (3)

Step 6 (the onlookers stage) If rand lt 119901119894 generate a new

solution V119894119895by (6) if 119891(V

119894119895) lt 119891(119909

119894119895) 119909119894119895

= V119894119895 trail119894= 1

else trail119894= trail

119894+ 1

Step 7 (the scouts stage) If trail gt Limit replace 119909119894119895with a

new produced solution by (1)

Step 8 If cycle number gt Maxcycle stop and memorize thebest solution achieved so far otherwise go to Step 2

4 Numerical Functionsand Experimental Results

41 Numerical Functions It is applied to minimize a set of 11scalable benchmark functions in Table 1 to test the efficiencyof the novel ABC algorithm The smaller the final result thebetter it is

42 Parameter Settings A set of experiments tested on 11numerical benchmark functions are performed in this sectionto compare the performance of the novel ABC algorithmwiththat of other improved ABC algorithms Functions 119891

1ndash1198914are

unimodal Function 1198915is a discontinuous step function The

Rosenbrock function of 1198916which is unimodal when 119863 =

2 and 3 may have multiple minima in the high dimensioncase [18] Functions 119891

7ndash11989111are multimodal To compare with

other variance algorithms the dimension119863 of every functionis 30 60 or 100 respectively the same as [15]The dimension119863 of 119891

11is 150 300 or 500 The maximum number of gen-

erations is 1500 3000 or 5000 respectively The populationsize is 100 namely SN = 50 and Limit is 200 which is thesame as [15] Each of the experiments is repeated 30 timesindependently

43 Experimental Results Figure 1 graphically presents theconvergence curves of 119891

1ndash11989110functions respectively We can

see from Figure 1 that in the later stage of evolution thestandard ABC algorithm enters a long period of stagnation

4 Mathematical Problems in Engineering

1010

100

10minus10

10minus20

10minus30

10minus40

10minus50

10minus60

0 500 1000 1500

Mea

n

Sphere Sumsquare10

10

100

10minus10

10minus20

10minus30

10minus40

10minus50

10minus60

0 500 1000 1500

Mea

n

Schwefel222

0 500 1000 1500

Mea

n

1020

1010

100

10minus10

10minus20

10minus30

0 500 1000 1500

Mea

n

Schwefel22110

2

101

100

10minus1

Mea

n

105

104

103

102

101

100

0 20 40 60 80 100 120 140 160 180

Step

0 500 1000 1500

Mea

n

Rosenbrock10

8

106

104

102

100

10minus2

10minus4

ABCNABC

Number of generations0 500 1000 1500

Mea

n

Rastrigin10

5

100

10minus5

10minus10

10minus15

0 500 1000 1500

Mea

n

Griewank10

5

100

10minus5

10minus10

10minus15

10minus20

Number of generations

ABCNABC

Figure 1 Continued

Mathematical Problems in Engineering 5

Number of generations0 500 1000 1500

Mea

n

Schwefel22610

5

100

10minus5

10minus10

10minus15

Number of generations0 500 1000 1500

Mea

n

Ackley10

2

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

ABCNABC

ABCNABC

Figure 1 Convergence curves of ABC and NABC algorithms for 1198911ndash11989110functions (119863 = 30)

However NABC still keeps fast convergence and best perfor-mance

The results of NABC compared with the standard ABCalgorithm are shown in Table 2 in terms of the best worstmedian mean and standard deviation of the solutionsTheseresults show that the convergence rate of NABC is betterthan the standard ABC algorithm for most test functionsThe rate of convergence obviously increases on 119891

1ndash1198913 1198915

1198917ndash11989111

functions from Figure 1 and Table 2 NABC can findthe optimal solutions on functions119891

511989171198918The convergence

rate of NABC is the same order of magnitude as the standardABC algorithm on 119891

6 So the superiority of standard ABC

algorithm is not very obvious From the results in Table 2NABC can obtain better and closer-to-optimal solutions thanthe standard ABC algorithm for most of the functions In aword NABC increases the exploitation greatly NABC alsoretains good exploration Here ldquo+rdquo indicates that NABCis statistically significantly better than its correspondingcompetitor algorithm ldquoasymprdquo stands for that the result of thecorresponding algorithm is statistically similar with that ofNABC

NABC is further compared with GABC [13] and EABC[15] in Tables 3 and 4 The results of GABC and EABCare gained from [15] directly It is clear that NABC worksbetter in almost all the cases and achieves better performancethan GABC The rate of convergence of NABC increasesobviously on all functions in Tables 3 and 4The convergencerate of NABC is a little worse than that of EABC fromTable 3 In particular the convergence rate of Schwefel221and Rosenbrock is better than EABC in Table 4

At the same time NABC is compared with four variantsof DE (DE [17] jDE [18] JADE [19] and SaDE [20]) and fivevariants of PSO (FIPS [21] HPSO-TVAC [22] CLPSO [23]FPSO [24] OLPSO-G [25]) The results of DEs and PSOs aregained from [15] directly The maximum number of functionevaluations (FEs) is shown in Tables 5 and 6 Here FEs isMaxgenlowastSN It can be seen that NABC is better on almostall the test functions except Ackley in JADE and OLPSO-G

In Table 7 we can find the execution time of ABC issimilar with that of NABC So NABC is largely faster thanABC and does not add the execution time

5 Application

51 Knapsack Problems We know that Knapsack problem[26] is an engineering optimization problem It can be definedan unconstrained optimization problem and described asfollows

min119891 (119909) = minus

119899

sum

119894=1

119901119894119909119894+ 120572 lowastmax(0

119899

sum

119894=1

119908119894119909119894minus 119888) (7)

where 119908119894is each itemrsquos weight and 119901

119894is each itemrsquos profit

respectively The knapsack has a limited weight capabilityof 119888 The objective function of this problem is to pack theknapsack so that the items in it have the maximal totalprofit The decision variable 119909

119894is the value one if the item

119894 is packed otherwise it is the value zero And we assumethat all profits and weights are positive and all weights aresmaller than the capacity of 119888 120572 is a penalty factor which isset to 10119890 minus 20 in this paper Tables 8 and 9 are parametersof five standard knapsack problems The experiments testedon five benchmark 0-1 knapsack problems are performed tocompare the performance of NABC with that of standardABC The maximum number of generations is set to 2000The population size is 100 namely SN = 50 and Limit is 10Each of the experiments is repeated 30 times independently

52 Data Analysis In Table 10 we can see that NABC caneasily solve the optimum of Knapsack problems And itcan find the optimum in most cases except 119891

5 ABC has

the worst performance because we can see that it can notfind the optimums for 119891

2and 119891

5 In short NABC has a

better performance than ABC on solving the five knapsackproblems

6 Mathematical Problems in Engineering

Table 2 Best worst medianmean and standard deviation values obtained byABC andNABC through 30 independent runs on 11 functions

Function Dim Best Worst Median Mean SD Significant

1198911

30 ABC 504119890 minus 16 725119890 minus 16 504119890 minus 16 606119890 minus 16 934119890 minus 17 +NABC 113119890 minus 55 966119890 minus 54 666119890 minus 55 110119890 minus 54 171119890 minus 54

60 ABC 140119890 minus 15 187119890 minus 15 152119890 minus 15 162119890 minus 15 202119890 minus 16 +NABC 103119890 minus 53 466119890 minus 52 906119890 minus 53 148119890 minus 52 137119890 minus 52

100 ABC 220119890 minus 15 932119890 minus 15 521119890 minus 15 336119890 minus 15 201119890 minus 15 +NABC 161119890 minus 52 121119890 minus 51 369119890 minus 51 145119890 minus 51 891119890 minus 52

1198912

30 ABC 430119890 minus 16 634119890 minus 16 450119890 minus 16 503119890 minus 16 711119890 minus 17 +NABC 164119890 minus 57 782119890 minus 56 116119890 minus 56 270119890 minus 56 248119890 minus 56

60 ABC 143119890 minus 15 200119890 minus 15 162119890 minus 15 167119890 minus 15 184119890 minus 16 +NABC 443119890 minus 55 782119890 minus 53 801119890 minus 54 208119890 minus 53 233119890 minus 53

100 ABC 247119890 minus 15 366119890 minus 15 343119890 minus 05 319119890 minus 15 421119890 minus 16 +NABC 351119890 minus 53 575119890 minus 52 266119890 minus 52 301119890 minus 52 156119890 minus 52

1198913

30 ABC 238119890 minus 11 134119890 minus 10 642119890 minus 11 705119890 minus 11 519119890 minus 11 +NABC 506119890 minus 30 411119890 minus 29 173119890 minus 29 202119890 minus 29 110119890 minus 29

60 ABC 902119890 minus 11 823119890 minus 10 512119890 minus 10 408119890 minus 10 157119890 minus 10 +NABC 712119890 minus 29 572119890 minus 28 257119890 minus 28 292119890 minus 28 154119890 minus 28

100 ABC 205119890 minus 09 536119890 minus 08 484119890 minus 09 367119890 minus 09 480119890 minus 09 +NABC 285119890 minus 28 281119890 minus 27 735119890 minus 28 104119890 minus 27 688119890 minus 28

1198914

30 ABC 868119890 + 00 120119890 + 01 184119890 + 01 110119890 + 01 129119890 minus 00 +NABC 177119890 minus 01 318119890 minus 01 268119890 minus 01 267119890 minus 01 355119890 minus 02

60 ABC 453119890 + 01 508119890 + 01 502119890 + 01 486119890 + 01 190119890 minus 00 +NABC 190119890 + 00 316119890 + 00 274119890 + 00 271119890 + 00 314119890 minus 01

100 ABC 692119890 + 01 743119890 + 01 726119890 + 01 725119890 + 01 186119890 minus 00 +NABC 832119890 + 00 110119890 + 01 979119890 + 00 992119890 + 00 704119890 minus 01

1198915

30 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

60 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

100 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

1198916

30 ABC 812119890 minus 03 168119890 minus 01 133119890 minus 02 486119890 minus 02 484119890 minus 02

asymp

NABC 221119890 minus 03 231119890 minus 01 332119890 minus 02 599119890 minus 02 564119890 minus 02

60 ABC 108119890 minus 02 588119890 minus 01 156119890 minus 01 141119890 minus 01 165119890 minus 01

NABC 288119890 minus 04 388119890 minus 01 473119890 minus 02 102119890 minus 01 995119890 minus 02 asymp

100 ABC 357119890 minus 02 994119890 minus 01 145119890 minus 01 297119890 minus 01 364119890 minus 01

NABC 292119890 minus 04 103119890 + 00 216119890 minus 01 350119890 minus 01 328119890 minus 01 asymp

1198917

30 ABC 536119890 minus 15 146119890 minus 01 456119890 minus 09 958119890 minus 04 105119890 minus 03 +NABC 0 0 0 0 0

60 ABC 587119890 minus 12 199119890 minus 00 125119890 minus 09 268119890 minus 02 632119890 minus 02 +NABC 0 0 0 0 0

100 ABC 865119890 minus 13 199119890 minus 00 115119890 minus 06 497119890 minus 02 216119890 minus 01 +NABC 0 0 0 0 0

1198918

30 ABC 801119890 minus 15 130119890 minus 12 814119890 minus 13 237119890 minus 13 418119890 minus 13 +NABC 0 0 0 0 0

60 ABC 832119890 minus 14 134119890 minus 11 720119890 minus 13 345119890 minus 12 289119890 minus 12 +NABC 0 0 0 0 0

100 ABC 536119890 minus 14 158119890 minus 09 427119890 minus 10 180119890 minus 10 203119890 minus 10 +NABC 0 0 0 0 0

Mathematical Problems in Engineering 7

Table 2 Continued

Function Dim Best Worst Median Mean SD Significant

1198919

30 ABC 154119890 minus 06 237119890 + 02 376119890 minus 01 886119890 + 01 862119890 + 01 +NABC minus364119890 minus 12 minus182119890 minus 12 minus182119890 minus 12 minus200119890 minus 12 555119890 minus 13

60 ABC 355119890 + 02 769119890 + 02 769119890 + 02 540119890 + 02 141119890 + 02 +NABC 291119890 minus 11 364119890 minus 11 364119890 minus 11 354119890 minus 11 252119890 minus 12

100 ABC 781119890 + 02 155119890 + 03 151119890 + 03 129119890 + 03 223119890 + 02 +NABC 102119890 minus 10 116119890 minus 10 109119890 minus 10 110119890 minus 10 458119890 minus 12

11989110

30 ABC 536119890 minus 10 123119890 minus 08 367119890 minus 09 145119890 minus 09 237119890 minus 09 +NABC 222119890 minus 14 293119890 minus 14 293119890 minus 14 266119890 minus 14 332119890 minus 15

60 ABC 814119890 minus 09 754119890 minus 08 538119890 minus 08 460119890 minus 08 204119890 minus 08 +NABC 506119890 minus 14 684119890 minus 14 648119890 minus 14 644119890 minus 14 526119890 minus 15

100 ABC 535119890 minus 08 428119890 minus 07 104119890 minus 07 283119890 minus 07 157119890 minus 07 +NABC 104119890 minus 13 122119890 minus 13 115119890 minus 13 116119890 minus 13 475119890 minus 15

11989111

150 ABC minus774221 minus769602 minus770460 minus771658 211119890 minus 01 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

300 ABC minus770391 minus768430 minus770391 minus769258 676119890 minus 02 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

500 ABC minus768562 minus765849 minus768562 minus767220 952119890 minus 02 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

Table 3 Comparison among GABC EABC and NABC on optimizing 8 benchmark functions

Function MaxFEs Dim GABC EABC NABCMean SD Mean SD Mean SD

Sphere150000 30 137119890 minus 25 270119890 minus 25 442119890 minus 67 271119890 minus 67 110119890 minus 54 171119890 minus 54

300000 60 486119890 minus 23 514119890 minus 23 230119890 minus 64 103119890 minus 64 148119890 minus 52 137119890 minus 52

500000 100 905119890 minus 22 428119890 minus 22 637119890 minus 63 255119890 minus 63 145119890 minus 51 891119890 minus 52

Schwefel222150000 30 556119890 minus 15 979119890 minus 15 551119890 minus 35 702119890 minus 35 202119890 minus 29 110119890 minus 29

300000 60 561119890 minus 14 993119890 minus 15 508119890 minus 33 195119890 minus 34 292119890 minus 28 154119890 minus 28

500000 100 229119890 minus 13 397119890 minus 13 468119890 minus 32 142119890 minus 32 104119890 minus 27 688119890 minus 28

Schwefel221150000 30 307119890 minus 00 472119890 minus 01 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

300000 60 385119890 + 01 342119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

500000 100 704119890 + 01 120119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock150000 30 130119890 minus 00 164119890 minus 00 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

300000 60 166119890 + 01 328119890 + 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

500000 100 239119890 + 01 334119890 + 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Griewank100000 30 140119890 minus 08 116119890 minus 08 0 0 0 0150000 60 158119890 minus 06 100119890 minus 06 0 0 0 0250000 100 244119890 minus 06 250119890 minus 06 0 0 0 0

Rastrigin100000 30 708119890 minus 03 119119890 minus 03 0 0 0 0150000 60 347119890 minus 00 878119890 minus 01 0 0 0 0250000 100 978119890 minus 00 291119890 minus 00 0 0 0 0

Schwefel22650000 30 638119890 minus 00 238119890 + 01 0 0 171119890 minus 09 301119890 minus 10

100000 60 330119890 + 01 494119890 + 01 392119890 minus 11 356119890 minus 12 124119890 minus 09 502119890 minus 09

200000 100 605119890 + 01 805119890 + 01 112119890 minus 10 356119890 minus 12 122119890 minus 10 297119890 minus 11

Ackley50000 30 645119890 minus 03 268119890 minus 03 339119890 minus 10 695119890 minus 11 204119890 minus 08 472119890 minus 09

100000 60 296119890 minus 03 214119890 minus 03 181119890 minus 09 270119890 minus 10 445119890 minus 08 150119890 minus 08

150000 100 601119890 minus 02 960119890 minus 02 453119890 minus 08 123119890 minus 08 601119890 minus 07 323119890 minus 07

8 Mathematical Problems in Engineering

Table 4 Comparison among ABC EABC and NABC on optimizing 4 benchmark functions with119863 = 30 60 100

Function Dim ABC EABC NABCMean SD Mean SD Mean SD

Schwefel22130 110119890 + 01 129119890 minus 00 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

60 486119890 + 01 190119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

100 725119890 + 01 186119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock30 486119890 minus 02 484119890 minus 02 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

60 141119890 minus 01 165119890 minus 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

100 297119890 minus 01 364119890 minus 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Schwefel22630 886119890 + 01 862119890 + 01 minus123119890 minus 13 109119890 minus 13 minus200119890 minus 12 555119890 minus 13

60 540119890 + 02 141119890 + 02 291119890 minus 11 0 354119890 minus 11 252119890 minus 12

100 129119890 + 03 223119890 + 02 945119890 minus 11 0 110119890 minus 10 458119890 minus 12

Ackley30 145119890 minus 09 237119890 minus 09 136119890 minus 14 174119890 minus 15 266119890 minus 14 332119890 minus 15

60 460119890 minus 08 204119890 minus 08 449119890 minus 14 284119890 minus 15 644119890 minus 14 526119890 minus 15

100 283119890 minus 07 157119890 minus 07 953119890 minus 14 284119890 minus 15 116119890 minus 13 475119890 minus 15

Table 5 Comparison among NABC DE jDE JADE and SaDE on optimizing 8 benchmark functions with119863 = 30

Function MaxFEs DE jDE JADE SaDE NABC

Sphere 150000 Mean 98119890 minus 14 146119890 minus 28 132119890 minus 54 328119890 minus 20 110119890 minus 54

SD 84119890 minus 14 178119890 minus 28 922119890 minus 54 362119890 minus 20 171119890 minus 54

Schwefel222 200000 Mean 16119890 minus 09 902119890 minus 24 318119890 minus 25 351119890 minus 25 720119890 minus 40

SD 11119890 minus 09 601119890 minus 24 205119890 minus 25 274119890 minus 25 333119890 minus 40

Step 10000 Mean 47119890 + 03 613119890 + 02 562119890 + 00 507119890 + 01 0SD 11119890 + 03 172119890 + 02 187119890 + 00 134119890 + 01 0

Rosenbrock 300000 Mean 21119890 + 00 13119890 + 01 32119890 minus 01 21119890 + 01 422119890 minus 02

SD 15119890 + 00 14119890 + 01 11119890 + 00 78119890 + 00 493119890 minus 02

Griewank 50000 Mean 20119890 minus 01 729119890 minus 06 157119890 minus 08 252119890 minus 09 109119890 minus 08

SD 11119890 minus 01 105119890 minus 05 109119890 minus 07 124119890 minus 08 286119890 minus 08

Rastrigin 100000 Mean 18119890 + 02 332119890 minus 04 133119890 minus 01 243119890 + 00 0

SD 13119890 + 01 639119890 minus 04 974119890 minus 02 160119890 + 00 0

Ackley 50000 Mean 11119890 minus 01 237119890 minus 04 335119890 minus 09 381119890 minus 06 204119890 minus 08

SD 39119890 minus 02 710119890 minus 05 284119890 minus 09 826119890 minus 07 472119890 minus 09

Schwefel226 100000 Mean 59119890 + 03 170119890 minus 10 262119890 minus 04 113119890 minus 08 minus194119890 minus 12

SD 11119890 + 03 171119890 minus 10 359119890 minus 04 108119890 minus 08 461119890 minus 13

Table 6 Comparison among NABC FIPS HPSO-TVAC CLPSO FPSO and OLPSO-G on optimizing 8 benchmark functions with 200000FEs

Function FIPS HPSO-TVAC CLPSO FPSO OLPSO-G NABC

Sphere Mean 242119890 minus 13 283119890 minus 33 158119890 minus 12 240119890 minus 16 412119890 minus 54 545119890 minus 75

SD 173119890 minus 13 319119890 minus 33 770119890 minus 13 200119890 minus 31 634119890 minus 54 355119890 minus 75

Schwefel222 Mean 276119890 minus 08 903119890 minus 20 251119890 minus 08 158119890 minus 11 985119890 minus 30 720119890 minus 40

SD 904119890 minus 09 958119890 minus 20 584119890 minus 09 103119890 minus 22 101119890 minus 29 333119890 minus 40

Step Mean 0 0 0 0 0 0SD 0 0 0 0 0 0

Rosenbrock Mean 251119890 + 01 239119890 + 01 113119890 + 01 281119890 + 01 215119890 + 01 633119890 minus 02

SD 510119890 minus 01 265119890 + 01 985119890 minus 00 231119890 + 02 299119890 + 01 634119890 minus 02

Griewank Mean 901119890 minus 12 975119890 minus 03 902119890 minus 09 147119890 minus 03 483119890 minus 03 0

SD 184119890 minus 11 833119890 minus 03 857119890 minus 09 128119890 minus 05 863119890 minus 03 0

Rastrigin Mean 651119890 + 01 943119890 minus 00 909119890 minus 05 738119890 + 01 107119890 minus 00 0

SD 133119890 + 01 348119890 minus 00 125119890 minus 04 370119890 + 02 992119890 minus 01 0

Ackley Mean 233119890 minus 07 729119890 minus 14 366119890 minus 07 217119890 minus 09 798119890 minus 15 278119890 minus 14

SD 719119890 minus 08 300119890 minus 14 757119890 minus 08 171119890 minus 18 203119890 minus 15 259119890 minus 15

Schwefel226 Mean 993119890 + 02 159119890 + 03 382119890 minus 04 134119890 + 03 384119890 + 02 minus243119890 minus 12

SD 509119890 + 02 326119890 + 02 128119890 minus 05 277119890 + 02 217119890 + 02 872119890 minus 13

Mathematical Problems in Engineering 9

Table 7 Execution time of ABC and NABC algorithm with119863 = 30 60 100 (second)

Dim Algorithm 1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918

1198919

11989110

30 ABC 19850 25535 33363 18768 3969 32338 40942 78680 94332 47626NABC 17661 25697 36352 20056 2768 30612 18229 31331 77497 42660

60 ABC 43236 72503 104475 47454 12304 81891 185687 259944 322573 205303NABC 44900 73449 108912 47991 8202 80281 95126 129497 265051 186870

100 ABC 89402 163012 255115 92929 37074 180789 371414 574304 841482 410198NABC 88507 168964 255134 91418 21081 175828 334624 308107 687463 365939

Table 8 The five benchmark knapsack problems

119891 Dim Parameter (119908 119901 119888)1198911

10 119908 = (95 4 60 32 23 72 80 62 65 46) 119901 = (55 10 47 5 4 50 8 61 85 87) 119888 = 269

1198912

20 119908 = (92 4 43 83 84 68 92 82 6 44 32 18 56 83 25 96 70 48 14 58) 119901 = (44 46 90 72 91 40 75 35 8 54 78 4077 15 61 17 75 29 75 63) 119888 = 878

1198913

4 119908 = (6 5 9 7) 119901 = (9 11 13 15) 119888 = 201198914

4 119908 = (2 4 6 7) 119901 = (6 10 12 13) 119888 = 11

1198915

15119908 = (56358531 80874050 47987304 89596240 74660482 85894345 51353496 1498459 36445204 1658986244569231 0466933 37788018 57118442 60716575) 119901 = (0125126 19330424 58500931 35029145 8228400517410810 71050142 30399487 9140294 14731285 98852504 11908322 0891140 53166295 60176397) 119888 = 375

Table 9 The optimal solutions of the five benchmark knapsack problems

119891 Optimal solution Optimal value Value of constraint1198911

(0 1 1 1 0 0 0 1 1 1) 295 01198912

(1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1) 1024 minus71198913

(1 1 0 1) 35 minus21198914

(0 1 0 1) 23 01198915

(0 0 1 0 1 0 1 1 0 1 1 1 0 1 1) 4810694 minus200392

Table 10 Best worst median mean and standard deviation values obtained by ABC and NABC

119891 Algorithm Best Worst Median Mean SD

1198911

ABC 295 295 295 295 0NDABC 295 295 295 295 0

1198912

ABC 1024 1013 1024 1021 452NDABC 1024 1024 1024 1024 0

1198913

ABC 35 35 35 35 0NDABC 35 35 35 35 0

1198914

ABC 23 23 23 23 0NDABC 23 23 23 23 0

1198915

ABC 481069 437935 475478 460729 1847NDABC 481069 435786 481069 470771 879

6 Conclusion

In this paper we develop a novel artificial bee colonyalgorithm named NABC We add the global best solutioninto the search equation to drive the new candidate solutiononly around the global best solution in order to improvethe exploitation And the search equation of the employedbees is improved to keep the exploration of algorithm Theexperimental results tested on 11 benchmark functions showthat the convergence of NABC is much faster than that of

other algorithms and the computing is more effective At thesame timewe applyNABCon solving five standardKnapsackproblems and get good optimums So it is fitted to solvemanyengineering practical problems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors are grateful to the editor and the anonymousreviewers for their valuable comments and suggestions

References

[1] K S Tang K FMan S Kwong andQ He ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal ProcessingMagazine vol 13no 6 pp 22ndash37 1996

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

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

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

[5] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 Erciyes University KayseriTurkey 2005

[6] B Basturk and D Karaboga ldquoAn artificial bee colony (ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium May 2006

[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

[10] D Karaboga B Akay and C Ozturk ldquoArtificial bee colony(ABC) optimization algorithm for training feed-forward neuralnetworksrdquo in Modeling Decisions for Artificial Intelligence vol4617 of LectureNotes in Computer Science pp 318ndash329 SpringerBerlin Germany 2007

[11] Y-F Liu and S-Y Liu ldquoA hybrid discrete artificial bee colonyalgorithm for permutation flowshop scheduling problemrdquoApplied Soft Computing Journal vol 13 no 3 pp 1459ndash14632013

[12] D Karaboga and B Basturk ldquoArtificial bee colony (ABC)optimization algorithm for solving constrained optimizationproblemsrdquo in Foundations of Fuzzy Logic and Soft Computingvol 4529 of Lecture Notes in Computer Science pp 789ndash798Springer Berlin Germany 2007

[13] G P Zhu and S Kwong ldquoGbest-guided artificial bee colonyalgorithm for numerical function optimizationrdquo Applied Math-ematics and Computation vol 217 no 7 pp 3166ndash3173 2010

[14] B Basturk and D Karaboga ldquoA modified artificial bee colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[15] W-F Gao S-Y Liu and L-L Huang ldquoEnhancing artificialbee colony algorithm using more information-based searchequationsrdquo Information Sciences vol 270 pp 112ndash133 2014

[16] D Karaboga and B Gorkemli ldquoA quick artificial bee colony(qABC) algorithm and its performance on optimization prob-lemsrdquo Applied Soft Computing vol 23 pp 227ndash238 2014

[17] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 23 pp 689ndash6942010

[18] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[19] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009

[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2008

[21] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[22] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[23] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[24] M A M de Oca T Stutzle M Birattari and M DorigoldquoFrankensteinrsquos PSO a composite particle swarm optimizationalgorithmrdquo IEEE Transactions on Evolutionary Computationvol 13 no 5 pp 1120ndash1132 2009

[25] Z-H Zhan J Zhang Y Li and Y-H Shi ldquoOrthogonal learningparticle swarm optimizationrdquo IEEE Transactions on Evolution-ary Computation vol 15 no 6 pp 832ndash847 2011

[26] D Zou L Gao S Li and J Wu ldquoSolving 0-1 knapsack problemby a novel global harmony search algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 1556ndash1564 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

1010

100

10minus10

10minus20

10minus30

10minus40

10minus50

10minus60

0 500 1000 1500

Mea

n

Sphere Sumsquare10

10

100

10minus10

10minus20

10minus30

10minus40

10minus50

10minus60

0 500 1000 1500

Mea

n

Schwefel222

0 500 1000 1500

Mea

n

1020

1010

100

10minus10

10minus20

10minus30

0 500 1000 1500

Mea

n

Schwefel22110

2

101

100

10minus1

Mea

n

105

104

103

102

101

100

0 20 40 60 80 100 120 140 160 180

Step

0 500 1000 1500

Mea

n

Rosenbrock10

8

106

104

102

100

10minus2

10minus4

ABCNABC

Number of generations0 500 1000 1500

Mea

n

Rastrigin10

5

100

10minus5

10minus10

10minus15

0 500 1000 1500

Mea

n

Griewank10

5

100

10minus5

10minus10

10minus15

10minus20

Number of generations

ABCNABC

Figure 1 Continued

Mathematical Problems in Engineering 5

Number of generations0 500 1000 1500

Mea

n

Schwefel22610

5

100

10minus5

10minus10

10minus15

Number of generations0 500 1000 1500

Mea

n

Ackley10

2

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

ABCNABC

ABCNABC

Figure 1 Convergence curves of ABC and NABC algorithms for 1198911ndash11989110functions (119863 = 30)

However NABC still keeps fast convergence and best perfor-mance

The results of NABC compared with the standard ABCalgorithm are shown in Table 2 in terms of the best worstmedian mean and standard deviation of the solutionsTheseresults show that the convergence rate of NABC is betterthan the standard ABC algorithm for most test functionsThe rate of convergence obviously increases on 119891

1ndash1198913 1198915

1198917ndash11989111

functions from Figure 1 and Table 2 NABC can findthe optimal solutions on functions119891

511989171198918The convergence

rate of NABC is the same order of magnitude as the standardABC algorithm on 119891

6 So the superiority of standard ABC

algorithm is not very obvious From the results in Table 2NABC can obtain better and closer-to-optimal solutions thanthe standard ABC algorithm for most of the functions In aword NABC increases the exploitation greatly NABC alsoretains good exploration Here ldquo+rdquo indicates that NABCis statistically significantly better than its correspondingcompetitor algorithm ldquoasymprdquo stands for that the result of thecorresponding algorithm is statistically similar with that ofNABC

NABC is further compared with GABC [13] and EABC[15] in Tables 3 and 4 The results of GABC and EABCare gained from [15] directly It is clear that NABC worksbetter in almost all the cases and achieves better performancethan GABC The rate of convergence of NABC increasesobviously on all functions in Tables 3 and 4The convergencerate of NABC is a little worse than that of EABC fromTable 3 In particular the convergence rate of Schwefel221and Rosenbrock is better than EABC in Table 4

At the same time NABC is compared with four variantsof DE (DE [17] jDE [18] JADE [19] and SaDE [20]) and fivevariants of PSO (FIPS [21] HPSO-TVAC [22] CLPSO [23]FPSO [24] OLPSO-G [25]) The results of DEs and PSOs aregained from [15] directly The maximum number of functionevaluations (FEs) is shown in Tables 5 and 6 Here FEs isMaxgenlowastSN It can be seen that NABC is better on almostall the test functions except Ackley in JADE and OLPSO-G

In Table 7 we can find the execution time of ABC issimilar with that of NABC So NABC is largely faster thanABC and does not add the execution time

5 Application

51 Knapsack Problems We know that Knapsack problem[26] is an engineering optimization problem It can be definedan unconstrained optimization problem and described asfollows

min119891 (119909) = minus

119899

sum

119894=1

119901119894119909119894+ 120572 lowastmax(0

119899

sum

119894=1

119908119894119909119894minus 119888) (7)

where 119908119894is each itemrsquos weight and 119901

119894is each itemrsquos profit

respectively The knapsack has a limited weight capabilityof 119888 The objective function of this problem is to pack theknapsack so that the items in it have the maximal totalprofit The decision variable 119909

119894is the value one if the item

119894 is packed otherwise it is the value zero And we assumethat all profits and weights are positive and all weights aresmaller than the capacity of 119888 120572 is a penalty factor which isset to 10119890 minus 20 in this paper Tables 8 and 9 are parametersof five standard knapsack problems The experiments testedon five benchmark 0-1 knapsack problems are performed tocompare the performance of NABC with that of standardABC The maximum number of generations is set to 2000The population size is 100 namely SN = 50 and Limit is 10Each of the experiments is repeated 30 times independently

52 Data Analysis In Table 10 we can see that NABC caneasily solve the optimum of Knapsack problems And itcan find the optimum in most cases except 119891

5 ABC has

the worst performance because we can see that it can notfind the optimums for 119891

2and 119891

5 In short NABC has a

better performance than ABC on solving the five knapsackproblems

6 Mathematical Problems in Engineering

Table 2 Best worst medianmean and standard deviation values obtained byABC andNABC through 30 independent runs on 11 functions

Function Dim Best Worst Median Mean SD Significant

1198911

30 ABC 504119890 minus 16 725119890 minus 16 504119890 minus 16 606119890 minus 16 934119890 minus 17 +NABC 113119890 minus 55 966119890 minus 54 666119890 minus 55 110119890 minus 54 171119890 minus 54

60 ABC 140119890 minus 15 187119890 minus 15 152119890 minus 15 162119890 minus 15 202119890 minus 16 +NABC 103119890 minus 53 466119890 minus 52 906119890 minus 53 148119890 minus 52 137119890 minus 52

100 ABC 220119890 minus 15 932119890 minus 15 521119890 minus 15 336119890 minus 15 201119890 minus 15 +NABC 161119890 minus 52 121119890 minus 51 369119890 minus 51 145119890 minus 51 891119890 minus 52

1198912

30 ABC 430119890 minus 16 634119890 minus 16 450119890 minus 16 503119890 minus 16 711119890 minus 17 +NABC 164119890 minus 57 782119890 minus 56 116119890 minus 56 270119890 minus 56 248119890 minus 56

60 ABC 143119890 minus 15 200119890 minus 15 162119890 minus 15 167119890 minus 15 184119890 minus 16 +NABC 443119890 minus 55 782119890 minus 53 801119890 minus 54 208119890 minus 53 233119890 minus 53

100 ABC 247119890 minus 15 366119890 minus 15 343119890 minus 05 319119890 minus 15 421119890 minus 16 +NABC 351119890 minus 53 575119890 minus 52 266119890 minus 52 301119890 minus 52 156119890 minus 52

1198913

30 ABC 238119890 minus 11 134119890 minus 10 642119890 minus 11 705119890 minus 11 519119890 minus 11 +NABC 506119890 minus 30 411119890 minus 29 173119890 minus 29 202119890 minus 29 110119890 minus 29

60 ABC 902119890 minus 11 823119890 minus 10 512119890 minus 10 408119890 minus 10 157119890 minus 10 +NABC 712119890 minus 29 572119890 minus 28 257119890 minus 28 292119890 minus 28 154119890 minus 28

100 ABC 205119890 minus 09 536119890 minus 08 484119890 minus 09 367119890 minus 09 480119890 minus 09 +NABC 285119890 minus 28 281119890 minus 27 735119890 minus 28 104119890 minus 27 688119890 minus 28

1198914

30 ABC 868119890 + 00 120119890 + 01 184119890 + 01 110119890 + 01 129119890 minus 00 +NABC 177119890 minus 01 318119890 minus 01 268119890 minus 01 267119890 minus 01 355119890 minus 02

60 ABC 453119890 + 01 508119890 + 01 502119890 + 01 486119890 + 01 190119890 minus 00 +NABC 190119890 + 00 316119890 + 00 274119890 + 00 271119890 + 00 314119890 minus 01

100 ABC 692119890 + 01 743119890 + 01 726119890 + 01 725119890 + 01 186119890 minus 00 +NABC 832119890 + 00 110119890 + 01 979119890 + 00 992119890 + 00 704119890 minus 01

1198915

30 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

60 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

100 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

1198916

30 ABC 812119890 minus 03 168119890 minus 01 133119890 minus 02 486119890 minus 02 484119890 minus 02

asymp

NABC 221119890 minus 03 231119890 minus 01 332119890 minus 02 599119890 minus 02 564119890 minus 02

60 ABC 108119890 minus 02 588119890 minus 01 156119890 minus 01 141119890 minus 01 165119890 minus 01

NABC 288119890 minus 04 388119890 minus 01 473119890 minus 02 102119890 minus 01 995119890 minus 02 asymp

100 ABC 357119890 minus 02 994119890 minus 01 145119890 minus 01 297119890 minus 01 364119890 minus 01

NABC 292119890 minus 04 103119890 + 00 216119890 minus 01 350119890 minus 01 328119890 minus 01 asymp

1198917

30 ABC 536119890 minus 15 146119890 minus 01 456119890 minus 09 958119890 minus 04 105119890 minus 03 +NABC 0 0 0 0 0

60 ABC 587119890 minus 12 199119890 minus 00 125119890 minus 09 268119890 minus 02 632119890 minus 02 +NABC 0 0 0 0 0

100 ABC 865119890 minus 13 199119890 minus 00 115119890 minus 06 497119890 minus 02 216119890 minus 01 +NABC 0 0 0 0 0

1198918

30 ABC 801119890 minus 15 130119890 minus 12 814119890 minus 13 237119890 minus 13 418119890 minus 13 +NABC 0 0 0 0 0

60 ABC 832119890 minus 14 134119890 minus 11 720119890 minus 13 345119890 minus 12 289119890 minus 12 +NABC 0 0 0 0 0

100 ABC 536119890 minus 14 158119890 minus 09 427119890 minus 10 180119890 minus 10 203119890 minus 10 +NABC 0 0 0 0 0

Mathematical Problems in Engineering 7

Table 2 Continued

Function Dim Best Worst Median Mean SD Significant

1198919

30 ABC 154119890 minus 06 237119890 + 02 376119890 minus 01 886119890 + 01 862119890 + 01 +NABC minus364119890 minus 12 minus182119890 minus 12 minus182119890 minus 12 minus200119890 minus 12 555119890 minus 13

60 ABC 355119890 + 02 769119890 + 02 769119890 + 02 540119890 + 02 141119890 + 02 +NABC 291119890 minus 11 364119890 minus 11 364119890 minus 11 354119890 minus 11 252119890 minus 12

100 ABC 781119890 + 02 155119890 + 03 151119890 + 03 129119890 + 03 223119890 + 02 +NABC 102119890 minus 10 116119890 minus 10 109119890 minus 10 110119890 minus 10 458119890 minus 12

11989110

30 ABC 536119890 minus 10 123119890 minus 08 367119890 minus 09 145119890 minus 09 237119890 minus 09 +NABC 222119890 minus 14 293119890 minus 14 293119890 minus 14 266119890 minus 14 332119890 minus 15

60 ABC 814119890 minus 09 754119890 minus 08 538119890 minus 08 460119890 minus 08 204119890 minus 08 +NABC 506119890 minus 14 684119890 minus 14 648119890 minus 14 644119890 minus 14 526119890 minus 15

100 ABC 535119890 minus 08 428119890 minus 07 104119890 minus 07 283119890 minus 07 157119890 minus 07 +NABC 104119890 minus 13 122119890 minus 13 115119890 minus 13 116119890 minus 13 475119890 minus 15

11989111

150 ABC minus774221 minus769602 minus770460 minus771658 211119890 minus 01 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

300 ABC minus770391 minus768430 minus770391 minus769258 676119890 minus 02 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

500 ABC minus768562 minus765849 minus768562 minus767220 952119890 minus 02 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

Table 3 Comparison among GABC EABC and NABC on optimizing 8 benchmark functions

Function MaxFEs Dim GABC EABC NABCMean SD Mean SD Mean SD

Sphere150000 30 137119890 minus 25 270119890 minus 25 442119890 minus 67 271119890 minus 67 110119890 minus 54 171119890 minus 54

300000 60 486119890 minus 23 514119890 minus 23 230119890 minus 64 103119890 minus 64 148119890 minus 52 137119890 minus 52

500000 100 905119890 minus 22 428119890 minus 22 637119890 minus 63 255119890 minus 63 145119890 minus 51 891119890 minus 52

Schwefel222150000 30 556119890 minus 15 979119890 minus 15 551119890 minus 35 702119890 minus 35 202119890 minus 29 110119890 minus 29

300000 60 561119890 minus 14 993119890 minus 15 508119890 minus 33 195119890 minus 34 292119890 minus 28 154119890 minus 28

500000 100 229119890 minus 13 397119890 minus 13 468119890 minus 32 142119890 minus 32 104119890 minus 27 688119890 minus 28

Schwefel221150000 30 307119890 minus 00 472119890 minus 01 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

300000 60 385119890 + 01 342119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

500000 100 704119890 + 01 120119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock150000 30 130119890 minus 00 164119890 minus 00 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

300000 60 166119890 + 01 328119890 + 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

500000 100 239119890 + 01 334119890 + 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Griewank100000 30 140119890 minus 08 116119890 minus 08 0 0 0 0150000 60 158119890 minus 06 100119890 minus 06 0 0 0 0250000 100 244119890 minus 06 250119890 minus 06 0 0 0 0

Rastrigin100000 30 708119890 minus 03 119119890 minus 03 0 0 0 0150000 60 347119890 minus 00 878119890 minus 01 0 0 0 0250000 100 978119890 minus 00 291119890 minus 00 0 0 0 0

Schwefel22650000 30 638119890 minus 00 238119890 + 01 0 0 171119890 minus 09 301119890 minus 10

100000 60 330119890 + 01 494119890 + 01 392119890 minus 11 356119890 minus 12 124119890 minus 09 502119890 minus 09

200000 100 605119890 + 01 805119890 + 01 112119890 minus 10 356119890 minus 12 122119890 minus 10 297119890 minus 11

Ackley50000 30 645119890 minus 03 268119890 minus 03 339119890 minus 10 695119890 minus 11 204119890 minus 08 472119890 minus 09

100000 60 296119890 minus 03 214119890 minus 03 181119890 minus 09 270119890 minus 10 445119890 minus 08 150119890 minus 08

150000 100 601119890 minus 02 960119890 minus 02 453119890 minus 08 123119890 minus 08 601119890 minus 07 323119890 minus 07

8 Mathematical Problems in Engineering

Table 4 Comparison among ABC EABC and NABC on optimizing 4 benchmark functions with119863 = 30 60 100

Function Dim ABC EABC NABCMean SD Mean SD Mean SD

Schwefel22130 110119890 + 01 129119890 minus 00 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

60 486119890 + 01 190119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

100 725119890 + 01 186119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock30 486119890 minus 02 484119890 minus 02 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

60 141119890 minus 01 165119890 minus 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

100 297119890 minus 01 364119890 minus 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Schwefel22630 886119890 + 01 862119890 + 01 minus123119890 minus 13 109119890 minus 13 minus200119890 minus 12 555119890 minus 13

60 540119890 + 02 141119890 + 02 291119890 minus 11 0 354119890 minus 11 252119890 minus 12

100 129119890 + 03 223119890 + 02 945119890 minus 11 0 110119890 minus 10 458119890 minus 12

Ackley30 145119890 minus 09 237119890 minus 09 136119890 minus 14 174119890 minus 15 266119890 minus 14 332119890 minus 15

60 460119890 minus 08 204119890 minus 08 449119890 minus 14 284119890 minus 15 644119890 minus 14 526119890 minus 15

100 283119890 minus 07 157119890 minus 07 953119890 minus 14 284119890 minus 15 116119890 minus 13 475119890 minus 15

Table 5 Comparison among NABC DE jDE JADE and SaDE on optimizing 8 benchmark functions with119863 = 30

Function MaxFEs DE jDE JADE SaDE NABC

Sphere 150000 Mean 98119890 minus 14 146119890 minus 28 132119890 minus 54 328119890 minus 20 110119890 minus 54

SD 84119890 minus 14 178119890 minus 28 922119890 minus 54 362119890 minus 20 171119890 minus 54

Schwefel222 200000 Mean 16119890 minus 09 902119890 minus 24 318119890 minus 25 351119890 minus 25 720119890 minus 40

SD 11119890 minus 09 601119890 minus 24 205119890 minus 25 274119890 minus 25 333119890 minus 40

Step 10000 Mean 47119890 + 03 613119890 + 02 562119890 + 00 507119890 + 01 0SD 11119890 + 03 172119890 + 02 187119890 + 00 134119890 + 01 0

Rosenbrock 300000 Mean 21119890 + 00 13119890 + 01 32119890 minus 01 21119890 + 01 422119890 minus 02

SD 15119890 + 00 14119890 + 01 11119890 + 00 78119890 + 00 493119890 minus 02

Griewank 50000 Mean 20119890 minus 01 729119890 minus 06 157119890 minus 08 252119890 minus 09 109119890 minus 08

SD 11119890 minus 01 105119890 minus 05 109119890 minus 07 124119890 minus 08 286119890 minus 08

Rastrigin 100000 Mean 18119890 + 02 332119890 minus 04 133119890 minus 01 243119890 + 00 0

SD 13119890 + 01 639119890 minus 04 974119890 minus 02 160119890 + 00 0

Ackley 50000 Mean 11119890 minus 01 237119890 minus 04 335119890 minus 09 381119890 minus 06 204119890 minus 08

SD 39119890 minus 02 710119890 minus 05 284119890 minus 09 826119890 minus 07 472119890 minus 09

Schwefel226 100000 Mean 59119890 + 03 170119890 minus 10 262119890 minus 04 113119890 minus 08 minus194119890 minus 12

SD 11119890 + 03 171119890 minus 10 359119890 minus 04 108119890 minus 08 461119890 minus 13

Table 6 Comparison among NABC FIPS HPSO-TVAC CLPSO FPSO and OLPSO-G on optimizing 8 benchmark functions with 200000FEs

Function FIPS HPSO-TVAC CLPSO FPSO OLPSO-G NABC

Sphere Mean 242119890 minus 13 283119890 minus 33 158119890 minus 12 240119890 minus 16 412119890 minus 54 545119890 minus 75

SD 173119890 minus 13 319119890 minus 33 770119890 minus 13 200119890 minus 31 634119890 minus 54 355119890 minus 75

Schwefel222 Mean 276119890 minus 08 903119890 minus 20 251119890 minus 08 158119890 minus 11 985119890 minus 30 720119890 minus 40

SD 904119890 minus 09 958119890 minus 20 584119890 minus 09 103119890 minus 22 101119890 minus 29 333119890 minus 40

Step Mean 0 0 0 0 0 0SD 0 0 0 0 0 0

Rosenbrock Mean 251119890 + 01 239119890 + 01 113119890 + 01 281119890 + 01 215119890 + 01 633119890 minus 02

SD 510119890 minus 01 265119890 + 01 985119890 minus 00 231119890 + 02 299119890 + 01 634119890 minus 02

Griewank Mean 901119890 minus 12 975119890 minus 03 902119890 minus 09 147119890 minus 03 483119890 minus 03 0

SD 184119890 minus 11 833119890 minus 03 857119890 minus 09 128119890 minus 05 863119890 minus 03 0

Rastrigin Mean 651119890 + 01 943119890 minus 00 909119890 minus 05 738119890 + 01 107119890 minus 00 0

SD 133119890 + 01 348119890 minus 00 125119890 minus 04 370119890 + 02 992119890 minus 01 0

Ackley Mean 233119890 minus 07 729119890 minus 14 366119890 minus 07 217119890 minus 09 798119890 minus 15 278119890 minus 14

SD 719119890 minus 08 300119890 minus 14 757119890 minus 08 171119890 minus 18 203119890 minus 15 259119890 minus 15

Schwefel226 Mean 993119890 + 02 159119890 + 03 382119890 minus 04 134119890 + 03 384119890 + 02 minus243119890 minus 12

SD 509119890 + 02 326119890 + 02 128119890 minus 05 277119890 + 02 217119890 + 02 872119890 minus 13

Mathematical Problems in Engineering 9

Table 7 Execution time of ABC and NABC algorithm with119863 = 30 60 100 (second)

Dim Algorithm 1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918

1198919

11989110

30 ABC 19850 25535 33363 18768 3969 32338 40942 78680 94332 47626NABC 17661 25697 36352 20056 2768 30612 18229 31331 77497 42660

60 ABC 43236 72503 104475 47454 12304 81891 185687 259944 322573 205303NABC 44900 73449 108912 47991 8202 80281 95126 129497 265051 186870

100 ABC 89402 163012 255115 92929 37074 180789 371414 574304 841482 410198NABC 88507 168964 255134 91418 21081 175828 334624 308107 687463 365939

Table 8 The five benchmark knapsack problems

119891 Dim Parameter (119908 119901 119888)1198911

10 119908 = (95 4 60 32 23 72 80 62 65 46) 119901 = (55 10 47 5 4 50 8 61 85 87) 119888 = 269

1198912

20 119908 = (92 4 43 83 84 68 92 82 6 44 32 18 56 83 25 96 70 48 14 58) 119901 = (44 46 90 72 91 40 75 35 8 54 78 4077 15 61 17 75 29 75 63) 119888 = 878

1198913

4 119908 = (6 5 9 7) 119901 = (9 11 13 15) 119888 = 201198914

4 119908 = (2 4 6 7) 119901 = (6 10 12 13) 119888 = 11

1198915

15119908 = (56358531 80874050 47987304 89596240 74660482 85894345 51353496 1498459 36445204 1658986244569231 0466933 37788018 57118442 60716575) 119901 = (0125126 19330424 58500931 35029145 8228400517410810 71050142 30399487 9140294 14731285 98852504 11908322 0891140 53166295 60176397) 119888 = 375

Table 9 The optimal solutions of the five benchmark knapsack problems

119891 Optimal solution Optimal value Value of constraint1198911

(0 1 1 1 0 0 0 1 1 1) 295 01198912

(1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1) 1024 minus71198913

(1 1 0 1) 35 minus21198914

(0 1 0 1) 23 01198915

(0 0 1 0 1 0 1 1 0 1 1 1 0 1 1) 4810694 minus200392

Table 10 Best worst median mean and standard deviation values obtained by ABC and NABC

119891 Algorithm Best Worst Median Mean SD

1198911

ABC 295 295 295 295 0NDABC 295 295 295 295 0

1198912

ABC 1024 1013 1024 1021 452NDABC 1024 1024 1024 1024 0

1198913

ABC 35 35 35 35 0NDABC 35 35 35 35 0

1198914

ABC 23 23 23 23 0NDABC 23 23 23 23 0

1198915

ABC 481069 437935 475478 460729 1847NDABC 481069 435786 481069 470771 879

6 Conclusion

In this paper we develop a novel artificial bee colonyalgorithm named NABC We add the global best solutioninto the search equation to drive the new candidate solutiononly around the global best solution in order to improvethe exploitation And the search equation of the employedbees is improved to keep the exploration of algorithm Theexperimental results tested on 11 benchmark functions showthat the convergence of NABC is much faster than that of

other algorithms and the computing is more effective At thesame timewe applyNABCon solving five standardKnapsackproblems and get good optimums So it is fitted to solvemanyengineering practical problems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors are grateful to the editor and the anonymousreviewers for their valuable comments and suggestions

References

[1] K S Tang K FMan S Kwong andQ He ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal ProcessingMagazine vol 13no 6 pp 22ndash37 1996

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

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

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

[5] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 Erciyes University KayseriTurkey 2005

[6] B Basturk and D Karaboga ldquoAn artificial bee colony (ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium May 2006

[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

[10] D Karaboga B Akay and C Ozturk ldquoArtificial bee colony(ABC) optimization algorithm for training feed-forward neuralnetworksrdquo in Modeling Decisions for Artificial Intelligence vol4617 of LectureNotes in Computer Science pp 318ndash329 SpringerBerlin Germany 2007

[11] Y-F Liu and S-Y Liu ldquoA hybrid discrete artificial bee colonyalgorithm for permutation flowshop scheduling problemrdquoApplied Soft Computing Journal vol 13 no 3 pp 1459ndash14632013

[12] D Karaboga and B Basturk ldquoArtificial bee colony (ABC)optimization algorithm for solving constrained optimizationproblemsrdquo in Foundations of Fuzzy Logic and Soft Computingvol 4529 of Lecture Notes in Computer Science pp 789ndash798Springer Berlin Germany 2007

[13] G P Zhu and S Kwong ldquoGbest-guided artificial bee colonyalgorithm for numerical function optimizationrdquo Applied Math-ematics and Computation vol 217 no 7 pp 3166ndash3173 2010

[14] B Basturk and D Karaboga ldquoA modified artificial bee colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[15] W-F Gao S-Y Liu and L-L Huang ldquoEnhancing artificialbee colony algorithm using more information-based searchequationsrdquo Information Sciences vol 270 pp 112ndash133 2014

[16] D Karaboga and B Gorkemli ldquoA quick artificial bee colony(qABC) algorithm and its performance on optimization prob-lemsrdquo Applied Soft Computing vol 23 pp 227ndash238 2014

[17] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 23 pp 689ndash6942010

[18] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[19] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009

[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2008

[21] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[22] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[23] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[24] M A M de Oca T Stutzle M Birattari and M DorigoldquoFrankensteinrsquos PSO a composite particle swarm optimizationalgorithmrdquo IEEE Transactions on Evolutionary Computationvol 13 no 5 pp 1120ndash1132 2009

[25] Z-H Zhan J Zhang Y Li and Y-H Shi ldquoOrthogonal learningparticle swarm optimizationrdquo IEEE Transactions on Evolution-ary Computation vol 15 no 6 pp 832ndash847 2011

[26] D Zou L Gao S Li and J Wu ldquoSolving 0-1 knapsack problemby a novel global harmony search algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 1556ndash1564 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Number of generations0 500 1000 1500

Mea

n

Schwefel22610

5

100

10minus5

10minus10

10minus15

Number of generations0 500 1000 1500

Mea

n

Ackley10

2

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

ABCNABC

ABCNABC

Figure 1 Convergence curves of ABC and NABC algorithms for 1198911ndash11989110functions (119863 = 30)

However NABC still keeps fast convergence and best perfor-mance

The results of NABC compared with the standard ABCalgorithm are shown in Table 2 in terms of the best worstmedian mean and standard deviation of the solutionsTheseresults show that the convergence rate of NABC is betterthan the standard ABC algorithm for most test functionsThe rate of convergence obviously increases on 119891

1ndash1198913 1198915

1198917ndash11989111

functions from Figure 1 and Table 2 NABC can findthe optimal solutions on functions119891

511989171198918The convergence

rate of NABC is the same order of magnitude as the standardABC algorithm on 119891

6 So the superiority of standard ABC

algorithm is not very obvious From the results in Table 2NABC can obtain better and closer-to-optimal solutions thanthe standard ABC algorithm for most of the functions In aword NABC increases the exploitation greatly NABC alsoretains good exploration Here ldquo+rdquo indicates that NABCis statistically significantly better than its correspondingcompetitor algorithm ldquoasymprdquo stands for that the result of thecorresponding algorithm is statistically similar with that ofNABC

NABC is further compared with GABC [13] and EABC[15] in Tables 3 and 4 The results of GABC and EABCare gained from [15] directly It is clear that NABC worksbetter in almost all the cases and achieves better performancethan GABC The rate of convergence of NABC increasesobviously on all functions in Tables 3 and 4The convergencerate of NABC is a little worse than that of EABC fromTable 3 In particular the convergence rate of Schwefel221and Rosenbrock is better than EABC in Table 4

At the same time NABC is compared with four variantsof DE (DE [17] jDE [18] JADE [19] and SaDE [20]) and fivevariants of PSO (FIPS [21] HPSO-TVAC [22] CLPSO [23]FPSO [24] OLPSO-G [25]) The results of DEs and PSOs aregained from [15] directly The maximum number of functionevaluations (FEs) is shown in Tables 5 and 6 Here FEs isMaxgenlowastSN It can be seen that NABC is better on almostall the test functions except Ackley in JADE and OLPSO-G

In Table 7 we can find the execution time of ABC issimilar with that of NABC So NABC is largely faster thanABC and does not add the execution time

5 Application

51 Knapsack Problems We know that Knapsack problem[26] is an engineering optimization problem It can be definedan unconstrained optimization problem and described asfollows

min119891 (119909) = minus

119899

sum

119894=1

119901119894119909119894+ 120572 lowastmax(0

119899

sum

119894=1

119908119894119909119894minus 119888) (7)

where 119908119894is each itemrsquos weight and 119901

119894is each itemrsquos profit

respectively The knapsack has a limited weight capabilityof 119888 The objective function of this problem is to pack theknapsack so that the items in it have the maximal totalprofit The decision variable 119909

119894is the value one if the item

119894 is packed otherwise it is the value zero And we assumethat all profits and weights are positive and all weights aresmaller than the capacity of 119888 120572 is a penalty factor which isset to 10119890 minus 20 in this paper Tables 8 and 9 are parametersof five standard knapsack problems The experiments testedon five benchmark 0-1 knapsack problems are performed tocompare the performance of NABC with that of standardABC The maximum number of generations is set to 2000The population size is 100 namely SN = 50 and Limit is 10Each of the experiments is repeated 30 times independently

52 Data Analysis In Table 10 we can see that NABC caneasily solve the optimum of Knapsack problems And itcan find the optimum in most cases except 119891

5 ABC has

the worst performance because we can see that it can notfind the optimums for 119891

2and 119891

5 In short NABC has a

better performance than ABC on solving the five knapsackproblems

6 Mathematical Problems in Engineering

Table 2 Best worst medianmean and standard deviation values obtained byABC andNABC through 30 independent runs on 11 functions

Function Dim Best Worst Median Mean SD Significant

1198911

30 ABC 504119890 minus 16 725119890 minus 16 504119890 minus 16 606119890 minus 16 934119890 minus 17 +NABC 113119890 minus 55 966119890 minus 54 666119890 minus 55 110119890 minus 54 171119890 minus 54

60 ABC 140119890 minus 15 187119890 minus 15 152119890 minus 15 162119890 minus 15 202119890 minus 16 +NABC 103119890 minus 53 466119890 minus 52 906119890 minus 53 148119890 minus 52 137119890 minus 52

100 ABC 220119890 minus 15 932119890 minus 15 521119890 minus 15 336119890 minus 15 201119890 minus 15 +NABC 161119890 minus 52 121119890 minus 51 369119890 minus 51 145119890 minus 51 891119890 minus 52

1198912

30 ABC 430119890 minus 16 634119890 minus 16 450119890 minus 16 503119890 minus 16 711119890 minus 17 +NABC 164119890 minus 57 782119890 minus 56 116119890 minus 56 270119890 minus 56 248119890 minus 56

60 ABC 143119890 minus 15 200119890 minus 15 162119890 minus 15 167119890 minus 15 184119890 minus 16 +NABC 443119890 minus 55 782119890 minus 53 801119890 minus 54 208119890 minus 53 233119890 minus 53

100 ABC 247119890 minus 15 366119890 minus 15 343119890 minus 05 319119890 minus 15 421119890 minus 16 +NABC 351119890 minus 53 575119890 minus 52 266119890 minus 52 301119890 minus 52 156119890 minus 52

1198913

30 ABC 238119890 minus 11 134119890 minus 10 642119890 minus 11 705119890 minus 11 519119890 minus 11 +NABC 506119890 minus 30 411119890 minus 29 173119890 minus 29 202119890 minus 29 110119890 minus 29

60 ABC 902119890 minus 11 823119890 minus 10 512119890 minus 10 408119890 minus 10 157119890 minus 10 +NABC 712119890 minus 29 572119890 minus 28 257119890 minus 28 292119890 minus 28 154119890 minus 28

100 ABC 205119890 minus 09 536119890 minus 08 484119890 minus 09 367119890 minus 09 480119890 minus 09 +NABC 285119890 minus 28 281119890 minus 27 735119890 minus 28 104119890 minus 27 688119890 minus 28

1198914

30 ABC 868119890 + 00 120119890 + 01 184119890 + 01 110119890 + 01 129119890 minus 00 +NABC 177119890 minus 01 318119890 minus 01 268119890 minus 01 267119890 minus 01 355119890 minus 02

60 ABC 453119890 + 01 508119890 + 01 502119890 + 01 486119890 + 01 190119890 minus 00 +NABC 190119890 + 00 316119890 + 00 274119890 + 00 271119890 + 00 314119890 minus 01

100 ABC 692119890 + 01 743119890 + 01 726119890 + 01 725119890 + 01 186119890 minus 00 +NABC 832119890 + 00 110119890 + 01 979119890 + 00 992119890 + 00 704119890 minus 01

1198915

30 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

60 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

100 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

1198916

30 ABC 812119890 minus 03 168119890 minus 01 133119890 minus 02 486119890 minus 02 484119890 minus 02

asymp

NABC 221119890 minus 03 231119890 minus 01 332119890 minus 02 599119890 minus 02 564119890 minus 02

60 ABC 108119890 minus 02 588119890 minus 01 156119890 minus 01 141119890 minus 01 165119890 minus 01

NABC 288119890 minus 04 388119890 minus 01 473119890 minus 02 102119890 minus 01 995119890 minus 02 asymp

100 ABC 357119890 minus 02 994119890 minus 01 145119890 minus 01 297119890 minus 01 364119890 minus 01

NABC 292119890 minus 04 103119890 + 00 216119890 minus 01 350119890 minus 01 328119890 minus 01 asymp

1198917

30 ABC 536119890 minus 15 146119890 minus 01 456119890 minus 09 958119890 minus 04 105119890 minus 03 +NABC 0 0 0 0 0

60 ABC 587119890 minus 12 199119890 minus 00 125119890 minus 09 268119890 minus 02 632119890 minus 02 +NABC 0 0 0 0 0

100 ABC 865119890 minus 13 199119890 minus 00 115119890 minus 06 497119890 minus 02 216119890 minus 01 +NABC 0 0 0 0 0

1198918

30 ABC 801119890 minus 15 130119890 minus 12 814119890 minus 13 237119890 minus 13 418119890 minus 13 +NABC 0 0 0 0 0

60 ABC 832119890 minus 14 134119890 minus 11 720119890 minus 13 345119890 minus 12 289119890 minus 12 +NABC 0 0 0 0 0

100 ABC 536119890 minus 14 158119890 minus 09 427119890 minus 10 180119890 minus 10 203119890 minus 10 +NABC 0 0 0 0 0

Mathematical Problems in Engineering 7

Table 2 Continued

Function Dim Best Worst Median Mean SD Significant

1198919

30 ABC 154119890 minus 06 237119890 + 02 376119890 minus 01 886119890 + 01 862119890 + 01 +NABC minus364119890 minus 12 minus182119890 minus 12 minus182119890 minus 12 minus200119890 minus 12 555119890 minus 13

60 ABC 355119890 + 02 769119890 + 02 769119890 + 02 540119890 + 02 141119890 + 02 +NABC 291119890 minus 11 364119890 minus 11 364119890 minus 11 354119890 minus 11 252119890 minus 12

100 ABC 781119890 + 02 155119890 + 03 151119890 + 03 129119890 + 03 223119890 + 02 +NABC 102119890 minus 10 116119890 minus 10 109119890 minus 10 110119890 minus 10 458119890 minus 12

11989110

30 ABC 536119890 minus 10 123119890 minus 08 367119890 minus 09 145119890 minus 09 237119890 minus 09 +NABC 222119890 minus 14 293119890 minus 14 293119890 minus 14 266119890 minus 14 332119890 minus 15

60 ABC 814119890 minus 09 754119890 minus 08 538119890 minus 08 460119890 minus 08 204119890 minus 08 +NABC 506119890 minus 14 684119890 minus 14 648119890 minus 14 644119890 minus 14 526119890 minus 15

100 ABC 535119890 minus 08 428119890 minus 07 104119890 minus 07 283119890 minus 07 157119890 minus 07 +NABC 104119890 minus 13 122119890 minus 13 115119890 minus 13 116119890 minus 13 475119890 minus 15

11989111

150 ABC minus774221 minus769602 minus770460 minus771658 211119890 minus 01 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

300 ABC minus770391 minus768430 minus770391 minus769258 676119890 minus 02 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

500 ABC minus768562 minus765849 minus768562 minus767220 952119890 minus 02 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

Table 3 Comparison among GABC EABC and NABC on optimizing 8 benchmark functions

Function MaxFEs Dim GABC EABC NABCMean SD Mean SD Mean SD

Sphere150000 30 137119890 minus 25 270119890 minus 25 442119890 minus 67 271119890 minus 67 110119890 minus 54 171119890 minus 54

300000 60 486119890 minus 23 514119890 minus 23 230119890 minus 64 103119890 minus 64 148119890 minus 52 137119890 minus 52

500000 100 905119890 minus 22 428119890 minus 22 637119890 minus 63 255119890 minus 63 145119890 minus 51 891119890 minus 52

Schwefel222150000 30 556119890 minus 15 979119890 minus 15 551119890 minus 35 702119890 minus 35 202119890 minus 29 110119890 minus 29

300000 60 561119890 minus 14 993119890 minus 15 508119890 minus 33 195119890 minus 34 292119890 minus 28 154119890 minus 28

500000 100 229119890 minus 13 397119890 minus 13 468119890 minus 32 142119890 minus 32 104119890 minus 27 688119890 minus 28

Schwefel221150000 30 307119890 minus 00 472119890 minus 01 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

300000 60 385119890 + 01 342119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

500000 100 704119890 + 01 120119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock150000 30 130119890 minus 00 164119890 minus 00 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

300000 60 166119890 + 01 328119890 + 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

500000 100 239119890 + 01 334119890 + 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Griewank100000 30 140119890 minus 08 116119890 minus 08 0 0 0 0150000 60 158119890 minus 06 100119890 minus 06 0 0 0 0250000 100 244119890 minus 06 250119890 minus 06 0 0 0 0

Rastrigin100000 30 708119890 minus 03 119119890 minus 03 0 0 0 0150000 60 347119890 minus 00 878119890 minus 01 0 0 0 0250000 100 978119890 minus 00 291119890 minus 00 0 0 0 0

Schwefel22650000 30 638119890 minus 00 238119890 + 01 0 0 171119890 minus 09 301119890 minus 10

100000 60 330119890 + 01 494119890 + 01 392119890 minus 11 356119890 minus 12 124119890 minus 09 502119890 minus 09

200000 100 605119890 + 01 805119890 + 01 112119890 minus 10 356119890 minus 12 122119890 minus 10 297119890 minus 11

Ackley50000 30 645119890 minus 03 268119890 minus 03 339119890 minus 10 695119890 minus 11 204119890 minus 08 472119890 minus 09

100000 60 296119890 minus 03 214119890 minus 03 181119890 minus 09 270119890 minus 10 445119890 minus 08 150119890 minus 08

150000 100 601119890 minus 02 960119890 minus 02 453119890 minus 08 123119890 minus 08 601119890 minus 07 323119890 minus 07

8 Mathematical Problems in Engineering

Table 4 Comparison among ABC EABC and NABC on optimizing 4 benchmark functions with119863 = 30 60 100

Function Dim ABC EABC NABCMean SD Mean SD Mean SD

Schwefel22130 110119890 + 01 129119890 minus 00 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

60 486119890 + 01 190119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

100 725119890 + 01 186119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock30 486119890 minus 02 484119890 minus 02 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

60 141119890 minus 01 165119890 minus 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

100 297119890 minus 01 364119890 minus 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Schwefel22630 886119890 + 01 862119890 + 01 minus123119890 minus 13 109119890 minus 13 minus200119890 minus 12 555119890 minus 13

60 540119890 + 02 141119890 + 02 291119890 minus 11 0 354119890 minus 11 252119890 minus 12

100 129119890 + 03 223119890 + 02 945119890 minus 11 0 110119890 minus 10 458119890 minus 12

Ackley30 145119890 minus 09 237119890 minus 09 136119890 minus 14 174119890 minus 15 266119890 minus 14 332119890 minus 15

60 460119890 minus 08 204119890 minus 08 449119890 minus 14 284119890 minus 15 644119890 minus 14 526119890 minus 15

100 283119890 minus 07 157119890 minus 07 953119890 minus 14 284119890 minus 15 116119890 minus 13 475119890 minus 15

Table 5 Comparison among NABC DE jDE JADE and SaDE on optimizing 8 benchmark functions with119863 = 30

Function MaxFEs DE jDE JADE SaDE NABC

Sphere 150000 Mean 98119890 minus 14 146119890 minus 28 132119890 minus 54 328119890 minus 20 110119890 minus 54

SD 84119890 minus 14 178119890 minus 28 922119890 minus 54 362119890 minus 20 171119890 minus 54

Schwefel222 200000 Mean 16119890 minus 09 902119890 minus 24 318119890 minus 25 351119890 minus 25 720119890 minus 40

SD 11119890 minus 09 601119890 minus 24 205119890 minus 25 274119890 minus 25 333119890 minus 40

Step 10000 Mean 47119890 + 03 613119890 + 02 562119890 + 00 507119890 + 01 0SD 11119890 + 03 172119890 + 02 187119890 + 00 134119890 + 01 0

Rosenbrock 300000 Mean 21119890 + 00 13119890 + 01 32119890 minus 01 21119890 + 01 422119890 minus 02

SD 15119890 + 00 14119890 + 01 11119890 + 00 78119890 + 00 493119890 minus 02

Griewank 50000 Mean 20119890 minus 01 729119890 minus 06 157119890 minus 08 252119890 minus 09 109119890 minus 08

SD 11119890 minus 01 105119890 minus 05 109119890 minus 07 124119890 minus 08 286119890 minus 08

Rastrigin 100000 Mean 18119890 + 02 332119890 minus 04 133119890 minus 01 243119890 + 00 0

SD 13119890 + 01 639119890 minus 04 974119890 minus 02 160119890 + 00 0

Ackley 50000 Mean 11119890 minus 01 237119890 minus 04 335119890 minus 09 381119890 minus 06 204119890 minus 08

SD 39119890 minus 02 710119890 minus 05 284119890 minus 09 826119890 minus 07 472119890 minus 09

Schwefel226 100000 Mean 59119890 + 03 170119890 minus 10 262119890 minus 04 113119890 minus 08 minus194119890 minus 12

SD 11119890 + 03 171119890 minus 10 359119890 minus 04 108119890 minus 08 461119890 minus 13

Table 6 Comparison among NABC FIPS HPSO-TVAC CLPSO FPSO and OLPSO-G on optimizing 8 benchmark functions with 200000FEs

Function FIPS HPSO-TVAC CLPSO FPSO OLPSO-G NABC

Sphere Mean 242119890 minus 13 283119890 minus 33 158119890 minus 12 240119890 minus 16 412119890 minus 54 545119890 minus 75

SD 173119890 minus 13 319119890 minus 33 770119890 minus 13 200119890 minus 31 634119890 minus 54 355119890 minus 75

Schwefel222 Mean 276119890 minus 08 903119890 minus 20 251119890 minus 08 158119890 minus 11 985119890 minus 30 720119890 minus 40

SD 904119890 minus 09 958119890 minus 20 584119890 minus 09 103119890 minus 22 101119890 minus 29 333119890 minus 40

Step Mean 0 0 0 0 0 0SD 0 0 0 0 0 0

Rosenbrock Mean 251119890 + 01 239119890 + 01 113119890 + 01 281119890 + 01 215119890 + 01 633119890 minus 02

SD 510119890 minus 01 265119890 + 01 985119890 minus 00 231119890 + 02 299119890 + 01 634119890 minus 02

Griewank Mean 901119890 minus 12 975119890 minus 03 902119890 minus 09 147119890 minus 03 483119890 minus 03 0

SD 184119890 minus 11 833119890 minus 03 857119890 minus 09 128119890 minus 05 863119890 minus 03 0

Rastrigin Mean 651119890 + 01 943119890 minus 00 909119890 minus 05 738119890 + 01 107119890 minus 00 0

SD 133119890 + 01 348119890 minus 00 125119890 minus 04 370119890 + 02 992119890 minus 01 0

Ackley Mean 233119890 minus 07 729119890 minus 14 366119890 minus 07 217119890 minus 09 798119890 minus 15 278119890 minus 14

SD 719119890 minus 08 300119890 minus 14 757119890 minus 08 171119890 minus 18 203119890 minus 15 259119890 minus 15

Schwefel226 Mean 993119890 + 02 159119890 + 03 382119890 minus 04 134119890 + 03 384119890 + 02 minus243119890 minus 12

SD 509119890 + 02 326119890 + 02 128119890 minus 05 277119890 + 02 217119890 + 02 872119890 minus 13

Mathematical Problems in Engineering 9

Table 7 Execution time of ABC and NABC algorithm with119863 = 30 60 100 (second)

Dim Algorithm 1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918

1198919

11989110

30 ABC 19850 25535 33363 18768 3969 32338 40942 78680 94332 47626NABC 17661 25697 36352 20056 2768 30612 18229 31331 77497 42660

60 ABC 43236 72503 104475 47454 12304 81891 185687 259944 322573 205303NABC 44900 73449 108912 47991 8202 80281 95126 129497 265051 186870

100 ABC 89402 163012 255115 92929 37074 180789 371414 574304 841482 410198NABC 88507 168964 255134 91418 21081 175828 334624 308107 687463 365939

Table 8 The five benchmark knapsack problems

119891 Dim Parameter (119908 119901 119888)1198911

10 119908 = (95 4 60 32 23 72 80 62 65 46) 119901 = (55 10 47 5 4 50 8 61 85 87) 119888 = 269

1198912

20 119908 = (92 4 43 83 84 68 92 82 6 44 32 18 56 83 25 96 70 48 14 58) 119901 = (44 46 90 72 91 40 75 35 8 54 78 4077 15 61 17 75 29 75 63) 119888 = 878

1198913

4 119908 = (6 5 9 7) 119901 = (9 11 13 15) 119888 = 201198914

4 119908 = (2 4 6 7) 119901 = (6 10 12 13) 119888 = 11

1198915

15119908 = (56358531 80874050 47987304 89596240 74660482 85894345 51353496 1498459 36445204 1658986244569231 0466933 37788018 57118442 60716575) 119901 = (0125126 19330424 58500931 35029145 8228400517410810 71050142 30399487 9140294 14731285 98852504 11908322 0891140 53166295 60176397) 119888 = 375

Table 9 The optimal solutions of the five benchmark knapsack problems

119891 Optimal solution Optimal value Value of constraint1198911

(0 1 1 1 0 0 0 1 1 1) 295 01198912

(1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1) 1024 minus71198913

(1 1 0 1) 35 minus21198914

(0 1 0 1) 23 01198915

(0 0 1 0 1 0 1 1 0 1 1 1 0 1 1) 4810694 minus200392

Table 10 Best worst median mean and standard deviation values obtained by ABC and NABC

119891 Algorithm Best Worst Median Mean SD

1198911

ABC 295 295 295 295 0NDABC 295 295 295 295 0

1198912

ABC 1024 1013 1024 1021 452NDABC 1024 1024 1024 1024 0

1198913

ABC 35 35 35 35 0NDABC 35 35 35 35 0

1198914

ABC 23 23 23 23 0NDABC 23 23 23 23 0

1198915

ABC 481069 437935 475478 460729 1847NDABC 481069 435786 481069 470771 879

6 Conclusion

In this paper we develop a novel artificial bee colonyalgorithm named NABC We add the global best solutioninto the search equation to drive the new candidate solutiononly around the global best solution in order to improvethe exploitation And the search equation of the employedbees is improved to keep the exploration of algorithm Theexperimental results tested on 11 benchmark functions showthat the convergence of NABC is much faster than that of

other algorithms and the computing is more effective At thesame timewe applyNABCon solving five standardKnapsackproblems and get good optimums So it is fitted to solvemanyengineering practical problems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors are grateful to the editor and the anonymousreviewers for their valuable comments and suggestions

References

[1] K S Tang K FMan S Kwong andQ He ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal ProcessingMagazine vol 13no 6 pp 22ndash37 1996

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

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

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

[5] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 Erciyes University KayseriTurkey 2005

[6] B Basturk and D Karaboga ldquoAn artificial bee colony (ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium May 2006

[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

[10] D Karaboga B Akay and C Ozturk ldquoArtificial bee colony(ABC) optimization algorithm for training feed-forward neuralnetworksrdquo in Modeling Decisions for Artificial Intelligence vol4617 of LectureNotes in Computer Science pp 318ndash329 SpringerBerlin Germany 2007

[11] Y-F Liu and S-Y Liu ldquoA hybrid discrete artificial bee colonyalgorithm for permutation flowshop scheduling problemrdquoApplied Soft Computing Journal vol 13 no 3 pp 1459ndash14632013

[12] D Karaboga and B Basturk ldquoArtificial bee colony (ABC)optimization algorithm for solving constrained optimizationproblemsrdquo in Foundations of Fuzzy Logic and Soft Computingvol 4529 of Lecture Notes in Computer Science pp 789ndash798Springer Berlin Germany 2007

[13] G P Zhu and S Kwong ldquoGbest-guided artificial bee colonyalgorithm for numerical function optimizationrdquo Applied Math-ematics and Computation vol 217 no 7 pp 3166ndash3173 2010

[14] B Basturk and D Karaboga ldquoA modified artificial bee colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[15] W-F Gao S-Y Liu and L-L Huang ldquoEnhancing artificialbee colony algorithm using more information-based searchequationsrdquo Information Sciences vol 270 pp 112ndash133 2014

[16] D Karaboga and B Gorkemli ldquoA quick artificial bee colony(qABC) algorithm and its performance on optimization prob-lemsrdquo Applied Soft Computing vol 23 pp 227ndash238 2014

[17] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 23 pp 689ndash6942010

[18] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[19] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009

[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2008

[21] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[22] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[23] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[24] M A M de Oca T Stutzle M Birattari and M DorigoldquoFrankensteinrsquos PSO a composite particle swarm optimizationalgorithmrdquo IEEE Transactions on Evolutionary Computationvol 13 no 5 pp 1120ndash1132 2009

[25] Z-H Zhan J Zhang Y Li and Y-H Shi ldquoOrthogonal learningparticle swarm optimizationrdquo IEEE Transactions on Evolution-ary Computation vol 15 no 6 pp 832ndash847 2011

[26] D Zou L Gao S Li and J Wu ldquoSolving 0-1 knapsack problemby a novel global harmony search algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 1556ndash1564 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Table 2 Best worst medianmean and standard deviation values obtained byABC andNABC through 30 independent runs on 11 functions

Function Dim Best Worst Median Mean SD Significant

1198911

30 ABC 504119890 minus 16 725119890 minus 16 504119890 minus 16 606119890 minus 16 934119890 minus 17 +NABC 113119890 minus 55 966119890 minus 54 666119890 minus 55 110119890 minus 54 171119890 minus 54

60 ABC 140119890 minus 15 187119890 minus 15 152119890 minus 15 162119890 minus 15 202119890 minus 16 +NABC 103119890 minus 53 466119890 minus 52 906119890 minus 53 148119890 minus 52 137119890 minus 52

100 ABC 220119890 minus 15 932119890 minus 15 521119890 minus 15 336119890 minus 15 201119890 minus 15 +NABC 161119890 minus 52 121119890 minus 51 369119890 minus 51 145119890 minus 51 891119890 minus 52

1198912

30 ABC 430119890 minus 16 634119890 minus 16 450119890 minus 16 503119890 minus 16 711119890 minus 17 +NABC 164119890 minus 57 782119890 minus 56 116119890 minus 56 270119890 minus 56 248119890 minus 56

60 ABC 143119890 minus 15 200119890 minus 15 162119890 minus 15 167119890 minus 15 184119890 minus 16 +NABC 443119890 minus 55 782119890 minus 53 801119890 minus 54 208119890 minus 53 233119890 minus 53

100 ABC 247119890 minus 15 366119890 minus 15 343119890 minus 05 319119890 minus 15 421119890 minus 16 +NABC 351119890 minus 53 575119890 minus 52 266119890 minus 52 301119890 minus 52 156119890 minus 52

1198913

30 ABC 238119890 minus 11 134119890 minus 10 642119890 minus 11 705119890 minus 11 519119890 minus 11 +NABC 506119890 minus 30 411119890 minus 29 173119890 minus 29 202119890 minus 29 110119890 minus 29

60 ABC 902119890 minus 11 823119890 minus 10 512119890 minus 10 408119890 minus 10 157119890 minus 10 +NABC 712119890 minus 29 572119890 minus 28 257119890 minus 28 292119890 minus 28 154119890 minus 28

100 ABC 205119890 minus 09 536119890 minus 08 484119890 minus 09 367119890 minus 09 480119890 minus 09 +NABC 285119890 minus 28 281119890 minus 27 735119890 minus 28 104119890 minus 27 688119890 minus 28

1198914

30 ABC 868119890 + 00 120119890 + 01 184119890 + 01 110119890 + 01 129119890 minus 00 +NABC 177119890 minus 01 318119890 minus 01 268119890 minus 01 267119890 minus 01 355119890 minus 02

60 ABC 453119890 + 01 508119890 + 01 502119890 + 01 486119890 + 01 190119890 minus 00 +NABC 190119890 + 00 316119890 + 00 274119890 + 00 271119890 + 00 314119890 minus 01

100 ABC 692119890 + 01 743119890 + 01 726119890 + 01 725119890 + 01 186119890 minus 00 +NABC 832119890 + 00 110119890 + 01 979119890 + 00 992119890 + 00 704119890 minus 01

1198915

30 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

60 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

100 ABC 0 0 0 0 0NABC 0 0 0 0 0 asymp

1198916

30 ABC 812119890 minus 03 168119890 minus 01 133119890 minus 02 486119890 minus 02 484119890 minus 02

asymp

NABC 221119890 minus 03 231119890 minus 01 332119890 minus 02 599119890 minus 02 564119890 minus 02

60 ABC 108119890 minus 02 588119890 minus 01 156119890 minus 01 141119890 minus 01 165119890 minus 01

NABC 288119890 minus 04 388119890 minus 01 473119890 minus 02 102119890 minus 01 995119890 minus 02 asymp

100 ABC 357119890 minus 02 994119890 minus 01 145119890 minus 01 297119890 minus 01 364119890 minus 01

NABC 292119890 minus 04 103119890 + 00 216119890 minus 01 350119890 minus 01 328119890 minus 01 asymp

1198917

30 ABC 536119890 minus 15 146119890 minus 01 456119890 minus 09 958119890 minus 04 105119890 minus 03 +NABC 0 0 0 0 0

60 ABC 587119890 minus 12 199119890 minus 00 125119890 minus 09 268119890 minus 02 632119890 minus 02 +NABC 0 0 0 0 0

100 ABC 865119890 minus 13 199119890 minus 00 115119890 minus 06 497119890 minus 02 216119890 minus 01 +NABC 0 0 0 0 0

1198918

30 ABC 801119890 minus 15 130119890 minus 12 814119890 minus 13 237119890 minus 13 418119890 minus 13 +NABC 0 0 0 0 0

60 ABC 832119890 minus 14 134119890 minus 11 720119890 minus 13 345119890 minus 12 289119890 minus 12 +NABC 0 0 0 0 0

100 ABC 536119890 minus 14 158119890 minus 09 427119890 minus 10 180119890 minus 10 203119890 minus 10 +NABC 0 0 0 0 0

Mathematical Problems in Engineering 7

Table 2 Continued

Function Dim Best Worst Median Mean SD Significant

1198919

30 ABC 154119890 minus 06 237119890 + 02 376119890 minus 01 886119890 + 01 862119890 + 01 +NABC minus364119890 minus 12 minus182119890 minus 12 minus182119890 minus 12 minus200119890 minus 12 555119890 minus 13

60 ABC 355119890 + 02 769119890 + 02 769119890 + 02 540119890 + 02 141119890 + 02 +NABC 291119890 minus 11 364119890 minus 11 364119890 minus 11 354119890 minus 11 252119890 minus 12

100 ABC 781119890 + 02 155119890 + 03 151119890 + 03 129119890 + 03 223119890 + 02 +NABC 102119890 minus 10 116119890 minus 10 109119890 minus 10 110119890 minus 10 458119890 minus 12

11989110

30 ABC 536119890 minus 10 123119890 minus 08 367119890 minus 09 145119890 minus 09 237119890 minus 09 +NABC 222119890 minus 14 293119890 minus 14 293119890 minus 14 266119890 minus 14 332119890 minus 15

60 ABC 814119890 minus 09 754119890 minus 08 538119890 minus 08 460119890 minus 08 204119890 minus 08 +NABC 506119890 minus 14 684119890 minus 14 648119890 minus 14 644119890 minus 14 526119890 minus 15

100 ABC 535119890 minus 08 428119890 minus 07 104119890 minus 07 283119890 minus 07 157119890 minus 07 +NABC 104119890 minus 13 122119890 minus 13 115119890 minus 13 116119890 minus 13 475119890 minus 15

11989111

150 ABC minus774221 minus769602 minus770460 minus771658 211119890 minus 01 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

300 ABC minus770391 minus768430 minus770391 minus769258 676119890 minus 02 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

500 ABC minus768562 minus765849 minus768562 minus767220 952119890 minus 02 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

Table 3 Comparison among GABC EABC and NABC on optimizing 8 benchmark functions

Function MaxFEs Dim GABC EABC NABCMean SD Mean SD Mean SD

Sphere150000 30 137119890 minus 25 270119890 minus 25 442119890 minus 67 271119890 minus 67 110119890 minus 54 171119890 minus 54

300000 60 486119890 minus 23 514119890 minus 23 230119890 minus 64 103119890 minus 64 148119890 minus 52 137119890 minus 52

500000 100 905119890 minus 22 428119890 minus 22 637119890 minus 63 255119890 minus 63 145119890 minus 51 891119890 minus 52

Schwefel222150000 30 556119890 minus 15 979119890 minus 15 551119890 minus 35 702119890 minus 35 202119890 minus 29 110119890 minus 29

300000 60 561119890 minus 14 993119890 minus 15 508119890 minus 33 195119890 minus 34 292119890 minus 28 154119890 minus 28

500000 100 229119890 minus 13 397119890 minus 13 468119890 minus 32 142119890 minus 32 104119890 minus 27 688119890 minus 28

Schwefel221150000 30 307119890 minus 00 472119890 minus 01 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

300000 60 385119890 + 01 342119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

500000 100 704119890 + 01 120119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock150000 30 130119890 minus 00 164119890 minus 00 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

300000 60 166119890 + 01 328119890 + 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

500000 100 239119890 + 01 334119890 + 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Griewank100000 30 140119890 minus 08 116119890 minus 08 0 0 0 0150000 60 158119890 minus 06 100119890 minus 06 0 0 0 0250000 100 244119890 minus 06 250119890 minus 06 0 0 0 0

Rastrigin100000 30 708119890 minus 03 119119890 minus 03 0 0 0 0150000 60 347119890 minus 00 878119890 minus 01 0 0 0 0250000 100 978119890 minus 00 291119890 minus 00 0 0 0 0

Schwefel22650000 30 638119890 minus 00 238119890 + 01 0 0 171119890 minus 09 301119890 minus 10

100000 60 330119890 + 01 494119890 + 01 392119890 minus 11 356119890 minus 12 124119890 minus 09 502119890 minus 09

200000 100 605119890 + 01 805119890 + 01 112119890 minus 10 356119890 minus 12 122119890 minus 10 297119890 minus 11

Ackley50000 30 645119890 minus 03 268119890 minus 03 339119890 minus 10 695119890 minus 11 204119890 minus 08 472119890 minus 09

100000 60 296119890 minus 03 214119890 minus 03 181119890 minus 09 270119890 minus 10 445119890 minus 08 150119890 minus 08

150000 100 601119890 minus 02 960119890 minus 02 453119890 minus 08 123119890 minus 08 601119890 minus 07 323119890 minus 07

8 Mathematical Problems in Engineering

Table 4 Comparison among ABC EABC and NABC on optimizing 4 benchmark functions with119863 = 30 60 100

Function Dim ABC EABC NABCMean SD Mean SD Mean SD

Schwefel22130 110119890 + 01 129119890 minus 00 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

60 486119890 + 01 190119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

100 725119890 + 01 186119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock30 486119890 minus 02 484119890 minus 02 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

60 141119890 minus 01 165119890 minus 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

100 297119890 minus 01 364119890 minus 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Schwefel22630 886119890 + 01 862119890 + 01 minus123119890 minus 13 109119890 minus 13 minus200119890 minus 12 555119890 minus 13

60 540119890 + 02 141119890 + 02 291119890 minus 11 0 354119890 minus 11 252119890 minus 12

100 129119890 + 03 223119890 + 02 945119890 minus 11 0 110119890 minus 10 458119890 minus 12

Ackley30 145119890 minus 09 237119890 minus 09 136119890 minus 14 174119890 minus 15 266119890 minus 14 332119890 minus 15

60 460119890 minus 08 204119890 minus 08 449119890 minus 14 284119890 minus 15 644119890 minus 14 526119890 minus 15

100 283119890 minus 07 157119890 minus 07 953119890 minus 14 284119890 minus 15 116119890 minus 13 475119890 minus 15

Table 5 Comparison among NABC DE jDE JADE and SaDE on optimizing 8 benchmark functions with119863 = 30

Function MaxFEs DE jDE JADE SaDE NABC

Sphere 150000 Mean 98119890 minus 14 146119890 minus 28 132119890 minus 54 328119890 minus 20 110119890 minus 54

SD 84119890 minus 14 178119890 minus 28 922119890 minus 54 362119890 minus 20 171119890 minus 54

Schwefel222 200000 Mean 16119890 minus 09 902119890 minus 24 318119890 minus 25 351119890 minus 25 720119890 minus 40

SD 11119890 minus 09 601119890 minus 24 205119890 minus 25 274119890 minus 25 333119890 minus 40

Step 10000 Mean 47119890 + 03 613119890 + 02 562119890 + 00 507119890 + 01 0SD 11119890 + 03 172119890 + 02 187119890 + 00 134119890 + 01 0

Rosenbrock 300000 Mean 21119890 + 00 13119890 + 01 32119890 minus 01 21119890 + 01 422119890 minus 02

SD 15119890 + 00 14119890 + 01 11119890 + 00 78119890 + 00 493119890 minus 02

Griewank 50000 Mean 20119890 minus 01 729119890 minus 06 157119890 minus 08 252119890 minus 09 109119890 minus 08

SD 11119890 minus 01 105119890 minus 05 109119890 minus 07 124119890 minus 08 286119890 minus 08

Rastrigin 100000 Mean 18119890 + 02 332119890 minus 04 133119890 minus 01 243119890 + 00 0

SD 13119890 + 01 639119890 minus 04 974119890 minus 02 160119890 + 00 0

Ackley 50000 Mean 11119890 minus 01 237119890 minus 04 335119890 minus 09 381119890 minus 06 204119890 minus 08

SD 39119890 minus 02 710119890 minus 05 284119890 minus 09 826119890 minus 07 472119890 minus 09

Schwefel226 100000 Mean 59119890 + 03 170119890 minus 10 262119890 minus 04 113119890 minus 08 minus194119890 minus 12

SD 11119890 + 03 171119890 minus 10 359119890 minus 04 108119890 minus 08 461119890 minus 13

Table 6 Comparison among NABC FIPS HPSO-TVAC CLPSO FPSO and OLPSO-G on optimizing 8 benchmark functions with 200000FEs

Function FIPS HPSO-TVAC CLPSO FPSO OLPSO-G NABC

Sphere Mean 242119890 minus 13 283119890 minus 33 158119890 minus 12 240119890 minus 16 412119890 minus 54 545119890 minus 75

SD 173119890 minus 13 319119890 minus 33 770119890 minus 13 200119890 minus 31 634119890 minus 54 355119890 minus 75

Schwefel222 Mean 276119890 minus 08 903119890 minus 20 251119890 minus 08 158119890 minus 11 985119890 minus 30 720119890 minus 40

SD 904119890 minus 09 958119890 minus 20 584119890 minus 09 103119890 minus 22 101119890 minus 29 333119890 minus 40

Step Mean 0 0 0 0 0 0SD 0 0 0 0 0 0

Rosenbrock Mean 251119890 + 01 239119890 + 01 113119890 + 01 281119890 + 01 215119890 + 01 633119890 minus 02

SD 510119890 minus 01 265119890 + 01 985119890 minus 00 231119890 + 02 299119890 + 01 634119890 minus 02

Griewank Mean 901119890 minus 12 975119890 minus 03 902119890 minus 09 147119890 minus 03 483119890 minus 03 0

SD 184119890 minus 11 833119890 minus 03 857119890 minus 09 128119890 minus 05 863119890 minus 03 0

Rastrigin Mean 651119890 + 01 943119890 minus 00 909119890 minus 05 738119890 + 01 107119890 minus 00 0

SD 133119890 + 01 348119890 minus 00 125119890 minus 04 370119890 + 02 992119890 minus 01 0

Ackley Mean 233119890 minus 07 729119890 minus 14 366119890 minus 07 217119890 minus 09 798119890 minus 15 278119890 minus 14

SD 719119890 minus 08 300119890 minus 14 757119890 minus 08 171119890 minus 18 203119890 minus 15 259119890 minus 15

Schwefel226 Mean 993119890 + 02 159119890 + 03 382119890 minus 04 134119890 + 03 384119890 + 02 minus243119890 minus 12

SD 509119890 + 02 326119890 + 02 128119890 minus 05 277119890 + 02 217119890 + 02 872119890 minus 13

Mathematical Problems in Engineering 9

Table 7 Execution time of ABC and NABC algorithm with119863 = 30 60 100 (second)

Dim Algorithm 1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918

1198919

11989110

30 ABC 19850 25535 33363 18768 3969 32338 40942 78680 94332 47626NABC 17661 25697 36352 20056 2768 30612 18229 31331 77497 42660

60 ABC 43236 72503 104475 47454 12304 81891 185687 259944 322573 205303NABC 44900 73449 108912 47991 8202 80281 95126 129497 265051 186870

100 ABC 89402 163012 255115 92929 37074 180789 371414 574304 841482 410198NABC 88507 168964 255134 91418 21081 175828 334624 308107 687463 365939

Table 8 The five benchmark knapsack problems

119891 Dim Parameter (119908 119901 119888)1198911

10 119908 = (95 4 60 32 23 72 80 62 65 46) 119901 = (55 10 47 5 4 50 8 61 85 87) 119888 = 269

1198912

20 119908 = (92 4 43 83 84 68 92 82 6 44 32 18 56 83 25 96 70 48 14 58) 119901 = (44 46 90 72 91 40 75 35 8 54 78 4077 15 61 17 75 29 75 63) 119888 = 878

1198913

4 119908 = (6 5 9 7) 119901 = (9 11 13 15) 119888 = 201198914

4 119908 = (2 4 6 7) 119901 = (6 10 12 13) 119888 = 11

1198915

15119908 = (56358531 80874050 47987304 89596240 74660482 85894345 51353496 1498459 36445204 1658986244569231 0466933 37788018 57118442 60716575) 119901 = (0125126 19330424 58500931 35029145 8228400517410810 71050142 30399487 9140294 14731285 98852504 11908322 0891140 53166295 60176397) 119888 = 375

Table 9 The optimal solutions of the five benchmark knapsack problems

119891 Optimal solution Optimal value Value of constraint1198911

(0 1 1 1 0 0 0 1 1 1) 295 01198912

(1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1) 1024 minus71198913

(1 1 0 1) 35 minus21198914

(0 1 0 1) 23 01198915

(0 0 1 0 1 0 1 1 0 1 1 1 0 1 1) 4810694 minus200392

Table 10 Best worst median mean and standard deviation values obtained by ABC and NABC

119891 Algorithm Best Worst Median Mean SD

1198911

ABC 295 295 295 295 0NDABC 295 295 295 295 0

1198912

ABC 1024 1013 1024 1021 452NDABC 1024 1024 1024 1024 0

1198913

ABC 35 35 35 35 0NDABC 35 35 35 35 0

1198914

ABC 23 23 23 23 0NDABC 23 23 23 23 0

1198915

ABC 481069 437935 475478 460729 1847NDABC 481069 435786 481069 470771 879

6 Conclusion

In this paper we develop a novel artificial bee colonyalgorithm named NABC We add the global best solutioninto the search equation to drive the new candidate solutiononly around the global best solution in order to improvethe exploitation And the search equation of the employedbees is improved to keep the exploration of algorithm Theexperimental results tested on 11 benchmark functions showthat the convergence of NABC is much faster than that of

other algorithms and the computing is more effective At thesame timewe applyNABCon solving five standardKnapsackproblems and get good optimums So it is fitted to solvemanyengineering practical problems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors are grateful to the editor and the anonymousreviewers for their valuable comments and suggestions

References

[1] K S Tang K FMan S Kwong andQ He ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal ProcessingMagazine vol 13no 6 pp 22ndash37 1996

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

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

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

[5] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 Erciyes University KayseriTurkey 2005

[6] B Basturk and D Karaboga ldquoAn artificial bee colony (ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium May 2006

[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

[10] D Karaboga B Akay and C Ozturk ldquoArtificial bee colony(ABC) optimization algorithm for training feed-forward neuralnetworksrdquo in Modeling Decisions for Artificial Intelligence vol4617 of LectureNotes in Computer Science pp 318ndash329 SpringerBerlin Germany 2007

[11] Y-F Liu and S-Y Liu ldquoA hybrid discrete artificial bee colonyalgorithm for permutation flowshop scheduling problemrdquoApplied Soft Computing Journal vol 13 no 3 pp 1459ndash14632013

[12] D Karaboga and B Basturk ldquoArtificial bee colony (ABC)optimization algorithm for solving constrained optimizationproblemsrdquo in Foundations of Fuzzy Logic and Soft Computingvol 4529 of Lecture Notes in Computer Science pp 789ndash798Springer Berlin Germany 2007

[13] G P Zhu and S Kwong ldquoGbest-guided artificial bee colonyalgorithm for numerical function optimizationrdquo Applied Math-ematics and Computation vol 217 no 7 pp 3166ndash3173 2010

[14] B Basturk and D Karaboga ldquoA modified artificial bee colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[15] W-F Gao S-Y Liu and L-L Huang ldquoEnhancing artificialbee colony algorithm using more information-based searchequationsrdquo Information Sciences vol 270 pp 112ndash133 2014

[16] D Karaboga and B Gorkemli ldquoA quick artificial bee colony(qABC) algorithm and its performance on optimization prob-lemsrdquo Applied Soft Computing vol 23 pp 227ndash238 2014

[17] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 23 pp 689ndash6942010

[18] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[19] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009

[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2008

[21] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[22] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[23] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[24] M A M de Oca T Stutzle M Birattari and M DorigoldquoFrankensteinrsquos PSO a composite particle swarm optimizationalgorithmrdquo IEEE Transactions on Evolutionary Computationvol 13 no 5 pp 1120ndash1132 2009

[25] Z-H Zhan J Zhang Y Li and Y-H Shi ldquoOrthogonal learningparticle swarm optimizationrdquo IEEE Transactions on Evolution-ary Computation vol 15 no 6 pp 832ndash847 2011

[26] D Zou L Gao S Li and J Wu ldquoSolving 0-1 knapsack problemby a novel global harmony search algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 1556ndash1564 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 2 Continued

Function Dim Best Worst Median Mean SD Significant

1198919

30 ABC 154119890 minus 06 237119890 + 02 376119890 minus 01 886119890 + 01 862119890 + 01 +NABC minus364119890 minus 12 minus182119890 minus 12 minus182119890 minus 12 minus200119890 minus 12 555119890 minus 13

60 ABC 355119890 + 02 769119890 + 02 769119890 + 02 540119890 + 02 141119890 + 02 +NABC 291119890 minus 11 364119890 minus 11 364119890 minus 11 354119890 minus 11 252119890 minus 12

100 ABC 781119890 + 02 155119890 + 03 151119890 + 03 129119890 + 03 223119890 + 02 +NABC 102119890 minus 10 116119890 minus 10 109119890 minus 10 110119890 minus 10 458119890 minus 12

11989110

30 ABC 536119890 minus 10 123119890 minus 08 367119890 minus 09 145119890 minus 09 237119890 minus 09 +NABC 222119890 minus 14 293119890 minus 14 293119890 minus 14 266119890 minus 14 332119890 minus 15

60 ABC 814119890 minus 09 754119890 minus 08 538119890 minus 08 460119890 minus 08 204119890 minus 08 +NABC 506119890 minus 14 684119890 minus 14 648119890 minus 14 644119890 minus 14 526119890 minus 15

100 ABC 535119890 minus 08 428119890 minus 07 104119890 minus 07 283119890 minus 07 157119890 minus 07 +NABC 104119890 minus 13 122119890 minus 13 115119890 minus 13 116119890 minus 13 475119890 minus 15

11989111

150 ABC minus774221 minus769602 minus770460 minus771658 211119890 minus 01 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

300 ABC minus770391 minus768430 minus770391 minus769258 676119890 minus 02 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

500 ABC minus768562 minus765849 minus768562 minus767220 952119890 minus 02 +NABC minus783323 minus783323 minus783323 minus783323 145119890 minus 14

Table 3 Comparison among GABC EABC and NABC on optimizing 8 benchmark functions

Function MaxFEs Dim GABC EABC NABCMean SD Mean SD Mean SD

Sphere150000 30 137119890 minus 25 270119890 minus 25 442119890 minus 67 271119890 minus 67 110119890 minus 54 171119890 minus 54

300000 60 486119890 minus 23 514119890 minus 23 230119890 minus 64 103119890 minus 64 148119890 minus 52 137119890 minus 52

500000 100 905119890 minus 22 428119890 minus 22 637119890 minus 63 255119890 minus 63 145119890 minus 51 891119890 minus 52

Schwefel222150000 30 556119890 minus 15 979119890 minus 15 551119890 minus 35 702119890 minus 35 202119890 minus 29 110119890 minus 29

300000 60 561119890 minus 14 993119890 minus 15 508119890 minus 33 195119890 minus 34 292119890 minus 28 154119890 minus 28

500000 100 229119890 minus 13 397119890 minus 13 468119890 minus 32 142119890 minus 32 104119890 minus 27 688119890 minus 28

Schwefel221150000 30 307119890 minus 00 472119890 minus 01 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

300000 60 385119890 + 01 342119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

500000 100 704119890 + 01 120119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock150000 30 130119890 minus 00 164119890 minus 00 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

300000 60 166119890 + 01 328119890 + 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

500000 100 239119890 + 01 334119890 + 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Griewank100000 30 140119890 minus 08 116119890 minus 08 0 0 0 0150000 60 158119890 minus 06 100119890 minus 06 0 0 0 0250000 100 244119890 minus 06 250119890 minus 06 0 0 0 0

Rastrigin100000 30 708119890 minus 03 119119890 minus 03 0 0 0 0150000 60 347119890 minus 00 878119890 minus 01 0 0 0 0250000 100 978119890 minus 00 291119890 minus 00 0 0 0 0

Schwefel22650000 30 638119890 minus 00 238119890 + 01 0 0 171119890 minus 09 301119890 minus 10

100000 60 330119890 + 01 494119890 + 01 392119890 minus 11 356119890 minus 12 124119890 minus 09 502119890 minus 09

200000 100 605119890 + 01 805119890 + 01 112119890 minus 10 356119890 minus 12 122119890 minus 10 297119890 minus 11

Ackley50000 30 645119890 minus 03 268119890 minus 03 339119890 minus 10 695119890 minus 11 204119890 minus 08 472119890 minus 09

100000 60 296119890 minus 03 214119890 minus 03 181119890 minus 09 270119890 minus 10 445119890 minus 08 150119890 minus 08

150000 100 601119890 minus 02 960119890 minus 02 453119890 minus 08 123119890 minus 08 601119890 minus 07 323119890 minus 07

8 Mathematical Problems in Engineering

Table 4 Comparison among ABC EABC and NABC on optimizing 4 benchmark functions with119863 = 30 60 100

Function Dim ABC EABC NABCMean SD Mean SD Mean SD

Schwefel22130 110119890 + 01 129119890 minus 00 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

60 486119890 + 01 190119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

100 725119890 + 01 186119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock30 486119890 minus 02 484119890 minus 02 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

60 141119890 minus 01 165119890 minus 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

100 297119890 minus 01 364119890 minus 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Schwefel22630 886119890 + 01 862119890 + 01 minus123119890 minus 13 109119890 minus 13 minus200119890 minus 12 555119890 minus 13

60 540119890 + 02 141119890 + 02 291119890 minus 11 0 354119890 minus 11 252119890 minus 12

100 129119890 + 03 223119890 + 02 945119890 minus 11 0 110119890 minus 10 458119890 minus 12

Ackley30 145119890 minus 09 237119890 minus 09 136119890 minus 14 174119890 minus 15 266119890 minus 14 332119890 minus 15

60 460119890 minus 08 204119890 minus 08 449119890 minus 14 284119890 minus 15 644119890 minus 14 526119890 minus 15

100 283119890 minus 07 157119890 minus 07 953119890 minus 14 284119890 minus 15 116119890 minus 13 475119890 minus 15

Table 5 Comparison among NABC DE jDE JADE and SaDE on optimizing 8 benchmark functions with119863 = 30

Function MaxFEs DE jDE JADE SaDE NABC

Sphere 150000 Mean 98119890 minus 14 146119890 minus 28 132119890 minus 54 328119890 minus 20 110119890 minus 54

SD 84119890 minus 14 178119890 minus 28 922119890 minus 54 362119890 minus 20 171119890 minus 54

Schwefel222 200000 Mean 16119890 minus 09 902119890 minus 24 318119890 minus 25 351119890 minus 25 720119890 minus 40

SD 11119890 minus 09 601119890 minus 24 205119890 minus 25 274119890 minus 25 333119890 minus 40

Step 10000 Mean 47119890 + 03 613119890 + 02 562119890 + 00 507119890 + 01 0SD 11119890 + 03 172119890 + 02 187119890 + 00 134119890 + 01 0

Rosenbrock 300000 Mean 21119890 + 00 13119890 + 01 32119890 minus 01 21119890 + 01 422119890 minus 02

SD 15119890 + 00 14119890 + 01 11119890 + 00 78119890 + 00 493119890 minus 02

Griewank 50000 Mean 20119890 minus 01 729119890 minus 06 157119890 minus 08 252119890 minus 09 109119890 minus 08

SD 11119890 minus 01 105119890 minus 05 109119890 minus 07 124119890 minus 08 286119890 minus 08

Rastrigin 100000 Mean 18119890 + 02 332119890 minus 04 133119890 minus 01 243119890 + 00 0

SD 13119890 + 01 639119890 minus 04 974119890 minus 02 160119890 + 00 0

Ackley 50000 Mean 11119890 minus 01 237119890 minus 04 335119890 minus 09 381119890 minus 06 204119890 minus 08

SD 39119890 minus 02 710119890 minus 05 284119890 minus 09 826119890 minus 07 472119890 minus 09

Schwefel226 100000 Mean 59119890 + 03 170119890 minus 10 262119890 minus 04 113119890 minus 08 minus194119890 minus 12

SD 11119890 + 03 171119890 minus 10 359119890 minus 04 108119890 minus 08 461119890 minus 13

Table 6 Comparison among NABC FIPS HPSO-TVAC CLPSO FPSO and OLPSO-G on optimizing 8 benchmark functions with 200000FEs

Function FIPS HPSO-TVAC CLPSO FPSO OLPSO-G NABC

Sphere Mean 242119890 minus 13 283119890 minus 33 158119890 minus 12 240119890 minus 16 412119890 minus 54 545119890 minus 75

SD 173119890 minus 13 319119890 minus 33 770119890 minus 13 200119890 minus 31 634119890 minus 54 355119890 minus 75

Schwefel222 Mean 276119890 minus 08 903119890 minus 20 251119890 minus 08 158119890 minus 11 985119890 minus 30 720119890 minus 40

SD 904119890 minus 09 958119890 minus 20 584119890 minus 09 103119890 minus 22 101119890 minus 29 333119890 minus 40

Step Mean 0 0 0 0 0 0SD 0 0 0 0 0 0

Rosenbrock Mean 251119890 + 01 239119890 + 01 113119890 + 01 281119890 + 01 215119890 + 01 633119890 minus 02

SD 510119890 minus 01 265119890 + 01 985119890 minus 00 231119890 + 02 299119890 + 01 634119890 minus 02

Griewank Mean 901119890 minus 12 975119890 minus 03 902119890 minus 09 147119890 minus 03 483119890 minus 03 0

SD 184119890 minus 11 833119890 minus 03 857119890 minus 09 128119890 minus 05 863119890 minus 03 0

Rastrigin Mean 651119890 + 01 943119890 minus 00 909119890 minus 05 738119890 + 01 107119890 minus 00 0

SD 133119890 + 01 348119890 minus 00 125119890 minus 04 370119890 + 02 992119890 minus 01 0

Ackley Mean 233119890 minus 07 729119890 minus 14 366119890 minus 07 217119890 minus 09 798119890 minus 15 278119890 minus 14

SD 719119890 minus 08 300119890 minus 14 757119890 minus 08 171119890 minus 18 203119890 minus 15 259119890 minus 15

Schwefel226 Mean 993119890 + 02 159119890 + 03 382119890 minus 04 134119890 + 03 384119890 + 02 minus243119890 minus 12

SD 509119890 + 02 326119890 + 02 128119890 minus 05 277119890 + 02 217119890 + 02 872119890 minus 13

Mathematical Problems in Engineering 9

Table 7 Execution time of ABC and NABC algorithm with119863 = 30 60 100 (second)

Dim Algorithm 1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918

1198919

11989110

30 ABC 19850 25535 33363 18768 3969 32338 40942 78680 94332 47626NABC 17661 25697 36352 20056 2768 30612 18229 31331 77497 42660

60 ABC 43236 72503 104475 47454 12304 81891 185687 259944 322573 205303NABC 44900 73449 108912 47991 8202 80281 95126 129497 265051 186870

100 ABC 89402 163012 255115 92929 37074 180789 371414 574304 841482 410198NABC 88507 168964 255134 91418 21081 175828 334624 308107 687463 365939

Table 8 The five benchmark knapsack problems

119891 Dim Parameter (119908 119901 119888)1198911

10 119908 = (95 4 60 32 23 72 80 62 65 46) 119901 = (55 10 47 5 4 50 8 61 85 87) 119888 = 269

1198912

20 119908 = (92 4 43 83 84 68 92 82 6 44 32 18 56 83 25 96 70 48 14 58) 119901 = (44 46 90 72 91 40 75 35 8 54 78 4077 15 61 17 75 29 75 63) 119888 = 878

1198913

4 119908 = (6 5 9 7) 119901 = (9 11 13 15) 119888 = 201198914

4 119908 = (2 4 6 7) 119901 = (6 10 12 13) 119888 = 11

1198915

15119908 = (56358531 80874050 47987304 89596240 74660482 85894345 51353496 1498459 36445204 1658986244569231 0466933 37788018 57118442 60716575) 119901 = (0125126 19330424 58500931 35029145 8228400517410810 71050142 30399487 9140294 14731285 98852504 11908322 0891140 53166295 60176397) 119888 = 375

Table 9 The optimal solutions of the five benchmark knapsack problems

119891 Optimal solution Optimal value Value of constraint1198911

(0 1 1 1 0 0 0 1 1 1) 295 01198912

(1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1) 1024 minus71198913

(1 1 0 1) 35 minus21198914

(0 1 0 1) 23 01198915

(0 0 1 0 1 0 1 1 0 1 1 1 0 1 1) 4810694 minus200392

Table 10 Best worst median mean and standard deviation values obtained by ABC and NABC

119891 Algorithm Best Worst Median Mean SD

1198911

ABC 295 295 295 295 0NDABC 295 295 295 295 0

1198912

ABC 1024 1013 1024 1021 452NDABC 1024 1024 1024 1024 0

1198913

ABC 35 35 35 35 0NDABC 35 35 35 35 0

1198914

ABC 23 23 23 23 0NDABC 23 23 23 23 0

1198915

ABC 481069 437935 475478 460729 1847NDABC 481069 435786 481069 470771 879

6 Conclusion

In this paper we develop a novel artificial bee colonyalgorithm named NABC We add the global best solutioninto the search equation to drive the new candidate solutiononly around the global best solution in order to improvethe exploitation And the search equation of the employedbees is improved to keep the exploration of algorithm Theexperimental results tested on 11 benchmark functions showthat the convergence of NABC is much faster than that of

other algorithms and the computing is more effective At thesame timewe applyNABCon solving five standardKnapsackproblems and get good optimums So it is fitted to solvemanyengineering practical problems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors are grateful to the editor and the anonymousreviewers for their valuable comments and suggestions

References

[1] K S Tang K FMan S Kwong andQ He ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal ProcessingMagazine vol 13no 6 pp 22ndash37 1996

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

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

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

[5] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 Erciyes University KayseriTurkey 2005

[6] B Basturk and D Karaboga ldquoAn artificial bee colony (ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium May 2006

[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

[10] D Karaboga B Akay and C Ozturk ldquoArtificial bee colony(ABC) optimization algorithm for training feed-forward neuralnetworksrdquo in Modeling Decisions for Artificial Intelligence vol4617 of LectureNotes in Computer Science pp 318ndash329 SpringerBerlin Germany 2007

[11] Y-F Liu and S-Y Liu ldquoA hybrid discrete artificial bee colonyalgorithm for permutation flowshop scheduling problemrdquoApplied Soft Computing Journal vol 13 no 3 pp 1459ndash14632013

[12] D Karaboga and B Basturk ldquoArtificial bee colony (ABC)optimization algorithm for solving constrained optimizationproblemsrdquo in Foundations of Fuzzy Logic and Soft Computingvol 4529 of Lecture Notes in Computer Science pp 789ndash798Springer Berlin Germany 2007

[13] G P Zhu and S Kwong ldquoGbest-guided artificial bee colonyalgorithm for numerical function optimizationrdquo Applied Math-ematics and Computation vol 217 no 7 pp 3166ndash3173 2010

[14] B Basturk and D Karaboga ldquoA modified artificial bee colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[15] W-F Gao S-Y Liu and L-L Huang ldquoEnhancing artificialbee colony algorithm using more information-based searchequationsrdquo Information Sciences vol 270 pp 112ndash133 2014

[16] D Karaboga and B Gorkemli ldquoA quick artificial bee colony(qABC) algorithm and its performance on optimization prob-lemsrdquo Applied Soft Computing vol 23 pp 227ndash238 2014

[17] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 23 pp 689ndash6942010

[18] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[19] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009

[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2008

[21] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[22] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[23] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[24] M A M de Oca T Stutzle M Birattari and M DorigoldquoFrankensteinrsquos PSO a composite particle swarm optimizationalgorithmrdquo IEEE Transactions on Evolutionary Computationvol 13 no 5 pp 1120ndash1132 2009

[25] Z-H Zhan J Zhang Y Li and Y-H Shi ldquoOrthogonal learningparticle swarm optimizationrdquo IEEE Transactions on Evolution-ary Computation vol 15 no 6 pp 832ndash847 2011

[26] D Zou L Gao S Li and J Wu ldquoSolving 0-1 knapsack problemby a novel global harmony search algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 1556ndash1564 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 4 Comparison among ABC EABC and NABC on optimizing 4 benchmark functions with119863 = 30 60 100

Function Dim ABC EABC NABCMean SD Mean SD Mean SD

Schwefel22130 110119890 + 01 129119890 minus 00 646119890 minus 01 103119890 minus 01 267119890 minus 01 355119890 minus 02

60 486119890 + 01 190119890 minus 00 249119890 + 01 204119890 minus 00 271119890 + 00 314119890 minus 01

100 725119890 + 01 186119890 minus 00 619119890 + 01 124119890 minus 00 992119890 + 00 704119890 minus 01

Rosenbrock30 486119890 minus 02 484119890 minus 02 867119890 minus 02 748119890 minus 02 599119890 minus 02 564119890 minus 02

60 141119890 minus 01 165119890 minus 01 203119890 minus 01 132119890 minus 01 102119890 minus 01 995119890 minus 02

100 297119890 minus 01 364119890 minus 01 503119890 minus 01 882119890 minus 01 350119890 minus 01 328119890 minus 01

Schwefel22630 886119890 + 01 862119890 + 01 minus123119890 minus 13 109119890 minus 13 minus200119890 minus 12 555119890 minus 13

60 540119890 + 02 141119890 + 02 291119890 minus 11 0 354119890 minus 11 252119890 minus 12

100 129119890 + 03 223119890 + 02 945119890 minus 11 0 110119890 minus 10 458119890 minus 12

Ackley30 145119890 minus 09 237119890 minus 09 136119890 minus 14 174119890 minus 15 266119890 minus 14 332119890 minus 15

60 460119890 minus 08 204119890 minus 08 449119890 minus 14 284119890 minus 15 644119890 minus 14 526119890 minus 15

100 283119890 minus 07 157119890 minus 07 953119890 minus 14 284119890 minus 15 116119890 minus 13 475119890 minus 15

Table 5 Comparison among NABC DE jDE JADE and SaDE on optimizing 8 benchmark functions with119863 = 30

Function MaxFEs DE jDE JADE SaDE NABC

Sphere 150000 Mean 98119890 minus 14 146119890 minus 28 132119890 minus 54 328119890 minus 20 110119890 minus 54

SD 84119890 minus 14 178119890 minus 28 922119890 minus 54 362119890 minus 20 171119890 minus 54

Schwefel222 200000 Mean 16119890 minus 09 902119890 minus 24 318119890 minus 25 351119890 minus 25 720119890 minus 40

SD 11119890 minus 09 601119890 minus 24 205119890 minus 25 274119890 minus 25 333119890 minus 40

Step 10000 Mean 47119890 + 03 613119890 + 02 562119890 + 00 507119890 + 01 0SD 11119890 + 03 172119890 + 02 187119890 + 00 134119890 + 01 0

Rosenbrock 300000 Mean 21119890 + 00 13119890 + 01 32119890 minus 01 21119890 + 01 422119890 minus 02

SD 15119890 + 00 14119890 + 01 11119890 + 00 78119890 + 00 493119890 minus 02

Griewank 50000 Mean 20119890 minus 01 729119890 minus 06 157119890 minus 08 252119890 minus 09 109119890 minus 08

SD 11119890 minus 01 105119890 minus 05 109119890 minus 07 124119890 minus 08 286119890 minus 08

Rastrigin 100000 Mean 18119890 + 02 332119890 minus 04 133119890 minus 01 243119890 + 00 0

SD 13119890 + 01 639119890 minus 04 974119890 minus 02 160119890 + 00 0

Ackley 50000 Mean 11119890 minus 01 237119890 minus 04 335119890 minus 09 381119890 minus 06 204119890 minus 08

SD 39119890 minus 02 710119890 minus 05 284119890 minus 09 826119890 minus 07 472119890 minus 09

Schwefel226 100000 Mean 59119890 + 03 170119890 minus 10 262119890 minus 04 113119890 minus 08 minus194119890 minus 12

SD 11119890 + 03 171119890 minus 10 359119890 minus 04 108119890 minus 08 461119890 minus 13

Table 6 Comparison among NABC FIPS HPSO-TVAC CLPSO FPSO and OLPSO-G on optimizing 8 benchmark functions with 200000FEs

Function FIPS HPSO-TVAC CLPSO FPSO OLPSO-G NABC

Sphere Mean 242119890 minus 13 283119890 minus 33 158119890 minus 12 240119890 minus 16 412119890 minus 54 545119890 minus 75

SD 173119890 minus 13 319119890 minus 33 770119890 minus 13 200119890 minus 31 634119890 minus 54 355119890 minus 75

Schwefel222 Mean 276119890 minus 08 903119890 minus 20 251119890 minus 08 158119890 minus 11 985119890 minus 30 720119890 minus 40

SD 904119890 minus 09 958119890 minus 20 584119890 minus 09 103119890 minus 22 101119890 minus 29 333119890 minus 40

Step Mean 0 0 0 0 0 0SD 0 0 0 0 0 0

Rosenbrock Mean 251119890 + 01 239119890 + 01 113119890 + 01 281119890 + 01 215119890 + 01 633119890 minus 02

SD 510119890 minus 01 265119890 + 01 985119890 minus 00 231119890 + 02 299119890 + 01 634119890 minus 02

Griewank Mean 901119890 minus 12 975119890 minus 03 902119890 minus 09 147119890 minus 03 483119890 minus 03 0

SD 184119890 minus 11 833119890 minus 03 857119890 minus 09 128119890 minus 05 863119890 minus 03 0

Rastrigin Mean 651119890 + 01 943119890 minus 00 909119890 minus 05 738119890 + 01 107119890 minus 00 0

SD 133119890 + 01 348119890 minus 00 125119890 minus 04 370119890 + 02 992119890 minus 01 0

Ackley Mean 233119890 minus 07 729119890 minus 14 366119890 minus 07 217119890 minus 09 798119890 minus 15 278119890 minus 14

SD 719119890 minus 08 300119890 minus 14 757119890 minus 08 171119890 minus 18 203119890 minus 15 259119890 minus 15

Schwefel226 Mean 993119890 + 02 159119890 + 03 382119890 minus 04 134119890 + 03 384119890 + 02 minus243119890 minus 12

SD 509119890 + 02 326119890 + 02 128119890 minus 05 277119890 + 02 217119890 + 02 872119890 minus 13

Mathematical Problems in Engineering 9

Table 7 Execution time of ABC and NABC algorithm with119863 = 30 60 100 (second)

Dim Algorithm 1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918

1198919

11989110

30 ABC 19850 25535 33363 18768 3969 32338 40942 78680 94332 47626NABC 17661 25697 36352 20056 2768 30612 18229 31331 77497 42660

60 ABC 43236 72503 104475 47454 12304 81891 185687 259944 322573 205303NABC 44900 73449 108912 47991 8202 80281 95126 129497 265051 186870

100 ABC 89402 163012 255115 92929 37074 180789 371414 574304 841482 410198NABC 88507 168964 255134 91418 21081 175828 334624 308107 687463 365939

Table 8 The five benchmark knapsack problems

119891 Dim Parameter (119908 119901 119888)1198911

10 119908 = (95 4 60 32 23 72 80 62 65 46) 119901 = (55 10 47 5 4 50 8 61 85 87) 119888 = 269

1198912

20 119908 = (92 4 43 83 84 68 92 82 6 44 32 18 56 83 25 96 70 48 14 58) 119901 = (44 46 90 72 91 40 75 35 8 54 78 4077 15 61 17 75 29 75 63) 119888 = 878

1198913

4 119908 = (6 5 9 7) 119901 = (9 11 13 15) 119888 = 201198914

4 119908 = (2 4 6 7) 119901 = (6 10 12 13) 119888 = 11

1198915

15119908 = (56358531 80874050 47987304 89596240 74660482 85894345 51353496 1498459 36445204 1658986244569231 0466933 37788018 57118442 60716575) 119901 = (0125126 19330424 58500931 35029145 8228400517410810 71050142 30399487 9140294 14731285 98852504 11908322 0891140 53166295 60176397) 119888 = 375

Table 9 The optimal solutions of the five benchmark knapsack problems

119891 Optimal solution Optimal value Value of constraint1198911

(0 1 1 1 0 0 0 1 1 1) 295 01198912

(1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1) 1024 minus71198913

(1 1 0 1) 35 minus21198914

(0 1 0 1) 23 01198915

(0 0 1 0 1 0 1 1 0 1 1 1 0 1 1) 4810694 minus200392

Table 10 Best worst median mean and standard deviation values obtained by ABC and NABC

119891 Algorithm Best Worst Median Mean SD

1198911

ABC 295 295 295 295 0NDABC 295 295 295 295 0

1198912

ABC 1024 1013 1024 1021 452NDABC 1024 1024 1024 1024 0

1198913

ABC 35 35 35 35 0NDABC 35 35 35 35 0

1198914

ABC 23 23 23 23 0NDABC 23 23 23 23 0

1198915

ABC 481069 437935 475478 460729 1847NDABC 481069 435786 481069 470771 879

6 Conclusion

In this paper we develop a novel artificial bee colonyalgorithm named NABC We add the global best solutioninto the search equation to drive the new candidate solutiononly around the global best solution in order to improvethe exploitation And the search equation of the employedbees is improved to keep the exploration of algorithm Theexperimental results tested on 11 benchmark functions showthat the convergence of NABC is much faster than that of

other algorithms and the computing is more effective At thesame timewe applyNABCon solving five standardKnapsackproblems and get good optimums So it is fitted to solvemanyengineering practical problems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors are grateful to the editor and the anonymousreviewers for their valuable comments and suggestions

References

[1] K S Tang K FMan S Kwong andQ He ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal ProcessingMagazine vol 13no 6 pp 22ndash37 1996

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

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

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

[5] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 Erciyes University KayseriTurkey 2005

[6] B Basturk and D Karaboga ldquoAn artificial bee colony (ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium May 2006

[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

[10] D Karaboga B Akay and C Ozturk ldquoArtificial bee colony(ABC) optimization algorithm for training feed-forward neuralnetworksrdquo in Modeling Decisions for Artificial Intelligence vol4617 of LectureNotes in Computer Science pp 318ndash329 SpringerBerlin Germany 2007

[11] Y-F Liu and S-Y Liu ldquoA hybrid discrete artificial bee colonyalgorithm for permutation flowshop scheduling problemrdquoApplied Soft Computing Journal vol 13 no 3 pp 1459ndash14632013

[12] D Karaboga and B Basturk ldquoArtificial bee colony (ABC)optimization algorithm for solving constrained optimizationproblemsrdquo in Foundations of Fuzzy Logic and Soft Computingvol 4529 of Lecture Notes in Computer Science pp 789ndash798Springer Berlin Germany 2007

[13] G P Zhu and S Kwong ldquoGbest-guided artificial bee colonyalgorithm for numerical function optimizationrdquo Applied Math-ematics and Computation vol 217 no 7 pp 3166ndash3173 2010

[14] B Basturk and D Karaboga ldquoA modified artificial bee colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[15] W-F Gao S-Y Liu and L-L Huang ldquoEnhancing artificialbee colony algorithm using more information-based searchequationsrdquo Information Sciences vol 270 pp 112ndash133 2014

[16] D Karaboga and B Gorkemli ldquoA quick artificial bee colony(qABC) algorithm and its performance on optimization prob-lemsrdquo Applied Soft Computing vol 23 pp 227ndash238 2014

[17] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 23 pp 689ndash6942010

[18] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[19] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009

[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2008

[21] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[22] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[23] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[24] M A M de Oca T Stutzle M Birattari and M DorigoldquoFrankensteinrsquos PSO a composite particle swarm optimizationalgorithmrdquo IEEE Transactions on Evolutionary Computationvol 13 no 5 pp 1120ndash1132 2009

[25] Z-H Zhan J Zhang Y Li and Y-H Shi ldquoOrthogonal learningparticle swarm optimizationrdquo IEEE Transactions on Evolution-ary Computation vol 15 no 6 pp 832ndash847 2011

[26] D Zou L Gao S Li and J Wu ldquoSolving 0-1 knapsack problemby a novel global harmony search algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 1556ndash1564 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 7 Execution time of ABC and NABC algorithm with119863 = 30 60 100 (second)

Dim Algorithm 1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918

1198919

11989110

30 ABC 19850 25535 33363 18768 3969 32338 40942 78680 94332 47626NABC 17661 25697 36352 20056 2768 30612 18229 31331 77497 42660

60 ABC 43236 72503 104475 47454 12304 81891 185687 259944 322573 205303NABC 44900 73449 108912 47991 8202 80281 95126 129497 265051 186870

100 ABC 89402 163012 255115 92929 37074 180789 371414 574304 841482 410198NABC 88507 168964 255134 91418 21081 175828 334624 308107 687463 365939

Table 8 The five benchmark knapsack problems

119891 Dim Parameter (119908 119901 119888)1198911

10 119908 = (95 4 60 32 23 72 80 62 65 46) 119901 = (55 10 47 5 4 50 8 61 85 87) 119888 = 269

1198912

20 119908 = (92 4 43 83 84 68 92 82 6 44 32 18 56 83 25 96 70 48 14 58) 119901 = (44 46 90 72 91 40 75 35 8 54 78 4077 15 61 17 75 29 75 63) 119888 = 878

1198913

4 119908 = (6 5 9 7) 119901 = (9 11 13 15) 119888 = 201198914

4 119908 = (2 4 6 7) 119901 = (6 10 12 13) 119888 = 11

1198915

15119908 = (56358531 80874050 47987304 89596240 74660482 85894345 51353496 1498459 36445204 1658986244569231 0466933 37788018 57118442 60716575) 119901 = (0125126 19330424 58500931 35029145 8228400517410810 71050142 30399487 9140294 14731285 98852504 11908322 0891140 53166295 60176397) 119888 = 375

Table 9 The optimal solutions of the five benchmark knapsack problems

119891 Optimal solution Optimal value Value of constraint1198911

(0 1 1 1 0 0 0 1 1 1) 295 01198912

(1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1) 1024 minus71198913

(1 1 0 1) 35 minus21198914

(0 1 0 1) 23 01198915

(0 0 1 0 1 0 1 1 0 1 1 1 0 1 1) 4810694 minus200392

Table 10 Best worst median mean and standard deviation values obtained by ABC and NABC

119891 Algorithm Best Worst Median Mean SD

1198911

ABC 295 295 295 295 0NDABC 295 295 295 295 0

1198912

ABC 1024 1013 1024 1021 452NDABC 1024 1024 1024 1024 0

1198913

ABC 35 35 35 35 0NDABC 35 35 35 35 0

1198914

ABC 23 23 23 23 0NDABC 23 23 23 23 0

1198915

ABC 481069 437935 475478 460729 1847NDABC 481069 435786 481069 470771 879

6 Conclusion

In this paper we develop a novel artificial bee colonyalgorithm named NABC We add the global best solutioninto the search equation to drive the new candidate solutiononly around the global best solution in order to improvethe exploitation And the search equation of the employedbees is improved to keep the exploration of algorithm Theexperimental results tested on 11 benchmark functions showthat the convergence of NABC is much faster than that of

other algorithms and the computing is more effective At thesame timewe applyNABCon solving five standardKnapsackproblems and get good optimums So it is fitted to solvemanyengineering practical problems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 Mathematical Problems in Engineering

Acknowledgments

The authors are grateful to the editor and the anonymousreviewers for their valuable comments and suggestions

References

[1] K S Tang K FMan S Kwong andQ He ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal ProcessingMagazine vol 13no 6 pp 22ndash37 1996

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

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

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

[5] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 Erciyes University KayseriTurkey 2005

[6] B Basturk and D Karaboga ldquoAn artificial bee colony (ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium May 2006

[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

[10] D Karaboga B Akay and C Ozturk ldquoArtificial bee colony(ABC) optimization algorithm for training feed-forward neuralnetworksrdquo in Modeling Decisions for Artificial Intelligence vol4617 of LectureNotes in Computer Science pp 318ndash329 SpringerBerlin Germany 2007

[11] Y-F Liu and S-Y Liu ldquoA hybrid discrete artificial bee colonyalgorithm for permutation flowshop scheduling problemrdquoApplied Soft Computing Journal vol 13 no 3 pp 1459ndash14632013

[12] D Karaboga and B Basturk ldquoArtificial bee colony (ABC)optimization algorithm for solving constrained optimizationproblemsrdquo in Foundations of Fuzzy Logic and Soft Computingvol 4529 of Lecture Notes in Computer Science pp 789ndash798Springer Berlin Germany 2007

[13] G P Zhu and S Kwong ldquoGbest-guided artificial bee colonyalgorithm for numerical function optimizationrdquo Applied Math-ematics and Computation vol 217 no 7 pp 3166ndash3173 2010

[14] B Basturk and D Karaboga ldquoA modified artificial bee colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[15] W-F Gao S-Y Liu and L-L Huang ldquoEnhancing artificialbee colony algorithm using more information-based searchequationsrdquo Information Sciences vol 270 pp 112ndash133 2014

[16] D Karaboga and B Gorkemli ldquoA quick artificial bee colony(qABC) algorithm and its performance on optimization prob-lemsrdquo Applied Soft Computing vol 23 pp 227ndash238 2014

[17] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 23 pp 689ndash6942010

[18] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[19] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009

[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2008

[21] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[22] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[23] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[24] M A M de Oca T Stutzle M Birattari and M DorigoldquoFrankensteinrsquos PSO a composite particle swarm optimizationalgorithmrdquo IEEE Transactions on Evolutionary Computationvol 13 no 5 pp 1120ndash1132 2009

[25] Z-H Zhan J Zhang Y Li and Y-H Shi ldquoOrthogonal learningparticle swarm optimizationrdquo IEEE Transactions on Evolution-ary Computation vol 15 no 6 pp 832ndash847 2011

[26] D Zou L Gao S Li and J Wu ldquoSolving 0-1 knapsack problemby a novel global harmony search algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 1556ndash1564 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Acknowledgments

The authors are grateful to the editor and the anonymousreviewers for their valuable comments and suggestions

References

[1] K S Tang K FMan S Kwong andQ He ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal ProcessingMagazine vol 13no 6 pp 22ndash37 1996

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

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

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

[5] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 Erciyes University KayseriTurkey 2005

[6] B Basturk and D Karaboga ldquoAn artificial bee colony (ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium May 2006

[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[8] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

[10] D Karaboga B Akay and C Ozturk ldquoArtificial bee colony(ABC) optimization algorithm for training feed-forward neuralnetworksrdquo in Modeling Decisions for Artificial Intelligence vol4617 of LectureNotes in Computer Science pp 318ndash329 SpringerBerlin Germany 2007

[11] Y-F Liu and S-Y Liu ldquoA hybrid discrete artificial bee colonyalgorithm for permutation flowshop scheduling problemrdquoApplied Soft Computing Journal vol 13 no 3 pp 1459ndash14632013

[12] D Karaboga and B Basturk ldquoArtificial bee colony (ABC)optimization algorithm for solving constrained optimizationproblemsrdquo in Foundations of Fuzzy Logic and Soft Computingvol 4529 of Lecture Notes in Computer Science pp 789ndash798Springer Berlin Germany 2007

[13] G P Zhu and S Kwong ldquoGbest-guided artificial bee colonyalgorithm for numerical function optimizationrdquo Applied Math-ematics and Computation vol 217 no 7 pp 3166ndash3173 2010

[14] B Basturk and D Karaboga ldquoA modified artificial bee colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[15] W-F Gao S-Y Liu and L-L Huang ldquoEnhancing artificialbee colony algorithm using more information-based searchequationsrdquo Information Sciences vol 270 pp 112ndash133 2014

[16] D Karaboga and B Gorkemli ldquoA quick artificial bee colony(qABC) algorithm and its performance on optimization prob-lemsrdquo Applied Soft Computing vol 23 pp 227ndash238 2014

[17] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 23 pp 689ndash6942010

[18] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006

[19] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009

[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2008

[21] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[22] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[23] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[24] M A M de Oca T Stutzle M Birattari and M DorigoldquoFrankensteinrsquos PSO a composite particle swarm optimizationalgorithmrdquo IEEE Transactions on Evolutionary Computationvol 13 no 5 pp 1120ndash1132 2009

[25] Z-H Zhan J Zhang Y Li and Y-H Shi ldquoOrthogonal learningparticle swarm optimizationrdquo IEEE Transactions on Evolution-ary Computation vol 15 no 6 pp 832ndash847 2011

[26] D Zou L Gao S Li and J Wu ldquoSolving 0-1 knapsack problemby a novel global harmony search algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 1556ndash1564 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of