Research ArticleWolf Pack Algorithm for Unconstrained Global Optimization

Hu-Sheng Wu12 and Feng-Ming Zhang1

1 Materiel Management and Safety Engineering Institute Air Force Engineering University Xirsquoan 710051 China2Materiel Engineering Institute Armed Police Force Engineering University Xirsquoan 710086 China

Correspondence should be addressed to Hu-Sheng Wu wuhusheng0421gmailcom

Received 28 June 2013 Revised 13 January 2014 Accepted 27 January 2014 Published 9 March 2014

Academic Editor Orwa Jaber Housheya

Copyright copy 2014 H-S Wu and F-M Zhang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The wolf pack unites and cooperates closely to hunt for the prey in the Tibetan Plateau which shows wonderful skills and amazingstrategies Inspired by their prey hunting behaviors and distribution mode we abstracted three intelligent behaviors scoutingcalling and besieging and two intelligent rules winner-take-all generation rule of lead wolf and stronger-survive renewing ruleof wolf pack Then we proposed a new heuristic swarm intelligent method named wolf pack algorithm (WPA) Experimentsare conducted on a suit of benchmark functions with different characteristics unimodalmultimodal separablenonseparableand the impact of several distance measurements and parameters on WPA is discussed What is more the compared simulationexperiments with other five typical intelligent algorithms genetic algorithm particle swarm optimization algorithm artificial fishswarm algorithm artificial bee colony algorithm and firefly algorithm show that WPA has better convergence and robustnessespecially for high-dimensional functions

1 Introduction

Global optimization is a hot topic with applications in manyareas such as science economy and engineering Generallyunconstrained global optimization problems can be formu-lated as follows

min or max119891 (119883) 119883 = (1199091 1199092 119909

119899) (1)

where 119891 119877119899 rarr 119877 is a real-valued objective function 119883 isin119877119899 and 119899 is the number of parameters to be optimized

As many real-world problems are becoming increasinglycomplex global optimization especially using traditionalmethods is becoming a challenging task [1] Because ofits great search space high-dimensional global optimizationproblems aremore difficult [2] Fortunately many algorithmsinspired by nature have become powerful tools for theseproblems [3ndash5] Since with long time of biological evolutionand natural selection there are many marvelous swarmintelligence phenomenons in nature which are wonderfuland can give us endless inspiration The remarkable swarmbehavior of animals such as swarming ants schooling fishand flocking birds has for long captivated the attention ofnaturalists and scientists [6] People have developed many

intelligent optimization methods to solve complex globalproblems in recent decades In 1995 inspired by social behav-ior and movement dynamics of birds Kennedy proposedthe particle swarm optimization algorithm (PSO) [7] In1996 inspired by social division and foraging behavior ofant colonies Dorigo proposed the ant colony optimizationalgorithm (ACO) [8] In 2002 inspired by foraging behaviorof fish schools Li proposed the artificial fish swarm algorithm(AFSA) [9] In 2005 motivated by the intelligent foragingbehavior of honeybee swarms Karaboga proposed the arti-ficial bee colony (ABC) algorithm [10] In 2008 based onthe flashing behavior of fireflies Doctor Yang proposed fireflyalgorithm (FA) [11] Researchers even give some conceptionsof swarm intelligent algorithms such as rats herds algorithmmosquito swarms algorithm and dolphins herds algorithm[12] Birds fishes ants and bees do not have any humancomplex intelligence such as logical reasoning and syntheticjudgment but under the same aim food they stand outpowerful swarm intelligence through constantly adaptingenvironment and mutual cooperation which give us manynew ideas for solving complex problems

The wolf pack is marvelous Harsh living environmentand constant evolution for centuries have created their

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 465082 17 pageshttpdxdoiorg1011552014465082

2 Mathematical Problems in Engineering

rigorous organization system and subtle hunting behaviorWolves tactics of Mongolia cavalry in Genghis Khan periodsubmarine tactics of Nazi Admiral Doenitz in World WarII and US military wolves attack system for electroniccountermeasures all highlight great charm of their swarmintelligence [13] proposes a wolf colony algorithm (WCA)to solve the optimization problem But the accuracy andefficiency of WCA are not good enough and easily fall intolocal optima especially for high-dimensional functions Soin this paper we reanalyzed collaborative predation behaviorand prey distribution mode of wolves and proposed a newswarm intelligence algorithm called wolf pack algorithm(WPA) Moreover the efficiency and robustness of the newalgorithm were tested by compared experiments

The remainder of this paper is structured as followsIn Section 2 the predation behaviors and prey distributionof wolves are analyzed In Section 3 WPA is describedSection 4 describes the experimental setup followed byexperimental results and analysis Finally conclusion andfuture work are presented in Section 5

2 System Analyzing of Wolf Pack

Wolves are gregarious animals and have clearly social workdivision There is a lead wolf some elite wolves act as scoutsand some ferociouswolves in awolf packThey cooperatewellwith each other and take their respective responsibility for thesurvival and thriving of wolf pack

Firstly the lead wolf as a leader under the law of thejungle is always the smartest and most ferocious one Itis responsible for commanding the wolves and constantlymaking decision by evaluating surrounding situation andperceiving information from other wolves These can avoidthe wolves in danger and command the wolves to smoothlycapture prey as soon as possible

Secondly the lead wolf sends some elite wolves to huntaround and look for prey in the probable scope Those elitewolves are scoutsTheywalk around and independentlymakedecision according to the concentration of smell left by preyand higher concentration means the prey is closer to thewolves So they always move towards the direction of gettingstronger smell

Thirdly once a scout wolf finds the trace of prey itwill howl and report that to lead wolf Then the lead wolfwill evaluate this situation and make a decision whether tosummon the ferocious wolves to round up the prey or notIf they are summoned the ferocious wolves will move fasttowards the direction of the scout wolf

Fourthly after capturing the prey the prey is not dis-tributed equitably but in an order from the strong to theweakThat is to say that the stronger the wolf is the more thefood it will get is Although this distribution rule will makesome weak wolf dead for lack of food it makes sure that thewolves that have the ability to capture prey getmore food so asto keep being strong and can capture more prey successfullyin the next timeThe rule avoids that thewhole pack starves todeath and ensures its continuance and proliferating In whatfollows the author made detailed description and realizationfor the above intelligent behaviors and rules

3 Wolf Pack Algorithm

31 Some Definitions If the predatory space of the artificialwolves is a 119873times119863 Euclidean space 119873 is the number of wolves119863 is the number of variables The position of one wolf 119894 isa vector X

119894= (1199091198941

1199091198942

119909119894119863

) and 119909119894119889is the 119889th variable

value of the 119894th artificial wolf 119884 = 119891(X) represents theconcentration of preyrsquos smell perceived by artificial wolveswhich is also the objective function value

The distance between two wolves 119901 and 119902 is describedas 119871(119901 119902) Several distance measurements can be selectedaccording to specific problems For example hamming dis-tance can be used in WPA for 0-1 discrete optimizationwhileManhattan distance (MD) and Euclidean distance (ED)can be used in WPA for continuous numerical functionoptimization In this paper we mainly discuss the latterproblem and the selection of distance measurements will bediscussed in Section 421 Moreover because the problemsof maximum value and minimal value can convert to eachother only the maximum value problem is discussed in whatfollows

32 The Description of Intelligent Behaviors and Rules Thecooperation between lead wolf scout wolves and ferociouswolves makes nearly perfect predation while prey distribu-tion from the strong to the weak makes the wolf pack thrivestowards the direction of the prey that it most probably can beable to capture The whole predation behavior of wolf pack isabstracted three intelligent behaviors scouting calling andbesieging behavior and two intelligent rules winner-take-all generating rule for the lead wolf and the stronger-surviverenewing rule for the wolf pack

(1) The winner-take-all generating rule for the lead wolfthe artificial wolf with the best objective function value is leadwolf During each iteration compare the function value of thelead wolf with the best one of other wolves if the value oflead wolf is not better it will be replaced Then the best wolfbecomes lead wolf Rather than acting the three intelligentbehaviors the lead wolf directly goes into the next iterationuntil it is replaced by other better wolf

(2) Scouting behavior S num elite wolves except the leadwolf are considered as the scout wolves they search thesolution in predatory space 119884

119894is the concentration of prey

smell perceived by the scout wolf 119894 119884lead is the concentrationof prey smell perceived by the lead wolf

If 119884119894

gt 119884lead that means the scout wolf is nearer to theprey and probably captures prey so the scout wolf 119894 becomeslead wolf and 119884lead = 119884

119894

If 119884119894

lt 119884lead the scout wolf 119894 respectively takes asteptowards ℎ different directions the step length is 119904119905119890119901

119886 After

taking a step towards the 119901th direction the state of the scoutwolf 119894 is formulated below

119909119901

119894119889= 119909119894119889

+ sin(2120587 times119901

ℎ) times step119889

119886 119901 = 1 2 ℎ (2)

It should be noted that ℎ is different for each wolf becauseof their different seeking ways So ℎ is randomly selected in[ℎmin ℎmax] and it must be an integer 119884

1198940is the concentration

of prey smell perceived by the scout wolf 119894 and 119884119894119901represents

Mathematical Problems in Engineering 3

the one after it took a step towards the 119901th direction Ifmax119884

1198941 1198841198942

119884119894ℎ

gt 1198841198940 the wolf 119894 steps forward and its

position 119883119894is updated Then repeat the above until 119884

119894gt 119884lead

or the maximum number of repetitions 119879max is reached(3)Calling behavior the lead wolf will howl and summon

119872 119899119906119898 ferocious wolves to gather around the prey Here theposition of the lead wolf is considered as the one of the preyso that the ferocious wolves aggregate towards the position ofleadwolf 119904119905119890119901

119887is the step length119892119896

119889is the position of artificial

lead wolf in the 119889th variable space at the 119896th iteration Theposition of the ferocious wolf 119894 in the 119896th iterative calculationis updated according to the following equation

119909119896+1119894119889

= 119909119896119894119889

+ step119889119887

sdot(119892119896119889

minus 119909119896119894119889

)1003816100381610038161003816119892119896

119889minus 119909119896119894119889

1003816100381610038161003816 (3)

This formula consists of two parts the former is thecurrent position of wolf 119894 which represents the foundationfor prey hunting the latter represents the aggregate tendencyof other wolves towards the lead wolf which shows the leadwolf rsquos leadership to the wolf pack

If 119884119894

gt 119884lead the ferocious wolf 119894 becomes lead wolfand 119884lead = 119884

119894 then the wolf 119894 takes the calling behavior If

119884119894

lt 119884lead the ferocious wolf 119894 keeps on aggregating towardsthe lead wolf with a fast speed until 119871(119894 119897) lt 119871near the wolftakes besieging behavior 119871(119894 119897) is the distance between thewolf 119894 and the lead wolf 119897 119871near is the distance determinantcoefficient as a judging condition which determine whetherwolf 119894 changes state from aggregating towards the lead wolfto besieging behavior The different value of 119871near will affectalgorithmic convergence rate There will be a discussion inSection 422

Calling behavior shows information transferring andsharing mechanism in wolf pack and blends the idea of socialcognition

(4) Besieging behavior after large-steps running towardsthe lead wolf the wolves are close to the prey then all wolvesexcept the leadwolf will take besieging behavior for capturingprey Now the position of lead wolf is considered as theposition of prey In particular 119866119896

119889reprensents the position of

prey in the119889th variable space at the 119896th iterationThepositionof wolf 119894 is updated according to the following equation

119909119896+1119894119889

= 119909119896119894119889

+ 120582 sdot step119889119888

sdot10038161003816100381610038161003816119866119896

119889minus 119909119896119894119889

10038161003816100381610038161003816 (4)

120582 is a random number uniformly distributed at theinterval [minus1 1] 119904119905119890119901

119888is the step length of wolf 119894 when it

takes besieging behavior1198841198940is the concentration of prey smell

perceived by the wolf 119894 and 119884119894119896represents the one after it

took this behavior If 1198841198940

lt 119884119894119896 the position X

119894is updated

otherwise it not changedThere are 119904119905119890119901

119886 119904119905119890119901119887 and 119904119905119890119901

119888in the three intelligent

behaviors and the three-step length in 119889th variable spaceshould have the following relationship

step119889119886

=step119889119887

2= 2 sdot step119889

119888= 119878 (5)

119878 is step coefficient and represents the fineness degree ofartificial wolf searching for prey in resolution space

(5) The stronger-survive renewing rule for the wolf packthe prey is distributed from the strong to the weak which willresult in some weak wolves deadThe algorithm will generate119877 wolves while deleting 119877 wolves with bad objective functionvalues Specifically with the help of the lead wolf rsquos huntingexperience in the 119889th variable space position of the 119894th oneof 119877 wolves is defined as follows

119909119894119889

= 119892119889

sdot rand 119894 = 1 2 119877 (6)

119892119889is the position of artificial lead wolf in the 119889th variable

space rand is a random number uniformly distributed at theinterval [minus01 01]

When the value of 119877 is larger it is better for sustainingwolf rsquos diversity and making the algorithm have the abilityto open up new resolution space But if 119877 is too large thealgorithm will nearly be a random search approach Becausethe number and scale of prey captured by wolves are differentin natural word which will lead to different number ofweak wolf dead 119877 is an integer and randomly selected atthe interval [119899(2 lowast 120573) 119899120573] 120573 is the population renewingproportional coefficient

33 Algorithm Description As described in the previoussection WPA has three artificial intelligent behaviors andtwo intelligent rules There are scouting behavior callingbehavior and besieging behavior and winner-take-all rule forgenerating lead wolf and the stronger-survive renewing rulefor wolf pack

Firstly the scouting behavior accelerates the possibilitythat WPA can fully traverse the solution space Secondly thewinner-take-all rule for generating lead wolf and the callingbehavior make the wolves move towards the lead wolf whoseposition is the nearest to the prey and most likely capturingpreyThe winner-take-all rule and calling behavior also makewolves arrive at the neighborhood of the global optimumonly after a few iterations elapsed since the step of wolvesin calling behavior is the largest one Thirdly with a smallstep step

119888 besieging behavior makes WPA algorithm have

the ability to open up new solution space and carefully searchthe global optima in good solution area Fourthly with thehelp of stronger-survive renewing rule for the wolf pack thealgorithm can get several new wolves whose positions arenear the best wolf lead wolf which allows for more latitudeof search space to anchor the global optimum while keepingpopulation diversity in each iteration

All the abovemakeWPA possesses superior performancein accuracy and robustness which will be seen in Section 4

Having discussed all the components ofWPA the impor-tant computation steps are detailed below

Step 1 (initialization) Initialize the following parametersthe initial position of artificial wolf 119894 (X

119894) the number of

the wolves (119873) the maximum number of iterations (119896max)the step coefficient (119878) the distance determinant coefficient(119871near) the maximum number of repetitions in scoutingbehavior (119879max) and the population renewing proportionalcoefficient (120573)

4 Mathematical Problems in Engineering

Table 1 Benchmark functions in experiments

No Functions Formulation Global extremum 119863 C Range1 Rosenbrock 119891() = 100(119909

2minus 11990921)2 + (1 minus 119909

1)2 119891min() = 0 2 UN (minus2048 2048)

2 Colville119891() = 100(1199092

1minus 1199092)2

+ (1199091

minus 1)2

+ (1199093

minus 1)2

+ 90(11990923

minus 1199094)2

+ 101(1199092

minus 1)2

+ (1199094

minus 1)2

+ 198(1199092

minus 1)(1199094

minus 1)119891min() = 0 4 UN (minus10 10)

3 Sphere 119891 () =119863

sum119894=1

1199092119894

119891min() = 0 200 US (minus100 100)

4 Sumsquares 119891 () =119863

sum119894=1

1198941199092119894

119891min() = 0 150 US (minus10 10)

5 Booth 119891() = (1199091

+ 21199092

minus 7)2 + (21199091

+ 1199092

minus 5)2 119891min() = 0 2 MS (minus10 10)

6 Bridge 119891 () =sinradic1199092

1+ 11990922

radic11990921

+ 11990922

+ exp(cos 2120587119909

1+ cos 2120587119909

2

2) minus 07129 119891max() = 30054 2 MN (minus15 15)

7 Ackley 119891() = minus20 exp(minus02radic1

119863

119863

sum119894=1

1199092119894) minus exp(

1

119863

119863

sum119894=1

cos 2120587119909119894) + 20 + 119890 119891min() = 0 50 MN (minus32 32)

8 Griewank 119891() =1

4000

119863

sum119894=1

1199092119894

minus119863

prod119894=1

cos(119909119894

radic119894) + 1 119891min() = 0 100 MN (minus600 600)

119863 dimension C characteristic U unimodal M multimodal S separable N nonseparable

Step 2 The wolf with best function value is considered aslead wolf In practical computation 119878 num = 119872 num =119899 minus 1 which means that wolves except for lead wolf actwith different behavior as different status So here exceptfor lead wolf according to formula (2) the rest of the 119899 minus 1wolves firstly act as the artificial scout wolves to take scoutingbehavior until 119884

119894gt 119884lead or the maximum number of

repetition 119879max is reached and then go to Step 3

Step 3 Except for the lead wolf the rest of the 119899 minus 1 wolvessecondly act as the artificial ferocious wolves and gathertowards the lead wolf according to (3) 119884

119894is the smell

concentration of prey perceived by wolf 119894 if 119884119894

ge 119884lead go toStep 2 otherwise the wolf 119894 continues running until 119871(119894 119897) le119871near then go to Step 4

Step 4 The position of artificial wolves who take besiegingbehavior is updated according to (4)

Step 5 Update the position of lead wolf under the winner-take-all generating rule and update the wolf pack under thepopulation renewing rule according to (6)

Step 6 If the program reaches the precision requirement orthemaximumnumber of iterations the position and functionvalue of lead wolf the problem optimal solution will beoutputted otherwise go to Step 2

So the flow chart of WPA can be shown as Figure 1

4 Experimental Results

The ingredients of the WPA method have been describedin Section 3 In this section the design of experimentsis explained sensitivity analysis of parameters on WPAis explored and the empirical results are reported which

Initialization

Scouting behavior

Yi gt Ylead

Yi gt Ylead

orT gt Tmax

Calling behavior

L(i l) gt Lnear

Besieging behavior

Renew the position of lead wolf

Renew wolf pack

Terminate

Output resultsYes

Yes

Yes

Yes

No

No

No

No

Figure 1 The flow chart of WPA

compare the WPA approach with those of GA PSO ASFAABC and FA

41 Design of the Experiments

411 Benchmark Functions In order to evaluate the perfor-mance of these algorithms eight classical benchmark func-tions are presented inTable 1Though only eight functions areused in this test they are enough to include some differentkinds of problems such as unimodal multimodal regularirregular separable nonseparable and multidimensional

If a function has more than one local optimum thisfunction is calledmultimodalMultimodal functions are usedto test the ability of algorithms to get rid of local minima

Mathematical Problems in Engineering 5

Table 2 The list of various methods used in the paper

Method Authors and referencesGenetic algorithm (GA) Goldberg [14]Particle swarm optimization algorithm(PSO) Kennedy and Eberhart [7]

Artificial fish school algorithm (ASFA) Li et al [9]Artificial bee colony algorithm (ABC) Karaboga [10]Firefly algorithm (FA) Yang [11]

Another group of test problems is separable or nonseparablefunctions A 119901-variable separable function can be expressedas the sum of 119901 functions of one variable such as Sumsquaresand Rastrigin Nonseparable functions cannot be written inthis form such as Bridge Rosenbrock Ackley andGriewankBecause nonseparable functions have interrelation amongtheir variable these functions are more difficult than theseparable functions

In Table 1 characteristics of each function are given underthe column titled 119862 In this column 119872 means that thefunction is multimodal while 119880 means that the functionis unimodal If the function is separable abbreviation 119878 isused to indicate this specification Letter 119873 refers to that thefunction is nonseparable As seen from Table 1 4 functionsare multimodal 4 functions are unimodal 3 functions areseparable and 5 functions are nonseparable

The variety of functions forms and dimensions makeit possible to fairly assess the robustness of the proposedalgorithms within limit iteration Many of these functionsallow a choice of dimension and an input dimension rangingfrom 2 to 200 for test functions is given Dimensions of theproblems that we used can be found under the column titled119863 Besides initial ranges formulas and global optimumvalues of these functions are also given in Table 1

412 Experimental Settings In this subsection experimentalsettings are given Firstly in order to fully compare the perfor-mance of different algorithms we take the simulation underthe same situation So the values of the common parametersused in each algorithm such as population size and evaluationnumber were chosen to be the same Population size was100 and the maximum evaluation number was 2000 forall algorithms on all functions Additionally we follow theparameter settings in the original paper of GA PSO AFSAABC and FA see Table 2

For each experiment 50 independent runs were con-ducted with different initial random seeds To evaluate theperformance of these algorithms six criteria are given inTable 3

Accelerating convergence speed and avoiding the localoptima have become two important and appealing goals inswarm intelligent search algorithms So as seen in Table 3we adopted criteria best mean and standard deviation toevaluate efficiency and accuracy of algorithms and adoptedcriteria Art Worst and SR to evaluate convergence speedeffectiveness and robustness of six algorithms

Table 3 Six criteria and their abbreviations

Criteria AbbreviationThe best value of optima found in 50 runs BestThe worst value of optima found in 50 runs WorstThe average value of optima found in 50 runs MeanThe standard deviations StdDevThe success rate of the results SRThe average reaching time Art

Specifically speaking SR provides very useful informa-tion about how stable an algorithm is Success is claimed ifan algorithm successfully gets a solution below a prespecifiedthreshold value with the maximum number of functionevaluations [15] So to calculate the success rate an erroraccuracy level 120576 = 10minus6 must be set (120576 = 10minus6 also usedin [16]) Thus we compared the result 119865 with the knownanalytical optima 119865lowast and consider 119865 to be ldquosuccessfulrdquo if thefollowing inequality holds

1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816

119865lowastlt 120576 119865lowast = 0

1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816 lt 120576 119865lowast = 0

(7)

The SR is a percentage value that is calculated as

SR =successful runs

runs (8)

Art is the average value of time once an algorithm gets asolution satisfying the formula (7) in 50-run computationsArt also provides very useful information about how fastan algorithm converges to certain accuracy or under thesame termination criterion which has important practicalsignificance

All algorithms have been tested in Matlab 2008a over thesame Lenovo A4600R computer with a Dual-Core 260GHzprocessor running Windows XP operating system over199Gb of memory

42 Experiments 1 Effect of Distance Measurements and FourParameters on WPA In order to study the effect of twodistance measures and four parameters on WPA differentmeasures and values of parameters were tested on typicalfunctions listed in Table 1 Each experiment WPA algorithmthat runs 50 times on each function and several criteriadescribed in Section 412 are used The experiment is con-ducted with the original coefficients shown in Table 9

421 Effect of Distance Measurements on the Performance ofWPA This subsection will investigate the performance ofdifferent distance measurements using functions with dif-ferent characteristics As is known to all Euclidean distance(ED) and Manhattan distance (MD) are the two most com-mon distance metrics in practical continuous optimizationIn the proposedWPA MD or ED can be adopted to measurethe distance between two wolves in the candidate solution

6 Mathematical Problems in Engineering

Table 4 Sensitivity analysis of distance measurements

Function Global extremum 119863 Distance Best Worst Mean StdDev SR Arts

Rosenbrock 119891min() = 0 2 MD 921119890 minus 11 324119890 minus 8 112119890 minus 8 118119890 minus 8 100 105165ED 426119890 minus 9 271119890 minus 7 127119890 minus 7 681119890 minus 8 100 371053

Colville 119891min() = 0 4 MD 562119890 minus 8 528119890 minus 7 249119890 minus 7 223119890 minus 7 100 468619ED 174119890 minus 7 170119890 minus 6 574119890 minus 7 370119890 minus 7 90 683220

Sphere 119891min() = 0 200 MD 320119890 minus 161 329119890 minus 144 207119890 minus 145 749119890 minus 145 100 115494ED 176119890 minus 160 336119890 minus 143 168119890 minus 144 751119890 minus 144 100 116825

Sumsquares 119891min() = 0 150 MD 156119890 minus 161 309119890 minus 144 179119890 minus 145 695119890 minus 145 100 85565ED 397119890 minus 160 224119890 minus 144 113119890 minus 145 500119890 minus 145 100 87109

Booth 119891min() = 0 2 MD 563119890 minus 12 115119890 minus 10 419119890 minus 11 332119890 minus 11 100 111074ED 108119890 minus 9 264119890 minus 8 116119890 minus 8 693119890 minus 9 100 405546

Bridge 119891max() = 30054 2 MD 30054 30054 30054 456119890 minus 16 100 11093ED 30054 30054 30054 456119890 minus 16 100 19541

Ackley 119891min() = 0 50 MD 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 193648ED 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 436884

Griewank 119891min() = 0 100 MD 0 01507 301119890 minus 3 00213 98 gt8771198903

ED 0 08350 00167 01181 92 gt135119890 + 4

space Therefore a discussion about their impacts on theperformance of WPA is needed

There are two wolves X119901

= (1199091199011

1199091199012

119909119901119863

) is theposition of wolf 119901X

119902= (1199091199021

1199091199022

119909119902119863

) is the positionof wolf 119902 and the ED and MD between them can berespectively calculated as formula (9) 119863 is the dimensionnumber of solution space

119871 ED (119901 119902) =119863

sum119889=1

(119909119901119889

minus 119909119902119889

)2

119871MD (119901 119902) =119863

sum119889=1

10038161003816100381610038161003816119909119901119889 minus 119909119902119889

10038161003816100381610038161003816

(9)

The statistical results obtained by WPA after 50-runcomputation are shown in Table 4 Firstly we note that WPAwithEuclidean distance (WPA ED)does not get 100 successrate on Colville (119863 = 4) and Griewank functions (119863 = 100)while WPA with Manhattan distance (WPA MD) does notget 100 success rate on Griewank functions (119863 = 100)which means that WPA ED and WPA MD with originalcoefficients still have the risk of premature convergence tolocal optima

As seen from Table 4 WPA is not very sensitive to twodistance measurements on most functions (RosenbrockSphere Sumsquares Booth and Ackley) and no matterwhich metric is used WPA can always get a good resultwith SR = 100 But for these functions comparing theresults between WPA MD and WPA ED in detail we canfind that WPA MD has shorter average reaching time (ARt)which means faster convergence speed to a certain accuracyThe reason may be that ED has the higher computationalcomplexityMeanwhileWPA MDhas better performance onother four criteria (best worst mean and StdDev) whichmeans better solution accuracy and robustness

Naturally because of its better efficiency precision androbustness WD is more suitable for WPA So the WPAalgorithm used in what follows is WPA MD

422 Effect of Four Parameters on the Performance of WPAIn this subsectionwe investigate the impact of the parameters119878 119871near 119879max and 120573 on the new algorithm 119878 is the stepcoefficient 119871near is the distance determinant coefficient 119879maxis the maximum number of repetitions in scouting behaviorand 120573 is the population renewing proportional coefficientThe parameters selection procedure is performed in a one-factor-at-a-time manner For each sensitivity analysis in thissection only one parameter is varied each time and theremaining parameters are kept at the values suggested by theoriginal estimate listed in Table 9 The interaction relationbetween parameters is assumed unimportant

Each time one of the WPA parameters is varied in a cer-tain interval to see which value within this internal will resultin the best performance Specifically theWPA algorithm alsoruns 50 times on each case

Table 5 shows the sensitivity analysis of the step coef-ficient 119878 All results are shown in the form of Mean plusmnStd (SR) The choice of interval [004 016] used in thisanalysis was motivated by the original Nelder-Mead simplexsearch procedure where a step coefficient greater than 004was suggested for general usage

Meanwhile based on detailed comparison of the resultson Rosenbrock Sphere and Bridge functions step coefficientis not sensitive to WPA and for Booth function there is atendency of better results with larger 119878 From Table 5 it isfound that a step coefficient setting at 012 returns the bestresult which has better Mean small Std and SR = 100 forall functions

Tables 6ndash8 analyze sensitivity of 119871near 119879max and 120573 Gen-erally speaking 119871near 119879max and 120573 are not sensitive to mostfunctions exceptGriewank function sinceGriewanknot only

Mathematical Problems in Engineering 7

Table5Sensitivityanalysisof

stepcoeffi

cient(

119878)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

69119890

minus8

plusmn4

3119890minus

82

7119890minus

8plusmn

35119890

minus8

11119890

minus8

plusmn9

1119890minus

93

2119890minus

9plusmn

27119890

minus9

50119890

minus9

plusmn5

7119890minus

93

2119890minus

9plusmn

37119890

minus9

12119890

minus9

plusmn1

6119890minus

9Colville

13119890

minus7

plusmn7

1119890minus

83

3119890minus

7plusmn

28119890

minus7(90)

26119890

minus7

plusmn1

9119890minus

72

3119890minus

7plusmn

14119890

minus7

35119890

minus7

plusmn2

5119890minus

79

5119890minus

7plusmn

10119890

minus6(80)

14119890

minus6

plusmn1

5119890minus

6(50)

Sphere

23119890

minus14

5plusmn

71119890

minus14

56

6119890minus

152

plusmn2

1119890minus

151

21119890

minus14

6plusmn

45119890

minus14

63

9119890minus

146

plusmn1

2119890minus

145

12119890

minus14

5plusmn

34119890

minus14

51

7119890minus

146

plusmn5

3119890minus

146

22119890

minus14

9plusmn

68119890

minus14

9Sumsquares

98119890

minus14

5plusmn

31119890

minus14

43

1119890minus

146

plusmn8

4119890minus

146

81119890

minus14

7plusmn

26119890

minus14

64

8119890minus

146

plusmn1

0119890minus

145

38119890

minus15

2plusmn

79119890

minus15

23

4119890minus

147

plusmn1

1119890minus

146

12119890

minus14

7plusmn

39119890

minus14

7Bo

oth

54119890

minus7

plusmn3

3119890minus

71

6119890minus

9plusmn

11119890

minus9

32119890

minus11

plusmn1

6119890minus

111

3119890minus

12plusmn

91119890

minus13

13119890

minus13

plusmn1

2119890minus

133

9119890minus

15plusmn

18119890

minus15

12119890

minus16

plusmn5

8119890minus

17Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

025

plusmn0

53(80)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

0plusmn

00

06plusmn

019

(92)

020

plusmn0

42(86)

8 Mathematical Problems in Engineering

Table6Sensitivityanalysisof

distance

determ

inantcoefficient(

119871near)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

44119890

minus8

plusmn6

5119890minus

82

3119890minus

8plusmn

37119890

minus8

34119890

minus9

plusmn4

8119890minus

93

0119890minus

8plusmn

29119890

minus8

19119890

minus8

plusmn2

4119890minus

82

4119890minus

8plusmn

47119890

minus8

29119890

minus8

plusmn5

3119890minus

8Colville

20119890

minus7

plusmn9

9119890minus

82

6119890minus

7plusmn

16119890

minus7

35119890

minus7

plusmn2

6119890minus

72

3119890minus

7plusmn

15119890

minus7

12119890

minus7

plusmn3

4119890minus

82

8119890minus

7plusmn

19119890

minus7

14119890

minus7

plusmn6

9119890minus

8Sphere

68119890

minus14

6plusmn

20119890

minus14

51

9119890minus

146

plusmn6

2119890minus

146

17119890

minus14

5plusmn

43119890

minus14

52

6119890minus

148

plusmn8

3119890minus

148

36119890

minus14

6plusmn

11119890

minus14

53

7119890minus

151

plusmn1

1119890minus

150

53119890

minus14

9plusmn

17119890

minus14

8Sumsquares1

1119890

minus14

7plusmn

34119890

minus14

71

0119890minus

146

plusmn3

3119890minus

146

37119890

minus15

1plusmn

89119890

minus15

16

2119890minus

146

plusmn1

9119890minus

145

62119890

minus15

2plusmn

19119890

minus15

11

22119890

minus14

5plusmn

29119890

minus14

51

3119890minus

148

plusmn4

0119890minus

148

Booth

26119890

minus11

plusmn1

3119890minus

112

9119890minus

11plusmn

19119890

minus11

24119890

minus11

plusmn1

6119890minus

113

1119890minus

11plusmn

18119890

minus01

12

4119890minus

11plusmn

13119890

minus11

31119890

minus11

plusmn2

1119890minus

111

0119890minus

10plusmn

13119890

minus10

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

014

plusmn0

43(90)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

159

119890minus

15plusmn

149

119890minus

158

9119890minus

16plusmn

0Grie

wank

008

plusmn0

26(90)

10119890

minus3

plusmn0

02(96)

0plusmn

00

plusmn0

0plusmn

00

10plusmn

033

(92)

0plusmn

0

Mathematical Problems in Engineering 9

Table7Sensitivityanalysisof

them

axim

umnu

mbero

frepetition

sinscou

tingbehavior

(119879max)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)6

810

1214

1618

Rosenb

rock

24119890

minus8

plusmn2

6119890minus

88

4119890minus

9plusmn

80119890

minus9

13119890

minus8

plusmn1

3119890minus

81

4119890minus

8plusmn

10119890

minus8

20119890

minus8

plusmn1

9119890minus

82

1119890minus

8plusmn

25119890

minus8

12119890

minus8

plusmn8

9119890minus

9Colville

48119890

minus7

plusmn2

2119890minus

73

4119890minus

7plusmn

18119890

minus7

15119890

minus7

plusmn1

2119890minus

73

8119890minus

7plusmn

20119890

minus7

36119890

minus7

plusmn3

7119890minus

7(96)

34119890

minus7

plusmn2

5119890minus

72

6119890minus

7plusmn

15119890

minus7

Sphere

71119890

minus14

7plusmn

22119890

minus14

64

5119890minus

146

plusmn9

0119890minus

146

78119890

minus14

6plusmn

23119890

minus14

51

9119890minus

148

plusmn5

3119890minus

148

57119890

minus14

8plusmn

13119890

minus14

76

9119890minus

145

plusmn2

2119890minus

144

36119890

minus14

7plusmn

11119890

minus14

6Sumsquares

41119890

minus14

6plusmn

13119890

minus14

52

4119890minus

149

plusmn4

8119890minus

149

42119890

minus14

9plusmn

13119890

minus14

88

3119890minus

150

plusmn2

6119890minus

149

85119890

minus14

7plusmn

27119890

minus14

65

4119890minus

146

plusmn9

0119890minus

146

14119890

minus15

1plusmn

44119890

minus15

1Bo

oth

32119890

minus11

plusmn2

9119890minus

114

2119890minus

11plusmn

27119890

minus11

25119890

minus11

plusmn1

5119890minus

112

1119890minus

11plusmn

15119890

minus11

32119890

minus11

plusmn2

5119890minus

112

6119890minus

11plusmn

18119890

minus11

26119890

minus11

plusmn2

7119890minus

11Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

08

9119890minus

16plusmn

0Grie

wank

010

plusmn0

33(92)

0plusmn

01

0119890minus

3plusmn

002

(98)

009

plusmn0

31(88)

0plusmn

00

09plusmn

029

(94)

83119890

minus4

plusmn0

02(98)

10 Mathematical Problems in Engineering

Table8Sensitivityanalysisof

popu

latio

nrenewingprop

ortio

nalcoefficient(

120573)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)2

34

56

78

Rosenb

rock

10119890

minus8

plusmn9

2119890minus

98

7119890minus

9plusmn

76119890

minus9

12119890

minus8

plusmn1

0119890minus

88

6119890minus

9plusmn

83119890

minus9

14119890

minus8

plusmn1

3119890minus

89

9119890minus

9plusmn

98119890

minus9

11119890

minus8

plusmn1

2119890minus

9Colville

32119890

minus8

plusmn1

8119890

minus8

14119890

minus7

plusmn1

3119890minus

71

2119890minus

7plusmn

59119890

minus8

14119890

minus7

plusmn9

4119890minus

83

0119890minus

7plusmn

69119890

minus8

39119890

minus7

plusmn1

7119890minus

78

6119890minus

7plusmn

40119890

minus7(80)

Sphere

19119890

minus16

6plusmn

05

2119890minus

158

plusmn1

6119890minus

157

29119890

minus15

3plusmn

92119890

minus15

34

3119890minus

149

plusmn1

3119890minus

148

79119890

minus13

9plusmn

25119890

minus13

88

3119890minus

134

plusmn1

8119890minus

133

34119890

minus12

6plusmn

80119890

minus12

6Sumsquares

28119890

minus16

7plusmn

01

4119890minus

157

plusmn4

3119890minus

157

28119890

minus15

5plusmn

45119890

minus15

58

3119890minus

146

plusmn1

8119890minus

145

69119890

minus14

3plusmn

17119890

minus14

25

3119890minus

143

plusmn1

3119890minus

142

33119890

minus12

7plusmn

10119890

minus12

6Bo

oth

81119890

minus11

plusmn1

3119890minus

102

5119890minus

11plusmn

17119890

minus11

19119890

minus11

plusmn1

2119890minus

112

5119890minus

11plusmn

17119890

minus01

12

5119890minus

11plusmn

15119890

minus11

23119890

minus11

plusmn1

5119890minus

112

3119890minus

11plusmn

14119890

minus11

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

019

plusmn0

41(86)

0plusmn

01

2119890minus

3plusmn

031

(96)

Mathematical Problems in Engineering 11

Table 9 Best suggestions for WPA parameters

No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2

020

2

0

xy

minusf(xy) minus1000

minus2000

minus3000

minus4000

minus2minus2

minus4minus4

(a)

x

y

0 1 2

0

05

1

15

2

minus05

minus15

minus2minus2

minus1

minus1

(b)

Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines

is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized

Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions

Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions

So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2

43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface

As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2

As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909

1 1199092) = (1 1)

But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction

On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909

1 1199092 1199093 1199094) = (1 1 1 1)

Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function

Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909

1 1199092 119909

119898) =

(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5

As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and

12 Mathematical Problems in Engineering

Table10Statistic

alresults

of50

runs

obtained

byGAP

SOA

FSAA

BCFAand

WPA

algorithm

s

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Rosenb

rock

119891min

(119909)

=0

2UN

GA

178

119890minus

1000373

000

91000

9210

gt7598

323

PSO

226

119890minus

115

89119890

minus7

107

119890minus

71

30119890

minus7

100

07444

AFS

A110

eminus13

111

119890minus

92

34119890

minus10

262

119890minus

10100

20578

ABC

599

119890minus

6000

998

61119890

minus4

00015

0gt3910

297

FA6

28119890

minus13

629

eminus10

186eminus

10162eminus

10100

3312

56WPA

349

119890minus

112

34119890

minus8

509

119890minus

94

34119890

minus9

100

66333

Colville

119891min

(119909)

=0

4UN

GA

00022

03343

01272

01062

0gt12

2119890+

3PS

O1

29119890

minus6

346

119890minus

45

06119890

minus5

671

119890minus

50

gt114

0869

AFS

A366

eminus8

891

119890minus

73

16119890

minus7

232

119890minus

7100

4018

07ABC

00103

05337

01871

01232

0gt3844193

FA2

41119890

minus7

369

119890minus

56

62119890

minus6

807

119890minus

68

gt3

14119890

+3

WPA

471

119890minus

8372

eminus7

125eminus

7697eminus

8100

27405

4

Sphere

119891min

(119909)

=0

200

US

GA

156

119890+

51

81119890

+5

171

119890+

55

78119890

+3

0gt4

44119890

+4

PSO

10361

15520

12883

01206

0gt2719

201

AFS

A5

12119890

+5

579

119890+

55

51119890

+5

163

119890+

40

gt7

41119890

+3

ABC

000

4112

521

004

4401773

0gt44

29045

FA01432

02327

01865

00199

0gt8

34119890

+3

WPA

149eminus

172

241

eminus165

156eminus

166

0100

61729

Sumsquares

119891min

(119909)

=0

150

US

GA

593

119890+

47

15+

46

63119890

+4

288

119890+

30

gt3

16119890

+4

PSO

397098

911145

559050

104165

0gt2325464

AFS

A1

43119890

+5

179

119890+

51

64119890

+5

958

119890+

30

gt7

36119890

+3

ABC

171

119890minus

500017

199

119890minus

43

36119890

minus4

0gt4351848

FA89920

998861

405721

192743

0gt6

88119890

+3

WPA

268

eminus172

547

eminus166

262eminus

167

0100

65954

Booth

119891min

(119909)

=0

2MS

GA

455

119890minus

114

55119890

minus11

455

119890minus

110

100

12621

PSO

122

119890minus

122

41119890

minus8

280

119890minus

94

52119890

minus9

100

020

79AFS

A3

02119890

minus12

145

119890minus

94

61119890

minus10

408

119890minus

10100

44329

ABC

605

eminus20

141

eminus17

463eminus

18414eminus

18100

04175

FA1

80119890

minus12

439

119890minus

91

18119890

minus9

111

119890minus

9100

379191

WPA

822

119890minus

157

05119890

minus13

121

119890minus

131

19119890

minus13

100

69339

Bridge

119891max

(119909)

=3

0054

2MN

GA

300

54300

54300

541

35119890

minus15

100

01927

PSO

300

54300

54300

544

84119890

minus8

100

009

29AFS

A300

54300

4730052

169

119890minus

412

gt8

01119890

+3

ABC

300

54300

54300

543

59119890

minus15

100

00932

FA300

54300

54300

543

11119890

minus10

100

227230

WPA

300

54300

54300

54358eminus

15100

01742

Ackley

119891min

(119909)

=0

50MN

GA

114570

126095

1216

1202719

0gt10

4119890+

4PS

O004

6917401

06846

06344

0gt1925522

AFS

A2016

0020600

9204229

01009

0gt9

80119890

+3

ABC

200085

200025

200061

00014

0gt5963841

FA00101

00209

00160

00021

0gt4

28119890

+3

WPA

888

eminus16

444

eminus15

110eminus

15852eminus

16100

79476

Mathematical Problems in Engineering 13

Table10C

ontin

ued

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Grie

wank

119891min

(119909)

=0

100

MN

GA

3174

525

3996

376

3634174

172922

0gt2

07119890

+4

PSO

00029

00082

00052

00011

0gt3670

080

AFS

A2

05119890

+3

255

119890+

32

33119890

+3

1096

821

0gt6

51119890

+3

ABC

895

119890minus

7000

432

26119890

minus4

781

119890minus

42

gt6209561

FA000

6800118

000

9100011

0gt5

72119890

+3

WPA

00

00

100

145338

14 Mathematical Problems in Engineering

0

100

10

0

1

xy

minusf(xy)

minus1

minus2

minus3

minus10minus10

times106

(a)

minus5

minus5

minus10minus10

x

y

0 5 10

0

5

10

(b)

Figure 3 Colville function (1199091

= 1199093 1199092

= 1199094) (a) surface plot and (b) contour lines

0100

0

1000

05

1

15

2

xy

minus100minus100

f(xy)

times104

(a)

x

y

0 50 100

0

50

100

minus100minus100

minus50

minus50

(b)

Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines

minus100

minus200

minus300

minus10minus10

0

100

10

0

xy

minusf(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines

Mathematical Problems in Engineering 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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

2 Mathematical Problems in Engineering

rigorous organization system and subtle hunting behaviorWolves tactics of Mongolia cavalry in Genghis Khan periodsubmarine tactics of Nazi Admiral Doenitz in World WarII and US military wolves attack system for electroniccountermeasures all highlight great charm of their swarmintelligence [13] proposes a wolf colony algorithm (WCA)to solve the optimization problem But the accuracy andefficiency of WCA are not good enough and easily fall intolocal optima especially for high-dimensional functions Soin this paper we reanalyzed collaborative predation behaviorand prey distribution mode of wolves and proposed a newswarm intelligence algorithm called wolf pack algorithm(WPA) Moreover the efficiency and robustness of the newalgorithm were tested by compared experiments

The remainder of this paper is structured as followsIn Section 2 the predation behaviors and prey distributionof wolves are analyzed In Section 3 WPA is describedSection 4 describes the experimental setup followed byexperimental results and analysis Finally conclusion andfuture work are presented in Section 5

2 System Analyzing of Wolf Pack

Wolves are gregarious animals and have clearly social workdivision There is a lead wolf some elite wolves act as scoutsand some ferociouswolves in awolf packThey cooperatewellwith each other and take their respective responsibility for thesurvival and thriving of wolf pack

Firstly the lead wolf as a leader under the law of thejungle is always the smartest and most ferocious one Itis responsible for commanding the wolves and constantlymaking decision by evaluating surrounding situation andperceiving information from other wolves These can avoidthe wolves in danger and command the wolves to smoothlycapture prey as soon as possible

Secondly the lead wolf sends some elite wolves to huntaround and look for prey in the probable scope Those elitewolves are scoutsTheywalk around and independentlymakedecision according to the concentration of smell left by preyand higher concentration means the prey is closer to thewolves So they always move towards the direction of gettingstronger smell

Thirdly once a scout wolf finds the trace of prey itwill howl and report that to lead wolf Then the lead wolfwill evaluate this situation and make a decision whether tosummon the ferocious wolves to round up the prey or notIf they are summoned the ferocious wolves will move fasttowards the direction of the scout wolf

Fourthly after capturing the prey the prey is not dis-tributed equitably but in an order from the strong to theweakThat is to say that the stronger the wolf is the more thefood it will get is Although this distribution rule will makesome weak wolf dead for lack of food it makes sure that thewolves that have the ability to capture prey getmore food so asto keep being strong and can capture more prey successfullyin the next timeThe rule avoids that thewhole pack starves todeath and ensures its continuance and proliferating In whatfollows the author made detailed description and realizationfor the above intelligent behaviors and rules

3 Wolf Pack Algorithm

31 Some Definitions If the predatory space of the artificialwolves is a 119873times119863 Euclidean space 119873 is the number of wolves119863 is the number of variables The position of one wolf 119894 isa vector X

119894= (1199091198941

1199091198942

119909119894119863

) and 119909119894119889is the 119889th variable

value of the 119894th artificial wolf 119884 = 119891(X) represents theconcentration of preyrsquos smell perceived by artificial wolveswhich is also the objective function value

The distance between two wolves 119901 and 119902 is describedas 119871(119901 119902) Several distance measurements can be selectedaccording to specific problems For example hamming dis-tance can be used in WPA for 0-1 discrete optimizationwhileManhattan distance (MD) and Euclidean distance (ED)can be used in WPA for continuous numerical functionoptimization In this paper we mainly discuss the latterproblem and the selection of distance measurements will bediscussed in Section 421 Moreover because the problemsof maximum value and minimal value can convert to eachother only the maximum value problem is discussed in whatfollows

32 The Description of Intelligent Behaviors and Rules Thecooperation between lead wolf scout wolves and ferociouswolves makes nearly perfect predation while prey distribu-tion from the strong to the weak makes the wolf pack thrivestowards the direction of the prey that it most probably can beable to capture The whole predation behavior of wolf pack isabstracted three intelligent behaviors scouting calling andbesieging behavior and two intelligent rules winner-take-all generating rule for the lead wolf and the stronger-surviverenewing rule for the wolf pack

(1) The winner-take-all generating rule for the lead wolfthe artificial wolf with the best objective function value is leadwolf During each iteration compare the function value of thelead wolf with the best one of other wolves if the value oflead wolf is not better it will be replaced Then the best wolfbecomes lead wolf Rather than acting the three intelligentbehaviors the lead wolf directly goes into the next iterationuntil it is replaced by other better wolf

(2) Scouting behavior S num elite wolves except the leadwolf are considered as the scout wolves they search thesolution in predatory space 119884

119894is the concentration of prey

smell perceived by the scout wolf 119894 119884lead is the concentrationof prey smell perceived by the lead wolf

If 119884119894

gt 119884lead that means the scout wolf is nearer to theprey and probably captures prey so the scout wolf 119894 becomeslead wolf and 119884lead = 119884

119894

If 119884119894

lt 119884lead the scout wolf 119894 respectively takes asteptowards ℎ different directions the step length is 119904119905119890119901

119886 After

taking a step towards the 119901th direction the state of the scoutwolf 119894 is formulated below

119909119901

119894119889= 119909119894119889

+ sin(2120587 times119901

ℎ) times step119889

119886 119901 = 1 2 ℎ (2)

It should be noted that ℎ is different for each wolf becauseof their different seeking ways So ℎ is randomly selected in[ℎmin ℎmax] and it must be an integer 119884

1198940is the concentration

of prey smell perceived by the scout wolf 119894 and 119884119894119901represents

Mathematical Problems in Engineering 3

the one after it took a step towards the 119901th direction Ifmax119884

1198941 1198841198942

119884119894ℎ

gt 1198841198940 the wolf 119894 steps forward and its

position 119883119894is updated Then repeat the above until 119884

119894gt 119884lead

or the maximum number of repetitions 119879max is reached(3)Calling behavior the lead wolf will howl and summon

119872 119899119906119898 ferocious wolves to gather around the prey Here theposition of the lead wolf is considered as the one of the preyso that the ferocious wolves aggregate towards the position ofleadwolf 119904119905119890119901

119887is the step length119892119896

119889is the position of artificial

lead wolf in the 119889th variable space at the 119896th iteration Theposition of the ferocious wolf 119894 in the 119896th iterative calculationis updated according to the following equation

119909119896+1119894119889

= 119909119896119894119889

+ step119889119887

sdot(119892119896119889

minus 119909119896119894119889

)1003816100381610038161003816119892119896

119889minus 119909119896119894119889

1003816100381610038161003816 (3)

This formula consists of two parts the former is thecurrent position of wolf 119894 which represents the foundationfor prey hunting the latter represents the aggregate tendencyof other wolves towards the lead wolf which shows the leadwolf rsquos leadership to the wolf pack

If 119884119894

gt 119884lead the ferocious wolf 119894 becomes lead wolfand 119884lead = 119884

119894 then the wolf 119894 takes the calling behavior If

119884119894

lt 119884lead the ferocious wolf 119894 keeps on aggregating towardsthe lead wolf with a fast speed until 119871(119894 119897) lt 119871near the wolftakes besieging behavior 119871(119894 119897) is the distance between thewolf 119894 and the lead wolf 119897 119871near is the distance determinantcoefficient as a judging condition which determine whetherwolf 119894 changes state from aggregating towards the lead wolfto besieging behavior The different value of 119871near will affectalgorithmic convergence rate There will be a discussion inSection 422

Calling behavior shows information transferring andsharing mechanism in wolf pack and blends the idea of socialcognition

(4) Besieging behavior after large-steps running towardsthe lead wolf the wolves are close to the prey then all wolvesexcept the leadwolf will take besieging behavior for capturingprey Now the position of lead wolf is considered as theposition of prey In particular 119866119896

119889reprensents the position of

prey in the119889th variable space at the 119896th iterationThepositionof wolf 119894 is updated according to the following equation

119909119896+1119894119889

= 119909119896119894119889

+ 120582 sdot step119889119888

sdot10038161003816100381610038161003816119866119896

119889minus 119909119896119894119889

10038161003816100381610038161003816 (4)

120582 is a random number uniformly distributed at theinterval [minus1 1] 119904119905119890119901

119888is the step length of wolf 119894 when it

takes besieging behavior1198841198940is the concentration of prey smell

perceived by the wolf 119894 and 119884119894119896represents the one after it

took this behavior If 1198841198940

lt 119884119894119896 the position X

119894is updated

otherwise it not changedThere are 119904119905119890119901

119886 119904119905119890119901119887 and 119904119905119890119901

119888in the three intelligent

behaviors and the three-step length in 119889th variable spaceshould have the following relationship

step119889119886

=step119889119887

2= 2 sdot step119889

119888= 119878 (5)

119878 is step coefficient and represents the fineness degree ofartificial wolf searching for prey in resolution space

(5) The stronger-survive renewing rule for the wolf packthe prey is distributed from the strong to the weak which willresult in some weak wolves deadThe algorithm will generate119877 wolves while deleting 119877 wolves with bad objective functionvalues Specifically with the help of the lead wolf rsquos huntingexperience in the 119889th variable space position of the 119894th oneof 119877 wolves is defined as follows

119909119894119889

= 119892119889

sdot rand 119894 = 1 2 119877 (6)

119892119889is the position of artificial lead wolf in the 119889th variable

space rand is a random number uniformly distributed at theinterval [minus01 01]

When the value of 119877 is larger it is better for sustainingwolf rsquos diversity and making the algorithm have the abilityto open up new resolution space But if 119877 is too large thealgorithm will nearly be a random search approach Becausethe number and scale of prey captured by wolves are differentin natural word which will lead to different number ofweak wolf dead 119877 is an integer and randomly selected atthe interval [119899(2 lowast 120573) 119899120573] 120573 is the population renewingproportional coefficient

33 Algorithm Description As described in the previoussection WPA has three artificial intelligent behaviors andtwo intelligent rules There are scouting behavior callingbehavior and besieging behavior and winner-take-all rule forgenerating lead wolf and the stronger-survive renewing rulefor wolf pack

Firstly the scouting behavior accelerates the possibilitythat WPA can fully traverse the solution space Secondly thewinner-take-all rule for generating lead wolf and the callingbehavior make the wolves move towards the lead wolf whoseposition is the nearest to the prey and most likely capturingpreyThe winner-take-all rule and calling behavior also makewolves arrive at the neighborhood of the global optimumonly after a few iterations elapsed since the step of wolvesin calling behavior is the largest one Thirdly with a smallstep step

119888 besieging behavior makes WPA algorithm have

the ability to open up new solution space and carefully searchthe global optima in good solution area Fourthly with thehelp of stronger-survive renewing rule for the wolf pack thealgorithm can get several new wolves whose positions arenear the best wolf lead wolf which allows for more latitudeof search space to anchor the global optimum while keepingpopulation diversity in each iteration

All the abovemakeWPA possesses superior performancein accuracy and robustness which will be seen in Section 4

Having discussed all the components ofWPA the impor-tant computation steps are detailed below

Step 1 (initialization) Initialize the following parametersthe initial position of artificial wolf 119894 (X

119894) the number of

the wolves (119873) the maximum number of iterations (119896max)the step coefficient (119878) the distance determinant coefficient(119871near) the maximum number of repetitions in scoutingbehavior (119879max) and the population renewing proportionalcoefficient (120573)

4 Mathematical Problems in Engineering

Table 1 Benchmark functions in experiments

No Functions Formulation Global extremum 119863 C Range1 Rosenbrock 119891() = 100(119909

2minus 11990921)2 + (1 minus 119909

1)2 119891min() = 0 2 UN (minus2048 2048)

2 Colville119891() = 100(1199092

1minus 1199092)2

+ (1199091

minus 1)2

+ (1199093

minus 1)2

+ 90(11990923

minus 1199094)2

+ 101(1199092

minus 1)2

+ (1199094

minus 1)2

+ 198(1199092

minus 1)(1199094

minus 1)119891min() = 0 4 UN (minus10 10)

3 Sphere 119891 () =119863

sum119894=1

1199092119894

119891min() = 0 200 US (minus100 100)

4 Sumsquares 119891 () =119863

sum119894=1

1198941199092119894

119891min() = 0 150 US (minus10 10)

5 Booth 119891() = (1199091

+ 21199092

minus 7)2 + (21199091

+ 1199092

minus 5)2 119891min() = 0 2 MS (minus10 10)

6 Bridge 119891 () =sinradic1199092

1+ 11990922

radic11990921

+ 11990922

+ exp(cos 2120587119909

1+ cos 2120587119909

2

2) minus 07129 119891max() = 30054 2 MN (minus15 15)

7 Ackley 119891() = minus20 exp(minus02radic1

119863

119863

sum119894=1

1199092119894) minus exp(

1

119863

119863

sum119894=1

cos 2120587119909119894) + 20 + 119890 119891min() = 0 50 MN (minus32 32)

8 Griewank 119891() =1

4000

119863

sum119894=1

1199092119894

minus119863

prod119894=1

cos(119909119894

radic119894) + 1 119891min() = 0 100 MN (minus600 600)

119863 dimension C characteristic U unimodal M multimodal S separable N nonseparable

Step 2 The wolf with best function value is considered aslead wolf In practical computation 119878 num = 119872 num =119899 minus 1 which means that wolves except for lead wolf actwith different behavior as different status So here exceptfor lead wolf according to formula (2) the rest of the 119899 minus 1wolves firstly act as the artificial scout wolves to take scoutingbehavior until 119884

119894gt 119884lead or the maximum number of

repetition 119879max is reached and then go to Step 3

Step 3 Except for the lead wolf the rest of the 119899 minus 1 wolvessecondly act as the artificial ferocious wolves and gathertowards the lead wolf according to (3) 119884

119894is the smell

concentration of prey perceived by wolf 119894 if 119884119894

ge 119884lead go toStep 2 otherwise the wolf 119894 continues running until 119871(119894 119897) le119871near then go to Step 4

Step 4 The position of artificial wolves who take besiegingbehavior is updated according to (4)

Step 5 Update the position of lead wolf under the winner-take-all generating rule and update the wolf pack under thepopulation renewing rule according to (6)

Step 6 If the program reaches the precision requirement orthemaximumnumber of iterations the position and functionvalue of lead wolf the problem optimal solution will beoutputted otherwise go to Step 2

So the flow chart of WPA can be shown as Figure 1

4 Experimental Results

The ingredients of the WPA method have been describedin Section 3 In this section the design of experimentsis explained sensitivity analysis of parameters on WPAis explored and the empirical results are reported which

Initialization

Scouting behavior

Yi gt Ylead

Yi gt Ylead

orT gt Tmax

Calling behavior

L(i l) gt Lnear

Besieging behavior

Renew the position of lead wolf

Renew wolf pack

Terminate

Output resultsYes

Yes

Yes

Yes

No

No

No

No

Figure 1 The flow chart of WPA

compare the WPA approach with those of GA PSO ASFAABC and FA

41 Design of the Experiments

411 Benchmark Functions In order to evaluate the perfor-mance of these algorithms eight classical benchmark func-tions are presented inTable 1Though only eight functions areused in this test they are enough to include some differentkinds of problems such as unimodal multimodal regularirregular separable nonseparable and multidimensional

If a function has more than one local optimum thisfunction is calledmultimodalMultimodal functions are usedto test the ability of algorithms to get rid of local minima

Mathematical Problems in Engineering 5

Table 2 The list of various methods used in the paper

Method Authors and referencesGenetic algorithm (GA) Goldberg [14]Particle swarm optimization algorithm(PSO) Kennedy and Eberhart [7]

Artificial fish school algorithm (ASFA) Li et al [9]Artificial bee colony algorithm (ABC) Karaboga [10]Firefly algorithm (FA) Yang [11]

Another group of test problems is separable or nonseparablefunctions A 119901-variable separable function can be expressedas the sum of 119901 functions of one variable such as Sumsquaresand Rastrigin Nonseparable functions cannot be written inthis form such as Bridge Rosenbrock Ackley andGriewankBecause nonseparable functions have interrelation amongtheir variable these functions are more difficult than theseparable functions

In Table 1 characteristics of each function are given underthe column titled 119862 In this column 119872 means that thefunction is multimodal while 119880 means that the functionis unimodal If the function is separable abbreviation 119878 isused to indicate this specification Letter 119873 refers to that thefunction is nonseparable As seen from Table 1 4 functionsare multimodal 4 functions are unimodal 3 functions areseparable and 5 functions are nonseparable

The variety of functions forms and dimensions makeit possible to fairly assess the robustness of the proposedalgorithms within limit iteration Many of these functionsallow a choice of dimension and an input dimension rangingfrom 2 to 200 for test functions is given Dimensions of theproblems that we used can be found under the column titled119863 Besides initial ranges formulas and global optimumvalues of these functions are also given in Table 1

412 Experimental Settings In this subsection experimentalsettings are given Firstly in order to fully compare the perfor-mance of different algorithms we take the simulation underthe same situation So the values of the common parametersused in each algorithm such as population size and evaluationnumber were chosen to be the same Population size was100 and the maximum evaluation number was 2000 forall algorithms on all functions Additionally we follow theparameter settings in the original paper of GA PSO AFSAABC and FA see Table 2

For each experiment 50 independent runs were con-ducted with different initial random seeds To evaluate theperformance of these algorithms six criteria are given inTable 3

Accelerating convergence speed and avoiding the localoptima have become two important and appealing goals inswarm intelligent search algorithms So as seen in Table 3we adopted criteria best mean and standard deviation toevaluate efficiency and accuracy of algorithms and adoptedcriteria Art Worst and SR to evaluate convergence speedeffectiveness and robustness of six algorithms

Table 3 Six criteria and their abbreviations

Criteria AbbreviationThe best value of optima found in 50 runs BestThe worst value of optima found in 50 runs WorstThe average value of optima found in 50 runs MeanThe standard deviations StdDevThe success rate of the results SRThe average reaching time Art

Specifically speaking SR provides very useful informa-tion about how stable an algorithm is Success is claimed ifan algorithm successfully gets a solution below a prespecifiedthreshold value with the maximum number of functionevaluations [15] So to calculate the success rate an erroraccuracy level 120576 = 10minus6 must be set (120576 = 10minus6 also usedin [16]) Thus we compared the result 119865 with the knownanalytical optima 119865lowast and consider 119865 to be ldquosuccessfulrdquo if thefollowing inequality holds

1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816

119865lowastlt 120576 119865lowast = 0

1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816 lt 120576 119865lowast = 0

(7)

The SR is a percentage value that is calculated as

SR =successful runs

runs (8)

Art is the average value of time once an algorithm gets asolution satisfying the formula (7) in 50-run computationsArt also provides very useful information about how fastan algorithm converges to certain accuracy or under thesame termination criterion which has important practicalsignificance

All algorithms have been tested in Matlab 2008a over thesame Lenovo A4600R computer with a Dual-Core 260GHzprocessor running Windows XP operating system over199Gb of memory

42 Experiments 1 Effect of Distance Measurements and FourParameters on WPA In order to study the effect of twodistance measures and four parameters on WPA differentmeasures and values of parameters were tested on typicalfunctions listed in Table 1 Each experiment WPA algorithmthat runs 50 times on each function and several criteriadescribed in Section 412 are used The experiment is con-ducted with the original coefficients shown in Table 9

421 Effect of Distance Measurements on the Performance ofWPA This subsection will investigate the performance ofdifferent distance measurements using functions with dif-ferent characteristics As is known to all Euclidean distance(ED) and Manhattan distance (MD) are the two most com-mon distance metrics in practical continuous optimizationIn the proposedWPA MD or ED can be adopted to measurethe distance between two wolves in the candidate solution

6 Mathematical Problems in Engineering

Table 4 Sensitivity analysis of distance measurements

Function Global extremum 119863 Distance Best Worst Mean StdDev SR Arts

Rosenbrock 119891min() = 0 2 MD 921119890 minus 11 324119890 minus 8 112119890 minus 8 118119890 minus 8 100 105165ED 426119890 minus 9 271119890 minus 7 127119890 minus 7 681119890 minus 8 100 371053

Colville 119891min() = 0 4 MD 562119890 minus 8 528119890 minus 7 249119890 minus 7 223119890 minus 7 100 468619ED 174119890 minus 7 170119890 minus 6 574119890 minus 7 370119890 minus 7 90 683220

Sphere 119891min() = 0 200 MD 320119890 minus 161 329119890 minus 144 207119890 minus 145 749119890 minus 145 100 115494ED 176119890 minus 160 336119890 minus 143 168119890 minus 144 751119890 minus 144 100 116825

Sumsquares 119891min() = 0 150 MD 156119890 minus 161 309119890 minus 144 179119890 minus 145 695119890 minus 145 100 85565ED 397119890 minus 160 224119890 minus 144 113119890 minus 145 500119890 minus 145 100 87109

Booth 119891min() = 0 2 MD 563119890 minus 12 115119890 minus 10 419119890 minus 11 332119890 minus 11 100 111074ED 108119890 minus 9 264119890 minus 8 116119890 minus 8 693119890 minus 9 100 405546

Bridge 119891max() = 30054 2 MD 30054 30054 30054 456119890 minus 16 100 11093ED 30054 30054 30054 456119890 minus 16 100 19541

Ackley 119891min() = 0 50 MD 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 193648ED 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 436884

Griewank 119891min() = 0 100 MD 0 01507 301119890 minus 3 00213 98 gt8771198903

ED 0 08350 00167 01181 92 gt135119890 + 4

space Therefore a discussion about their impacts on theperformance of WPA is needed

There are two wolves X119901

= (1199091199011

1199091199012

119909119901119863

) is theposition of wolf 119901X

119902= (1199091199021

1199091199022

119909119902119863

) is the positionof wolf 119902 and the ED and MD between them can berespectively calculated as formula (9) 119863 is the dimensionnumber of solution space

119871 ED (119901 119902) =119863

sum119889=1

(119909119901119889

minus 119909119902119889

)2

119871MD (119901 119902) =119863

sum119889=1

10038161003816100381610038161003816119909119901119889 minus 119909119902119889

10038161003816100381610038161003816

(9)

The statistical results obtained by WPA after 50-runcomputation are shown in Table 4 Firstly we note that WPAwithEuclidean distance (WPA ED)does not get 100 successrate on Colville (119863 = 4) and Griewank functions (119863 = 100)while WPA with Manhattan distance (WPA MD) does notget 100 success rate on Griewank functions (119863 = 100)which means that WPA ED and WPA MD with originalcoefficients still have the risk of premature convergence tolocal optima

As seen from Table 4 WPA is not very sensitive to twodistance measurements on most functions (RosenbrockSphere Sumsquares Booth and Ackley) and no matterwhich metric is used WPA can always get a good resultwith SR = 100 But for these functions comparing theresults between WPA MD and WPA ED in detail we canfind that WPA MD has shorter average reaching time (ARt)which means faster convergence speed to a certain accuracyThe reason may be that ED has the higher computationalcomplexityMeanwhileWPA MDhas better performance onother four criteria (best worst mean and StdDev) whichmeans better solution accuracy and robustness

Naturally because of its better efficiency precision androbustness WD is more suitable for WPA So the WPAalgorithm used in what follows is WPA MD

422 Effect of Four Parameters on the Performance of WPAIn this subsectionwe investigate the impact of the parameters119878 119871near 119879max and 120573 on the new algorithm 119878 is the stepcoefficient 119871near is the distance determinant coefficient 119879maxis the maximum number of repetitions in scouting behaviorand 120573 is the population renewing proportional coefficientThe parameters selection procedure is performed in a one-factor-at-a-time manner For each sensitivity analysis in thissection only one parameter is varied each time and theremaining parameters are kept at the values suggested by theoriginal estimate listed in Table 9 The interaction relationbetween parameters is assumed unimportant

Each time one of the WPA parameters is varied in a cer-tain interval to see which value within this internal will resultin the best performance Specifically theWPA algorithm alsoruns 50 times on each case

Table 5 shows the sensitivity analysis of the step coef-ficient 119878 All results are shown in the form of Mean plusmnStd (SR) The choice of interval [004 016] used in thisanalysis was motivated by the original Nelder-Mead simplexsearch procedure where a step coefficient greater than 004was suggested for general usage

Meanwhile based on detailed comparison of the resultson Rosenbrock Sphere and Bridge functions step coefficientis not sensitive to WPA and for Booth function there is atendency of better results with larger 119878 From Table 5 it isfound that a step coefficient setting at 012 returns the bestresult which has better Mean small Std and SR = 100 forall functions

Tables 6ndash8 analyze sensitivity of 119871near 119879max and 120573 Gen-erally speaking 119871near 119879max and 120573 are not sensitive to mostfunctions exceptGriewank function sinceGriewanknot only

Mathematical Problems in Engineering 7

Table5Sensitivityanalysisof

stepcoeffi

cient(

119878)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

69119890

minus8

plusmn4

3119890minus

82

7119890minus

8plusmn

35119890

minus8

11119890

minus8

plusmn9

1119890minus

93

2119890minus

9plusmn

27119890

minus9

50119890

minus9

plusmn5

7119890minus

93

2119890minus

9plusmn

37119890

minus9

12119890

minus9

plusmn1

6119890minus

9Colville

13119890

minus7

plusmn7

1119890minus

83

3119890minus

7plusmn

28119890

minus7(90)

26119890

minus7

plusmn1

9119890minus

72

3119890minus

7plusmn

14119890

minus7

35119890

minus7

plusmn2

5119890minus

79

5119890minus

7plusmn

10119890

minus6(80)

14119890

minus6

plusmn1

5119890minus

6(50)

Sphere

23119890

minus14

5plusmn

71119890

minus14

56

6119890minus

152

plusmn2

1119890minus

151

21119890

minus14

6plusmn

45119890

minus14

63

9119890minus

146

plusmn1

2119890minus

145

12119890

minus14

5plusmn

34119890

minus14

51

7119890minus

146

plusmn5

3119890minus

146

22119890

minus14

9plusmn

68119890

minus14

9Sumsquares

98119890

minus14

5plusmn

31119890

minus14

43

1119890minus

146

plusmn8

4119890minus

146

81119890

minus14

7plusmn

26119890

minus14

64

8119890minus

146

plusmn1

0119890minus

145

38119890

minus15

2plusmn

79119890

minus15

23

4119890minus

147

plusmn1

1119890minus

146

12119890

minus14

7plusmn

39119890

minus14

7Bo

oth

54119890

minus7

plusmn3

3119890minus

71

6119890minus

9plusmn

11119890

minus9

32119890

minus11

plusmn1

6119890minus

111

3119890minus

12plusmn

91119890

minus13

13119890

minus13

plusmn1

2119890minus

133

9119890minus

15plusmn

18119890

minus15

12119890

minus16

plusmn5

8119890minus

17Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

025

plusmn0

53(80)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

0plusmn

00

06plusmn

019

(92)

020

plusmn0

42(86)

8 Mathematical Problems in Engineering

Table6Sensitivityanalysisof

distance

determ

inantcoefficient(

119871near)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

44119890

minus8

plusmn6

5119890minus

82

3119890minus

8plusmn

37119890

minus8

34119890

minus9

plusmn4

8119890minus

93

0119890minus

8plusmn

29119890

minus8

19119890

minus8

plusmn2

4119890minus

82

4119890minus

8plusmn

47119890

minus8

29119890

minus8

plusmn5

3119890minus

8Colville

20119890

minus7

plusmn9

9119890minus

82

6119890minus

7plusmn

16119890

minus7

35119890

minus7

plusmn2

6119890minus

72

3119890minus

7plusmn

15119890

minus7

12119890

minus7

plusmn3

4119890minus

82

8119890minus

7plusmn

19119890

minus7

14119890

minus7

plusmn6

9119890minus

8Sphere

68119890

minus14

6plusmn

20119890

minus14

51

9119890minus

146

plusmn6

2119890minus

146

17119890

minus14

5plusmn

43119890

minus14

52

6119890minus

148

plusmn8

3119890minus

148

36119890

minus14

6plusmn

11119890

minus14

53

7119890minus

151

plusmn1

1119890minus

150

53119890

minus14

9plusmn

17119890

minus14

8Sumsquares1

1119890

minus14

7plusmn

34119890

minus14

71

0119890minus

146

plusmn3

3119890minus

146

37119890

minus15

1plusmn

89119890

minus15

16

2119890minus

146

plusmn1

9119890minus

145

62119890

minus15

2plusmn

19119890

minus15

11

22119890

minus14

5plusmn

29119890

minus14

51

3119890minus

148

plusmn4

0119890minus

148

Booth

26119890

minus11

plusmn1

3119890minus

112

9119890minus

11plusmn

19119890

minus11

24119890

minus11

plusmn1

6119890minus

113

1119890minus

11plusmn

18119890

minus01

12

4119890minus

11plusmn

13119890

minus11

31119890

minus11

plusmn2

1119890minus

111

0119890minus

10plusmn

13119890

minus10

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

014

plusmn0

43(90)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

159

119890minus

15plusmn

149

119890minus

158

9119890minus

16plusmn

0Grie

wank

008

plusmn0

26(90)

10119890

minus3

plusmn0

02(96)

0plusmn

00

plusmn0

0plusmn

00

10plusmn

033

(92)

0plusmn

0

Mathematical Problems in Engineering 9

Table7Sensitivityanalysisof

them

axim

umnu

mbero

frepetition

sinscou

tingbehavior

(119879max)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)6

810

1214

1618

Rosenb

rock

24119890

minus8

plusmn2

6119890minus

88

4119890minus

9plusmn

80119890

minus9

13119890

minus8

plusmn1

3119890minus

81

4119890minus

8plusmn

10119890

minus8

20119890

minus8

plusmn1

9119890minus

82

1119890minus

8plusmn

25119890

minus8

12119890

minus8

plusmn8

9119890minus

9Colville

48119890

minus7

plusmn2

2119890minus

73

4119890minus

7plusmn

18119890

minus7

15119890

minus7

plusmn1

2119890minus

73

8119890minus

7plusmn

20119890

minus7

36119890

minus7

plusmn3

7119890minus

7(96)

34119890

minus7

plusmn2

5119890minus

72

6119890minus

7plusmn

15119890

minus7

Sphere

71119890

minus14

7plusmn

22119890

minus14

64

5119890minus

146

plusmn9

0119890minus

146

78119890

minus14

6plusmn

23119890

minus14

51

9119890minus

148

plusmn5

3119890minus

148

57119890

minus14

8plusmn

13119890

minus14

76

9119890minus

145

plusmn2

2119890minus

144

36119890

minus14

7plusmn

11119890

minus14

6Sumsquares

41119890

minus14

6plusmn

13119890

minus14

52

4119890minus

149

plusmn4

8119890minus

149

42119890

minus14

9plusmn

13119890

minus14

88

3119890minus

150

plusmn2

6119890minus

149

85119890

minus14

7plusmn

27119890

minus14

65

4119890minus

146

plusmn9

0119890minus

146

14119890

minus15

1plusmn

44119890

minus15

1Bo

oth

32119890

minus11

plusmn2

9119890minus

114

2119890minus

11plusmn

27119890

minus11

25119890

minus11

plusmn1

5119890minus

112

1119890minus

11plusmn

15119890

minus11

32119890

minus11

plusmn2

5119890minus

112

6119890minus

11plusmn

18119890

minus11

26119890

minus11

plusmn2

7119890minus

11Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

08

9119890minus

16plusmn

0Grie

wank

010

plusmn0

33(92)

0plusmn

01

0119890minus

3plusmn

002

(98)

009

plusmn0

31(88)

0plusmn

00

09plusmn

029

(94)

83119890

minus4

plusmn0

02(98)

10 Mathematical Problems in Engineering

Table8Sensitivityanalysisof

popu

latio

nrenewingprop

ortio

nalcoefficient(

120573)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)2

34

56

78

Rosenb

rock

10119890

minus8

plusmn9

2119890minus

98

7119890minus

9plusmn

76119890

minus9

12119890

minus8

plusmn1

0119890minus

88

6119890minus

9plusmn

83119890

minus9

14119890

minus8

plusmn1

3119890minus

89

9119890minus

9plusmn

98119890

minus9

11119890

minus8

plusmn1

2119890minus

9Colville

32119890

minus8

plusmn1

8119890

minus8

14119890

minus7

plusmn1

3119890minus

71

2119890minus

7plusmn

59119890

minus8

14119890

minus7

plusmn9

4119890minus

83

0119890minus

7plusmn

69119890

minus8

39119890

minus7

plusmn1

7119890minus

78

6119890minus

7plusmn

40119890

minus7(80)

Sphere

19119890

minus16

6plusmn

05

2119890minus

158

plusmn1

6119890minus

157

29119890

minus15

3plusmn

92119890

minus15

34

3119890minus

149

plusmn1

3119890minus

148

79119890

minus13

9plusmn

25119890

minus13

88

3119890minus

134

plusmn1

8119890minus

133

34119890

minus12

6plusmn

80119890

minus12

6Sumsquares

28119890

minus16

7plusmn

01

4119890minus

157

plusmn4

3119890minus

157

28119890

minus15

5plusmn

45119890

minus15

58

3119890minus

146

plusmn1

8119890minus

145

69119890

minus14

3plusmn

17119890

minus14

25

3119890minus

143

plusmn1

3119890minus

142

33119890

minus12

7plusmn

10119890

minus12

6Bo

oth

81119890

minus11

plusmn1

3119890minus

102

5119890minus

11plusmn

17119890

minus11

19119890

minus11

plusmn1

2119890minus

112

5119890minus

11plusmn

17119890

minus01

12

5119890minus

11plusmn

15119890

minus11

23119890

minus11

plusmn1

5119890minus

112

3119890minus

11plusmn

14119890

minus11

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

019

plusmn0

41(86)

0plusmn

01

2119890minus

3plusmn

031

(96)

Mathematical Problems in Engineering 11

Table 9 Best suggestions for WPA parameters

No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2

020

2

0

xy

minusf(xy) minus1000

minus2000

minus3000

minus4000

minus2minus2

minus4minus4

(a)

x

y

0 1 2

0

05

1

15

2

minus05

minus15

minus2minus2

minus1

minus1

(b)

Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines

is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized

Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions

Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions

So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2

43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface

As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2

As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909

1 1199092) = (1 1)

But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction

On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909

1 1199092 1199093 1199094) = (1 1 1 1)

Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function

Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909

1 1199092 119909

119898) =

(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5

As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and

12 Mathematical Problems in Engineering

Table10Statistic

alresults

of50

runs

obtained

byGAP

SOA

FSAA

BCFAand

WPA

algorithm

s

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Rosenb

rock

119891min

(119909)

=0

2UN

GA

178

119890minus

1000373

000

91000

9210

gt7598

323

PSO

226

119890minus

115

89119890

minus7

107

119890minus

71

30119890

minus7

100

07444

AFS

A110

eminus13

111

119890minus

92

34119890

minus10

262

119890minus

10100

20578

ABC

599

119890minus

6000

998

61119890

minus4

00015

0gt3910

297

FA6

28119890

minus13

629

eminus10

186eminus

10162eminus

10100

3312

56WPA

349

119890minus

112

34119890

minus8

509

119890minus

94

34119890

minus9

100

66333

Colville

119891min

(119909)

=0

4UN

GA

00022

03343

01272

01062

0gt12

2119890+

3PS

O1

29119890

minus6

346

119890minus

45

06119890

minus5

671

119890minus

50

gt114

0869

AFS

A366

eminus8

891

119890minus

73

16119890

minus7

232

119890minus

7100

4018

07ABC

00103

05337

01871

01232

0gt3844193

FA2

41119890

minus7

369

119890minus

56

62119890

minus6

807

119890minus

68

gt3

14119890

+3

WPA

471

119890minus

8372

eminus7

125eminus

7697eminus

8100

27405

4

Sphere

119891min

(119909)

=0

200

US

GA

156

119890+

51

81119890

+5

171

119890+

55

78119890

+3

0gt4

44119890

+4

PSO

10361

15520

12883

01206

0gt2719

201

AFS

A5

12119890

+5

579

119890+

55

51119890

+5

163

119890+

40

gt7

41119890

+3

ABC

000

4112

521

004

4401773

0gt44

29045

FA01432

02327

01865

00199

0gt8

34119890

+3

WPA

149eminus

172

241

eminus165

156eminus

166

0100

61729

Sumsquares

119891min

(119909)

=0

150

US

GA

593

119890+

47

15+

46

63119890

+4

288

119890+

30

gt3

16119890

+4

PSO

397098

911145

559050

104165

0gt2325464

AFS

A1

43119890

+5

179

119890+

51

64119890

+5

958

119890+

30

gt7

36119890

+3

ABC

171

119890minus

500017

199

119890minus

43

36119890

minus4

0gt4351848

FA89920

998861

405721

192743

0gt6

88119890

+3

WPA

268

eminus172

547

eminus166

262eminus

167

0100

65954

Booth

119891min

(119909)

=0

2MS

GA

455

119890minus

114

55119890

minus11

455

119890minus

110

100

12621

PSO

122

119890minus

122

41119890

minus8

280

119890minus

94

52119890

minus9

100

020

79AFS

A3

02119890

minus12

145

119890minus

94

61119890

minus10

408

119890minus

10100

44329

ABC

605

eminus20

141

eminus17

463eminus

18414eminus

18100

04175

FA1

80119890

minus12

439

119890minus

91

18119890

minus9

111

119890minus

9100

379191

WPA

822

119890minus

157

05119890

minus13

121

119890minus

131

19119890

minus13

100

69339

Bridge

119891max

(119909)

=3

0054

2MN

GA

300

54300

54300

541

35119890

minus15

100

01927

PSO

300

54300

54300

544

84119890

minus8

100

009

29AFS

A300

54300

4730052

169

119890minus

412

gt8

01119890

+3

ABC

300

54300

54300

543

59119890

minus15

100

00932

FA300

54300

54300

543

11119890

minus10

100

227230

WPA

300

54300

54300

54358eminus

15100

01742

Ackley

119891min

(119909)

=0

50MN

GA

114570

126095

1216

1202719

0gt10

4119890+

4PS

O004

6917401

06846

06344

0gt1925522

AFS

A2016

0020600

9204229

01009

0gt9

80119890

+3

ABC

200085

200025

200061

00014

0gt5963841

FA00101

00209

00160

00021

0gt4

28119890

+3

WPA

888

eminus16

444

eminus15

110eminus

15852eminus

16100

79476

Mathematical Problems in Engineering 13

Table10C

ontin

ued

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Grie

wank

119891min

(119909)

=0

100

MN

GA

3174

525

3996

376

3634174

172922

0gt2

07119890

+4

PSO

00029

00082

00052

00011

0gt3670

080

AFS

A2

05119890

+3

255

119890+

32

33119890

+3

1096

821

0gt6

51119890

+3

ABC

895

119890minus

7000

432

26119890

minus4

781

119890minus

42

gt6209561

FA000

6800118

000

9100011

0gt5

72119890

+3

WPA

00

00

100

145338

14 Mathematical Problems in Engineering

0

100

10

0

1

xy

minusf(xy)

minus1

minus2

minus3

minus10minus10

times106

(a)

minus5

minus5

minus10minus10

x

y

0 5 10

0

5

10

(b)

Figure 3 Colville function (1199091

= 1199093 1199092

= 1199094) (a) surface plot and (b) contour lines

0100

0

1000

05

1

15

2

xy

minus100minus100

f(xy)

times104

(a)

x

y

0 50 100

0

50

100

minus100minus100

minus50

minus50

(b)

Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines

minus100

minus200

minus300

minus10minus10

0

100

10

0

xy

minusf(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines

Mathematical Problems in Engineering 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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 3

the one after it took a step towards the 119901th direction Ifmax119884

1198941 1198841198942

119884119894ℎ

gt 1198841198940 the wolf 119894 steps forward and its

position 119883119894is updated Then repeat the above until 119884

119894gt 119884lead

or the maximum number of repetitions 119879max is reached(3)Calling behavior the lead wolf will howl and summon

119872 119899119906119898 ferocious wolves to gather around the prey Here theposition of the lead wolf is considered as the one of the preyso that the ferocious wolves aggregate towards the position ofleadwolf 119904119905119890119901

119887is the step length119892119896

119889is the position of artificial

lead wolf in the 119889th variable space at the 119896th iteration Theposition of the ferocious wolf 119894 in the 119896th iterative calculationis updated according to the following equation

119909119896+1119894119889

= 119909119896119894119889

+ step119889119887

sdot(119892119896119889

minus 119909119896119894119889

)1003816100381610038161003816119892119896

119889minus 119909119896119894119889

1003816100381610038161003816 (3)

This formula consists of two parts the former is thecurrent position of wolf 119894 which represents the foundationfor prey hunting the latter represents the aggregate tendencyof other wolves towards the lead wolf which shows the leadwolf rsquos leadership to the wolf pack

If 119884119894

gt 119884lead the ferocious wolf 119894 becomes lead wolfand 119884lead = 119884

119894 then the wolf 119894 takes the calling behavior If

119884119894

lt 119884lead the ferocious wolf 119894 keeps on aggregating towardsthe lead wolf with a fast speed until 119871(119894 119897) lt 119871near the wolftakes besieging behavior 119871(119894 119897) is the distance between thewolf 119894 and the lead wolf 119897 119871near is the distance determinantcoefficient as a judging condition which determine whetherwolf 119894 changes state from aggregating towards the lead wolfto besieging behavior The different value of 119871near will affectalgorithmic convergence rate There will be a discussion inSection 422

Calling behavior shows information transferring andsharing mechanism in wolf pack and blends the idea of socialcognition

(4) Besieging behavior after large-steps running towardsthe lead wolf the wolves are close to the prey then all wolvesexcept the leadwolf will take besieging behavior for capturingprey Now the position of lead wolf is considered as theposition of prey In particular 119866119896

119889reprensents the position of

prey in the119889th variable space at the 119896th iterationThepositionof wolf 119894 is updated according to the following equation

119909119896+1119894119889

= 119909119896119894119889

+ 120582 sdot step119889119888

sdot10038161003816100381610038161003816119866119896

119889minus 119909119896119894119889

10038161003816100381610038161003816 (4)

120582 is a random number uniformly distributed at theinterval [minus1 1] 119904119905119890119901

119888is the step length of wolf 119894 when it

takes besieging behavior1198841198940is the concentration of prey smell

perceived by the wolf 119894 and 119884119894119896represents the one after it

took this behavior If 1198841198940

lt 119884119894119896 the position X

119894is updated

otherwise it not changedThere are 119904119905119890119901

119886 119904119905119890119901119887 and 119904119905119890119901

119888in the three intelligent

behaviors and the three-step length in 119889th variable spaceshould have the following relationship

step119889119886

=step119889119887

2= 2 sdot step119889

119888= 119878 (5)

119878 is step coefficient and represents the fineness degree ofartificial wolf searching for prey in resolution space

(5) The stronger-survive renewing rule for the wolf packthe prey is distributed from the strong to the weak which willresult in some weak wolves deadThe algorithm will generate119877 wolves while deleting 119877 wolves with bad objective functionvalues Specifically with the help of the lead wolf rsquos huntingexperience in the 119889th variable space position of the 119894th oneof 119877 wolves is defined as follows

119909119894119889

= 119892119889

sdot rand 119894 = 1 2 119877 (6)

119892119889is the position of artificial lead wolf in the 119889th variable

space rand is a random number uniformly distributed at theinterval [minus01 01]

When the value of 119877 is larger it is better for sustainingwolf rsquos diversity and making the algorithm have the abilityto open up new resolution space But if 119877 is too large thealgorithm will nearly be a random search approach Becausethe number and scale of prey captured by wolves are differentin natural word which will lead to different number ofweak wolf dead 119877 is an integer and randomly selected atthe interval [119899(2 lowast 120573) 119899120573] 120573 is the population renewingproportional coefficient

33 Algorithm Description As described in the previoussection WPA has three artificial intelligent behaviors andtwo intelligent rules There are scouting behavior callingbehavior and besieging behavior and winner-take-all rule forgenerating lead wolf and the stronger-survive renewing rulefor wolf pack

Firstly the scouting behavior accelerates the possibilitythat WPA can fully traverse the solution space Secondly thewinner-take-all rule for generating lead wolf and the callingbehavior make the wolves move towards the lead wolf whoseposition is the nearest to the prey and most likely capturingpreyThe winner-take-all rule and calling behavior also makewolves arrive at the neighborhood of the global optimumonly after a few iterations elapsed since the step of wolvesin calling behavior is the largest one Thirdly with a smallstep step

119888 besieging behavior makes WPA algorithm have

the ability to open up new solution space and carefully searchthe global optima in good solution area Fourthly with thehelp of stronger-survive renewing rule for the wolf pack thealgorithm can get several new wolves whose positions arenear the best wolf lead wolf which allows for more latitudeof search space to anchor the global optimum while keepingpopulation diversity in each iteration

All the abovemakeWPA possesses superior performancein accuracy and robustness which will be seen in Section 4

Having discussed all the components ofWPA the impor-tant computation steps are detailed below

Step 1 (initialization) Initialize the following parametersthe initial position of artificial wolf 119894 (X

119894) the number of

the wolves (119873) the maximum number of iterations (119896max)the step coefficient (119878) the distance determinant coefficient(119871near) the maximum number of repetitions in scoutingbehavior (119879max) and the population renewing proportionalcoefficient (120573)

4 Mathematical Problems in Engineering

Table 1 Benchmark functions in experiments

No Functions Formulation Global extremum 119863 C Range1 Rosenbrock 119891() = 100(119909

2minus 11990921)2 + (1 minus 119909

1)2 119891min() = 0 2 UN (minus2048 2048)

2 Colville119891() = 100(1199092

1minus 1199092)2

+ (1199091

minus 1)2

+ (1199093

minus 1)2

+ 90(11990923

minus 1199094)2

+ 101(1199092

minus 1)2

+ (1199094

minus 1)2

+ 198(1199092

minus 1)(1199094

minus 1)119891min() = 0 4 UN (minus10 10)

3 Sphere 119891 () =119863

sum119894=1

1199092119894

119891min() = 0 200 US (minus100 100)

4 Sumsquares 119891 () =119863

sum119894=1

1198941199092119894

119891min() = 0 150 US (minus10 10)

5 Booth 119891() = (1199091

+ 21199092

minus 7)2 + (21199091

+ 1199092

minus 5)2 119891min() = 0 2 MS (minus10 10)

6 Bridge 119891 () =sinradic1199092

1+ 11990922

radic11990921

+ 11990922

+ exp(cos 2120587119909

1+ cos 2120587119909

2

2) minus 07129 119891max() = 30054 2 MN (minus15 15)

7 Ackley 119891() = minus20 exp(minus02radic1

119863

119863

sum119894=1

1199092119894) minus exp(

1

119863

119863

sum119894=1

cos 2120587119909119894) + 20 + 119890 119891min() = 0 50 MN (minus32 32)

8 Griewank 119891() =1

4000

119863

sum119894=1

1199092119894

minus119863

prod119894=1

cos(119909119894

radic119894) + 1 119891min() = 0 100 MN (minus600 600)

119863 dimension C characteristic U unimodal M multimodal S separable N nonseparable

Step 2 The wolf with best function value is considered aslead wolf In practical computation 119878 num = 119872 num =119899 minus 1 which means that wolves except for lead wolf actwith different behavior as different status So here exceptfor lead wolf according to formula (2) the rest of the 119899 minus 1wolves firstly act as the artificial scout wolves to take scoutingbehavior until 119884

119894gt 119884lead or the maximum number of

repetition 119879max is reached and then go to Step 3

Step 3 Except for the lead wolf the rest of the 119899 minus 1 wolvessecondly act as the artificial ferocious wolves and gathertowards the lead wolf according to (3) 119884

119894is the smell

concentration of prey perceived by wolf 119894 if 119884119894

ge 119884lead go toStep 2 otherwise the wolf 119894 continues running until 119871(119894 119897) le119871near then go to Step 4

Step 4 The position of artificial wolves who take besiegingbehavior is updated according to (4)

Step 5 Update the position of lead wolf under the winner-take-all generating rule and update the wolf pack under thepopulation renewing rule according to (6)

Step 6 If the program reaches the precision requirement orthemaximumnumber of iterations the position and functionvalue of lead wolf the problem optimal solution will beoutputted otherwise go to Step 2

So the flow chart of WPA can be shown as Figure 1

4 Experimental Results

The ingredients of the WPA method have been describedin Section 3 In this section the design of experimentsis explained sensitivity analysis of parameters on WPAis explored and the empirical results are reported which

Initialization

Scouting behavior

Yi gt Ylead

Yi gt Ylead

orT gt Tmax

Calling behavior

L(i l) gt Lnear

Besieging behavior

Renew the position of lead wolf

Renew wolf pack

Terminate

Output resultsYes

Yes

Yes

Yes

No

No

No

No

Figure 1 The flow chart of WPA

compare the WPA approach with those of GA PSO ASFAABC and FA

41 Design of the Experiments

411 Benchmark Functions In order to evaluate the perfor-mance of these algorithms eight classical benchmark func-tions are presented inTable 1Though only eight functions areused in this test they are enough to include some differentkinds of problems such as unimodal multimodal regularirregular separable nonseparable and multidimensional

If a function has more than one local optimum thisfunction is calledmultimodalMultimodal functions are usedto test the ability of algorithms to get rid of local minima

Mathematical Problems in Engineering 5

Table 2 The list of various methods used in the paper

Method Authors and referencesGenetic algorithm (GA) Goldberg [14]Particle swarm optimization algorithm(PSO) Kennedy and Eberhart [7]

Artificial fish school algorithm (ASFA) Li et al [9]Artificial bee colony algorithm (ABC) Karaboga [10]Firefly algorithm (FA) Yang [11]

Another group of test problems is separable or nonseparablefunctions A 119901-variable separable function can be expressedas the sum of 119901 functions of one variable such as Sumsquaresand Rastrigin Nonseparable functions cannot be written inthis form such as Bridge Rosenbrock Ackley andGriewankBecause nonseparable functions have interrelation amongtheir variable these functions are more difficult than theseparable functions

In Table 1 characteristics of each function are given underthe column titled 119862 In this column 119872 means that thefunction is multimodal while 119880 means that the functionis unimodal If the function is separable abbreviation 119878 isused to indicate this specification Letter 119873 refers to that thefunction is nonseparable As seen from Table 1 4 functionsare multimodal 4 functions are unimodal 3 functions areseparable and 5 functions are nonseparable

The variety of functions forms and dimensions makeit possible to fairly assess the robustness of the proposedalgorithms within limit iteration Many of these functionsallow a choice of dimension and an input dimension rangingfrom 2 to 200 for test functions is given Dimensions of theproblems that we used can be found under the column titled119863 Besides initial ranges formulas and global optimumvalues of these functions are also given in Table 1

412 Experimental Settings In this subsection experimentalsettings are given Firstly in order to fully compare the perfor-mance of different algorithms we take the simulation underthe same situation So the values of the common parametersused in each algorithm such as population size and evaluationnumber were chosen to be the same Population size was100 and the maximum evaluation number was 2000 forall algorithms on all functions Additionally we follow theparameter settings in the original paper of GA PSO AFSAABC and FA see Table 2

For each experiment 50 independent runs were con-ducted with different initial random seeds To evaluate theperformance of these algorithms six criteria are given inTable 3

Accelerating convergence speed and avoiding the localoptima have become two important and appealing goals inswarm intelligent search algorithms So as seen in Table 3we adopted criteria best mean and standard deviation toevaluate efficiency and accuracy of algorithms and adoptedcriteria Art Worst and SR to evaluate convergence speedeffectiveness and robustness of six algorithms

Table 3 Six criteria and their abbreviations

Criteria AbbreviationThe best value of optima found in 50 runs BestThe worst value of optima found in 50 runs WorstThe average value of optima found in 50 runs MeanThe standard deviations StdDevThe success rate of the results SRThe average reaching time Art

Specifically speaking SR provides very useful informa-tion about how stable an algorithm is Success is claimed ifan algorithm successfully gets a solution below a prespecifiedthreshold value with the maximum number of functionevaluations [15] So to calculate the success rate an erroraccuracy level 120576 = 10minus6 must be set (120576 = 10minus6 also usedin [16]) Thus we compared the result 119865 with the knownanalytical optima 119865lowast and consider 119865 to be ldquosuccessfulrdquo if thefollowing inequality holds

1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816

119865lowastlt 120576 119865lowast = 0

1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816 lt 120576 119865lowast = 0

(7)

The SR is a percentage value that is calculated as

SR =successful runs

runs (8)

Art is the average value of time once an algorithm gets asolution satisfying the formula (7) in 50-run computationsArt also provides very useful information about how fastan algorithm converges to certain accuracy or under thesame termination criterion which has important practicalsignificance

All algorithms have been tested in Matlab 2008a over thesame Lenovo A4600R computer with a Dual-Core 260GHzprocessor running Windows XP operating system over199Gb of memory

42 Experiments 1 Effect of Distance Measurements and FourParameters on WPA In order to study the effect of twodistance measures and four parameters on WPA differentmeasures and values of parameters were tested on typicalfunctions listed in Table 1 Each experiment WPA algorithmthat runs 50 times on each function and several criteriadescribed in Section 412 are used The experiment is con-ducted with the original coefficients shown in Table 9

421 Effect of Distance Measurements on the Performance ofWPA This subsection will investigate the performance ofdifferent distance measurements using functions with dif-ferent characteristics As is known to all Euclidean distance(ED) and Manhattan distance (MD) are the two most com-mon distance metrics in practical continuous optimizationIn the proposedWPA MD or ED can be adopted to measurethe distance between two wolves in the candidate solution

6 Mathematical Problems in Engineering

Table 4 Sensitivity analysis of distance measurements

Function Global extremum 119863 Distance Best Worst Mean StdDev SR Arts

Rosenbrock 119891min() = 0 2 MD 921119890 minus 11 324119890 minus 8 112119890 minus 8 118119890 minus 8 100 105165ED 426119890 minus 9 271119890 minus 7 127119890 minus 7 681119890 minus 8 100 371053

Colville 119891min() = 0 4 MD 562119890 minus 8 528119890 minus 7 249119890 minus 7 223119890 minus 7 100 468619ED 174119890 minus 7 170119890 minus 6 574119890 minus 7 370119890 minus 7 90 683220

Sphere 119891min() = 0 200 MD 320119890 minus 161 329119890 minus 144 207119890 minus 145 749119890 minus 145 100 115494ED 176119890 minus 160 336119890 minus 143 168119890 minus 144 751119890 minus 144 100 116825

Sumsquares 119891min() = 0 150 MD 156119890 minus 161 309119890 minus 144 179119890 minus 145 695119890 minus 145 100 85565ED 397119890 minus 160 224119890 minus 144 113119890 minus 145 500119890 minus 145 100 87109

Booth 119891min() = 0 2 MD 563119890 minus 12 115119890 minus 10 419119890 minus 11 332119890 minus 11 100 111074ED 108119890 minus 9 264119890 minus 8 116119890 minus 8 693119890 minus 9 100 405546

Bridge 119891max() = 30054 2 MD 30054 30054 30054 456119890 minus 16 100 11093ED 30054 30054 30054 456119890 minus 16 100 19541

Ackley 119891min() = 0 50 MD 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 193648ED 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 436884

Griewank 119891min() = 0 100 MD 0 01507 301119890 minus 3 00213 98 gt8771198903

ED 0 08350 00167 01181 92 gt135119890 + 4

space Therefore a discussion about their impacts on theperformance of WPA is needed

There are two wolves X119901

= (1199091199011

1199091199012

119909119901119863

) is theposition of wolf 119901X

119902= (1199091199021

1199091199022

119909119902119863

) is the positionof wolf 119902 and the ED and MD between them can berespectively calculated as formula (9) 119863 is the dimensionnumber of solution space

119871 ED (119901 119902) =119863

sum119889=1

(119909119901119889

minus 119909119902119889

)2

119871MD (119901 119902) =119863

sum119889=1

10038161003816100381610038161003816119909119901119889 minus 119909119902119889

10038161003816100381610038161003816

(9)

The statistical results obtained by WPA after 50-runcomputation are shown in Table 4 Firstly we note that WPAwithEuclidean distance (WPA ED)does not get 100 successrate on Colville (119863 = 4) and Griewank functions (119863 = 100)while WPA with Manhattan distance (WPA MD) does notget 100 success rate on Griewank functions (119863 = 100)which means that WPA ED and WPA MD with originalcoefficients still have the risk of premature convergence tolocal optima

As seen from Table 4 WPA is not very sensitive to twodistance measurements on most functions (RosenbrockSphere Sumsquares Booth and Ackley) and no matterwhich metric is used WPA can always get a good resultwith SR = 100 But for these functions comparing theresults between WPA MD and WPA ED in detail we canfind that WPA MD has shorter average reaching time (ARt)which means faster convergence speed to a certain accuracyThe reason may be that ED has the higher computationalcomplexityMeanwhileWPA MDhas better performance onother four criteria (best worst mean and StdDev) whichmeans better solution accuracy and robustness

Naturally because of its better efficiency precision androbustness WD is more suitable for WPA So the WPAalgorithm used in what follows is WPA MD

422 Effect of Four Parameters on the Performance of WPAIn this subsectionwe investigate the impact of the parameters119878 119871near 119879max and 120573 on the new algorithm 119878 is the stepcoefficient 119871near is the distance determinant coefficient 119879maxis the maximum number of repetitions in scouting behaviorand 120573 is the population renewing proportional coefficientThe parameters selection procedure is performed in a one-factor-at-a-time manner For each sensitivity analysis in thissection only one parameter is varied each time and theremaining parameters are kept at the values suggested by theoriginal estimate listed in Table 9 The interaction relationbetween parameters is assumed unimportant

Each time one of the WPA parameters is varied in a cer-tain interval to see which value within this internal will resultin the best performance Specifically theWPA algorithm alsoruns 50 times on each case

Table 5 shows the sensitivity analysis of the step coef-ficient 119878 All results are shown in the form of Mean plusmnStd (SR) The choice of interval [004 016] used in thisanalysis was motivated by the original Nelder-Mead simplexsearch procedure where a step coefficient greater than 004was suggested for general usage

Meanwhile based on detailed comparison of the resultson Rosenbrock Sphere and Bridge functions step coefficientis not sensitive to WPA and for Booth function there is atendency of better results with larger 119878 From Table 5 it isfound that a step coefficient setting at 012 returns the bestresult which has better Mean small Std and SR = 100 forall functions

Tables 6ndash8 analyze sensitivity of 119871near 119879max and 120573 Gen-erally speaking 119871near 119879max and 120573 are not sensitive to mostfunctions exceptGriewank function sinceGriewanknot only

Mathematical Problems in Engineering 7

Table5Sensitivityanalysisof

stepcoeffi

cient(

119878)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

69119890

minus8

plusmn4

3119890minus

82

7119890minus

8plusmn

35119890

minus8

11119890

minus8

plusmn9

1119890minus

93

2119890minus

9plusmn

27119890

minus9

50119890

minus9

plusmn5

7119890minus

93

2119890minus

9plusmn

37119890

minus9

12119890

minus9

plusmn1

6119890minus

9Colville

13119890

minus7

plusmn7

1119890minus

83

3119890minus

7plusmn

28119890

minus7(90)

26119890

minus7

plusmn1

9119890minus

72

3119890minus

7plusmn

14119890

minus7

35119890

minus7

plusmn2

5119890minus

79

5119890minus

7plusmn

10119890

minus6(80)

14119890

minus6

plusmn1

5119890minus

6(50)

Sphere

23119890

minus14

5plusmn

71119890

minus14

56

6119890minus

152

plusmn2

1119890minus

151

21119890

minus14

6plusmn

45119890

minus14

63

9119890minus

146

plusmn1

2119890minus

145

12119890

minus14

5plusmn

34119890

minus14

51

7119890minus

146

plusmn5

3119890minus

146

22119890

minus14

9plusmn

68119890

minus14

9Sumsquares

98119890

minus14

5plusmn

31119890

minus14

43

1119890minus

146

plusmn8

4119890minus

146

81119890

minus14

7plusmn

26119890

minus14

64

8119890minus

146

plusmn1

0119890minus

145

38119890

minus15

2plusmn

79119890

minus15

23

4119890minus

147

plusmn1

1119890minus

146

12119890

minus14

7plusmn

39119890

minus14

7Bo

oth

54119890

minus7

plusmn3

3119890minus

71

6119890minus

9plusmn

11119890

minus9

32119890

minus11

plusmn1

6119890minus

111

3119890minus

12plusmn

91119890

minus13

13119890

minus13

plusmn1

2119890minus

133

9119890minus

15plusmn

18119890

minus15

12119890

minus16

plusmn5

8119890minus

17Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

025

plusmn0

53(80)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

0plusmn

00

06plusmn

019

(92)

020

plusmn0

42(86)

8 Mathematical Problems in Engineering

Table6Sensitivityanalysisof

distance

determ

inantcoefficient(

119871near)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

44119890

minus8

plusmn6

5119890minus

82

3119890minus

8plusmn

37119890

minus8

34119890

minus9

plusmn4

8119890minus

93

0119890minus

8plusmn

29119890

minus8

19119890

minus8

plusmn2

4119890minus

82

4119890minus

8plusmn

47119890

minus8

29119890

minus8

plusmn5

3119890minus

8Colville

20119890

minus7

plusmn9

9119890minus

82

6119890minus

7plusmn

16119890

minus7

35119890

minus7

plusmn2

6119890minus

72

3119890minus

7plusmn

15119890

minus7

12119890

minus7

plusmn3

4119890minus

82

8119890minus

7plusmn

19119890

minus7

14119890

minus7

plusmn6

9119890minus

8Sphere

68119890

minus14

6plusmn

20119890

minus14

51

9119890minus

146

plusmn6

2119890minus

146

17119890

minus14

5plusmn

43119890

minus14

52

6119890minus

148

plusmn8

3119890minus

148

36119890

minus14

6plusmn

11119890

minus14

53

7119890minus

151

plusmn1

1119890minus

150

53119890

minus14

9plusmn

17119890

minus14

8Sumsquares1

1119890

minus14

7plusmn

34119890

minus14

71

0119890minus

146

plusmn3

3119890minus

146

37119890

minus15

1plusmn

89119890

minus15

16

2119890minus

146

plusmn1

9119890minus

145

62119890

minus15

2plusmn

19119890

minus15

11

22119890

minus14

5plusmn

29119890

minus14

51

3119890minus

148

plusmn4

0119890minus

148

Booth

26119890

minus11

plusmn1

3119890minus

112

9119890minus

11plusmn

19119890

minus11

24119890

minus11

plusmn1

6119890minus

113

1119890minus

11plusmn

18119890

minus01

12

4119890minus

11plusmn

13119890

minus11

31119890

minus11

plusmn2

1119890minus

111

0119890minus

10plusmn

13119890

minus10

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

014

plusmn0

43(90)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

159

119890minus

15plusmn

149

119890minus

158

9119890minus

16plusmn

0Grie

wank

008

plusmn0

26(90)

10119890

minus3

plusmn0

02(96)

0plusmn

00

plusmn0

0plusmn

00

10plusmn

033

(92)

0plusmn

0

Mathematical Problems in Engineering 9

Table7Sensitivityanalysisof

them

axim

umnu

mbero

frepetition

sinscou

tingbehavior

(119879max)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)6

810

1214

1618

Rosenb

rock

24119890

minus8

plusmn2

6119890minus

88

4119890minus

9plusmn

80119890

minus9

13119890

minus8

plusmn1

3119890minus

81

4119890minus

8plusmn

10119890

minus8

20119890

minus8

plusmn1

9119890minus

82

1119890minus

8plusmn

25119890

minus8

12119890

minus8

plusmn8

9119890minus

9Colville

48119890

minus7

plusmn2

2119890minus

73

4119890minus

7plusmn

18119890

minus7

15119890

minus7

plusmn1

2119890minus

73

8119890minus

7plusmn

20119890

minus7

36119890

minus7

plusmn3

7119890minus

7(96)

34119890

minus7

plusmn2

5119890minus

72

6119890minus

7plusmn

15119890

minus7

Sphere

71119890

minus14

7plusmn

22119890

minus14

64

5119890minus

146

plusmn9

0119890minus

146

78119890

minus14

6plusmn

23119890

minus14

51

9119890minus

148

plusmn5

3119890minus

148

57119890

minus14

8plusmn

13119890

minus14

76

9119890minus

145

plusmn2

2119890minus

144

36119890

minus14

7plusmn

11119890

minus14

6Sumsquares

41119890

minus14

6plusmn

13119890

minus14

52

4119890minus

149

plusmn4

8119890minus

149

42119890

minus14

9plusmn

13119890

minus14

88

3119890minus

150

plusmn2

6119890minus

149

85119890

minus14

7plusmn

27119890

minus14

65

4119890minus

146

plusmn9

0119890minus

146

14119890

minus15

1plusmn

44119890

minus15

1Bo

oth

32119890

minus11

plusmn2

9119890minus

114

2119890minus

11plusmn

27119890

minus11

25119890

minus11

plusmn1

5119890minus

112

1119890minus

11plusmn

15119890

minus11

32119890

minus11

plusmn2

5119890minus

112

6119890minus

11plusmn

18119890

minus11

26119890

minus11

plusmn2

7119890minus

11Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

08

9119890minus

16plusmn

0Grie

wank

010

plusmn0

33(92)

0plusmn

01

0119890minus

3plusmn

002

(98)

009

plusmn0

31(88)

0plusmn

00

09plusmn

029

(94)

83119890

minus4

plusmn0

02(98)

10 Mathematical Problems in Engineering

Table8Sensitivityanalysisof

popu

latio

nrenewingprop

ortio

nalcoefficient(

120573)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)2

34

56

78

Rosenb

rock

10119890

minus8

plusmn9

2119890minus

98

7119890minus

9plusmn

76119890

minus9

12119890

minus8

plusmn1

0119890minus

88

6119890minus

9plusmn

83119890

minus9

14119890

minus8

plusmn1

3119890minus

89

9119890minus

9plusmn

98119890

minus9

11119890

minus8

plusmn1

2119890minus

9Colville

32119890

minus8

plusmn1

8119890

minus8

14119890

minus7

plusmn1

3119890minus

71

2119890minus

7plusmn

59119890

minus8

14119890

minus7

plusmn9

4119890minus

83

0119890minus

7plusmn

69119890

minus8

39119890

minus7

plusmn1

7119890minus

78

6119890minus

7plusmn

40119890

minus7(80)

Sphere

19119890

minus16

6plusmn

05

2119890minus

158

plusmn1

6119890minus

157

29119890

minus15

3plusmn

92119890

minus15

34

3119890minus

149

plusmn1

3119890minus

148

79119890

minus13

9plusmn

25119890

minus13

88

3119890minus

134

plusmn1

8119890minus

133

34119890

minus12

6plusmn

80119890

minus12

6Sumsquares

28119890

minus16

7plusmn

01

4119890minus

157

plusmn4

3119890minus

157

28119890

minus15

5plusmn

45119890

minus15

58

3119890minus

146

plusmn1

8119890minus

145

69119890

minus14

3plusmn

17119890

minus14

25

3119890minus

143

plusmn1

3119890minus

142

33119890

minus12

7plusmn

10119890

minus12

6Bo

oth

81119890

minus11

plusmn1

3119890minus

102

5119890minus

11plusmn

17119890

minus11

19119890

minus11

plusmn1

2119890minus

112

5119890minus

11plusmn

17119890

minus01

12

5119890minus

11plusmn

15119890

minus11

23119890

minus11

plusmn1

5119890minus

112

3119890minus

11plusmn

14119890

minus11

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

019

plusmn0

41(86)

0plusmn

01

2119890minus

3plusmn

031

(96)

Mathematical Problems in Engineering 11

Table 9 Best suggestions for WPA parameters

No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2

020

2

0

xy

minusf(xy) minus1000

minus2000

minus3000

minus4000

minus2minus2

minus4minus4

(a)

x

y

0 1 2

0

05

1

15

2

minus05

minus15

minus2minus2

minus1

minus1

(b)

Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines

is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized

Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions

Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions

So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2

43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface

As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2

As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909

1 1199092) = (1 1)

But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction

On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909

1 1199092 1199093 1199094) = (1 1 1 1)

Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function

Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909

1 1199092 119909

119898) =

(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5

As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and

12 Mathematical Problems in Engineering

Table10Statistic

alresults

of50

runs

obtained

byGAP

SOA

FSAA

BCFAand

WPA

algorithm

s

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Rosenb

rock

119891min

(119909)

=0

2UN

GA

178

119890minus

1000373

000

91000

9210

gt7598

323

PSO

226

119890minus

115

89119890

minus7

107

119890minus

71

30119890

minus7

100

07444

AFS

A110

eminus13

111

119890minus

92

34119890

minus10

262

119890minus

10100

20578

ABC

599

119890minus

6000

998

61119890

minus4

00015

0gt3910

297

FA6

28119890

minus13

629

eminus10

186eminus

10162eminus

10100

3312

56WPA

349

119890minus

112

34119890

minus8

509

119890minus

94

34119890

minus9

100

66333

Colville

119891min

(119909)

=0

4UN

GA

00022

03343

01272

01062

0gt12

2119890+

3PS

O1

29119890

minus6

346

119890minus

45

06119890

minus5

671

119890minus

50

gt114

0869

AFS

A366

eminus8

891

119890minus

73

16119890

minus7

232

119890minus

7100

4018

07ABC

00103

05337

01871

01232

0gt3844193

FA2

41119890

minus7

369

119890minus

56

62119890

minus6

807

119890minus

68

gt3

14119890

+3

WPA

471

119890minus

8372

eminus7

125eminus

7697eminus

8100

27405

4

Sphere

119891min

(119909)

=0

200

US

GA

156

119890+

51

81119890

+5

171

119890+

55

78119890

+3

0gt4

44119890

+4

PSO

10361

15520

12883

01206

0gt2719

201

AFS

A5

12119890

+5

579

119890+

55

51119890

+5

163

119890+

40

gt7

41119890

+3

ABC

000

4112

521

004

4401773

0gt44

29045

FA01432

02327

01865

00199

0gt8

34119890

+3

WPA

149eminus

172

241

eminus165

156eminus

166

0100

61729

Sumsquares

119891min

(119909)

=0

150

US

GA

593

119890+

47

15+

46

63119890

+4

288

119890+

30

gt3

16119890

+4

PSO

397098

911145

559050

104165

0gt2325464

AFS

A1

43119890

+5

179

119890+

51

64119890

+5

958

119890+

30

gt7

36119890

+3

ABC

171

119890minus

500017

199

119890minus

43

36119890

minus4

0gt4351848

FA89920

998861

405721

192743

0gt6

88119890

+3

WPA

268

eminus172

547

eminus166

262eminus

167

0100

65954

Booth

119891min

(119909)

=0

2MS

GA

455

119890minus

114

55119890

minus11

455

119890minus

110

100

12621

PSO

122

119890minus

122

41119890

minus8

280

119890minus

94

52119890

minus9

100

020

79AFS

A3

02119890

minus12

145

119890minus

94

61119890

minus10

408

119890minus

10100

44329

ABC

605

eminus20

141

eminus17

463eminus

18414eminus

18100

04175

FA1

80119890

minus12

439

119890minus

91

18119890

minus9

111

119890minus

9100

379191

WPA

822

119890minus

157

05119890

minus13

121

119890minus

131

19119890

minus13

100

69339

Bridge

119891max

(119909)

=3

0054

2MN

GA

300

54300

54300

541

35119890

minus15

100

01927

PSO

300

54300

54300

544

84119890

minus8

100

009

29AFS

A300

54300

4730052

169

119890minus

412

gt8

01119890

+3

ABC

300

54300

54300

543

59119890

minus15

100

00932

FA300

54300

54300

543

11119890

minus10

100

227230

WPA

300

54300

54300

54358eminus

15100

01742

Ackley

119891min

(119909)

=0

50MN

GA

114570

126095

1216

1202719

0gt10

4119890+

4PS

O004

6917401

06846

06344

0gt1925522

AFS

A2016

0020600

9204229

01009

0gt9

80119890

+3

ABC

200085

200025

200061

00014

0gt5963841

FA00101

00209

00160

00021

0gt4

28119890

+3

WPA

888

eminus16

444

eminus15

110eminus

15852eminus

16100

79476

Mathematical Problems in Engineering 13

Table10C

ontin

ued

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Grie

wank

119891min

(119909)

=0

100

MN

GA

3174

525

3996

376

3634174

172922

0gt2

07119890

+4

PSO

00029

00082

00052

00011

0gt3670

080

AFS

A2

05119890

+3

255

119890+

32

33119890

+3

1096

821

0gt6

51119890

+3

ABC

895

119890minus

7000

432

26119890

minus4

781

119890minus

42

gt6209561

FA000

6800118

000

9100011

0gt5

72119890

+3

WPA

00

00

100

145338

14 Mathematical Problems in Engineering

0

100

10

0

1

xy

minusf(xy)

minus1

minus2

minus3

minus10minus10

times106

(a)

minus5

minus5

minus10minus10

x

y

0 5 10

0

5

10

(b)

Figure 3 Colville function (1199091

= 1199093 1199092

= 1199094) (a) surface plot and (b) contour lines

0100

0

1000

05

1

15

2

xy

minus100minus100

f(xy)

times104

(a)

x

y

0 50 100

0

50

100

minus100minus100

minus50

minus50

(b)

Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines

minus100

minus200

minus300

minus10minus10

0

100

10

0

xy

minusf(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines

Mathematical Problems in Engineering 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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

4 Mathematical Problems in Engineering

Table 1 Benchmark functions in experiments

No Functions Formulation Global extremum 119863 C Range1 Rosenbrock 119891() = 100(119909

2minus 11990921)2 + (1 minus 119909

1)2 119891min() = 0 2 UN (minus2048 2048)

2 Colville119891() = 100(1199092

1minus 1199092)2

+ (1199091

minus 1)2

+ (1199093

minus 1)2

+ 90(11990923

minus 1199094)2

+ 101(1199092

minus 1)2

+ (1199094

minus 1)2

+ 198(1199092

minus 1)(1199094

minus 1)119891min() = 0 4 UN (minus10 10)

3 Sphere 119891 () =119863

sum119894=1

1199092119894

119891min() = 0 200 US (minus100 100)

4 Sumsquares 119891 () =119863

sum119894=1

1198941199092119894

119891min() = 0 150 US (minus10 10)

5 Booth 119891() = (1199091

+ 21199092

minus 7)2 + (21199091

+ 1199092

minus 5)2 119891min() = 0 2 MS (minus10 10)

6 Bridge 119891 () =sinradic1199092

1+ 11990922

radic11990921

+ 11990922

+ exp(cos 2120587119909

1+ cos 2120587119909

2

2) minus 07129 119891max() = 30054 2 MN (minus15 15)

7 Ackley 119891() = minus20 exp(minus02radic1

119863

119863

sum119894=1

1199092119894) minus exp(

1

119863

119863

sum119894=1

cos 2120587119909119894) + 20 + 119890 119891min() = 0 50 MN (minus32 32)

8 Griewank 119891() =1

4000

119863

sum119894=1

1199092119894

minus119863

prod119894=1

cos(119909119894

radic119894) + 1 119891min() = 0 100 MN (minus600 600)

119863 dimension C characteristic U unimodal M multimodal S separable N nonseparable

Step 2 The wolf with best function value is considered aslead wolf In practical computation 119878 num = 119872 num =119899 minus 1 which means that wolves except for lead wolf actwith different behavior as different status So here exceptfor lead wolf according to formula (2) the rest of the 119899 minus 1wolves firstly act as the artificial scout wolves to take scoutingbehavior until 119884

119894gt 119884lead or the maximum number of

repetition 119879max is reached and then go to Step 3

Step 3 Except for the lead wolf the rest of the 119899 minus 1 wolvessecondly act as the artificial ferocious wolves and gathertowards the lead wolf according to (3) 119884

119894is the smell

concentration of prey perceived by wolf 119894 if 119884119894

ge 119884lead go toStep 2 otherwise the wolf 119894 continues running until 119871(119894 119897) le119871near then go to Step 4

Step 4 The position of artificial wolves who take besiegingbehavior is updated according to (4)

Step 5 Update the position of lead wolf under the winner-take-all generating rule and update the wolf pack under thepopulation renewing rule according to (6)

Step 6 If the program reaches the precision requirement orthemaximumnumber of iterations the position and functionvalue of lead wolf the problem optimal solution will beoutputted otherwise go to Step 2

So the flow chart of WPA can be shown as Figure 1

4 Experimental Results

The ingredients of the WPA method have been describedin Section 3 In this section the design of experimentsis explained sensitivity analysis of parameters on WPAis explored and the empirical results are reported which

Initialization

Scouting behavior

Yi gt Ylead

Yi gt Ylead

orT gt Tmax

Calling behavior

L(i l) gt Lnear

Besieging behavior

Renew the position of lead wolf

Renew wolf pack

Terminate

Output resultsYes

Yes

Yes

Yes

No

No

No

No

Figure 1 The flow chart of WPA

compare the WPA approach with those of GA PSO ASFAABC and FA

41 Design of the Experiments

411 Benchmark Functions In order to evaluate the perfor-mance of these algorithms eight classical benchmark func-tions are presented inTable 1Though only eight functions areused in this test they are enough to include some differentkinds of problems such as unimodal multimodal regularirregular separable nonseparable and multidimensional

If a function has more than one local optimum thisfunction is calledmultimodalMultimodal functions are usedto test the ability of algorithms to get rid of local minima

Mathematical Problems in Engineering 5

Table 2 The list of various methods used in the paper

Method Authors and referencesGenetic algorithm (GA) Goldberg [14]Particle swarm optimization algorithm(PSO) Kennedy and Eberhart [7]

Artificial fish school algorithm (ASFA) Li et al [9]Artificial bee colony algorithm (ABC) Karaboga [10]Firefly algorithm (FA) Yang [11]

Another group of test problems is separable or nonseparablefunctions A 119901-variable separable function can be expressedas the sum of 119901 functions of one variable such as Sumsquaresand Rastrigin Nonseparable functions cannot be written inthis form such as Bridge Rosenbrock Ackley andGriewankBecause nonseparable functions have interrelation amongtheir variable these functions are more difficult than theseparable functions

In Table 1 characteristics of each function are given underthe column titled 119862 In this column 119872 means that thefunction is multimodal while 119880 means that the functionis unimodal If the function is separable abbreviation 119878 isused to indicate this specification Letter 119873 refers to that thefunction is nonseparable As seen from Table 1 4 functionsare multimodal 4 functions are unimodal 3 functions areseparable and 5 functions are nonseparable

The variety of functions forms and dimensions makeit possible to fairly assess the robustness of the proposedalgorithms within limit iteration Many of these functionsallow a choice of dimension and an input dimension rangingfrom 2 to 200 for test functions is given Dimensions of theproblems that we used can be found under the column titled119863 Besides initial ranges formulas and global optimumvalues of these functions are also given in Table 1

412 Experimental Settings In this subsection experimentalsettings are given Firstly in order to fully compare the perfor-mance of different algorithms we take the simulation underthe same situation So the values of the common parametersused in each algorithm such as population size and evaluationnumber were chosen to be the same Population size was100 and the maximum evaluation number was 2000 forall algorithms on all functions Additionally we follow theparameter settings in the original paper of GA PSO AFSAABC and FA see Table 2

For each experiment 50 independent runs were con-ducted with different initial random seeds To evaluate theperformance of these algorithms six criteria are given inTable 3

Accelerating convergence speed and avoiding the localoptima have become two important and appealing goals inswarm intelligent search algorithms So as seen in Table 3we adopted criteria best mean and standard deviation toevaluate efficiency and accuracy of algorithms and adoptedcriteria Art Worst and SR to evaluate convergence speedeffectiveness and robustness of six algorithms

Table 3 Six criteria and their abbreviations

Criteria AbbreviationThe best value of optima found in 50 runs BestThe worst value of optima found in 50 runs WorstThe average value of optima found in 50 runs MeanThe standard deviations StdDevThe success rate of the results SRThe average reaching time Art

Specifically speaking SR provides very useful informa-tion about how stable an algorithm is Success is claimed ifan algorithm successfully gets a solution below a prespecifiedthreshold value with the maximum number of functionevaluations [15] So to calculate the success rate an erroraccuracy level 120576 = 10minus6 must be set (120576 = 10minus6 also usedin [16]) Thus we compared the result 119865 with the knownanalytical optima 119865lowast and consider 119865 to be ldquosuccessfulrdquo if thefollowing inequality holds

1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816

119865lowastlt 120576 119865lowast = 0

1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816 lt 120576 119865lowast = 0

(7)

The SR is a percentage value that is calculated as

SR =successful runs

runs (8)

Art is the average value of time once an algorithm gets asolution satisfying the formula (7) in 50-run computationsArt also provides very useful information about how fastan algorithm converges to certain accuracy or under thesame termination criterion which has important practicalsignificance

All algorithms have been tested in Matlab 2008a over thesame Lenovo A4600R computer with a Dual-Core 260GHzprocessor running Windows XP operating system over199Gb of memory

42 Experiments 1 Effect of Distance Measurements and FourParameters on WPA In order to study the effect of twodistance measures and four parameters on WPA differentmeasures and values of parameters were tested on typicalfunctions listed in Table 1 Each experiment WPA algorithmthat runs 50 times on each function and several criteriadescribed in Section 412 are used The experiment is con-ducted with the original coefficients shown in Table 9

421 Effect of Distance Measurements on the Performance ofWPA This subsection will investigate the performance ofdifferent distance measurements using functions with dif-ferent characteristics As is known to all Euclidean distance(ED) and Manhattan distance (MD) are the two most com-mon distance metrics in practical continuous optimizationIn the proposedWPA MD or ED can be adopted to measurethe distance between two wolves in the candidate solution

6 Mathematical Problems in Engineering

Table 4 Sensitivity analysis of distance measurements

Function Global extremum 119863 Distance Best Worst Mean StdDev SR Arts

Rosenbrock 119891min() = 0 2 MD 921119890 minus 11 324119890 minus 8 112119890 minus 8 118119890 minus 8 100 105165ED 426119890 minus 9 271119890 minus 7 127119890 minus 7 681119890 minus 8 100 371053

Colville 119891min() = 0 4 MD 562119890 minus 8 528119890 minus 7 249119890 minus 7 223119890 minus 7 100 468619ED 174119890 minus 7 170119890 minus 6 574119890 minus 7 370119890 minus 7 90 683220

Sphere 119891min() = 0 200 MD 320119890 minus 161 329119890 minus 144 207119890 minus 145 749119890 minus 145 100 115494ED 176119890 minus 160 336119890 minus 143 168119890 minus 144 751119890 minus 144 100 116825

Sumsquares 119891min() = 0 150 MD 156119890 minus 161 309119890 minus 144 179119890 minus 145 695119890 minus 145 100 85565ED 397119890 minus 160 224119890 minus 144 113119890 minus 145 500119890 minus 145 100 87109

Booth 119891min() = 0 2 MD 563119890 minus 12 115119890 minus 10 419119890 minus 11 332119890 minus 11 100 111074ED 108119890 minus 9 264119890 minus 8 116119890 minus 8 693119890 minus 9 100 405546

Bridge 119891max() = 30054 2 MD 30054 30054 30054 456119890 minus 16 100 11093ED 30054 30054 30054 456119890 minus 16 100 19541

Ackley 119891min() = 0 50 MD 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 193648ED 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 436884

Griewank 119891min() = 0 100 MD 0 01507 301119890 minus 3 00213 98 gt8771198903

ED 0 08350 00167 01181 92 gt135119890 + 4

space Therefore a discussion about their impacts on theperformance of WPA is needed

There are two wolves X119901

= (1199091199011

1199091199012

119909119901119863

) is theposition of wolf 119901X

119902= (1199091199021

1199091199022

119909119902119863

) is the positionof wolf 119902 and the ED and MD between them can berespectively calculated as formula (9) 119863 is the dimensionnumber of solution space

119871 ED (119901 119902) =119863

sum119889=1

(119909119901119889

minus 119909119902119889

)2

119871MD (119901 119902) =119863

sum119889=1

10038161003816100381610038161003816119909119901119889 minus 119909119902119889

10038161003816100381610038161003816

(9)

The statistical results obtained by WPA after 50-runcomputation are shown in Table 4 Firstly we note that WPAwithEuclidean distance (WPA ED)does not get 100 successrate on Colville (119863 = 4) and Griewank functions (119863 = 100)while WPA with Manhattan distance (WPA MD) does notget 100 success rate on Griewank functions (119863 = 100)which means that WPA ED and WPA MD with originalcoefficients still have the risk of premature convergence tolocal optima

As seen from Table 4 WPA is not very sensitive to twodistance measurements on most functions (RosenbrockSphere Sumsquares Booth and Ackley) and no matterwhich metric is used WPA can always get a good resultwith SR = 100 But for these functions comparing theresults between WPA MD and WPA ED in detail we canfind that WPA MD has shorter average reaching time (ARt)which means faster convergence speed to a certain accuracyThe reason may be that ED has the higher computationalcomplexityMeanwhileWPA MDhas better performance onother four criteria (best worst mean and StdDev) whichmeans better solution accuracy and robustness

Naturally because of its better efficiency precision androbustness WD is more suitable for WPA So the WPAalgorithm used in what follows is WPA MD

422 Effect of Four Parameters on the Performance of WPAIn this subsectionwe investigate the impact of the parameters119878 119871near 119879max and 120573 on the new algorithm 119878 is the stepcoefficient 119871near is the distance determinant coefficient 119879maxis the maximum number of repetitions in scouting behaviorand 120573 is the population renewing proportional coefficientThe parameters selection procedure is performed in a one-factor-at-a-time manner For each sensitivity analysis in thissection only one parameter is varied each time and theremaining parameters are kept at the values suggested by theoriginal estimate listed in Table 9 The interaction relationbetween parameters is assumed unimportant

Each time one of the WPA parameters is varied in a cer-tain interval to see which value within this internal will resultin the best performance Specifically theWPA algorithm alsoruns 50 times on each case

Table 5 shows the sensitivity analysis of the step coef-ficient 119878 All results are shown in the form of Mean plusmnStd (SR) The choice of interval [004 016] used in thisanalysis was motivated by the original Nelder-Mead simplexsearch procedure where a step coefficient greater than 004was suggested for general usage

Meanwhile based on detailed comparison of the resultson Rosenbrock Sphere and Bridge functions step coefficientis not sensitive to WPA and for Booth function there is atendency of better results with larger 119878 From Table 5 it isfound that a step coefficient setting at 012 returns the bestresult which has better Mean small Std and SR = 100 forall functions

Tables 6ndash8 analyze sensitivity of 119871near 119879max and 120573 Gen-erally speaking 119871near 119879max and 120573 are not sensitive to mostfunctions exceptGriewank function sinceGriewanknot only

Mathematical Problems in Engineering 7

Table5Sensitivityanalysisof

stepcoeffi

cient(

119878)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

69119890

minus8

plusmn4

3119890minus

82

7119890minus

8plusmn

35119890

minus8

11119890

minus8

plusmn9

1119890minus

93

2119890minus

9plusmn

27119890

minus9

50119890

minus9

plusmn5

7119890minus

93

2119890minus

9plusmn

37119890

minus9

12119890

minus9

plusmn1

6119890minus

9Colville

13119890

minus7

plusmn7

1119890minus

83

3119890minus

7plusmn

28119890

minus7(90)

26119890

minus7

plusmn1

9119890minus

72

3119890minus

7plusmn

14119890

minus7

35119890

minus7

plusmn2

5119890minus

79

5119890minus

7plusmn

10119890

minus6(80)

14119890

minus6

plusmn1

5119890minus

6(50)

Sphere

23119890

minus14

5plusmn

71119890

minus14

56

6119890minus

152

plusmn2

1119890minus

151

21119890

minus14

6plusmn

45119890

minus14

63

9119890minus

146

plusmn1

2119890minus

145

12119890

minus14

5plusmn

34119890

minus14

51

7119890minus

146

plusmn5

3119890minus

146

22119890

minus14

9plusmn

68119890

minus14

9Sumsquares

98119890

minus14

5plusmn

31119890

minus14

43

1119890minus

146

plusmn8

4119890minus

146

81119890

minus14

7plusmn

26119890

minus14

64

8119890minus

146

plusmn1

0119890minus

145

38119890

minus15

2plusmn

79119890

minus15

23

4119890minus

147

plusmn1

1119890minus

146

12119890

minus14

7plusmn

39119890

minus14

7Bo

oth

54119890

minus7

plusmn3

3119890minus

71

6119890minus

9plusmn

11119890

minus9

32119890

minus11

plusmn1

6119890minus

111

3119890minus

12plusmn

91119890

minus13

13119890

minus13

plusmn1

2119890minus

133

9119890minus

15plusmn

18119890

minus15

12119890

minus16

plusmn5

8119890minus

17Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

025

plusmn0

53(80)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

0plusmn

00

06plusmn

019

(92)

020

plusmn0

42(86)

8 Mathematical Problems in Engineering

Table6Sensitivityanalysisof

distance

determ

inantcoefficient(

119871near)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

44119890

minus8

plusmn6

5119890minus

82

3119890minus

8plusmn

37119890

minus8

34119890

minus9

plusmn4

8119890minus

93

0119890minus

8plusmn

29119890

minus8

19119890

minus8

plusmn2

4119890minus

82

4119890minus

8plusmn

47119890

minus8

29119890

minus8

plusmn5

3119890minus

8Colville

20119890

minus7

plusmn9

9119890minus

82

6119890minus

7plusmn

16119890

minus7

35119890

minus7

plusmn2

6119890minus

72

3119890minus

7plusmn

15119890

minus7

12119890

minus7

plusmn3

4119890minus

82

8119890minus

7plusmn

19119890

minus7

14119890

minus7

plusmn6

9119890minus

8Sphere

68119890

minus14

6plusmn

20119890

minus14

51

9119890minus

146

plusmn6

2119890minus

146

17119890

minus14

5plusmn

43119890

minus14

52

6119890minus

148

plusmn8

3119890minus

148

36119890

minus14

6plusmn

11119890

minus14

53

7119890minus

151

plusmn1

1119890minus

150

53119890

minus14

9plusmn

17119890

minus14

8Sumsquares1

1119890

minus14

7plusmn

34119890

minus14

71

0119890minus

146

plusmn3

3119890minus

146

37119890

minus15

1plusmn

89119890

minus15

16

2119890minus

146

plusmn1

9119890minus

145

62119890

minus15

2plusmn

19119890

minus15

11

22119890

minus14

5plusmn

29119890

minus14

51

3119890minus

148

plusmn4

0119890minus

148

Booth

26119890

minus11

plusmn1

3119890minus

112

9119890minus

11plusmn

19119890

minus11

24119890

minus11

plusmn1

6119890minus

113

1119890minus

11plusmn

18119890

minus01

12

4119890minus

11plusmn

13119890

minus11

31119890

minus11

plusmn2

1119890minus

111

0119890minus

10plusmn

13119890

minus10

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

014

plusmn0

43(90)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

159

119890minus

15plusmn

149

119890minus

158

9119890minus

16plusmn

0Grie

wank

008

plusmn0

26(90)

10119890

minus3

plusmn0

02(96)

0plusmn

00

plusmn0

0plusmn

00

10plusmn

033

(92)

0plusmn

0

Mathematical Problems in Engineering 9

Table7Sensitivityanalysisof

them

axim

umnu

mbero

frepetition

sinscou

tingbehavior

(119879max)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)6

810

1214

1618

Rosenb

rock

24119890

minus8

plusmn2

6119890minus

88

4119890minus

9plusmn

80119890

minus9

13119890

minus8

plusmn1

3119890minus

81

4119890minus

8plusmn

10119890

minus8

20119890

minus8

plusmn1

9119890minus

82

1119890minus

8plusmn

25119890

minus8

12119890

minus8

plusmn8

9119890minus

9Colville

48119890

minus7

plusmn2

2119890minus

73

4119890minus

7plusmn

18119890

minus7

15119890

minus7

plusmn1

2119890minus

73

8119890minus

7plusmn

20119890

minus7

36119890

minus7

plusmn3

7119890minus

7(96)

34119890

minus7

plusmn2

5119890minus

72

6119890minus

7plusmn

15119890

minus7

Sphere

71119890

minus14

7plusmn

22119890

minus14

64

5119890minus

146

plusmn9

0119890minus

146

78119890

minus14

6plusmn

23119890

minus14

51

9119890minus

148

plusmn5

3119890minus

148

57119890

minus14

8plusmn

13119890

minus14

76

9119890minus

145

plusmn2

2119890minus

144

36119890

minus14

7plusmn

11119890

minus14

6Sumsquares

41119890

minus14

6plusmn

13119890

minus14

52

4119890minus

149

plusmn4

8119890minus

149

42119890

minus14

9plusmn

13119890

minus14

88

3119890minus

150

plusmn2

6119890minus

149

85119890

minus14

7plusmn

27119890

minus14

65

4119890minus

146

plusmn9

0119890minus

146

14119890

minus15

1plusmn

44119890

minus15

1Bo

oth

32119890

minus11

plusmn2

9119890minus

114

2119890minus

11plusmn

27119890

minus11

25119890

minus11

plusmn1

5119890minus

112

1119890minus

11plusmn

15119890

minus11

32119890

minus11

plusmn2

5119890minus

112

6119890minus

11plusmn

18119890

minus11

26119890

minus11

plusmn2

7119890minus

11Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

08

9119890minus

16plusmn

0Grie

wank

010

plusmn0

33(92)

0plusmn

01

0119890minus

3plusmn

002

(98)

009

plusmn0

31(88)

0plusmn

00

09plusmn

029

(94)

83119890

minus4

plusmn0

02(98)

10 Mathematical Problems in Engineering

Table8Sensitivityanalysisof

popu

latio

nrenewingprop

ortio

nalcoefficient(

120573)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)2

34

56

78

Rosenb

rock

10119890

minus8

plusmn9

2119890minus

98

7119890minus

9plusmn

76119890

minus9

12119890

minus8

plusmn1

0119890minus

88

6119890minus

9plusmn

83119890

minus9

14119890

minus8

plusmn1

3119890minus

89

9119890minus

9plusmn

98119890

minus9

11119890

minus8

plusmn1

2119890minus

9Colville

32119890

minus8

plusmn1

8119890

minus8

14119890

minus7

plusmn1

3119890minus

71

2119890minus

7plusmn

59119890

minus8

14119890

minus7

plusmn9

4119890minus

83

0119890minus

7plusmn

69119890

minus8

39119890

minus7

plusmn1

7119890minus

78

6119890minus

7plusmn

40119890

minus7(80)

Sphere

19119890

minus16

6plusmn

05

2119890minus

158

plusmn1

6119890minus

157

29119890

minus15

3plusmn

92119890

minus15

34

3119890minus

149

plusmn1

3119890minus

148

79119890

minus13

9plusmn

25119890

minus13

88

3119890minus

134

plusmn1

8119890minus

133

34119890

minus12

6plusmn

80119890

minus12

6Sumsquares

28119890

minus16

7plusmn

01

4119890minus

157

plusmn4

3119890minus

157

28119890

minus15

5plusmn

45119890

minus15

58

3119890minus

146

plusmn1

8119890minus

145

69119890

minus14

3plusmn

17119890

minus14

25

3119890minus

143

plusmn1

3119890minus

142

33119890

minus12

7plusmn

10119890

minus12

6Bo

oth

81119890

minus11

plusmn1

3119890minus

102

5119890minus

11plusmn

17119890

minus11

19119890

minus11

plusmn1

2119890minus

112

5119890minus

11plusmn

17119890

minus01

12

5119890minus

11plusmn

15119890

minus11

23119890

minus11

plusmn1

5119890minus

112

3119890minus

11plusmn

14119890

minus11

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

019

plusmn0

41(86)

0plusmn

01

2119890minus

3plusmn

031

(96)

Mathematical Problems in Engineering 11

Table 9 Best suggestions for WPA parameters

No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2

020

2

0

xy

minusf(xy) minus1000

minus2000

minus3000

minus4000

minus2minus2

minus4minus4

(a)

x

y

0 1 2

0

05

1

15

2

minus05

minus15

minus2minus2

minus1

minus1

(b)

Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines

is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized

Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions

Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions

So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2

43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface

As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2

As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909

1 1199092) = (1 1)

But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction

On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909

1 1199092 1199093 1199094) = (1 1 1 1)

Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function

Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909

1 1199092 119909

119898) =

(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5

As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and

12 Mathematical Problems in Engineering

Table10Statistic

alresults

of50

runs

obtained

byGAP

SOA

FSAA

BCFAand

WPA

algorithm

s

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Rosenb

rock

119891min

(119909)

=0

2UN

GA

178

119890minus

1000373

000

91000

9210

gt7598

323

PSO

226

119890minus

115

89119890

minus7

107

119890minus

71

30119890

minus7

100

07444

AFS

A110

eminus13

111

119890minus

92

34119890

minus10

262

119890minus

10100

20578

ABC

599

119890minus

6000

998

61119890

minus4

00015

0gt3910

297

FA6

28119890

minus13

629

eminus10

186eminus

10162eminus

10100

3312

56WPA

349

119890minus

112

34119890

minus8

509

119890minus

94

34119890

minus9

100

66333

Colville

119891min

(119909)

=0

4UN

GA

00022

03343

01272

01062

0gt12

2119890+

3PS

O1

29119890

minus6

346

119890minus

45

06119890

minus5

671

119890minus

50

gt114

0869

AFS

A366

eminus8

891

119890minus

73

16119890

minus7

232

119890minus

7100

4018

07ABC

00103

05337

01871

01232

0gt3844193

FA2

41119890

minus7

369

119890minus

56

62119890

minus6

807

119890minus

68

gt3

14119890

+3

WPA

471

119890minus

8372

eminus7

125eminus

7697eminus

8100

27405

4

Sphere

119891min

(119909)

=0

200

US

GA

156

119890+

51

81119890

+5

171

119890+

55

78119890

+3

0gt4

44119890

+4

PSO

10361

15520

12883

01206

0gt2719

201

AFS

A5

12119890

+5

579

119890+

55

51119890

+5

163

119890+

40

gt7

41119890

+3

ABC

000

4112

521

004

4401773

0gt44

29045

FA01432

02327

01865

00199

0gt8

34119890

+3

WPA

149eminus

172

241

eminus165

156eminus

166

0100

61729

Sumsquares

119891min

(119909)

=0

150

US

GA

593

119890+

47

15+

46

63119890

+4

288

119890+

30

gt3

16119890

+4

PSO

397098

911145

559050

104165

0gt2325464

AFS

A1

43119890

+5

179

119890+

51

64119890

+5

958

119890+

30

gt7

36119890

+3

ABC

171

119890minus

500017

199

119890minus

43

36119890

minus4

0gt4351848

FA89920

998861

405721

192743

0gt6

88119890

+3

WPA

268

eminus172

547

eminus166

262eminus

167

0100

65954

Booth

119891min

(119909)

=0

2MS

GA

455

119890minus

114

55119890

minus11

455

119890minus

110

100

12621

PSO

122

119890minus

122

41119890

minus8

280

119890minus

94

52119890

minus9

100

020

79AFS

A3

02119890

minus12

145

119890minus

94

61119890

minus10

408

119890minus

10100

44329

ABC

605

eminus20

141

eminus17

463eminus

18414eminus

18100

04175

FA1

80119890

minus12

439

119890minus

91

18119890

minus9

111

119890minus

9100

379191

WPA

822

119890minus

157

05119890

minus13

121

119890minus

131

19119890

minus13

100

69339

Bridge

119891max

(119909)

=3

0054

2MN

GA

300

54300

54300

541

35119890

minus15

100

01927

PSO

300

54300

54300

544

84119890

minus8

100

009

29AFS

A300

54300

4730052

169

119890minus

412

gt8

01119890

+3

ABC

300

54300

54300

543

59119890

minus15

100

00932

FA300

54300

54300

543

11119890

minus10

100

227230

WPA

300

54300

54300

54358eminus

15100

01742

Ackley

119891min

(119909)

=0

50MN

GA

114570

126095

1216

1202719

0gt10

4119890+

4PS

O004

6917401

06846

06344

0gt1925522

AFS

A2016

0020600

9204229

01009

0gt9

80119890

+3

ABC

200085

200025

200061

00014

0gt5963841

FA00101

00209

00160

00021

0gt4

28119890

+3

WPA

888

eminus16

444

eminus15

110eminus

15852eminus

16100

79476

Mathematical Problems in Engineering 13

Table10C

ontin

ued

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Grie

wank

119891min

(119909)

=0

100

MN

GA

3174

525

3996

376

3634174

172922

0gt2

07119890

+4

PSO

00029

00082

00052

00011

0gt3670

080

AFS

A2

05119890

+3

255

119890+

32

33119890

+3

1096

821

0gt6

51119890

+3

ABC

895

119890minus

7000

432

26119890

minus4

781

119890minus

42

gt6209561

FA000

6800118

000

9100011

0gt5

72119890

+3

WPA

00

00

100

145338

14 Mathematical Problems in Engineering

0

100

10

0

1

xy

minusf(xy)

minus1

minus2

minus3

minus10minus10

times106

(a)

minus5

minus5

minus10minus10

x

y

0 5 10

0

5

10

(b)

Figure 3 Colville function (1199091

= 1199093 1199092

= 1199094) (a) surface plot and (b) contour lines

0100

0

1000

05

1

15

2

xy

minus100minus100

f(xy)

times104

(a)

x

y

0 50 100

0

50

100

minus100minus100

minus50

minus50

(b)

Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines

minus100

minus200

minus300

minus10minus10

0

100

10

0

xy

minusf(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines

Mathematical Problems in Engineering 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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 5

Table 2 The list of various methods used in the paper

Method Authors and referencesGenetic algorithm (GA) Goldberg [14]Particle swarm optimization algorithm(PSO) Kennedy and Eberhart [7]

Artificial fish school algorithm (ASFA) Li et al [9]Artificial bee colony algorithm (ABC) Karaboga [10]Firefly algorithm (FA) Yang [11]

Another group of test problems is separable or nonseparablefunctions A 119901-variable separable function can be expressedas the sum of 119901 functions of one variable such as Sumsquaresand Rastrigin Nonseparable functions cannot be written inthis form such as Bridge Rosenbrock Ackley andGriewankBecause nonseparable functions have interrelation amongtheir variable these functions are more difficult than theseparable functions

In Table 1 characteristics of each function are given underthe column titled 119862 In this column 119872 means that thefunction is multimodal while 119880 means that the functionis unimodal If the function is separable abbreviation 119878 isused to indicate this specification Letter 119873 refers to that thefunction is nonseparable As seen from Table 1 4 functionsare multimodal 4 functions are unimodal 3 functions areseparable and 5 functions are nonseparable

The variety of functions forms and dimensions makeit possible to fairly assess the robustness of the proposedalgorithms within limit iteration Many of these functionsallow a choice of dimension and an input dimension rangingfrom 2 to 200 for test functions is given Dimensions of theproblems that we used can be found under the column titled119863 Besides initial ranges formulas and global optimumvalues of these functions are also given in Table 1

412 Experimental Settings In this subsection experimentalsettings are given Firstly in order to fully compare the perfor-mance of different algorithms we take the simulation underthe same situation So the values of the common parametersused in each algorithm such as population size and evaluationnumber were chosen to be the same Population size was100 and the maximum evaluation number was 2000 forall algorithms on all functions Additionally we follow theparameter settings in the original paper of GA PSO AFSAABC and FA see Table 2

For each experiment 50 independent runs were con-ducted with different initial random seeds To evaluate theperformance of these algorithms six criteria are given inTable 3

Accelerating convergence speed and avoiding the localoptima have become two important and appealing goals inswarm intelligent search algorithms So as seen in Table 3we adopted criteria best mean and standard deviation toevaluate efficiency and accuracy of algorithms and adoptedcriteria Art Worst and SR to evaluate convergence speedeffectiveness and robustness of six algorithms

Table 3 Six criteria and their abbreviations

Criteria AbbreviationThe best value of optima found in 50 runs BestThe worst value of optima found in 50 runs WorstThe average value of optima found in 50 runs MeanThe standard deviations StdDevThe success rate of the results SRThe average reaching time Art

Specifically speaking SR provides very useful informa-tion about how stable an algorithm is Success is claimed ifan algorithm successfully gets a solution below a prespecifiedthreshold value with the maximum number of functionevaluations [15] So to calculate the success rate an erroraccuracy level 120576 = 10minus6 must be set (120576 = 10minus6 also usedin [16]) Thus we compared the result 119865 with the knownanalytical optima 119865lowast and consider 119865 to be ldquosuccessfulrdquo if thefollowing inequality holds

1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816

119865lowastlt 120576 119865lowast = 0

1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816 lt 120576 119865lowast = 0

(7)

The SR is a percentage value that is calculated as

SR =successful runs

runs (8)

Art is the average value of time once an algorithm gets asolution satisfying the formula (7) in 50-run computationsArt also provides very useful information about how fastan algorithm converges to certain accuracy or under thesame termination criterion which has important practicalsignificance

All algorithms have been tested in Matlab 2008a over thesame Lenovo A4600R computer with a Dual-Core 260GHzprocessor running Windows XP operating system over199Gb of memory

42 Experiments 1 Effect of Distance Measurements and FourParameters on WPA In order to study the effect of twodistance measures and four parameters on WPA differentmeasures and values of parameters were tested on typicalfunctions listed in Table 1 Each experiment WPA algorithmthat runs 50 times on each function and several criteriadescribed in Section 412 are used The experiment is con-ducted with the original coefficients shown in Table 9

421 Effect of Distance Measurements on the Performance ofWPA This subsection will investigate the performance ofdifferent distance measurements using functions with dif-ferent characteristics As is known to all Euclidean distance(ED) and Manhattan distance (MD) are the two most com-mon distance metrics in practical continuous optimizationIn the proposedWPA MD or ED can be adopted to measurethe distance between two wolves in the candidate solution

6 Mathematical Problems in Engineering

Table 4 Sensitivity analysis of distance measurements

Function Global extremum 119863 Distance Best Worst Mean StdDev SR Arts

Rosenbrock 119891min() = 0 2 MD 921119890 minus 11 324119890 minus 8 112119890 minus 8 118119890 minus 8 100 105165ED 426119890 minus 9 271119890 minus 7 127119890 minus 7 681119890 minus 8 100 371053

Colville 119891min() = 0 4 MD 562119890 minus 8 528119890 minus 7 249119890 minus 7 223119890 minus 7 100 468619ED 174119890 minus 7 170119890 minus 6 574119890 minus 7 370119890 minus 7 90 683220

Sphere 119891min() = 0 200 MD 320119890 minus 161 329119890 minus 144 207119890 minus 145 749119890 minus 145 100 115494ED 176119890 minus 160 336119890 minus 143 168119890 minus 144 751119890 minus 144 100 116825

Sumsquares 119891min() = 0 150 MD 156119890 minus 161 309119890 minus 144 179119890 minus 145 695119890 minus 145 100 85565ED 397119890 minus 160 224119890 minus 144 113119890 minus 145 500119890 minus 145 100 87109

Booth 119891min() = 0 2 MD 563119890 minus 12 115119890 minus 10 419119890 minus 11 332119890 minus 11 100 111074ED 108119890 minus 9 264119890 minus 8 116119890 minus 8 693119890 minus 9 100 405546

Bridge 119891max() = 30054 2 MD 30054 30054 30054 456119890 minus 16 100 11093ED 30054 30054 30054 456119890 minus 16 100 19541

Ackley 119891min() = 0 50 MD 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 193648ED 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 436884

Griewank 119891min() = 0 100 MD 0 01507 301119890 minus 3 00213 98 gt8771198903

ED 0 08350 00167 01181 92 gt135119890 + 4

space Therefore a discussion about their impacts on theperformance of WPA is needed

There are two wolves X119901

= (1199091199011

1199091199012

119909119901119863

) is theposition of wolf 119901X

119902= (1199091199021

1199091199022

119909119902119863

) is the positionof wolf 119902 and the ED and MD between them can berespectively calculated as formula (9) 119863 is the dimensionnumber of solution space

119871 ED (119901 119902) =119863

sum119889=1

(119909119901119889

minus 119909119902119889

)2

119871MD (119901 119902) =119863

sum119889=1

10038161003816100381610038161003816119909119901119889 minus 119909119902119889

10038161003816100381610038161003816

(9)

The statistical results obtained by WPA after 50-runcomputation are shown in Table 4 Firstly we note that WPAwithEuclidean distance (WPA ED)does not get 100 successrate on Colville (119863 = 4) and Griewank functions (119863 = 100)while WPA with Manhattan distance (WPA MD) does notget 100 success rate on Griewank functions (119863 = 100)which means that WPA ED and WPA MD with originalcoefficients still have the risk of premature convergence tolocal optima

As seen from Table 4 WPA is not very sensitive to twodistance measurements on most functions (RosenbrockSphere Sumsquares Booth and Ackley) and no matterwhich metric is used WPA can always get a good resultwith SR = 100 But for these functions comparing theresults between WPA MD and WPA ED in detail we canfind that WPA MD has shorter average reaching time (ARt)which means faster convergence speed to a certain accuracyThe reason may be that ED has the higher computationalcomplexityMeanwhileWPA MDhas better performance onother four criteria (best worst mean and StdDev) whichmeans better solution accuracy and robustness

Naturally because of its better efficiency precision androbustness WD is more suitable for WPA So the WPAalgorithm used in what follows is WPA MD

422 Effect of Four Parameters on the Performance of WPAIn this subsectionwe investigate the impact of the parameters119878 119871near 119879max and 120573 on the new algorithm 119878 is the stepcoefficient 119871near is the distance determinant coefficient 119879maxis the maximum number of repetitions in scouting behaviorand 120573 is the population renewing proportional coefficientThe parameters selection procedure is performed in a one-factor-at-a-time manner For each sensitivity analysis in thissection only one parameter is varied each time and theremaining parameters are kept at the values suggested by theoriginal estimate listed in Table 9 The interaction relationbetween parameters is assumed unimportant

Each time one of the WPA parameters is varied in a cer-tain interval to see which value within this internal will resultin the best performance Specifically theWPA algorithm alsoruns 50 times on each case

Table 5 shows the sensitivity analysis of the step coef-ficient 119878 All results are shown in the form of Mean plusmnStd (SR) The choice of interval [004 016] used in thisanalysis was motivated by the original Nelder-Mead simplexsearch procedure where a step coefficient greater than 004was suggested for general usage

Meanwhile based on detailed comparison of the resultson Rosenbrock Sphere and Bridge functions step coefficientis not sensitive to WPA and for Booth function there is atendency of better results with larger 119878 From Table 5 it isfound that a step coefficient setting at 012 returns the bestresult which has better Mean small Std and SR = 100 forall functions

Tables 6ndash8 analyze sensitivity of 119871near 119879max and 120573 Gen-erally speaking 119871near 119879max and 120573 are not sensitive to mostfunctions exceptGriewank function sinceGriewanknot only

Mathematical Problems in Engineering 7

Table5Sensitivityanalysisof

stepcoeffi

cient(

119878)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

69119890

minus8

plusmn4

3119890minus

82

7119890minus

8plusmn

35119890

minus8

11119890

minus8

plusmn9

1119890minus

93

2119890minus

9plusmn

27119890

minus9

50119890

minus9

plusmn5

7119890minus

93

2119890minus

9plusmn

37119890

minus9

12119890

minus9

plusmn1

6119890minus

9Colville

13119890

minus7

plusmn7

1119890minus

83

3119890minus

7plusmn

28119890

minus7(90)

26119890

minus7

plusmn1

9119890minus

72

3119890minus

7plusmn

14119890

minus7

35119890

minus7

plusmn2

5119890minus

79

5119890minus

7plusmn

10119890

minus6(80)

14119890

minus6

plusmn1

5119890minus

6(50)

Sphere

23119890

minus14

5plusmn

71119890

minus14

56

6119890minus

152

plusmn2

1119890minus

151

21119890

minus14

6plusmn

45119890

minus14

63

9119890minus

146

plusmn1

2119890minus

145

12119890

minus14

5plusmn

34119890

minus14

51

7119890minus

146

plusmn5

3119890minus

146

22119890

minus14

9plusmn

68119890

minus14

9Sumsquares

98119890

minus14

5plusmn

31119890

minus14

43

1119890minus

146

plusmn8

4119890minus

146

81119890

minus14

7plusmn

26119890

minus14

64

8119890minus

146

plusmn1

0119890minus

145

38119890

minus15

2plusmn

79119890

minus15

23

4119890minus

147

plusmn1

1119890minus

146

12119890

minus14

7plusmn

39119890

minus14

7Bo

oth

54119890

minus7

plusmn3

3119890minus

71

6119890minus

9plusmn

11119890

minus9

32119890

minus11

plusmn1

6119890minus

111

3119890minus

12plusmn

91119890

minus13

13119890

minus13

plusmn1

2119890minus

133

9119890minus

15plusmn

18119890

minus15

12119890

minus16

plusmn5

8119890minus

17Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

025

plusmn0

53(80)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

0plusmn

00

06plusmn

019

(92)

020

plusmn0

42(86)

8 Mathematical Problems in Engineering

Table6Sensitivityanalysisof

distance

determ

inantcoefficient(

119871near)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

44119890

minus8

plusmn6

5119890minus

82

3119890minus

8plusmn

37119890

minus8

34119890

minus9

plusmn4

8119890minus

93

0119890minus

8plusmn

29119890

minus8

19119890

minus8

plusmn2

4119890minus

82

4119890minus

8plusmn

47119890

minus8

29119890

minus8

plusmn5

3119890minus

8Colville

20119890

minus7

plusmn9

9119890minus

82

6119890minus

7plusmn

16119890

minus7

35119890

minus7

plusmn2

6119890minus

72

3119890minus

7plusmn

15119890

minus7

12119890

minus7

plusmn3

4119890minus

82

8119890minus

7plusmn

19119890

minus7

14119890

minus7

plusmn6

9119890minus

8Sphere

68119890

minus14

6plusmn

20119890

minus14

51

9119890minus

146

plusmn6

2119890minus

146

17119890

minus14

5plusmn

43119890

minus14

52

6119890minus

148

plusmn8

3119890minus

148

36119890

minus14

6plusmn

11119890

minus14

53

7119890minus

151

plusmn1

1119890minus

150

53119890

minus14

9plusmn

17119890

minus14

8Sumsquares1

1119890

minus14

7plusmn

34119890

minus14

71

0119890minus

146

plusmn3

3119890minus

146

37119890

minus15

1plusmn

89119890

minus15

16

2119890minus

146

plusmn1

9119890minus

145

62119890

minus15

2plusmn

19119890

minus15

11

22119890

minus14

5plusmn

29119890

minus14

51

3119890minus

148

plusmn4

0119890minus

148

Booth

26119890

minus11

plusmn1

3119890minus

112

9119890minus

11plusmn

19119890

minus11

24119890

minus11

plusmn1

6119890minus

113

1119890minus

11plusmn

18119890

minus01

12

4119890minus

11plusmn

13119890

minus11

31119890

minus11

plusmn2

1119890minus

111

0119890minus

10plusmn

13119890

minus10

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

014

plusmn0

43(90)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

159

119890minus

15plusmn

149

119890minus

158

9119890minus

16plusmn

0Grie

wank

008

plusmn0

26(90)

10119890

minus3

plusmn0

02(96)

0plusmn

00

plusmn0

0plusmn

00

10plusmn

033

(92)

0plusmn

0

Mathematical Problems in Engineering 9

Table7Sensitivityanalysisof

them

axim

umnu

mbero

frepetition

sinscou

tingbehavior

(119879max)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)6

810

1214

1618

Rosenb

rock

24119890

minus8

plusmn2

6119890minus

88

4119890minus

9plusmn

80119890

minus9

13119890

minus8

plusmn1

3119890minus

81

4119890minus

8plusmn

10119890

minus8

20119890

minus8

plusmn1

9119890minus

82

1119890minus

8plusmn

25119890

minus8

12119890

minus8

plusmn8

9119890minus

9Colville

48119890

minus7

plusmn2

2119890minus

73

4119890minus

7plusmn

18119890

minus7

15119890

minus7

plusmn1

2119890minus

73

8119890minus

7plusmn

20119890

minus7

36119890

minus7

plusmn3

7119890minus

7(96)

34119890

minus7

plusmn2

5119890minus

72

6119890minus

7plusmn

15119890

minus7

Sphere

71119890

minus14

7plusmn

22119890

minus14

64

5119890minus

146

plusmn9

0119890minus

146

78119890

minus14

6plusmn

23119890

minus14

51

9119890minus

148

plusmn5

3119890minus

148

57119890

minus14

8plusmn

13119890

minus14

76

9119890minus

145

plusmn2

2119890minus

144

36119890

minus14

7plusmn

11119890

minus14

6Sumsquares

41119890

minus14

6plusmn

13119890

minus14

52

4119890minus

149

plusmn4

8119890minus

149

42119890

minus14

9plusmn

13119890

minus14

88

3119890minus

150

plusmn2

6119890minus

149

85119890

minus14

7plusmn

27119890

minus14

65

4119890minus

146

plusmn9

0119890minus

146

14119890

minus15

1plusmn

44119890

minus15

1Bo

oth

32119890

minus11

plusmn2

9119890minus

114

2119890minus

11plusmn

27119890

minus11

25119890

minus11

plusmn1

5119890minus

112

1119890minus

11plusmn

15119890

minus11

32119890

minus11

plusmn2

5119890minus

112

6119890minus

11plusmn

18119890

minus11

26119890

minus11

plusmn2

7119890minus

11Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

08

9119890minus

16plusmn

0Grie

wank

010

plusmn0

33(92)

0plusmn

01

0119890minus

3plusmn

002

(98)

009

plusmn0

31(88)

0plusmn

00

09plusmn

029

(94)

83119890

minus4

plusmn0

02(98)

10 Mathematical Problems in Engineering

Table8Sensitivityanalysisof

popu

latio

nrenewingprop

ortio

nalcoefficient(

120573)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)2

34

56

78

Rosenb

rock

10119890

minus8

plusmn9

2119890minus

98

7119890minus

9plusmn

76119890

minus9

12119890

minus8

plusmn1

0119890minus

88

6119890minus

9plusmn

83119890

minus9

14119890

minus8

plusmn1

3119890minus

89

9119890minus

9plusmn

98119890

minus9

11119890

minus8

plusmn1

2119890minus

9Colville

32119890

minus8

plusmn1

8119890

minus8

14119890

minus7

plusmn1

3119890minus

71

2119890minus

7plusmn

59119890

minus8

14119890

minus7

plusmn9

4119890minus

83

0119890minus

7plusmn

69119890

minus8

39119890

minus7

plusmn1

7119890minus

78

6119890minus

7plusmn

40119890

minus7(80)

Sphere

19119890

minus16

6plusmn

05

2119890minus

158

plusmn1

6119890minus

157

29119890

minus15

3plusmn

92119890

minus15

34

3119890minus

149

plusmn1

3119890minus

148

79119890

minus13

9plusmn

25119890

minus13

88

3119890minus

134

plusmn1

8119890minus

133

34119890

minus12

6plusmn

80119890

minus12

6Sumsquares

28119890

minus16

7plusmn

01

4119890minus

157

plusmn4

3119890minus

157

28119890

minus15

5plusmn

45119890

minus15

58

3119890minus

146

plusmn1

8119890minus

145

69119890

minus14

3plusmn

17119890

minus14

25

3119890minus

143

plusmn1

3119890minus

142

33119890

minus12

7plusmn

10119890

minus12

6Bo

oth

81119890

minus11

plusmn1

3119890minus

102

5119890minus

11plusmn

17119890

minus11

19119890

minus11

plusmn1

2119890minus

112

5119890minus

11plusmn

17119890

minus01

12

5119890minus

11plusmn

15119890

minus11

23119890

minus11

plusmn1

5119890minus

112

3119890minus

11plusmn

14119890

minus11

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

019

plusmn0

41(86)

0plusmn

01

2119890minus

3plusmn

031

(96)

Mathematical Problems in Engineering 11

Table 9 Best suggestions for WPA parameters

No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2

020

2

0

xy

minusf(xy) minus1000

minus2000

minus3000

minus4000

minus2minus2

minus4minus4

(a)

x

y

0 1 2

0

05

1

15

2

minus05

minus15

minus2minus2

minus1

minus1

(b)

Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines

is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized

Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions

Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions

So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2

43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface

As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2

As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909

1 1199092) = (1 1)

But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction

On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909

1 1199092 1199093 1199094) = (1 1 1 1)

Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function

Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909

1 1199092 119909

119898) =

(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5

As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and

12 Mathematical Problems in Engineering

Table10Statistic

alresults

of50

runs

obtained

byGAP

SOA

FSAA

BCFAand

WPA

algorithm

s

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Rosenb

rock

119891min

(119909)

=0

2UN

GA

178

119890minus

1000373

000

91000

9210

gt7598

323

PSO

226

119890minus

115

89119890

minus7

107

119890minus

71

30119890

minus7

100

07444

AFS

A110

eminus13

111

119890minus

92

34119890

minus10

262

119890minus

10100

20578

ABC

599

119890minus

6000

998

61119890

minus4

00015

0gt3910

297

FA6

28119890

minus13

629

eminus10

186eminus

10162eminus

10100

3312

56WPA

349

119890minus

112

34119890

minus8

509

119890minus

94

34119890

minus9

100

66333

Colville

119891min

(119909)

=0

4UN

GA

00022

03343

01272

01062

0gt12

2119890+

3PS

O1

29119890

minus6

346

119890minus

45

06119890

minus5

671

119890minus

50

gt114

0869

AFS

A366

eminus8

891

119890minus

73

16119890

minus7

232

119890minus

7100

4018

07ABC

00103

05337

01871

01232

0gt3844193

FA2

41119890

minus7

369

119890minus

56

62119890

minus6

807

119890minus

68

gt3

14119890

+3

WPA

471

119890minus

8372

eminus7

125eminus

7697eminus

8100

27405

4

Sphere

119891min

(119909)

=0

200

US

GA

156

119890+

51

81119890

+5

171

119890+

55

78119890

+3

0gt4

44119890

+4

PSO

10361

15520

12883

01206

0gt2719

201

AFS

A5

12119890

+5

579

119890+

55

51119890

+5

163

119890+

40

gt7

41119890

+3

ABC

000

4112

521

004

4401773

0gt44

29045

FA01432

02327

01865

00199

0gt8

34119890

+3

WPA

149eminus

172

241

eminus165

156eminus

166

0100

61729

Sumsquares

119891min

(119909)

=0

150

US

GA

593

119890+

47

15+

46

63119890

+4

288

119890+

30

gt3

16119890

+4

PSO

397098

911145

559050

104165

0gt2325464

AFS

A1

43119890

+5

179

119890+

51

64119890

+5

958

119890+

30

gt7

36119890

+3

ABC

171

119890minus

500017

199

119890minus

43

36119890

minus4

0gt4351848

FA89920

998861

405721

192743

0gt6

88119890

+3

WPA

268

eminus172

547

eminus166

262eminus

167

0100

65954

Booth

119891min

(119909)

=0

2MS

GA

455

119890minus

114

55119890

minus11

455

119890minus

110

100

12621

PSO

122

119890minus

122

41119890

minus8

280

119890minus

94

52119890

minus9

100

020

79AFS

A3

02119890

minus12

145

119890minus

94

61119890

minus10

408

119890minus

10100

44329

ABC

605

eminus20

141

eminus17

463eminus

18414eminus

18100

04175

FA1

80119890

minus12

439

119890minus

91

18119890

minus9

111

119890minus

9100

379191

WPA

822

119890minus

157

05119890

minus13

121

119890minus

131

19119890

minus13

100

69339

Bridge

119891max

(119909)

=3

0054

2MN

GA

300

54300

54300

541

35119890

minus15

100

01927

PSO

300

54300

54300

544

84119890

minus8

100

009

29AFS

A300

54300

4730052

169

119890minus

412

gt8

01119890

+3

ABC

300

54300

54300

543

59119890

minus15

100

00932

FA300

54300

54300

543

11119890

minus10

100

227230

WPA

300

54300

54300

54358eminus

15100

01742

Ackley

119891min

(119909)

=0

50MN

GA

114570

126095

1216

1202719

0gt10

4119890+

4PS

O004

6917401

06846

06344

0gt1925522

AFS

A2016

0020600

9204229

01009

0gt9

80119890

+3

ABC

200085

200025

200061

00014

0gt5963841

FA00101

00209

00160

00021

0gt4

28119890

+3

WPA

888

eminus16

444

eminus15

110eminus

15852eminus

16100

79476

Mathematical Problems in Engineering 13

Table10C

ontin

ued

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Grie

wank

119891min

(119909)

=0

100

MN

GA

3174

525

3996

376

3634174

172922

0gt2

07119890

+4

PSO

00029

00082

00052

00011

0gt3670

080

AFS

A2

05119890

+3

255

119890+

32

33119890

+3

1096

821

0gt6

51119890

+3

ABC

895

119890minus

7000

432

26119890

minus4

781

119890minus

42

gt6209561

FA000

6800118

000

9100011

0gt5

72119890

+3

WPA

00

00

100

145338

14 Mathematical Problems in Engineering

0

100

10

0

1

xy

minusf(xy)

minus1

minus2

minus3

minus10minus10

times106

(a)

minus5

minus5

minus10minus10

x

y

0 5 10

0

5

10

(b)

Figure 3 Colville function (1199091

= 1199093 1199092

= 1199094) (a) surface plot and (b) contour lines

0100

0

1000

05

1

15

2

xy

minus100minus100

f(xy)

times104

(a)

x

y

0 50 100

0

50

100

minus100minus100

minus50

minus50

(b)

Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines

minus100

minus200

minus300

minus10minus10

0

100

10

0

xy

minusf(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines

Mathematical Problems in Engineering 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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

6 Mathematical Problems in Engineering

Table 4 Sensitivity analysis of distance measurements

Function Global extremum 119863 Distance Best Worst Mean StdDev SR Arts

Rosenbrock 119891min() = 0 2 MD 921119890 minus 11 324119890 minus 8 112119890 minus 8 118119890 minus 8 100 105165ED 426119890 minus 9 271119890 minus 7 127119890 minus 7 681119890 minus 8 100 371053

Colville 119891min() = 0 4 MD 562119890 minus 8 528119890 minus 7 249119890 minus 7 223119890 minus 7 100 468619ED 174119890 minus 7 170119890 minus 6 574119890 minus 7 370119890 minus 7 90 683220

Sphere 119891min() = 0 200 MD 320119890 minus 161 329119890 minus 144 207119890 minus 145 749119890 minus 145 100 115494ED 176119890 minus 160 336119890 minus 143 168119890 minus 144 751119890 minus 144 100 116825

Sumsquares 119891min() = 0 150 MD 156119890 minus 161 309119890 minus 144 179119890 minus 145 695119890 minus 145 100 85565ED 397119890 minus 160 224119890 minus 144 113119890 minus 145 500119890 minus 145 100 87109

Booth 119891min() = 0 2 MD 563119890 minus 12 115119890 minus 10 419119890 minus 11 332119890 minus 11 100 111074ED 108119890 minus 9 264119890 minus 8 116119890 minus 8 693119890 minus 9 100 405546

Bridge 119891max() = 30054 2 MD 30054 30054 30054 456119890 minus 16 100 11093ED 30054 30054 30054 456119890 minus 16 100 19541

Ackley 119891min() = 0 50 MD 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 193648ED 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 436884

Griewank 119891min() = 0 100 MD 0 01507 301119890 minus 3 00213 98 gt8771198903

ED 0 08350 00167 01181 92 gt135119890 + 4

space Therefore a discussion about their impacts on theperformance of WPA is needed

There are two wolves X119901

= (1199091199011

1199091199012

119909119901119863

) is theposition of wolf 119901X

119902= (1199091199021

1199091199022

119909119902119863

) is the positionof wolf 119902 and the ED and MD between them can berespectively calculated as formula (9) 119863 is the dimensionnumber of solution space

119871 ED (119901 119902) =119863

sum119889=1

(119909119901119889

minus 119909119902119889

)2

119871MD (119901 119902) =119863

sum119889=1

10038161003816100381610038161003816119909119901119889 minus 119909119902119889

10038161003816100381610038161003816

(9)

The statistical results obtained by WPA after 50-runcomputation are shown in Table 4 Firstly we note that WPAwithEuclidean distance (WPA ED)does not get 100 successrate on Colville (119863 = 4) and Griewank functions (119863 = 100)while WPA with Manhattan distance (WPA MD) does notget 100 success rate on Griewank functions (119863 = 100)which means that WPA ED and WPA MD with originalcoefficients still have the risk of premature convergence tolocal optima

As seen from Table 4 WPA is not very sensitive to twodistance measurements on most functions (RosenbrockSphere Sumsquares Booth and Ackley) and no matterwhich metric is used WPA can always get a good resultwith SR = 100 But for these functions comparing theresults between WPA MD and WPA ED in detail we canfind that WPA MD has shorter average reaching time (ARt)which means faster convergence speed to a certain accuracyThe reason may be that ED has the higher computationalcomplexityMeanwhileWPA MDhas better performance onother four criteria (best worst mean and StdDev) whichmeans better solution accuracy and robustness

Naturally because of its better efficiency precision androbustness WD is more suitable for WPA So the WPAalgorithm used in what follows is WPA MD

422 Effect of Four Parameters on the Performance of WPAIn this subsectionwe investigate the impact of the parameters119878 119871near 119879max and 120573 on the new algorithm 119878 is the stepcoefficient 119871near is the distance determinant coefficient 119879maxis the maximum number of repetitions in scouting behaviorand 120573 is the population renewing proportional coefficientThe parameters selection procedure is performed in a one-factor-at-a-time manner For each sensitivity analysis in thissection only one parameter is varied each time and theremaining parameters are kept at the values suggested by theoriginal estimate listed in Table 9 The interaction relationbetween parameters is assumed unimportant

Each time one of the WPA parameters is varied in a cer-tain interval to see which value within this internal will resultin the best performance Specifically theWPA algorithm alsoruns 50 times on each case

Table 5 shows the sensitivity analysis of the step coef-ficient 119878 All results are shown in the form of Mean plusmnStd (SR) The choice of interval [004 016] used in thisanalysis was motivated by the original Nelder-Mead simplexsearch procedure where a step coefficient greater than 004was suggested for general usage

Meanwhile based on detailed comparison of the resultson Rosenbrock Sphere and Bridge functions step coefficientis not sensitive to WPA and for Booth function there is atendency of better results with larger 119878 From Table 5 it isfound that a step coefficient setting at 012 returns the bestresult which has better Mean small Std and SR = 100 forall functions

Tables 6ndash8 analyze sensitivity of 119871near 119879max and 120573 Gen-erally speaking 119871near 119879max and 120573 are not sensitive to mostfunctions exceptGriewank function sinceGriewanknot only

Mathematical Problems in Engineering 7

Table5Sensitivityanalysisof

stepcoeffi

cient(

119878)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

69119890

minus8

plusmn4

3119890minus

82

7119890minus

8plusmn

35119890

minus8

11119890

minus8

plusmn9

1119890minus

93

2119890minus

9plusmn

27119890

minus9

50119890

minus9

plusmn5

7119890minus

93

2119890minus

9plusmn

37119890

minus9

12119890

minus9

plusmn1

6119890minus

9Colville

13119890

minus7

plusmn7

1119890minus

83

3119890minus

7plusmn

28119890

minus7(90)

26119890

minus7

plusmn1

9119890minus

72

3119890minus

7plusmn

14119890

minus7

35119890

minus7

plusmn2

5119890minus

79

5119890minus

7plusmn

10119890

minus6(80)

14119890

minus6

plusmn1

5119890minus

6(50)

Sphere

23119890

minus14

5plusmn

71119890

minus14

56

6119890minus

152

plusmn2

1119890minus

151

21119890

minus14

6plusmn

45119890

minus14

63

9119890minus

146

plusmn1

2119890minus

145

12119890

minus14

5plusmn

34119890

minus14

51

7119890minus

146

plusmn5

3119890minus

146

22119890

minus14

9plusmn

68119890

minus14

9Sumsquares

98119890

minus14

5plusmn

31119890

minus14

43

1119890minus

146

plusmn8

4119890minus

146

81119890

minus14

7plusmn

26119890

minus14

64

8119890minus

146

plusmn1

0119890minus

145

38119890

minus15

2plusmn

79119890

minus15

23

4119890minus

147

plusmn1

1119890minus

146

12119890

minus14

7plusmn

39119890

minus14

7Bo

oth

54119890

minus7

plusmn3

3119890minus

71

6119890minus

9plusmn

11119890

minus9

32119890

minus11

plusmn1

6119890minus

111

3119890minus

12plusmn

91119890

minus13

13119890

minus13

plusmn1

2119890minus

133

9119890minus

15plusmn

18119890

minus15

12119890

minus16

plusmn5

8119890minus

17Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

025

plusmn0

53(80)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

0plusmn

00

06plusmn

019

(92)

020

plusmn0

42(86)

8 Mathematical Problems in Engineering

Table6Sensitivityanalysisof

distance

determ

inantcoefficient(

119871near)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

44119890

minus8

plusmn6

5119890minus

82

3119890minus

8plusmn

37119890

minus8

34119890

minus9

plusmn4

8119890minus

93

0119890minus

8plusmn

29119890

minus8

19119890

minus8

plusmn2

4119890minus

82

4119890minus

8plusmn

47119890

minus8

29119890

minus8

plusmn5

3119890minus

8Colville

20119890

minus7

plusmn9

9119890minus

82

6119890minus

7plusmn

16119890

minus7

35119890

minus7

plusmn2

6119890minus

72

3119890minus

7plusmn

15119890

minus7

12119890

minus7

plusmn3

4119890minus

82

8119890minus

7plusmn

19119890

minus7

14119890

minus7

plusmn6

9119890minus

8Sphere

68119890

minus14

6plusmn

20119890

minus14

51

9119890minus

146

plusmn6

2119890minus

146

17119890

minus14

5plusmn

43119890

minus14

52

6119890minus

148

plusmn8

3119890minus

148

36119890

minus14

6plusmn

11119890

minus14

53

7119890minus

151

plusmn1

1119890minus

150

53119890

minus14

9plusmn

17119890

minus14

8Sumsquares1

1119890

minus14

7plusmn

34119890

minus14

71

0119890minus

146

plusmn3

3119890minus

146

37119890

minus15

1plusmn

89119890

minus15

16

2119890minus

146

plusmn1

9119890minus

145

62119890

minus15

2plusmn

19119890

minus15

11

22119890

minus14

5plusmn

29119890

minus14

51

3119890minus

148

plusmn4

0119890minus

148

Booth

26119890

minus11

plusmn1

3119890minus

112

9119890minus

11plusmn

19119890

minus11

24119890

minus11

plusmn1

6119890minus

113

1119890minus

11plusmn

18119890

minus01

12

4119890minus

11plusmn

13119890

minus11

31119890

minus11

plusmn2

1119890minus

111

0119890minus

10plusmn

13119890

minus10

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

014

plusmn0

43(90)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

159

119890minus

15plusmn

149

119890minus

158

9119890minus

16plusmn

0Grie

wank

008

plusmn0

26(90)

10119890

minus3

plusmn0

02(96)

0plusmn

00

plusmn0

0plusmn

00

10plusmn

033

(92)

0plusmn

0

Mathematical Problems in Engineering 9

Table7Sensitivityanalysisof

them

axim

umnu

mbero

frepetition

sinscou

tingbehavior

(119879max)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)6

810

1214

1618

Rosenb

rock

24119890

minus8

plusmn2

6119890minus

88

4119890minus

9plusmn

80119890

minus9

13119890

minus8

plusmn1

3119890minus

81

4119890minus

8plusmn

10119890

minus8

20119890

minus8

plusmn1

9119890minus

82

1119890minus

8plusmn

25119890

minus8

12119890

minus8

plusmn8

9119890minus

9Colville

48119890

minus7

plusmn2

2119890minus

73

4119890minus

7plusmn

18119890

minus7

15119890

minus7

plusmn1

2119890minus

73

8119890minus

7plusmn

20119890

minus7

36119890

minus7

plusmn3

7119890minus

7(96)

34119890

minus7

plusmn2

5119890minus

72

6119890minus

7plusmn

15119890

minus7

Sphere

71119890

minus14

7plusmn

22119890

minus14

64

5119890minus

146

plusmn9

0119890minus

146

78119890

minus14

6plusmn

23119890

minus14

51

9119890minus

148

plusmn5

3119890minus

148

57119890

minus14

8plusmn

13119890

minus14

76

9119890minus

145

plusmn2

2119890minus

144

36119890

minus14

7plusmn

11119890

minus14

6Sumsquares

41119890

minus14

6plusmn

13119890

minus14

52

4119890minus

149

plusmn4

8119890minus

149

42119890

minus14

9plusmn

13119890

minus14

88

3119890minus

150

plusmn2

6119890minus

149

85119890

minus14

7plusmn

27119890

minus14

65

4119890minus

146

plusmn9

0119890minus

146

14119890

minus15

1plusmn

44119890

minus15

1Bo

oth

32119890

minus11

plusmn2

9119890minus

114

2119890minus

11plusmn

27119890

minus11

25119890

minus11

plusmn1

5119890minus

112

1119890minus

11plusmn

15119890

minus11

32119890

minus11

plusmn2

5119890minus

112

6119890minus

11plusmn

18119890

minus11

26119890

minus11

plusmn2

7119890minus

11Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

08

9119890minus

16plusmn

0Grie

wank

010

plusmn0

33(92)

0plusmn

01

0119890minus

3plusmn

002

(98)

009

plusmn0

31(88)

0plusmn

00

09plusmn

029

(94)

83119890

minus4

plusmn0

02(98)

10 Mathematical Problems in Engineering

Table8Sensitivityanalysisof

popu

latio

nrenewingprop

ortio

nalcoefficient(

120573)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)2

34

56

78

Rosenb

rock

10119890

minus8

plusmn9

2119890minus

98

7119890minus

9plusmn

76119890

minus9

12119890

minus8

plusmn1

0119890minus

88

6119890minus

9plusmn

83119890

minus9

14119890

minus8

plusmn1

3119890minus

89

9119890minus

9plusmn

98119890

minus9

11119890

minus8

plusmn1

2119890minus

9Colville

32119890

minus8

plusmn1

8119890

minus8

14119890

minus7

plusmn1

3119890minus

71

2119890minus

7plusmn

59119890

minus8

14119890

minus7

plusmn9

4119890minus

83

0119890minus

7plusmn

69119890

minus8

39119890

minus7

plusmn1

7119890minus

78

6119890minus

7plusmn

40119890

minus7(80)

Sphere

19119890

minus16

6plusmn

05

2119890minus

158

plusmn1

6119890minus

157

29119890

minus15

3plusmn

92119890

minus15

34

3119890minus

149

plusmn1

3119890minus

148

79119890

minus13

9plusmn

25119890

minus13

88

3119890minus

134

plusmn1

8119890minus

133

34119890

minus12

6plusmn

80119890

minus12

6Sumsquares

28119890

minus16

7plusmn

01

4119890minus

157

plusmn4

3119890minus

157

28119890

minus15

5plusmn

45119890

minus15

58

3119890minus

146

plusmn1

8119890minus

145

69119890

minus14

3plusmn

17119890

minus14

25

3119890minus

143

plusmn1

3119890minus

142

33119890

minus12

7plusmn

10119890

minus12

6Bo

oth

81119890

minus11

plusmn1

3119890minus

102

5119890minus

11plusmn

17119890

minus11

19119890

minus11

plusmn1

2119890minus

112

5119890minus

11plusmn

17119890

minus01

12

5119890minus

11plusmn

15119890

minus11

23119890

minus11

plusmn1

5119890minus

112

3119890minus

11plusmn

14119890

minus11

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

019

plusmn0

41(86)

0plusmn

01

2119890minus

3plusmn

031

(96)

Mathematical Problems in Engineering 11

Table 9 Best suggestions for WPA parameters

No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2

020

2

0

xy

minusf(xy) minus1000

minus2000

minus3000

minus4000

minus2minus2

minus4minus4

(a)

x

y

0 1 2

0

05

1

15

2

minus05

minus15

minus2minus2

minus1

minus1

(b)

Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines

is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized

Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions

Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions

So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2

43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface

As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2

As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909

1 1199092) = (1 1)

But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction

On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909

1 1199092 1199093 1199094) = (1 1 1 1)

Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function

Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909

1 1199092 119909

119898) =

(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5

As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and

12 Mathematical Problems in Engineering

Table10Statistic

alresults

of50

runs

obtained

byGAP

SOA

FSAA

BCFAand

WPA

algorithm

s

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Rosenb

rock

119891min

(119909)

=0

2UN

GA

178

119890minus

1000373

000

91000

9210

gt7598

323

PSO

226

119890minus

115

89119890

minus7

107

119890minus

71

30119890

minus7

100

07444

AFS

A110

eminus13

111

119890minus

92

34119890

minus10

262

119890minus

10100

20578

ABC

599

119890minus

6000

998

61119890

minus4

00015

0gt3910

297

FA6

28119890

minus13

629

eminus10

186eminus

10162eminus

10100

3312

56WPA

349

119890minus

112

34119890

minus8

509

119890minus

94

34119890

minus9

100

66333

Colville

119891min

(119909)

=0

4UN

GA

00022

03343

01272

01062

0gt12

2119890+

3PS

O1

29119890

minus6

346

119890minus

45

06119890

minus5

671

119890minus

50

gt114

0869

AFS

A366

eminus8

891

119890minus

73

16119890

minus7

232

119890minus

7100

4018

07ABC

00103

05337

01871

01232

0gt3844193

FA2

41119890

minus7

369

119890minus

56

62119890

minus6

807

119890minus

68

gt3

14119890

+3

WPA

471

119890minus

8372

eminus7

125eminus

7697eminus

8100

27405

4

Sphere

119891min

(119909)

=0

200

US

GA

156

119890+

51

81119890

+5

171

119890+

55

78119890

+3

0gt4

44119890

+4

PSO

10361

15520

12883

01206

0gt2719

201

AFS

A5

12119890

+5

579

119890+

55

51119890

+5

163

119890+

40

gt7

41119890

+3

ABC

000

4112

521

004

4401773

0gt44

29045

FA01432

02327

01865

00199

0gt8

34119890

+3

WPA

149eminus

172

241

eminus165

156eminus

166

0100

61729

Sumsquares

119891min

(119909)

=0

150

US

GA

593

119890+

47

15+

46

63119890

+4

288

119890+

30

gt3

16119890

+4

PSO

397098

911145

559050

104165

0gt2325464

AFS

A1

43119890

+5

179

119890+

51

64119890

+5

958

119890+

30

gt7

36119890

+3

ABC

171

119890minus

500017

199

119890minus

43

36119890

minus4

0gt4351848

FA89920

998861

405721

192743

0gt6

88119890

+3

WPA

268

eminus172

547

eminus166

262eminus

167

0100

65954

Booth

119891min

(119909)

=0

2MS

GA

455

119890minus

114

55119890

minus11

455

119890minus

110

100

12621

PSO

122

119890minus

122

41119890

minus8

280

119890minus

94

52119890

minus9

100

020

79AFS

A3

02119890

minus12

145

119890minus

94

61119890

minus10

408

119890minus

10100

44329

ABC

605

eminus20

141

eminus17

463eminus

18414eminus

18100

04175

FA1

80119890

minus12

439

119890minus

91

18119890

minus9

111

119890minus

9100

379191

WPA

822

119890minus

157

05119890

minus13

121

119890minus

131

19119890

minus13

100

69339

Bridge

119891max

(119909)

=3

0054

2MN

GA

300

54300

54300

541

35119890

minus15

100

01927

PSO

300

54300

54300

544

84119890

minus8

100

009

29AFS

A300

54300

4730052

169

119890minus

412

gt8

01119890

+3

ABC

300

54300

54300

543

59119890

minus15

100

00932

FA300

54300

54300

543

11119890

minus10

100

227230

WPA

300

54300

54300

54358eminus

15100

01742

Ackley

119891min

(119909)

=0

50MN

GA

114570

126095

1216

1202719

0gt10

4119890+

4PS

O004

6917401

06846

06344

0gt1925522

AFS

A2016

0020600

9204229

01009

0gt9

80119890

+3

ABC

200085

200025

200061

00014

0gt5963841

FA00101

00209

00160

00021

0gt4

28119890

+3

WPA

888

eminus16

444

eminus15

110eminus

15852eminus

16100

79476

Mathematical Problems in Engineering 13

Table10C

ontin

ued

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Grie

wank

119891min

(119909)

=0

100

MN

GA

3174

525

3996

376

3634174

172922

0gt2

07119890

+4

PSO

00029

00082

00052

00011

0gt3670

080

AFS

A2

05119890

+3

255

119890+

32

33119890

+3

1096

821

0gt6

51119890

+3

ABC

895

119890minus

7000

432

26119890

minus4

781

119890minus

42

gt6209561

FA000

6800118

000

9100011

0gt5

72119890

+3

WPA

00

00

100

145338

14 Mathematical Problems in Engineering

0

100

10

0

1

xy

minusf(xy)

minus1

minus2

minus3

minus10minus10

times106

(a)

minus5

minus5

minus10minus10

x

y

0 5 10

0

5

10

(b)

Figure 3 Colville function (1199091

= 1199093 1199092

= 1199094) (a) surface plot and (b) contour lines

0100

0

1000

05

1

15

2

xy

minus100minus100

f(xy)

times104

(a)

x

y

0 50 100

0

50

100

minus100minus100

minus50

minus50

(b)

Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines

minus100

minus200

minus300

minus10minus10

0

100

10

0

xy

minusf(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines

Mathematical Problems in Engineering 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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 7

Table5Sensitivityanalysisof

stepcoeffi

cient(

119878)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

69119890

minus8

plusmn4

3119890minus

82

7119890minus

8plusmn

35119890

minus8

11119890

minus8

plusmn9

1119890minus

93

2119890minus

9plusmn

27119890

minus9

50119890

minus9

plusmn5

7119890minus

93

2119890minus

9plusmn

37119890

minus9

12119890

minus9

plusmn1

6119890minus

9Colville

13119890

minus7

plusmn7

1119890minus

83

3119890minus

7plusmn

28119890

minus7(90)

26119890

minus7

plusmn1

9119890minus

72

3119890minus

7plusmn

14119890

minus7

35119890

minus7

plusmn2

5119890minus

79

5119890minus

7plusmn

10119890

minus6(80)

14119890

minus6

plusmn1

5119890minus

6(50)

Sphere

23119890

minus14

5plusmn

71119890

minus14

56

6119890minus

152

plusmn2

1119890minus

151

21119890

minus14

6plusmn

45119890

minus14

63

9119890minus

146

plusmn1

2119890minus

145

12119890

minus14

5plusmn

34119890

minus14

51

7119890minus

146

plusmn5

3119890minus

146

22119890

minus14

9plusmn

68119890

minus14

9Sumsquares

98119890

minus14

5plusmn

31119890

minus14

43

1119890minus

146

plusmn8

4119890minus

146

81119890

minus14

7plusmn

26119890

minus14

64

8119890minus

146

plusmn1

0119890minus

145

38119890

minus15

2plusmn

79119890

minus15

23

4119890minus

147

plusmn1

1119890minus

146

12119890

minus14

7plusmn

39119890

minus14

7Bo

oth

54119890

minus7

plusmn3

3119890minus

71

6119890minus

9plusmn

11119890

minus9

32119890

minus11

plusmn1

6119890minus

111

3119890minus

12plusmn

91119890

minus13

13119890

minus13

plusmn1

2119890minus

133

9119890minus

15plusmn

18119890

minus15

12119890

minus16

plusmn5

8119890minus

17Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

025

plusmn0

53(80)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

0plusmn

00

06plusmn

019

(92)

020

plusmn0

42(86)

8 Mathematical Problems in Engineering

Table6Sensitivityanalysisof

distance

determ

inantcoefficient(

119871near)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

44119890

minus8

plusmn6

5119890minus

82

3119890minus

8plusmn

37119890

minus8

34119890

minus9

plusmn4

8119890minus

93

0119890minus

8plusmn

29119890

minus8

19119890

minus8

plusmn2

4119890minus

82

4119890minus

8plusmn

47119890

minus8

29119890

minus8

plusmn5

3119890minus

8Colville

20119890

minus7

plusmn9

9119890minus

82

6119890minus

7plusmn

16119890

minus7

35119890

minus7

plusmn2

6119890minus

72

3119890minus

7plusmn

15119890

minus7

12119890

minus7

plusmn3

4119890minus

82

8119890minus

7plusmn

19119890

minus7

14119890

minus7

plusmn6

9119890minus

8Sphere

68119890

minus14

6plusmn

20119890

minus14

51

9119890minus

146

plusmn6

2119890minus

146

17119890

minus14

5plusmn

43119890

minus14

52

6119890minus

148

plusmn8

3119890minus

148

36119890

minus14

6plusmn

11119890

minus14

53

7119890minus

151

plusmn1

1119890minus

150

53119890

minus14

9plusmn

17119890

minus14

8Sumsquares1

1119890

minus14

7plusmn

34119890

minus14

71

0119890minus

146

plusmn3

3119890minus

146

37119890

minus15

1plusmn

89119890

minus15

16

2119890minus

146

plusmn1

9119890minus

145

62119890

minus15

2plusmn

19119890

minus15

11

22119890

minus14

5plusmn

29119890

minus14

51

3119890minus

148

plusmn4

0119890minus

148

Booth

26119890

minus11

plusmn1

3119890minus

112

9119890minus

11plusmn

19119890

minus11

24119890

minus11

plusmn1

6119890minus

113

1119890minus

11plusmn

18119890

minus01

12

4119890minus

11plusmn

13119890

minus11

31119890

minus11

plusmn2

1119890minus

111

0119890minus

10plusmn

13119890

minus10

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

014

plusmn0

43(90)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

159

119890minus

15plusmn

149

119890minus

158

9119890minus

16plusmn

0Grie

wank

008

plusmn0

26(90)

10119890

minus3

plusmn0

02(96)

0plusmn

00

plusmn0

0plusmn

00

10plusmn

033

(92)

0plusmn

0

Mathematical Problems in Engineering 9

Table7Sensitivityanalysisof

them

axim

umnu

mbero

frepetition

sinscou

tingbehavior

(119879max)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)6

810

1214

1618

Rosenb

rock

24119890

minus8

plusmn2

6119890minus

88

4119890minus

9plusmn

80119890

minus9

13119890

minus8

plusmn1

3119890minus

81

4119890minus

8plusmn

10119890

minus8

20119890

minus8

plusmn1

9119890minus

82

1119890minus

8plusmn

25119890

minus8

12119890

minus8

plusmn8

9119890minus

9Colville

48119890

minus7

plusmn2

2119890minus

73

4119890minus

7plusmn

18119890

minus7

15119890

minus7

plusmn1

2119890minus

73

8119890minus

7plusmn

20119890

minus7

36119890

minus7

plusmn3

7119890minus

7(96)

34119890

minus7

plusmn2

5119890minus

72

6119890minus

7plusmn

15119890

minus7

Sphere

71119890

minus14

7plusmn

22119890

minus14

64

5119890minus

146

plusmn9

0119890minus

146

78119890

minus14

6plusmn

23119890

minus14

51

9119890minus

148

plusmn5

3119890minus

148

57119890

minus14

8plusmn

13119890

minus14

76

9119890minus

145

plusmn2

2119890minus

144

36119890

minus14

7plusmn

11119890

minus14

6Sumsquares

41119890

minus14

6plusmn

13119890

minus14

52

4119890minus

149

plusmn4

8119890minus

149

42119890

minus14

9plusmn

13119890

minus14

88

3119890minus

150

plusmn2

6119890minus

149

85119890

minus14

7plusmn

27119890

minus14

65

4119890minus

146

plusmn9

0119890minus

146

14119890

minus15

1plusmn

44119890

minus15

1Bo

oth

32119890

minus11

plusmn2

9119890minus

114

2119890minus

11plusmn

27119890

minus11

25119890

minus11

plusmn1

5119890minus

112

1119890minus

11plusmn

15119890

minus11

32119890

minus11

plusmn2

5119890minus

112

6119890minus

11plusmn

18119890

minus11

26119890

minus11

plusmn2

7119890minus

11Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

08

9119890minus

16plusmn

0Grie

wank

010

plusmn0

33(92)

0plusmn

01

0119890minus

3plusmn

002

(98)

009

plusmn0

31(88)

0plusmn

00

09plusmn

029

(94)

83119890

minus4

plusmn0

02(98)

10 Mathematical Problems in Engineering

Table8Sensitivityanalysisof

popu

latio

nrenewingprop

ortio

nalcoefficient(

120573)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)2

34

56

78

Rosenb

rock

10119890

minus8

plusmn9

2119890minus

98

7119890minus

9plusmn

76119890

minus9

12119890

minus8

plusmn1

0119890minus

88

6119890minus

9plusmn

83119890

minus9

14119890

minus8

plusmn1

3119890minus

89

9119890minus

9plusmn

98119890

minus9

11119890

minus8

plusmn1

2119890minus

9Colville

32119890

minus8

plusmn1

8119890

minus8

14119890

minus7

plusmn1

3119890minus

71

2119890minus

7plusmn

59119890

minus8

14119890

minus7

plusmn9

4119890minus

83

0119890minus

7plusmn

69119890

minus8

39119890

minus7

plusmn1

7119890minus

78

6119890minus

7plusmn

40119890

minus7(80)

Sphere

19119890

minus16

6plusmn

05

2119890minus

158

plusmn1

6119890minus

157

29119890

minus15

3plusmn

92119890

minus15

34

3119890minus

149

plusmn1

3119890minus

148

79119890

minus13

9plusmn

25119890

minus13

88

3119890minus

134

plusmn1

8119890minus

133

34119890

minus12

6plusmn

80119890

minus12

6Sumsquares

28119890

minus16

7plusmn

01

4119890minus

157

plusmn4

3119890minus

157

28119890

minus15

5plusmn

45119890

minus15

58

3119890minus

146

plusmn1

8119890minus

145

69119890

minus14

3plusmn

17119890

minus14

25

3119890minus

143

plusmn1

3119890minus

142

33119890

minus12

7plusmn

10119890

minus12

6Bo

oth

81119890

minus11

plusmn1

3119890minus

102

5119890minus

11plusmn

17119890

minus11

19119890

minus11

plusmn1

2119890minus

112

5119890minus

11plusmn

17119890

minus01

12

5119890minus

11plusmn

15119890

minus11

23119890

minus11

plusmn1

5119890minus

112

3119890minus

11plusmn

14119890

minus11

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

019

plusmn0

41(86)

0plusmn

01

2119890minus

3plusmn

031

(96)

Mathematical Problems in Engineering 11

Table 9 Best suggestions for WPA parameters

No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2

020

2

0

xy

minusf(xy) minus1000

minus2000

minus3000

minus4000

minus2minus2

minus4minus4

(a)

x

y

0 1 2

0

05

1

15

2

minus05

minus15

minus2minus2

minus1

minus1

(b)

Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines

is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized

Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions

Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions

So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2

43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface

As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2

As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909

1 1199092) = (1 1)

But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction

On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909

1 1199092 1199093 1199094) = (1 1 1 1)

Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function

Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909

1 1199092 119909

119898) =

(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5

As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and

12 Mathematical Problems in Engineering

Table10Statistic

alresults

of50

runs

obtained

byGAP

SOA

FSAA

BCFAand

WPA

algorithm

s

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Rosenb

rock

119891min

(119909)

=0

2UN

GA

178

119890minus

1000373

000

91000

9210

gt7598

323

PSO

226

119890minus

115

89119890

minus7

107

119890minus

71

30119890

minus7

100

07444

AFS

A110

eminus13

111

119890minus

92

34119890

minus10

262

119890minus

10100

20578

ABC

599

119890minus

6000

998

61119890

minus4

00015

0gt3910

297

FA6

28119890

minus13

629

eminus10

186eminus

10162eminus

10100

3312

56WPA

349

119890minus

112

34119890

minus8

509

119890minus

94

34119890

minus9

100

66333

Colville

119891min

(119909)

=0

4UN

GA

00022

03343

01272

01062

0gt12

2119890+

3PS

O1

29119890

minus6

346

119890minus

45

06119890

minus5

671

119890minus

50

gt114

0869

AFS

A366

eminus8

891

119890minus

73

16119890

minus7

232

119890minus

7100

4018

07ABC

00103

05337

01871

01232

0gt3844193

FA2

41119890

minus7

369

119890minus

56

62119890

minus6

807

119890minus

68

gt3

14119890

+3

WPA

471

119890minus

8372

eminus7

125eminus

7697eminus

8100

27405

4

Sphere

119891min

(119909)

=0

200

US

GA

156

119890+

51

81119890

+5

171

119890+

55

78119890

+3

0gt4

44119890

+4

PSO

10361

15520

12883

01206

0gt2719

201

AFS

A5

12119890

+5

579

119890+

55

51119890

+5

163

119890+

40

gt7

41119890

+3

ABC

000

4112

521

004

4401773

0gt44

29045

FA01432

02327

01865

00199

0gt8

34119890

+3

WPA

149eminus

172

241

eminus165

156eminus

166

0100

61729

Sumsquares

119891min

(119909)

=0

150

US

GA

593

119890+

47

15+

46

63119890

+4

288

119890+

30

gt3

16119890

+4

PSO

397098

911145

559050

104165

0gt2325464

AFS

A1

43119890

+5

179

119890+

51

64119890

+5

958

119890+

30

gt7

36119890

+3

ABC

171

119890minus

500017

199

119890minus

43

36119890

minus4

0gt4351848

FA89920

998861

405721

192743

0gt6

88119890

+3

WPA

268

eminus172

547

eminus166

262eminus

167

0100

65954

Booth

119891min

(119909)

=0

2MS

GA

455

119890minus

114

55119890

minus11

455

119890minus

110

100

12621

PSO

122

119890minus

122

41119890

minus8

280

119890minus

94

52119890

minus9

100

020

79AFS

A3

02119890

minus12

145

119890minus

94

61119890

minus10

408

119890minus

10100

44329

ABC

605

eminus20

141

eminus17

463eminus

18414eminus

18100

04175

FA1

80119890

minus12

439

119890minus

91

18119890

minus9

111

119890minus

9100

379191

WPA

822

119890minus

157

05119890

minus13

121

119890minus

131

19119890

minus13

100

69339

Bridge

119891max

(119909)

=3

0054

2MN

GA

300

54300

54300

541

35119890

minus15

100

01927

PSO

300

54300

54300

544

84119890

minus8

100

009

29AFS

A300

54300

4730052

169

119890minus

412

gt8

01119890

+3

ABC

300

54300

54300

543

59119890

minus15

100

00932

FA300

54300

54300

543

11119890

minus10

100

227230

WPA

300

54300

54300

54358eminus

15100

01742

Ackley

119891min

(119909)

=0

50MN

GA

114570

126095

1216

1202719

0gt10

4119890+

4PS

O004

6917401

06846

06344

0gt1925522

AFS

A2016

0020600

9204229

01009

0gt9

80119890

+3

ABC

200085

200025

200061

00014

0gt5963841

FA00101

00209

00160

00021

0gt4

28119890

+3

WPA

888

eminus16

444

eminus15

110eminus

15852eminus

16100

79476

Mathematical Problems in Engineering 13

Table10C

ontin

ued

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Grie

wank

119891min

(119909)

=0

100

MN

GA

3174

525

3996

376

3634174

172922

0gt2

07119890

+4

PSO

00029

00082

00052

00011

0gt3670

080

AFS

A2

05119890

+3

255

119890+

32

33119890

+3

1096

821

0gt6

51119890

+3

ABC

895

119890minus

7000

432

26119890

minus4

781

119890minus

42

gt6209561

FA000

6800118

000

9100011

0gt5

72119890

+3

WPA

00

00

100

145338

14 Mathematical Problems in Engineering

0

100

10

0

1

xy

minusf(xy)

minus1

minus2

minus3

minus10minus10

times106

(a)

minus5

minus5

minus10minus10

x

y

0 5 10

0

5

10

(b)

Figure 3 Colville function (1199091

= 1199093 1199092

= 1199094) (a) surface plot and (b) contour lines

0100

0

1000

05

1

15

2

xy

minus100minus100

f(xy)

times104

(a)

x

y

0 50 100

0

50

100

minus100minus100

minus50

minus50

(b)

Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines

minus100

minus200

minus300

minus10minus10

0

100

10

0

xy

minusf(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines

Mathematical Problems in Engineering 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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

Table6Sensitivityanalysisof

distance

determ

inantcoefficient(

119871near)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)004

006

008

010

012

014

016

Rosenb

rock

44119890

minus8

plusmn6

5119890minus

82

3119890minus

8plusmn

37119890

minus8

34119890

minus9

plusmn4

8119890minus

93

0119890minus

8plusmn

29119890

minus8

19119890

minus8

plusmn2

4119890minus

82

4119890minus

8plusmn

47119890

minus8

29119890

minus8

plusmn5

3119890minus

8Colville

20119890

minus7

plusmn9

9119890minus

82

6119890minus

7plusmn

16119890

minus7

35119890

minus7

plusmn2

6119890minus

72

3119890minus

7plusmn

15119890

minus7

12119890

minus7

plusmn3

4119890minus

82

8119890minus

7plusmn

19119890

minus7

14119890

minus7

plusmn6

9119890minus

8Sphere

68119890

minus14

6plusmn

20119890

minus14

51

9119890minus

146

plusmn6

2119890minus

146

17119890

minus14

5plusmn

43119890

minus14

52

6119890minus

148

plusmn8

3119890minus

148

36119890

minus14

6plusmn

11119890

minus14

53

7119890minus

151

plusmn1

1119890minus

150

53119890

minus14

9plusmn

17119890

minus14

8Sumsquares1

1119890

minus14

7plusmn

34119890

minus14

71

0119890minus

146

plusmn3

3119890minus

146

37119890

minus15

1plusmn

89119890

minus15

16

2119890minus

146

plusmn1

9119890minus

145

62119890

minus15

2plusmn

19119890

minus15

11

22119890

minus14

5plusmn

29119890

minus14

51

3119890minus

148

plusmn4

0119890minus

148

Booth

26119890

minus11

plusmn1

3119890minus

112

9119890minus

11plusmn

19119890

minus11

24119890

minus11

plusmn1

6119890minus

113

1119890minus

11plusmn

18119890

minus01

12

4119890minus

11plusmn

13119890

minus11

31119890

minus11

plusmn2

1119890minus

111

0119890minus

10plusmn

13119890

minus10

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

014

plusmn0

43(90)

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

01

2119890minus

15plusmn

11119890

minus15

89119890

minus16

plusmn0

159

119890minus

15plusmn

149

119890minus

158

9119890minus

16plusmn

0Grie

wank

008

plusmn0

26(90)

10119890

minus3

plusmn0

02(96)

0plusmn

00

plusmn0

0plusmn

00

10plusmn

033

(92)

0plusmn

0

Mathematical Problems in Engineering 9

Table7Sensitivityanalysisof

them

axim

umnu

mbero

frepetition

sinscou

tingbehavior

(119879max)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)6

810

1214

1618

Rosenb

rock

24119890

minus8

plusmn2

6119890minus

88

4119890minus

9plusmn

80119890

minus9

13119890

minus8

plusmn1

3119890minus

81

4119890minus

8plusmn

10119890

minus8

20119890

minus8

plusmn1

9119890minus

82

1119890minus

8plusmn

25119890

minus8

12119890

minus8

plusmn8

9119890minus

9Colville

48119890

minus7

plusmn2

2119890minus

73

4119890minus

7plusmn

18119890

minus7

15119890

minus7

plusmn1

2119890minus

73

8119890minus

7plusmn

20119890

minus7

36119890

minus7

plusmn3

7119890minus

7(96)

34119890

minus7

plusmn2

5119890minus

72

6119890minus

7plusmn

15119890

minus7

Sphere

71119890

minus14

7plusmn

22119890

minus14

64

5119890minus

146

plusmn9

0119890minus

146

78119890

minus14

6plusmn

23119890

minus14

51

9119890minus

148

plusmn5

3119890minus

148

57119890

minus14

8plusmn

13119890

minus14

76

9119890minus

145

plusmn2

2119890minus

144

36119890

minus14

7plusmn

11119890

minus14

6Sumsquares

41119890

minus14

6plusmn

13119890

minus14

52

4119890minus

149

plusmn4

8119890minus

149

42119890

minus14

9plusmn

13119890

minus14

88

3119890minus

150

plusmn2

6119890minus

149

85119890

minus14

7plusmn

27119890

minus14

65

4119890minus

146

plusmn9

0119890minus

146

14119890

minus15

1plusmn

44119890

minus15

1Bo

oth

32119890

minus11

plusmn2

9119890minus

114

2119890minus

11plusmn

27119890

minus11

25119890

minus11

plusmn1

5119890minus

112

1119890minus

11plusmn

15119890

minus11

32119890

minus11

plusmn2

5119890minus

112

6119890minus

11plusmn

18119890

minus11

26119890

minus11

plusmn2

7119890minus

11Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

08

9119890minus

16plusmn

0Grie

wank

010

plusmn0

33(92)

0plusmn

01

0119890minus

3plusmn

002

(98)

009

plusmn0

31(88)

0plusmn

00

09plusmn

029

(94)

83119890

minus4

plusmn0

02(98)

10 Mathematical Problems in Engineering

Table8Sensitivityanalysisof

popu

latio

nrenewingprop

ortio

nalcoefficient(

120573)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)2

34

56

78

Rosenb

rock

10119890

minus8

plusmn9

2119890minus

98

7119890minus

9plusmn

76119890

minus9

12119890

minus8

plusmn1

0119890minus

88

6119890minus

9plusmn

83119890

minus9

14119890

minus8

plusmn1

3119890minus

89

9119890minus

9plusmn

98119890

minus9

11119890

minus8

plusmn1

2119890minus

9Colville

32119890

minus8

plusmn1

8119890

minus8

14119890

minus7

plusmn1

3119890minus

71

2119890minus

7plusmn

59119890

minus8

14119890

minus7

plusmn9

4119890minus

83

0119890minus

7plusmn

69119890

minus8

39119890

minus7

plusmn1

7119890minus

78

6119890minus

7plusmn

40119890

minus7(80)

Sphere

19119890

minus16

6plusmn

05

2119890minus

158

plusmn1

6119890minus

157

29119890

minus15

3plusmn

92119890

minus15

34

3119890minus

149

plusmn1

3119890minus

148

79119890

minus13

9plusmn

25119890

minus13

88

3119890minus

134

plusmn1

8119890minus

133

34119890

minus12

6plusmn

80119890

minus12

6Sumsquares

28119890

minus16

7plusmn

01

4119890minus

157

plusmn4

3119890minus

157

28119890

minus15

5plusmn

45119890

minus15

58

3119890minus

146

plusmn1

8119890minus

145

69119890

minus14

3plusmn

17119890

minus14

25

3119890minus

143

plusmn1

3119890minus

142

33119890

minus12

7plusmn

10119890

minus12

6Bo

oth

81119890

minus11

plusmn1

3119890minus

102

5119890minus

11plusmn

17119890

minus11

19119890

minus11

plusmn1

2119890minus

112

5119890minus

11plusmn

17119890

minus01

12

5119890minus

11plusmn

15119890

minus11

23119890

minus11

plusmn1

5119890minus

112

3119890minus

11plusmn

14119890

minus11

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

019

plusmn0

41(86)

0plusmn

01

2119890minus

3plusmn

031

(96)

Mathematical Problems in Engineering 11

Table 9 Best suggestions for WPA parameters

No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2

020

2

0

xy

minusf(xy) minus1000

minus2000

minus3000

minus4000

minus2minus2

minus4minus4

(a)

x

y

0 1 2

0

05

1

15

2

minus05

minus15

minus2minus2

minus1

minus1

(b)

Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines

is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized

Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions

Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions

So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2

43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface

As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2

As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909

1 1199092) = (1 1)

But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction

On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909

1 1199092 1199093 1199094) = (1 1 1 1)

Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function

Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909

1 1199092 119909

119898) =

(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5

As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and

12 Mathematical Problems in Engineering

Table10Statistic

alresults

of50

runs

obtained

byGAP

SOA

FSAA

BCFAand

WPA

algorithm

s

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Rosenb

rock

119891min

(119909)

=0

2UN

GA

178

119890minus

1000373

000

91000

9210

gt7598

323

PSO

226

119890minus

115

89119890

minus7

107

119890minus

71

30119890

minus7

100

07444

AFS

A110

eminus13

111

119890minus

92

34119890

minus10

262

119890minus

10100

20578

ABC

599

119890minus

6000

998

61119890

minus4

00015

0gt3910

297

FA6

28119890

minus13

629

eminus10

186eminus

10162eminus

10100

3312

56WPA

349

119890minus

112

34119890

minus8

509

119890minus

94

34119890

minus9

100

66333

Colville

119891min

(119909)

=0

4UN

GA

00022

03343

01272

01062

0gt12

2119890+

3PS

O1

29119890

minus6

346

119890minus

45

06119890

minus5

671

119890minus

50

gt114

0869

AFS

A366

eminus8

891

119890minus

73

16119890

minus7

232

119890minus

7100

4018

07ABC

00103

05337

01871

01232

0gt3844193

FA2

41119890

minus7

369

119890minus

56

62119890

minus6

807

119890minus

68

gt3

14119890

+3

WPA

471

119890minus

8372

eminus7

125eminus

7697eminus

8100

27405

4

Sphere

119891min

(119909)

=0

200

US

GA

156

119890+

51

81119890

+5

171

119890+

55

78119890

+3

0gt4

44119890

+4

PSO

10361

15520

12883

01206

0gt2719

201

AFS

A5

12119890

+5

579

119890+

55

51119890

+5

163

119890+

40

gt7

41119890

+3

ABC

000

4112

521

004

4401773

0gt44

29045

FA01432

02327

01865

00199

0gt8

34119890

+3

WPA

149eminus

172

241

eminus165

156eminus

166

0100

61729

Sumsquares

119891min

(119909)

=0

150

US

GA

593

119890+

47

15+

46

63119890

+4

288

119890+

30

gt3

16119890

+4

PSO

397098

911145

559050

104165

0gt2325464

AFS

A1

43119890

+5

179

119890+

51

64119890

+5

958

119890+

30

gt7

36119890

+3

ABC

171

119890minus

500017

199

119890minus

43

36119890

minus4

0gt4351848

FA89920

998861

405721

192743

0gt6

88119890

+3

WPA

268

eminus172

547

eminus166

262eminus

167

0100

65954

Booth

119891min

(119909)

=0

2MS

GA

455

119890minus

114

55119890

minus11

455

119890minus

110

100

12621

PSO

122

119890minus

122

41119890

minus8

280

119890minus

94

52119890

minus9

100

020

79AFS

A3

02119890

minus12

145

119890minus

94

61119890

minus10

408

119890minus

10100

44329

ABC

605

eminus20

141

eminus17

463eminus

18414eminus

18100

04175

FA1

80119890

minus12

439

119890minus

91

18119890

minus9

111

119890minus

9100

379191

WPA

822

119890minus

157

05119890

minus13

121

119890minus

131

19119890

minus13

100

69339

Bridge

119891max

(119909)

=3

0054

2MN

GA

300

54300

54300

541

35119890

minus15

100

01927

PSO

300

54300

54300

544

84119890

minus8

100

009

29AFS

A300

54300

4730052

169

119890minus

412

gt8

01119890

+3

ABC

300

54300

54300

543

59119890

minus15

100

00932

FA300

54300

54300

543

11119890

minus10

100

227230

WPA

300

54300

54300

54358eminus

15100

01742

Ackley

119891min

(119909)

=0

50MN

GA

114570

126095

1216

1202719

0gt10

4119890+

4PS

O004

6917401

06846

06344

0gt1925522

AFS

A2016

0020600

9204229

01009

0gt9

80119890

+3

ABC

200085

200025

200061

00014

0gt5963841

FA00101

00209

00160

00021

0gt4

28119890

+3

WPA

888

eminus16

444

eminus15

110eminus

15852eminus

16100

79476

Mathematical Problems in Engineering 13

Table10C

ontin

ued

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Grie

wank

119891min

(119909)

=0

100

MN

GA

3174

525

3996

376

3634174

172922

0gt2

07119890

+4

PSO

00029

00082

00052

00011

0gt3670

080

AFS

A2

05119890

+3

255

119890+

32

33119890

+3

1096

821

0gt6

51119890

+3

ABC

895

119890minus

7000

432

26119890

minus4

781

119890minus

42

gt6209561

FA000

6800118

000

9100011

0gt5

72119890

+3

WPA

00

00

100

145338

14 Mathematical Problems in Engineering

0

100

10

0

1

xy

minusf(xy)

minus1

minus2

minus3

minus10minus10

times106

(a)

minus5

minus5

minus10minus10

x

y

0 5 10

0

5

10

(b)

Figure 3 Colville function (1199091

= 1199093 1199092

= 1199094) (a) surface plot and (b) contour lines

0100

0

1000

05

1

15

2

xy

minus100minus100

f(xy)

times104

(a)

x

y

0 50 100

0

50

100

minus100minus100

minus50

minus50

(b)

Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines

minus100

minus200

minus300

minus10minus10

0

100

10

0

xy

minusf(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines

Mathematical Problems in Engineering 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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

Table7Sensitivityanalysisof

them

axim

umnu

mbero

frepetition

sinscou

tingbehavior

(119879max)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)6

810

1214

1618

Rosenb

rock

24119890

minus8

plusmn2

6119890minus

88

4119890minus

9plusmn

80119890

minus9

13119890

minus8

plusmn1

3119890minus

81

4119890minus

8plusmn

10119890

minus8

20119890

minus8

plusmn1

9119890minus

82

1119890minus

8plusmn

25119890

minus8

12119890

minus8

plusmn8

9119890minus

9Colville

48119890

minus7

plusmn2

2119890minus

73

4119890minus

7plusmn

18119890

minus7

15119890

minus7

plusmn1

2119890minus

73

8119890minus

7plusmn

20119890

minus7

36119890

minus7

plusmn3

7119890minus

7(96)

34119890

minus7

plusmn2

5119890minus

72

6119890minus

7plusmn

15119890

minus7

Sphere

71119890

minus14

7plusmn

22119890

minus14

64

5119890minus

146

plusmn9

0119890minus

146

78119890

minus14

6plusmn

23119890

minus14

51

9119890minus

148

plusmn5

3119890minus

148

57119890

minus14

8plusmn

13119890

minus14

76

9119890minus

145

plusmn2

2119890minus

144

36119890

minus14

7plusmn

11119890

minus14

6Sumsquares

41119890

minus14

6plusmn

13119890

minus14

52

4119890minus

149

plusmn4

8119890minus

149

42119890

minus14

9plusmn

13119890

minus14

88

3119890minus

150

plusmn2

6119890minus

149

85119890

minus14

7plusmn

27119890

minus14

65

4119890minus

146

plusmn9

0119890minus

146

14119890

minus15

1plusmn

44119890

minus15

1Bo

oth

32119890

minus11

plusmn2

9119890minus

114

2119890minus

11plusmn

27119890

minus11

25119890

minus11

plusmn1

5119890minus

112

1119890minus

11plusmn

15119890

minus11

32119890

minus11

plusmn2

5119890minus

112

6119890minus

11plusmn

18119890

minus11

26119890

minus11

plusmn2

7119890minus

11Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

12119890

minus15

plusmn1

1119890minus

158

9119890minus

16plusmn

08

9119890minus

16plusmn

0Grie

wank

010

plusmn0

33(92)

0plusmn

01

0119890minus

3plusmn

002

(98)

009

plusmn0

31(88)

0plusmn

00

09plusmn

029

(94)

83119890

minus4

plusmn0

02(98)

10 Mathematical Problems in Engineering

Table8Sensitivityanalysisof

popu

latio

nrenewingprop

ortio

nalcoefficient(

120573)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)2

34

56

78

Rosenb

rock

10119890

minus8

plusmn9

2119890minus

98

7119890minus

9plusmn

76119890

minus9

12119890

minus8

plusmn1

0119890minus

88

6119890minus

9plusmn

83119890

minus9

14119890

minus8

plusmn1

3119890minus

89

9119890minus

9plusmn

98119890

minus9

11119890

minus8

plusmn1

2119890minus

9Colville

32119890

minus8

plusmn1

8119890

minus8

14119890

minus7

plusmn1

3119890minus

71

2119890minus

7plusmn

59119890

minus8

14119890

minus7

plusmn9

4119890minus

83

0119890minus

7plusmn

69119890

minus8

39119890

minus7

plusmn1

7119890minus

78

6119890minus

7plusmn

40119890

minus7(80)

Sphere

19119890

minus16

6plusmn

05

2119890minus

158

plusmn1

6119890minus

157

29119890

minus15

3plusmn

92119890

minus15

34

3119890minus

149

plusmn1

3119890minus

148

79119890

minus13

9plusmn

25119890

minus13

88

3119890minus

134

plusmn1

8119890minus

133

34119890

minus12

6plusmn

80119890

minus12

6Sumsquares

28119890

minus16

7plusmn

01

4119890minus

157

plusmn4

3119890minus

157

28119890

minus15

5plusmn

45119890

minus15

58

3119890minus

146

plusmn1

8119890minus

145

69119890

minus14

3plusmn

17119890

minus14

25

3119890minus

143

plusmn1

3119890minus

142

33119890

minus12

7plusmn

10119890

minus12

6Bo

oth

81119890

minus11

plusmn1

3119890minus

102

5119890minus

11plusmn

17119890

minus11

19119890

minus11

plusmn1

2119890minus

112

5119890minus

11plusmn

17119890

minus01

12

5119890minus

11plusmn

15119890

minus11

23119890

minus11

plusmn1

5119890minus

112

3119890minus

11plusmn

14119890

minus11

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

019

plusmn0

41(86)

0plusmn

01

2119890minus

3plusmn

031

(96)

Mathematical Problems in Engineering 11

Table 9 Best suggestions for WPA parameters

No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2

020

2

0

xy

minusf(xy) minus1000

minus2000

minus3000

minus4000

minus2minus2

minus4minus4

(a)

x

y

0 1 2

0

05

1

15

2

minus05

minus15

minus2minus2

minus1

minus1

(b)

Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines

is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized

Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions

Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions

So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2

43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface

As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2

As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909

1 1199092) = (1 1)

But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction

On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909

1 1199092 1199093 1199094) = (1 1 1 1)

Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function

Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909

1 1199092 119909

119898) =

(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5

As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and

12 Mathematical Problems in Engineering

Table10Statistic

alresults

of50

runs

obtained

byGAP

SOA

FSAA

BCFAand

WPA

algorithm

s

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Rosenb

rock

119891min

(119909)

=0

2UN

GA

178

119890minus

1000373

000

91000

9210

gt7598

323

PSO

226

119890minus

115

89119890

minus7

107

119890minus

71

30119890

minus7

100

07444

AFS

A110

eminus13

111

119890minus

92

34119890

minus10

262

119890minus

10100

20578

ABC

599

119890minus

6000

998

61119890

minus4

00015

0gt3910

297

FA6

28119890

minus13

629

eminus10

186eminus

10162eminus

10100

3312

56WPA

349

119890minus

112

34119890

minus8

509

119890minus

94

34119890

minus9

100

66333

Colville

119891min

(119909)

=0

4UN

GA

00022

03343

01272

01062

0gt12

2119890+

3PS

O1

29119890

minus6

346

119890minus

45

06119890

minus5

671

119890minus

50

gt114

0869

AFS

A366

eminus8

891

119890minus

73

16119890

minus7

232

119890minus

7100

4018

07ABC

00103

05337

01871

01232

0gt3844193

FA2

41119890

minus7

369

119890minus

56

62119890

minus6

807

119890minus

68

gt3

14119890

+3

WPA

471

119890minus

8372

eminus7

125eminus

7697eminus

8100

27405

4

Sphere

119891min

(119909)

=0

200

US

GA

156

119890+

51

81119890

+5

171

119890+

55

78119890

+3

0gt4

44119890

+4

PSO

10361

15520

12883

01206

0gt2719

201

AFS

A5

12119890

+5

579

119890+

55

51119890

+5

163

119890+

40

gt7

41119890

+3

ABC

000

4112

521

004

4401773

0gt44

29045

FA01432

02327

01865

00199

0gt8

34119890

+3

WPA

149eminus

172

241

eminus165

156eminus

166

0100

61729

Sumsquares

119891min

(119909)

=0

150

US

GA

593

119890+

47

15+

46

63119890

+4

288

119890+

30

gt3

16119890

+4

PSO

397098

911145

559050

104165

0gt2325464

AFS

A1

43119890

+5

179

119890+

51

64119890

+5

958

119890+

30

gt7

36119890

+3

ABC

171

119890minus

500017

199

119890minus

43

36119890

minus4

0gt4351848

FA89920

998861

405721

192743

0gt6

88119890

+3

WPA

268

eminus172

547

eminus166

262eminus

167

0100

65954

Booth

119891min

(119909)

=0

2MS

GA

455

119890minus

114

55119890

minus11

455

119890minus

110

100

12621

PSO

122

119890minus

122

41119890

minus8

280

119890minus

94

52119890

minus9

100

020

79AFS

A3

02119890

minus12

145

119890minus

94

61119890

minus10

408

119890minus

10100

44329

ABC

605

eminus20

141

eminus17

463eminus

18414eminus

18100

04175

FA1

80119890

minus12

439

119890minus

91

18119890

minus9

111

119890minus

9100

379191

WPA

822

119890minus

157

05119890

minus13

121

119890minus

131

19119890

minus13

100

69339

Bridge

119891max

(119909)

=3

0054

2MN

GA

300

54300

54300

541

35119890

minus15

100

01927

PSO

300

54300

54300

544

84119890

minus8

100

009

29AFS

A300

54300

4730052

169

119890minus

412

gt8

01119890

+3

ABC

300

54300

54300

543

59119890

minus15

100

00932

FA300

54300

54300

543

11119890

minus10

100

227230

WPA

300

54300

54300

54358eminus

15100

01742

Ackley

119891min

(119909)

=0

50MN

GA

114570

126095

1216

1202719

0gt10

4119890+

4PS

O004

6917401

06846

06344

0gt1925522

AFS

A2016

0020600

9204229

01009

0gt9

80119890

+3

ABC

200085

200025

200061

00014

0gt5963841

FA00101

00209

00160

00021

0gt4

28119890

+3

WPA

888

eminus16

444

eminus15

110eminus

15852eminus

16100

79476

Mathematical Problems in Engineering 13

Table10C

ontin

ued

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Grie

wank

119891min

(119909)

=0

100

MN

GA

3174

525

3996

376

3634174

172922

0gt2

07119890

+4

PSO

00029

00082

00052

00011

0gt3670

080

AFS

A2

05119890

+3

255

119890+

32

33119890

+3

1096

821

0gt6

51119890

+3

ABC

895

119890minus

7000

432

26119890

minus4

781

119890minus

42

gt6209561

FA000

6800118

000

9100011

0gt5

72119890

+3

WPA

00

00

100

145338

14 Mathematical Problems in Engineering

0

100

10

0

1

xy

minusf(xy)

minus1

minus2

minus3

minus10minus10

times106

(a)

minus5

minus5

minus10minus10

x

y

0 5 10

0

5

10

(b)

Figure 3 Colville function (1199091

= 1199093 1199092

= 1199094) (a) surface plot and (b) contour lines

0100

0

1000

05

1

15

2

xy

minus100minus100

f(xy)

times104

(a)

x

y

0 50 100

0

50

100

minus100minus100

minus50

minus50

(b)

Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines

minus100

minus200

minus300

minus10minus10

0

100

10

0

xy

minusf(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines

Mathematical Problems in Engineering 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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

Table8Sensitivityanalysisof

popu

latio

nrenewingprop

ortio

nalcoefficient(

120573)

Functio

nsMean

plusmnStd(SR)(thed

efaultof

SRis100

)2

34

56

78

Rosenb

rock

10119890

minus8

plusmn9

2119890minus

98

7119890minus

9plusmn

76119890

minus9

12119890

minus8

plusmn1

0119890minus

88

6119890minus

9plusmn

83119890

minus9

14119890

minus8

plusmn1

3119890minus

89

9119890minus

9plusmn

98119890

minus9

11119890

minus8

plusmn1

2119890minus

9Colville

32119890

minus8

plusmn1

8119890

minus8

14119890

minus7

plusmn1

3119890minus

71

2119890minus

7plusmn

59119890

minus8

14119890

minus7

plusmn9

4119890minus

83

0119890minus

7plusmn

69119890

minus8

39119890

minus7

plusmn1

7119890minus

78

6119890minus

7plusmn

40119890

minus7(80)

Sphere

19119890

minus16

6plusmn

05

2119890minus

158

plusmn1

6119890minus

157

29119890

minus15

3plusmn

92119890

minus15

34

3119890minus

149

plusmn1

3119890minus

148

79119890

minus13

9plusmn

25119890

minus13

88

3119890minus

134

plusmn1

8119890minus

133

34119890

minus12

6plusmn

80119890

minus12

6Sumsquares

28119890

minus16

7plusmn

01

4119890minus

157

plusmn4

3119890minus

157

28119890

minus15

5plusmn

45119890

minus15

58

3119890minus

146

plusmn1

8119890minus

145

69119890

minus14

3plusmn

17119890

minus14

25

3119890minus

143

plusmn1

3119890minus

142

33119890

minus12

7plusmn

10119890

minus12

6Bo

oth

81119890

minus11

plusmn1

3119890minus

102

5119890minus

11plusmn

17119890

minus11

19119890

minus11

plusmn1

2119890minus

112

5119890minus

11plusmn

17119890

minus01

12

5119890minus

11plusmn

15119890

minus11

23119890

minus11

plusmn1

5119890minus

112

3119890minus

11plusmn

14119890

minus11

Bridge

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

300

54plusmn

47119890

minus16

Ackley

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

89119890

minus16

plusmn0

Grie

wank

0plusmn

00

plusmn0

0plusmn

00

plusmn0

019

plusmn0

41(86)

0plusmn

01

2119890minus

3plusmn

031

(96)

Mathematical Problems in Engineering 11

Table 9 Best suggestions for WPA parameters

No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2

020

2

0

xy

minusf(xy) minus1000

minus2000

minus3000

minus4000

minus2minus2

minus4minus4

(a)

x

y

0 1 2

0

05

1

15

2

minus05

minus15

minus2minus2

minus1

minus1

(b)

Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines

is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized

Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions

Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions

So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2

43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface

As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2

As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909

1 1199092) = (1 1)

But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction

On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909

1 1199092 1199093 1199094) = (1 1 1 1)

Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function

Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909

1 1199092 119909

119898) =

(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5

As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and

12 Mathematical Problems in Engineering

Table10Statistic

alresults

of50

runs

obtained

byGAP

SOA

FSAA

BCFAand

WPA

algorithm

s

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Rosenb

rock

119891min

(119909)

=0

2UN

GA

178

119890minus

1000373

000

91000

9210

gt7598

323

PSO

226

119890minus

115

89119890

minus7

107

119890minus

71

30119890

minus7

100

07444

AFS

A110

eminus13

111

119890minus

92

34119890

minus10

262

119890minus

10100

20578

ABC

599

119890minus

6000

998

61119890

minus4

00015

0gt3910

297

FA6

28119890

minus13

629

eminus10

186eminus

10162eminus

10100

3312

56WPA

349

119890minus

112

34119890

minus8

509

119890minus

94

34119890

minus9

100

66333

Colville

119891min

(119909)

=0

4UN

GA

00022

03343

01272

01062

0gt12

2119890+

3PS

O1

29119890

minus6

346

119890minus

45

06119890

minus5

671

119890minus

50

gt114

0869

AFS

A366

eminus8

891

119890minus

73

16119890

minus7

232

119890minus

7100

4018

07ABC

00103

05337

01871

01232

0gt3844193

FA2

41119890

minus7

369

119890minus

56

62119890

minus6

807

119890minus

68

gt3

14119890

+3

WPA

471

119890minus

8372

eminus7

125eminus

7697eminus

8100

27405

4

Sphere

119891min

(119909)

=0

200

US

GA

156

119890+

51

81119890

+5

171

119890+

55

78119890

+3

0gt4

44119890

+4

PSO

10361

15520

12883

01206

0gt2719

201

AFS

A5

12119890

+5

579

119890+

55

51119890

+5

163

119890+

40

gt7

41119890

+3

ABC

000

4112

521

004

4401773

0gt44

29045

FA01432

02327

01865

00199

0gt8

34119890

+3

WPA

149eminus

172

241

eminus165

156eminus

166

0100

61729

Sumsquares

119891min

(119909)

=0

150

US

GA

593

119890+

47

15+

46

63119890

+4

288

119890+

30

gt3

16119890

+4

PSO

397098

911145

559050

104165

0gt2325464

AFS

A1

43119890

+5

179

119890+

51

64119890

+5

958

119890+

30

gt7

36119890

+3

ABC

171

119890minus

500017

199

119890minus

43

36119890

minus4

0gt4351848

FA89920

998861

405721

192743

0gt6

88119890

+3

WPA

268

eminus172

547

eminus166

262eminus

167

0100

65954

Booth

119891min

(119909)

=0

2MS

GA

455

119890minus

114

55119890

minus11

455

119890minus

110

100

12621

PSO

122

119890minus

122

41119890

minus8

280

119890minus

94

52119890

minus9

100

020

79AFS

A3

02119890

minus12

145

119890minus

94

61119890

minus10

408

119890minus

10100

44329

ABC

605

eminus20

141

eminus17

463eminus

18414eminus

18100

04175

FA1

80119890

minus12

439

119890minus

91

18119890

minus9

111

119890minus

9100

379191

WPA

822

119890minus

157

05119890

minus13

121

119890minus

131

19119890

minus13

100

69339

Bridge

119891max

(119909)

=3

0054

2MN

GA

300

54300

54300

541

35119890

minus15

100

01927

PSO

300

54300

54300

544

84119890

minus8

100

009

29AFS

A300

54300

4730052

169

119890minus

412

gt8

01119890

+3

ABC

300

54300

54300

543

59119890

minus15

100

00932

FA300

54300

54300

543

11119890

minus10

100

227230

WPA

300

54300

54300

54358eminus

15100

01742

Ackley

119891min

(119909)

=0

50MN

GA

114570

126095

1216

1202719

0gt10

4119890+

4PS

O004

6917401

06846

06344

0gt1925522

AFS

A2016

0020600

9204229

01009

0gt9

80119890

+3

ABC

200085

200025

200061

00014

0gt5963841

FA00101

00209

00160

00021

0gt4

28119890

+3

WPA

888

eminus16

444

eminus15

110eminus

15852eminus

16100

79476

Mathematical Problems in Engineering 13

Table10C

ontin

ued

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Grie

wank

119891min

(119909)

=0

100

MN

GA

3174

525

3996

376

3634174

172922

0gt2

07119890

+4

PSO

00029

00082

00052

00011

0gt3670

080

AFS

A2

05119890

+3

255

119890+

32

33119890

+3

1096

821

0gt6

51119890

+3

ABC

895

119890minus

7000

432

26119890

minus4

781

119890minus

42

gt6209561

FA000

6800118

000

9100011

0gt5

72119890

+3

WPA

00

00

100

145338

14 Mathematical Problems in Engineering

0

100

10

0

1

xy

minusf(xy)

minus1

minus2

minus3

minus10minus10

times106

(a)

minus5

minus5

minus10minus10

x

y

0 5 10

0

5

10

(b)

Figure 3 Colville function (1199091

= 1199093 1199092

= 1199094) (a) surface plot and (b) contour lines

0100

0

1000

05

1

15

2

xy

minus100minus100

f(xy)

times104

(a)

x

y

0 50 100

0

50

100

minus100minus100

minus50

minus50

(b)

Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines

minus100

minus200

minus300

minus10minus10

0

100

10

0

xy

minusf(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines

Mathematical Problems in Engineering 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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 11

Table 9 Best suggestions for WPA parameters

No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2

020

2

0

xy

minusf(xy) minus1000

minus2000

minus3000

minus4000

minus2minus2

minus4minus4

(a)

x

y

0 1 2

0

05

1

15

2

minus05

minus15

minus2minus2

minus1

minus1

(b)

Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines

is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized

Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions

Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions

So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2

43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface

As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2

As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909

1 1199092) = (1 1)

But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction

On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909

1 1199092 1199093 1199094) = (1 1 1 1)

Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function

Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909

1 1199092 119909

119898) =

(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5

As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and

12 Mathematical Problems in Engineering

Table10Statistic

alresults

of50

runs

obtained

byGAP

SOA

FSAA

BCFAand

WPA

algorithm

s

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Rosenb

rock

119891min

(119909)

=0

2UN

GA

178

119890minus

1000373

000

91000

9210

gt7598

323

PSO

226

119890minus

115

89119890

minus7

107

119890minus

71

30119890

minus7

100

07444

AFS

A110

eminus13

111

119890minus

92

34119890

minus10

262

119890minus

10100

20578

ABC

599

119890minus

6000

998

61119890

minus4

00015

0gt3910

297

FA6

28119890

minus13

629

eminus10

186eminus

10162eminus

10100

3312

56WPA

349

119890minus

112

34119890

minus8

509

119890minus

94

34119890

minus9

100

66333

Colville

119891min

(119909)

=0

4UN

GA

00022

03343

01272

01062

0gt12

2119890+

3PS

O1

29119890

minus6

346

119890minus

45

06119890

minus5

671

119890minus

50

gt114

0869

AFS

A366

eminus8

891

119890minus

73

16119890

minus7

232

119890minus

7100

4018

07ABC

00103

05337

01871

01232

0gt3844193

FA2

41119890

minus7

369

119890minus

56

62119890

minus6

807

119890minus

68

gt3

14119890

+3

WPA

471

119890minus

8372

eminus7

125eminus

7697eminus

8100

27405

4

Sphere

119891min

(119909)

=0

200

US

GA

156

119890+

51

81119890

+5

171

119890+

55

78119890

+3

0gt4

44119890

+4

PSO

10361

15520

12883

01206

0gt2719

201

AFS

A5

12119890

+5

579

119890+

55

51119890

+5

163

119890+

40

gt7

41119890

+3

ABC

000

4112

521

004

4401773

0gt44

29045

FA01432

02327

01865

00199

0gt8

34119890

+3

WPA

149eminus

172

241

eminus165

156eminus

166

0100

61729

Sumsquares

119891min

(119909)

=0

150

US

GA

593

119890+

47

15+

46

63119890

+4

288

119890+

30

gt3

16119890

+4

PSO

397098

911145

559050

104165

0gt2325464

AFS

A1

43119890

+5

179

119890+

51

64119890

+5

958

119890+

30

gt7

36119890

+3

ABC

171

119890minus

500017

199

119890minus

43

36119890

minus4

0gt4351848

FA89920

998861

405721

192743

0gt6

88119890

+3

WPA

268

eminus172

547

eminus166

262eminus

167

0100

65954

Booth

119891min

(119909)

=0

2MS

GA

455

119890minus

114

55119890

minus11

455

119890minus

110

100

12621

PSO

122

119890minus

122

41119890

minus8

280

119890minus

94

52119890

minus9

100

020

79AFS

A3

02119890

minus12

145

119890minus

94

61119890

minus10

408

119890minus

10100

44329

ABC

605

eminus20

141

eminus17

463eminus

18414eminus

18100

04175

FA1

80119890

minus12

439

119890minus

91

18119890

minus9

111

119890minus

9100

379191

WPA

822

119890minus

157

05119890

minus13

121

119890minus

131

19119890

minus13

100

69339

Bridge

119891max

(119909)

=3

0054

2MN

GA

300

54300

54300

541

35119890

minus15

100

01927

PSO

300

54300

54300

544

84119890

minus8

100

009

29AFS

A300

54300

4730052

169

119890minus

412

gt8

01119890

+3

ABC

300

54300

54300

543

59119890

minus15

100

00932

FA300

54300

54300

543

11119890

minus10

100

227230

WPA

300

54300

54300

54358eminus

15100

01742

Ackley

119891min

(119909)

=0

50MN

GA

114570

126095

1216

1202719

0gt10

4119890+

4PS

O004

6917401

06846

06344

0gt1925522

AFS

A2016

0020600

9204229

01009

0gt9

80119890

+3

ABC

200085

200025

200061

00014

0gt5963841

FA00101

00209

00160

00021

0gt4

28119890

+3

WPA

888

eminus16

444

eminus15

110eminus

15852eminus

16100

79476

Mathematical Problems in Engineering 13

Table10C

ontin

ued

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Grie

wank

119891min

(119909)

=0

100

MN

GA

3174

525

3996

376

3634174

172922

0gt2

07119890

+4

PSO

00029

00082

00052

00011

0gt3670

080

AFS

A2

05119890

+3

255

119890+

32

33119890

+3

1096

821

0gt6

51119890

+3

ABC

895

119890minus

7000

432

26119890

minus4

781

119890minus

42

gt6209561

FA000

6800118

000

9100011

0gt5

72119890

+3

WPA

00

00

100

145338

14 Mathematical Problems in Engineering

0

100

10

0

1

xy

minusf(xy)

minus1

minus2

minus3

minus10minus10

times106

(a)

minus5

minus5

minus10minus10

x

y

0 5 10

0

5

10

(b)

Figure 3 Colville function (1199091

= 1199093 1199092

= 1199094) (a) surface plot and (b) contour lines

0100

0

1000

05

1

15

2

xy

minus100minus100

f(xy)

times104

(a)

x

y

0 50 100

0

50

100

minus100minus100

minus50

minus50

(b)

Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines

minus100

minus200

minus300

minus10minus10

0

100

10

0

xy

minusf(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines

Mathematical Problems in Engineering 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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

12 Mathematical Problems in Engineering

Table10Statistic

alresults

of50

runs

obtained

byGAP

SOA

FSAA

BCFAand

WPA

algorithm

s

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Rosenb

rock

119891min

(119909)

=0

2UN

GA

178

119890minus

1000373

000

91000

9210

gt7598

323

PSO

226

119890minus

115

89119890

minus7

107

119890minus

71

30119890

minus7

100

07444

AFS

A110

eminus13

111

119890minus

92

34119890

minus10

262

119890minus

10100

20578

ABC

599

119890minus

6000

998

61119890

minus4

00015

0gt3910

297

FA6

28119890

minus13

629

eminus10

186eminus

10162eminus

10100

3312

56WPA

349

119890minus

112

34119890

minus8

509

119890minus

94

34119890

minus9

100

66333

Colville

119891min

(119909)

=0

4UN

GA

00022

03343

01272

01062

0gt12

2119890+

3PS

O1

29119890

minus6

346

119890minus

45

06119890

minus5

671

119890minus

50

gt114

0869

AFS

A366

eminus8

891

119890minus

73

16119890

minus7

232

119890minus

7100

4018

07ABC

00103

05337

01871

01232

0gt3844193

FA2

41119890

minus7

369

119890minus

56

62119890

minus6

807

119890minus

68

gt3

14119890

+3

WPA

471

119890minus

8372

eminus7

125eminus

7697eminus

8100

27405

4

Sphere

119891min

(119909)

=0

200

US

GA

156

119890+

51

81119890

+5

171

119890+

55

78119890

+3

0gt4

44119890

+4

PSO

10361

15520

12883

01206

0gt2719

201

AFS

A5

12119890

+5

579

119890+

55

51119890

+5

163

119890+

40

gt7

41119890

+3

ABC

000

4112

521

004

4401773

0gt44

29045

FA01432

02327

01865

00199

0gt8

34119890

+3

WPA

149eminus

172

241

eminus165

156eminus

166

0100

61729

Sumsquares

119891min

(119909)

=0

150

US

GA

593

119890+

47

15+

46

63119890

+4

288

119890+

30

gt3

16119890

+4

PSO

397098

911145

559050

104165

0gt2325464

AFS

A1

43119890

+5

179

119890+

51

64119890

+5

958

119890+

30

gt7

36119890

+3

ABC

171

119890minus

500017

199

119890minus

43

36119890

minus4

0gt4351848

FA89920

998861

405721

192743

0gt6

88119890

+3

WPA

268

eminus172

547

eminus166

262eminus

167

0100

65954

Booth

119891min

(119909)

=0

2MS

GA

455

119890minus

114

55119890

minus11

455

119890minus

110

100

12621

PSO

122

119890minus

122

41119890

minus8

280

119890minus

94

52119890

minus9

100

020

79AFS

A3

02119890

minus12

145

119890minus

94

61119890

minus10

408

119890minus

10100

44329

ABC

605

eminus20

141

eminus17

463eminus

18414eminus

18100

04175

FA1

80119890

minus12

439

119890minus

91

18119890

minus9

111

119890minus

9100

379191

WPA

822

119890minus

157

05119890

minus13

121

119890minus

131

19119890

minus13

100

69339

Bridge

119891max

(119909)

=3

0054

2MN

GA

300

54300

54300

541

35119890

minus15

100

01927

PSO

300

54300

54300

544

84119890

minus8

100

009

29AFS

A300

54300

4730052

169

119890minus

412

gt8

01119890

+3

ABC

300

54300

54300

543

59119890

minus15

100

00932

FA300

54300

54300

543

11119890

minus10

100

227230

WPA

300

54300

54300

54358eminus

15100

01742

Ackley

119891min

(119909)

=0

50MN

GA

114570

126095

1216

1202719

0gt10

4119890+

4PS

O004

6917401

06846

06344

0gt1925522

AFS

A2016

0020600

9204229

01009

0gt9

80119890

+3

ABC

200085

200025

200061

00014

0gt5963841

FA00101

00209

00160

00021

0gt4

28119890

+3

WPA

888

eminus16

444

eminus15

110eminus

15852eminus

16100

79476

Mathematical Problems in Engineering 13

Table10C

ontin

ued

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Grie

wank

119891min

(119909)

=0

100

MN

GA

3174

525

3996

376

3634174

172922

0gt2

07119890

+4

PSO

00029

00082

00052

00011

0gt3670

080

AFS

A2

05119890

+3

255

119890+

32

33119890

+3

1096

821

0gt6

51119890

+3

ABC

895

119890minus

7000

432

26119890

minus4

781

119890minus

42

gt6209561

FA000

6800118

000

9100011

0gt5

72119890

+3

WPA

00

00

100

145338

14 Mathematical Problems in Engineering

0

100

10

0

1

xy

minusf(xy)

minus1

minus2

minus3

minus10minus10

times106

(a)

minus5

minus5

minus10minus10

x

y

0 5 10

0

5

10

(b)

Figure 3 Colville function (1199091

= 1199093 1199092

= 1199094) (a) surface plot and (b) contour lines

0100

0

1000

05

1

15

2

xy

minus100minus100

f(xy)

times104

(a)

x

y

0 50 100

0

50

100

minus100minus100

minus50

minus50

(b)

Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines

minus100

minus200

minus300

minus10minus10

0

100

10

0

xy

minusf(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines

Mathematical Problems in Engineering 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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 13

Table10C

ontin

ued

Functio

nGlobalextremum

119863C

Algorith

ms

Best

Worst

Mean

StdD

evSR

Art119904

Grie

wank

119891min

(119909)

=0

100

MN

GA

3174

525

3996

376

3634174

172922

0gt2

07119890

+4

PSO

00029

00082

00052

00011

0gt3670

080

AFS

A2

05119890

+3

255

119890+

32

33119890

+3

1096

821

0gt6

51119890

+3

ABC

895

119890minus

7000

432

26119890

minus4

781

119890minus

42

gt6209561

FA000

6800118

000

9100011

0gt5

72119890

+3

WPA

00

00

100

145338

14 Mathematical Problems in Engineering

0

100

10

0

1

xy

minusf(xy)

minus1

minus2

minus3

minus10minus10

times106

(a)

minus5

minus5

minus10minus10

x

y

0 5 10

0

5

10

(b)

Figure 3 Colville function (1199091

= 1199093 1199092

= 1199094) (a) surface plot and (b) contour lines

0100

0

1000

05

1

15

2

xy

minus100minus100

f(xy)

times104

(a)

x

y

0 50 100

0

50

100

minus100minus100

minus50

minus50

(b)

Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines

minus100

minus200

minus300

minus10minus10

0

100

10

0

xy

minusf(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines

Mathematical Problems in Engineering 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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

14 Mathematical Problems in Engineering

0

100

10

0

1

xy

minusf(xy)

minus1

minus2

minus3

minus10minus10

times106

(a)

minus5

minus5

minus10minus10

x

y

0 5 10

0

5

10

(b)

Figure 3 Colville function (1199091

= 1199093 1199092

= 1199094) (a) surface plot and (b) contour lines

0100

0

1000

05

1

15

2

xy

minus100minus100

f(xy)

times104

(a)

x

y

0 50 100

0

50

100

minus100minus100

minus50

minus50

(b)

Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines

minus100

minus200

minus300

minus10minus10

0

100

10

0

xy

minusf(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines

Mathematical Problems in Engineering 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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 15

0

100

100

1000

2000

3000

xy

minus10minus10

f(xy)

(a)

x

y

0 5 10

0

5

10

minus10minus10

minus5

minus5

(b)

Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines

0

20

20

1

2

3

xy

minus2minus2

f(xy)

(a)

x

y

minus1 0 1

0

1

05

05

15

15

minus05

minus05

minus1

minus15minus15

(b)

Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines

Sumsquares functions when compared with the best resultsof the other algorithms

Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909

1 1199092) =

(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6

As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate

Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909

1 1199092) rarr (0 0)The global

minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909

119898) = (0 0 0) Surface plot and contour

lines of them are separately shown in Figures 7 and 8

As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms

Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909

1 1199092 119909

119898) = (0 0 0) Moreover

the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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

16 Mathematical Problems in Engineering

020

400

50

0

xy

minus10

minus20

minus20

minus30

minus40minus50

minusf(xy)

(a)

minus10

minus10

minus20

minus20

minus30

minus30

x

y

0 10 20 30

0

10

20

30

(b)

Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines

01000

0

1000

0

xy

minusf(xy)

minus50

minus100

minus150

minus200

minus1000 minus1000

(a)

x

y

0 200 400 600

0

200

400

600

minus200

minus200

minus400

minus400minus600

minus600

(b)

Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines

plot and contour lines of Griewank function are shown inFigure 9

WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation

As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness

In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions

(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]

It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve

Mathematical Problems in Engineering 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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 17

optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices

5 Conclusions

Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness

After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying

Conflict of Interests

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

References

[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009

[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012

[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013

[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013

[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013

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

[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005

[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009

[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012

[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011

[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989

[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007

[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013

[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005

[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013

[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013

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