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The Effects of Using Chaotic Map on Improving the Performance The Effects of Using Chaotic Map on Improving the Performance

of Multiobjective Evolutionary Algorithms of Multiobjective Evolutionary Algorithms

Hui Lu Beihang University China

Xiaoteng Wang Beihang Univeristy China

Zongming Fei University of Kentucky zongmingfeiukyedu

Meikang Qiu San Jose State University

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The Effects of Using Chaotic Map on Improving the Performance of The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms Multiobjective Evolutionary Algorithms

Digital Object Identifier (DOI) httpdxdoiorg1011552014924652

NotesCitation Information NotesCitation Information Published in Mathematical Problems in Engineering v 2014 article ID 924652 p 1-16

Copyright copy 2014 Hui Lu et al This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This article is available at UKnowledge httpsuknowledgeukyeducs_facpub1

Research ArticleThe Effects of Using Chaotic Map on Improving thePerformance of Multiobjective Evolutionary Algorithms

Hui Lu1 Xiaoteng Wang1 Zongming Fei2 and Meikang Qiu3

1 School of Electronic and Information Engineering Beihang University Beijing 100191 China2Department of Computer Science University of Kentucky Lexington KY 40506-0046 USA3Department of Computer Engineering San Jose State University San Jose CA 95192 USA

Correspondence should be addressed to Hui Lu mluhuibuaaeducn

Received 3 December 2013 Accepted 10 January 2014 Published 27 February 2014

Academic Editor Rongni Yang

Copyright copy 2014 Hui Lu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Chaotic maps play an important role in improving evolutionary algorithms (EAs) for avoiding the local optima and speeding upthe convergence However different chaotic maps in different phases have different effects on EAsThis paper focuses on exploringthe effects of chaotic maps and giving comprehensive guidance for improving multiobjective evolutionary algorithms (MOEAs)by series of experiments NSGA-II algorithm a representative of MOEAs using the nondominated sorting and elitist strategyis taken as the framework to study the effect of chaotic maps Ten chaotic maps are applied in MOEAs in three phases that isinitial population crossover and mutation operator Multiobjective problems (MOPs) adopted are ZDT series problems to showthe generality Since the scale of some sequences generated by chaotic maps is changed to fit for MOPs the correctness of scalingtransformation of chaotic sequences is proved bymeasuring the largest Lyapunov exponentThe convergencemetric 120574 and diversitymetricΔ are chosen to evaluate the performance of new algorithms with chaosThe results of experiments demonstrate that chaoticmaps can improve the performance of MOEAs especially in solving problems with convex and piecewise Pareto front In additioncat map has the best performance in solving problems with local optima

1 Introduction

Multiobjective evolutionary algorithms have attracted wides-pread attention and have been applied successfully in manyareas such as test task scheduling problem (TTSP) [1]reservoir operation [2] proportional integral and derivative(PID) controller [3] and distribution feeder reconfiguration(DFR) [4] One key challenge in multiobjective evolutionaryalgorithms is the problem of resolving local optima andthe speed of convergence There are different solutions forimproving evolutionary algorithms Some approaches havebeen devoted to propose new algorithms such as MOEAD[5] SPEA-2 [6] and NSGA-II [7] Other researchers haveproposed a variety of hybrid algorithms which combinedthe advantages of two different methods For example a newhybrid evolutionary algorithm (EA) based on the combina-tion of the honey bee mating optimization (HBMO) and thediscrete particle swarm optimization (DPSO) called DPSO-HBMO is applied to solve the multiobjective distribution

feeder reconfiguration (DFR) problem [4] Another approachhas focused on modifying original algorithms For examplenew particle swarm optimization (PSO) methods were pro-posed by using chaotic maps for parameter adaptation [8]The results showed that chaos embedded PSO can improvethe quality of results in some optimization problems Chaosvariables are loaded into the variable colony of the immunealgorithm in the immune evolutionary algorithm and theexperimental results indicate that the new immune evolu-tionary algorithm improves the convergence performanceand search efficiency [9] Due to the characteristics such asrandomness regularity ergodicity and initial value sensitiv-eness chaos has been widely applied in the original evolutio-nary algorithms to improve the performance

Recently researches have been done to the chaos embed inevolutionary algorithms For example Alatas et al [8] appliedseven chaotic maps to generate seven new chaotic artificialbee colony algorithms Three phases were adopted in gen-erating these algorithms to solve three different benchmark

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 924652 16 pageshttpdxdoiorg1011552014924652

2 Mathematical Problems in Engineering

single objective problems Results showed that thesemethodshave somewhat improved the solution quality Tavazoei andHaeri [10] introduced ten chaotic maps in weighted gradientdirection to solve two test functions Results showed thatnone of these maps transcends other maps for all of the prob-lems and desired criteria Those researches demonstratedthat chaotic sequences replacing the random parameters inthree phases including initial population crossover operatorand mutation operator can improve the performance ofevolutionary algorithms However questions remain that fora given MOP which chaotic map should be chosen in orderto achieve the best performance It is also not clean whatkinds of combination of chaotic maps used in a particularphase have the best property Therefore it is difficult togive comprehensive guidance to improve the performance ofevolutionary algorithms

In addition from the problems solved by COA it canbe seen that single objective optimization problems are thefocus Comparisons of different chaotic maps in improvingthe effects of COAs for solving single objective problems arecommon but it is rare in solving multiobjective optimizationproblems (MOPs) Yu et al [11] revealed that COA is noteffective for solvingMOPs whereas the experiments inAlatasand Akin [12] showed the opposite The results on theseforegoing researches demonstrate that COAs are successfuland competitive for solving single objective optimizationproblem but effects of COAs on solving MOPs are notconsistent

In summary although there have been many researchesabout the chaos and its application in COAs the effectsof different chaotic maps used in different phases on theperformance of evolutionary algorithms have not yet beenfully evaluated especially for the multiobjective evolutionaryalgorithms

In this paper we explore the relationships of chaoticmaps and phases on improving multiobjective evolutionaryalgorithms by a series of experiments We will answer thequestion whether chaotic maps are suitable to improve theevolutionary algorithms in solvingMOPsWe also investigatewhich phase should be chosen when one chaotic map is usedto improve a multiobjective evolutionary algorithm

In this research NSGA-II is chosen as the main opti-mization algorithm because it captures the core ideas andcharacteristics of MOEAs with the properties of a fastnondominated sorting procedure an elitist strategy a param-eterless approach and a simple yet efficient constraint-handling method [7] Despite these good aspects of NSGA-II for solving MOPs it may be entrapped into local optimalsolutions Thus the properties of chaos can help to improvethe performance of NSGA-II

In order to reflect the diversity of chaotic maps tenchaotic maps that have been widely used in pioneeringresearches are studied in this paper They are circle mapcubic map Gauss map ICMIC map logistic map sinusoidalmap tent map Bakerrsquos map cat map and Zaslavskii mapEach chaotic map has its own property and has its own effecton improving the performance of evolutionary algorithmsFor example logistic map has Chebyshev-type distributionbut not uniform distribution As a result it is necessary foroptimal solution to go through multiple iterations

Similar to past researches chaotic maps are used in threecommon phases in evolutionary algorithms in experimentsthat is chaotic sequences for initial population chaoticsequences for crossover operator and chaotic sequences formutation operator

Five benchmark MOPs including ZDT1 ZDT2 ZDT3ZDT4 and ZDT6 [7] are chosen as test problems TheseMOPs have different characteristics and can reflect theproperty of evolutionary algorithms from different aspectsFor example we can use problem ZDT4 to evaluate theperformance of evolutionary algorithms for resolving localoptimal because ZDT4 has different local Pareto-optimalsolutions in the search space

In addition ranges of chaotic maps are not always fitfor test problems Scaling transformation is needed to applychaotic sequences For example Coelho and Mariani [13]adopted Zaslavskiirsquos map by changing its range to (0 1) andAlatas [12 14 15] took a similar approach The problem iswhether the chaotic sequences through scaling transforma-tion still maintain the properties of chaos In this paper thecorrectness of scaling transformation of chaotic sequences isproved by measuring the largest Lyapunov exponent

Finally the criteria of convergence and distribution pro-posed by Deb et al [7] are adopted in this paper to evaluatethe effects of the combinations of phases and chaotic maps onimproving the performance of multiobjective evolutionaryalgorithms One is metric 120574 which measures the extent ofconvergence to a known set of Pareto-optimal solutionsThe other is metric Δ which measures the extent of spreadachieved among the obtained solutions

From the results of experiments it can be seen thatNSGA-II embedded with chaotic maps in most cases getbetter results with regard to themetrics 120574 andΔThe effects ofusing chaotic maps depend on which chaotic map is selectedand inwhich phase it is used In particular chaos can improvethe ability of NSGA-II in solving ZDT3 and ZDT6 which aredifficult for the original NSGA-II algorithm Besides cat mapis good at solving problems with local optima such as ZDT4

The rest of paper is organized as follows Section 2 givesa summary of related work on applying chaos to improveevolutionary algorithms Section 3 shows the phases in whichchaos can be embedded in evolutionary algorithms Section 4defines ten chaotic maps which are embedded in NSGA-II inthe experiments Section 5 proves that the chaotic sequencesthrough scaling transformation still hold the properties ofchaos Section 6 describes the test problems and metricsused in the experiments Section 7 presents the performanceresults of the experiments Section 8 concludes the paper

2 Related Work

Applying chaotic maps to improve evolutionary algorithmshas been studied for a whileThere are two different strategiesto apply the chaotic maps in the evolutionary algorithms

One is to use chaotic sequences generated by chaoticmaps to replace the random parameters needed by evolution-ary algorithms Coelho [16] proposed a quantum-behavedparticle swarm optimization (QPSO) Random sequences

Mathematical Problems in Engineering 3

of mutation operator in QPSO were replaced with chaoticsequences based on Zaslavskii mapThe results demonstratedthat it is a powerful strategy to diversify the population andimprove the performance in preventing premature conver-gence to local minima Dos Coelho and Alotto consideredthe chaotic crossover operator using the Zaslavskii map tosolve multiobjective optimization problems [17] Zhang et al[18] proposed three chaotic sequences based multiobjec-tive differential evolution (CS-MODE) to solve short-termhydrothermal optimal scheduling with economic emission(SHOSEE) In themodifiedmutation operator chaotic theoryis used to increase the population diversity and some adap-tive tuning parameters are produced by chaotic mappings tocontrol the evolution

The other strategy is to use the chaos optimization as anoperator For example Alatas [14] applied chaotic search incase that a solution does not obtain improvements in artificialbee colony (ABC) algorithm The results showed that thestrategy has better performance than that of ABC algorithmWang and Zhang [19] employed chaos analogously Whenthe value of objective function had no improvement incontinuous iterations one chaotic system was applied toreinitialize half of the population It replaced the worst halfpart of the population in order to jump out of the localoptimum whereas the best half part is kept unchanged

Since evolutionary algorithms have sensitive dependenceon their initial condition and parameters the improvementson these parameters can have a good effect That may be oneof the reasons that the first strategy is widely adopted In thefirst strategy it is necessary to consider the phases of replacingrandom sequences with chaotic sequences and the differentchaotic maps adopted

For the phases of the evolutionary algorithms Caponettoet al [20] introduced chaotic sequences instead of randomones during all the phases of the evolution process Resultsshowed that the behaviors of all operators were influenced bychaotic sequences Alatas [15] Ahmadi andMojallali [21] andMa [22] focused on random parameters in initial populationCoelho [16] and Zhang et al [18] did their research onmutation operator However which phase is the best choicewas not discussed

To study the performance of different chaotic mapssome researchers give the comparisons of different chaoticmaps solving both single objective optimization problemsand MOPs Talatahari et al [23] proposed a novel chaoticimproved imperialist competitive algorithm (CICA) forglobal optimization Seven chaotic maps were utilized toimprove the movement step of the algorithm and thelogistic and sinusoidal maps were found as the best choicesCaponetto et al [20] proposed an experimental analysison the convergence of evolutionary algorithms Six chaoticmaps four phases and single-objective statistical testsshowed an improvement of evolutionary algorithms whenchaotic sequences were used instead of random processes Luet al [1] proposed a chaotic nondominated sorting geneticalgorithm (CNSGA) to solve the automatic test task schedul-ing problem (TTSP) According to the different capabilitiesof the logistic and the cat chaotic operators the CNSGAapproach using the cat population initialization the cat or

logistic crossover operator and the logisticmutation operatorperforms well and is very suitable for solving the TTSP Thecomparisons of the performance of chaotic maps in theseresearches are based on solving one specific problem so theresults cannot be generalized to offer guidance on how tochoose a chaotic map for solving other problems Further-more most researches focus on single objective problems

In contrast this paper performs extensive experimentson genetic multiobjective evolutionary algorithms embed-ded with chaotic sequences It focuses on exploring therelationships of phases and chaotic maps on improvingmultiobjective evolutionary algorithms As mentioned aboveten chaoticmaps and three phases of evolutionary algorithmsare considered Five general benchmark problems are used todemonstrate that the conclusions can be generalized Finallythe guidance is presented to help researchers choose the suit-able chaotic map and phases in multiobjective evolutionaryalgorithms for different MOPs

3 Phases in Chaos EmbeddedEvolutionary Algorithms

With the ergodic property chaos is adopted to enrich thesearching behavior and to avoid solutions being trapped intolocal optimum in optimization problems In this sectionthree key phases in evolutionary algorithms initializationcrossover and mutation are chosen to be embedded withchaos Those three phases are described as follows

31 Initialization Initial population is the starting pointof iterations Ergodicity and diversity of initial populationare very important for making sure that the individuals inthe population spread in the search spaces uniformly asfar as possible In this case initial population is generatedby chaotic maps which can form a feasible solution spacewith good distribution by the properties of randomicity andergodicity of chaos Chaotic sequences can guarantee thediversity of the initial population speed up its convergenceand enhance global search capability

More specifically a chaotic map such as logistic mapor cat map is adopted instead of random population ini-tialization of evolutionary algorithms In the experiments ofmultiobjective evolutionary algorithms with chaos the initialpopulation is generated by chaos maps For example one ofthe individuals can be denoted by 119909

119904= 1199091

119904 1199092

119904 119909119894

119904 119909119899

119904

119904 = 1 2 119873 119894 = 1 2 119873 For the logistic mapinitialization 119909119894+1

119904= 4119909119894

119904(1 minus 119909

119894

119904)

32 Crossover Operator Crossover operator is most impor-tant for evolutionary algorithms Most of the offsprings aregenerated through the crossover operator It has a great influ-ence on the convergence speed A good crossover operatormay prevent premature convergence Ergodicity of chaoshelps search all the solutions avoid solutions from falling intolocal optimum and gain the global optimum

There are many different crossover operators such assimulated binary crossover operator [7] in NSGA-II algo-rithmandmultiparent arithmetic crossover operator Chaotic

4 Mathematical Problems in Engineering

sequences substitute random parameters in the crossoveroperators Chaotic sequences do not change the randomnessof the parameter but display better randomness and thereforeenhance the global performance of evolutionary algorithms

In this paper simulated binary crossover (SBX) opera-tor is adopted in the experiment According to SBX twochild individuals 119909

1198881= 119909

1

1198881 119909

119894

1198881 119909

119899

1198881 and 119909

1198882=

1199091

1198882 119909

119894

1198882 119909

119899

1198882 are generated by a pair of parents 119909

1199011=

1199091

1199011 119909

119894

1199011 119909

119899

1199011 and 119909

1199012= 1199091

1199012 119909

119894

1199012 119909

119899

1199012 as

follows

119909119894

1198881=1

2[(1 minus 120573) 119909

119894

1199011+ (1 + 120573) 119909

119894

1199012]

119909119894

1198882=1

2[(1 + 120573) 119909

119894

1199011+ (1 minus 120573) 119909

119894

1199012]

(1)

and 120573 is generated in the following manner

120573 =

(2119906)1(120578119888+1)

if 119906 le 05

(1

2 (1 minus 119906))

1(120578119888+1)

others(2)

where 119906 is a random number in the range [0 1] 120578119888is the

distribution index for the crossover operatorSince119906 is a randomnumber119906 can be generated by chaotic

maps For instance if the chaotic map is a logistic map and inthe 119894th iteration 119906 = 119906

119894 then in the (119894 + 1)th iteration 119906

119904=

119906119894+1= 4 times 119906

119894(1 minus 119906

119894)

33 Mutation Operator Mutation operator is indispensablein the process of evolutionary algorithms This mechanismavoids solutions from falling into local optimum and guar-antees more possibilities of obtaining global optimum Theproperties of chaos like randomness and sensitivity to initialconditions contribute to preventing solutions from beingtrapped into local optimum

Random parameters in mutation operators for instancepolynomial variation are replaced by chaotic sequences Fora solution 119909

119904 the polynomial mutation is described as

119909lowast

119904= 119909119904+ (119909119906

119904minus 119909119897

119904) times 120575119904 (3)

where 119909119906119904and 119909119897

119904are the upper and lower bounds of 119909

119904 and

120575119904= (2119906119904)1(120578119898+1)

minus 1 if 119906119904lt 05

1 minus (2 times (1 minus 119906119904))1(120578119898+1)

others(4)

where 119906119904is a random number ranging from 0 to 1 120578

119898is the

distribution index for the mutation operatorThe phase for mutation is that 119906

119904is calculated by chaotic

maps in iterations For example if the chaotic map is logisticmap and in the 119894th iteration 119906

119904= 119906119894 then in the (119894 + 1)th

iteration 119906119904= 119906119894+1= 4 times 119906

119894(1 minus 119906

119894)

As a representative of MOEAs the framework of NSGA-II algorithm is adopted in the experiments In order toeliminate the effect of NSGA-II algorithm other two differentmutation operators that is Gauss mutation and Cauchymutation are chosen to replace polynomial variation

331 Gauss Mutation If random variable 119883 has the proba-bility density function

119901 (119909) =1

radic2120587120590119890minus((119909minus120583)

2

21205902

)

minusinfin lt 119909 lt +infin (5)

then119883 obeys Gauss normal distribution with the parameters120583 120590 that is119883 sim 119873(120583 1205902)

Gaussmutationmeans that the randomnumbers obeyinggauss distribution substitute 120575

119904in polynomial mutation that

is 120575119904sim 119873(120583 120590

2

)

332 Cauchy Mutation The probability density function ofCauchy distribution concentrated near the origin It is definedas

119891 (119909) =1

120587

119905

1199052 + 1199092 minusinfin lt 119909 lt +infin 119905 gt 0 (6)

It is similar to Gauss probability density function Thedifference is that the value of Cauchy distribution is lowerthan the value of Gauss distribution in the vertical directionand Cauchy distribution is closer to the horizontal axis inthe horizontal direction Cauchy mutation means that therandom numbers obeying Cauchy distribution substitute 120575

119904

in polynomial mutation

4 Chaotic Maps

Chaotic maps generate chaotic sequences in the processof evolutionary algorithms Ten chaotic maps includingboth one-dimensional maps and two-dimensional maps areintroduced in this section They will be used to improve theperformance of MOP algorithms

41 One-Dimensional Maps

(1) Circle Map Circle map is a member of a family ofdynamical systems on the circle first defined by AndreyKolmogorov He proposed this family as a simplified modelfor driven mechanical rotors specifically a free-spinningwheel weakly coupled by a spring to a motor The circlemap equations also describe a simplified model of the phase-locked loop in electronics The circle map [24] is given byiterating the map

119909119896+1= 119909119896+ 119887 minus (

119886

2120587) sin (2120587119909

119896) mod (1) (7)

with 119886 = 05 and 119887 = 02 it generates chaotic sequence in(0 1)

(2) Cubic Map Cubic map is one of the most commonly usedmaps in generating chaotic sequences in various applicationsThis map is formally defined by the following equation [25]

119909119896+1= 120588119909119896(1 minus 119909

2

119896) 119909

119896isin (0 1) (8)

Cubic map generates chaotic sequences in (0 1) with 120588 =259

Mathematical Problems in Engineering 5

(3) Gauss Map Gauss map is also one of the well-known andcommonly employed maps in generating chaotic sequences[26] as follows

119909119896+1=

0 119909119896= 0

1

119909119896

mod (1) otherwise (9)

This map also generates chaotic sequences in (0 1)

(4) ICMIC Map The iterative chaotic map with infinitecollapses (ICMIC) [27] is defined by the following equation

119909119896+1= sin( 119886

119909119896

) 119886 isin (0infin) 119909119896isin (minus1 1) (10)

The parameter ldquo119886rdquo is an adjustable parameter This paperchooses 119886 = 2 Because the range of119909

119896is not (0 1) the chaotic

sequences need to be transformed to change the range

(5) LogisticMap As awell-known chaoticmap logisticmap isone of the simplest maps and was introduced by May in 2004[28] It is often cited as an example of how complex behaviorcan arise from a very simple nonlinear dynamical equationLogistic map generates chaotic sequences in (0 1) This mapis formally defined by the following equation

119909119896+1= 119886119909119896(1 minus 119909

119896) (11)

Parameter 119886 is set to 4 in the simulation

(6) Sinusoidal IteratorThe sinusoidal iterator [29] is formallydefined by the following equation

119909119896+1= 1198861199092

119896sin (120587119909

119896) 119909

119896isin (0 1) (12)

In this paper the simplified equation is used in the followingiteration

119909119896+1= sin (120587119909

119896) 119909

119896isin (0 1) (13)

(7) Tent Map Tent chaotic map is very similar to the logisticmap which displays specific chaotic effects [30] This map isdefined by the following equation

119909119896+1= 2119909119896 119909

119896lt 05

2 (1 minus 119909119896) 119909

119896ge 05

(14)

where 119909119896is ranging from 0 to 1

Tent map generates chaotic sequences in (0 1)

42 Two-Dimensional Maps

(1) Bakerrsquos Map The Baker map [31] is described by thefollowing formulas

119861 (119909 119910) =

(2119909 2119910) for 0 le 119909 lt 05(2 minus 2119909 1 minus

119910

2) for 05 le 119909 lt 1

(15)

In the following simulations one dimension of Bakerrsquosmap which is similar to tent map is adopted The equationis defined by

119909119896+1= 2119909119896 for 0 le 119909

119896lt 05

2 minus 2119909119896 for 05 le 119909

119896lt 1

(16)

This map generates chaotic sequences in (0 1)

(2) Arnoldrsquos Cat Map Arnoldrsquos cat map is named afterVladimir Arnold who demonstrated its effects in the 1960susing an image of a cat It is represented by [32]

119909119896+1= 119909119896+ 119910119896mod (1)

119910119896+1= 119909119896+ 2119910119896mod (1)

(17)

It is obvious that the sequences 119909119896isin (0 1) and 119910

119896isin (0 1)

(3) Zaslavskii Map Zaslavskii map [33] is an interestingdynamic system with chaotic behavior The discretized equa-tion is given by

119909119896+1= (119909119896+ V + 119886119910

119896+1) mod (1)

119910119896+1= cos (2120587119909

119896) + 119890minus119903

119910119896

(18)

The Zaslavskii map shows a strange attractor with thelargest Lyapunov exponent for V = 400 119903 = 3 and119886 = 126695 In this case it can be calculated that 119910

119896+1isin

[minus10512 10512] Only one dimension is chosen in thefollowing simulation Since the scale of 119910

119896+1is not [0 1] the

chaotic sequences generated need scale transformation

5 Chaotic Properties of Sequences Generatedby Scale Transformation

Asmentioned in the previous sections the scale of sequencesgenerated by chaotic maps is not always fit for the problemsto be solved Some sequences have to change their scale andsome sequences are generated by one dimension of a two-dimension chaoticmap Hence it is necessary to demonstratethe chaotic properties of sequences after these changes

Detecting the presence of chaos in a dynamical system isusually solved by measuring the largest Lyapunov exponentwhich describes quantitatively the speed of index divergenceor convergence between the adjacent phase space orbits Apositive largest Lyapunov exponent indicates chaos Sincethe chaotic sequences adopted in this paper are discrete theLyapunov exponent of discrete series can be calculated bysmall data sets arithmetic [34] This method makes full useof all the data obtains higher accuracy and has strongerrobustness for the amount of data the embedding dimensionand the time delay

51 Small Data Sets Arithmetic The reconstructed trajectory119883 can be expressed as a matrix where each row is a phase-space vector that is

119883 = (1198831 1198832 119883

119872)119879

(19)

6 Mathematical Problems in Engineering

where 119883119894is the state of the system at discrete time 119894 For an

119873-point time series 1199091 1199092 119909

119873 each119883

119894is given by

119883119894= (119909119894 119909119894+119869 119909

119894+(119898minus1)119869) (20)

where 119869 is the lag or reconstruction delay and 119898 is theembedding dimension Thus119883 is an119872times119898matrix and theconstants119898119872 119869 and119873 are related as

119872 = 119873 minus (119898 minus 1) 119869 (21)After reconstructing the dynamics the algorithm locates thenearest neighbor of each point on the trajectory The nearestneighbor 119883

119895 where 119895 isin 1 2 119872 is found by searching

for the point that minimizes the distance to the particularreference point119883

119895 This is expressed as

119889119895(0) = min

119883119895

10038171003817100381710038171003817119883119895minus 119883119895

10038171003817100381710038171003817 (22)

where 119889119895(0) is the initial distance from the 119895th point to its

nearest neighbor and denotes the Euclidean norm Weimpose an additional constraint that the nearest neighborshave a temporal separation greater than the mean period ofthe time series

10038161003816100381610038161003816119895 minus 11989510038161003816100381610038161003816gt 119901 (23)

where 119901 is the mean period of time series 119901 can be estimatedby the reciprocal of the mean frequency of the powerspectrum This allows us to consider each pair of neighborsas nearby initial conditions for different trajectories Thelargest Lyapunov exponent is estimated as the mean rate ofseparation of the nearest neighbors

For each reference point119883119895 119889119895(119894) is the distance between

the 119895th pair of nearest neighbors after 119894 discrete time

119889119895(119894) =

10038171003817100381710038171003817119883119895+119894minus 119883119895+119894

10038171003817100381710038171003817 119894 = 1 2 min (119872 minus 119895119872 minus 119895)

(24)Assume that reference point 119883

119895and its nearest neighbor

119883119895have index divergence rate 120582

1 then

119889119895(119894) = 119862

1198951198901205821(119894sdotΔ119905)

119862119895= 119889119895(0) (25)

where 119862119895is the initial separation By taking the logarithm of

both sides of (25) we getln 119889119895(119894) asymp ln119862

119895+ 1205821(119894 sdot Δ119905) (26)

Equation (26) represents a set of approximately parallel lines(for 119895 = 1 2 119872) each with a slope 119904 roughly proportionalto 1205821 The largest Lyapunov exponent is easily and accurately

calculated using a least square fit to the ldquoaveragerdquo line definedby

119910 (119894) =1

Δ119905⟨ln 119889119895(119894)⟩ (27)

where ⟨ ⟩ denotes the average over all values of 119895 So

119910 (119894) =1

119902Δ119905

119902

sum

119895=1

ln 119889119895(119894) (28)

where 119902 is the number of 119889119895(119894) with 119889

119895(119894) = 0

Choose a linear area of the curve 119910(119894) sim 119894 and apply theleast square method to get the regression straight line Thenthe slope of the regression straight line is the largest Lyapunovexponent 120582

1

52 The Lyapunov Exponent of Sequences In the calculationprocess the embedding dimension 119898 is calculated throughthe method of false nearest neighbors (FNN) For the timedelay 119869 a good approximation of 119869 is equal to the numberlagging where the autocorrelation function drops to 1 minus 1119890of its initial value

Since different test problems have different rangeschaotic sequences need to be changed to different scalesTwo kinds of sequences used in experiments need to beinvestigated sequences with scales changed and sequencesgenerated by one dimension of a two-dimension chaoticmap

521 Sequences with Scales Changed Since the sequence 1199091

to 119909100

generated by ICMIC is not in (0 1) the new sequence1199101to 119910100

has to be generated by the following function

119910119894=1

2(119909119894+ 1) 119894 isin [1 100] (29)

The sequence 1199101to 119910100

is in the range of (0 1) The Lya-punov exponent of the new sequence is calculated throughsmall data sets arithmetic The average Lyapunov exponentof 10 runs is 00744 Since it is a positive number the newsequence 119910

1to 119910100

conforms to the chaotic nature

522 Sequences Generated by One Dimension of a Two-Dimension ChaoticMap For the Zaslavskii map one dimen-sion119910

119896is chosen in the following simulationThe sequence119910

1

to 119910100

is generated by 100 iterations through Zaslavskii mapThe new sequence 119911

1to 119911100

is generated by the followingfunction

119911119894=(119910119894+ 10513)

21026 119894 isin [1 100] (30)

Then the sequence 1199111to 119911100

is in (0 1) By a similarprocessing with ICMIC the average Lyapunov exponent is000194 Then the new sequence 119911

1to 119911100

conforms to thechaotic nature

6 Test Problem and Performance Measures

61 Test Problems Two-objective optimization problems arechosen to test and measure the performance improvementof the evolutionary algorithms using chaotic maps in threephases We use well-defined benchmark functions as objec-tive functions Their properties are shown in Table 1

62 Performance Measures Two criteria are used to evaluatethe performance of multiobjective optimization (1) conver-gence to the Pareto-optimal set and (2) maintenance of diver-sity in solutions of the Pareto-optimal set [7] Twometrics areadopted to evaluate the effects of the combinations of phasesand chaotic maps

The first metric 120574 measures the extent of convergence toa known set of Pareto-optimal solutions It is defined as

120574 =1

119873

119873

sum

119894=1

119889119894 (31)

Mathematical Problems in Engineering 7

Table 1 Test problems

Problem 119899 Variable bounds Objective functions Optimal solutions

ZDT1 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909)]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT2 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus (119909

1119892(119909))

2

]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT3 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909) minus (119909

1119892(119909)) sin(10120587119909

1)]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT4 101199091isin [0 1]

119909119894isin [minus5 5]

119894 = 2 119899

1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909)]

119892(119909) = 1 + (10(119899 minus 1) + sum119899

119894=2[1199092

119894minus 10 cos(4120587119909

119894)])

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT6 10 [0 1]1198911(119909) = 1 minus exp(minus4119909

1)sin6(6120587119909

1)

1198912(119909) = 119892(119909)[1 minus (119891

1(119909)119892(119909))

2

]

119892(119909) = 1 + (9[(sum119899

119894=2119909119894) (119899 minus 1)]

025

)

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

where 119889119894is the minimum Euclidean distance of every

obtained solution to the Pareto-optimal front The smallerthe value of this metric is the nearer the convergence towardPareto-front is

The other metric Δ measures the extent of spreadachieved among the obtained solutions The metric Δ isdefined by

Δ =119889119891+ 119889119897+ sum119873minus1

119894=1

10038161003816100381610038161003816119889119894minus 11988910038161003816100381610038161003816

119889119891+ 119889119897+ (119873 minus 1) 119889

(32)

The parameter 119889119894is the Euclidean distance between consecu-

tive solutions in the obtained nondominated set of solutionsTheparameters119889

119897and119889119891are the Euclidean distances between

the extreme solutions and the boundary solutions of theobtained nondominated set The parameter 119889 is the averageof all distances 119889

119894 119894 = 1 2 119873 minus 1 assuming that there are

119873 solutions on the best nondominated front

7 Experiments and Results

To explore the relationship of phases and chaotic mapsto solve MOPs NSGA-II algorithm is chosen as the mainframeworkThe ten chaotic maps mentioned in Section 4 areembedded in three different phases in the original NSGA-II algorithm Each time only one parameter is modifiedFor example if initial population is generated by chaoticmap the crossover and mutation operator are not changedSimilarly if crossover operator is modified by a chaoticmap the initial population and mutation operator are notchanged The solutions generated by the chaos embeddedNSGA-II algorithm are evaluated by two metrics 120574 and ΔFor readerrsquos convenience the new algorithms with differentcombinations of chaotic maps and phases are named asldquocns [chaotic map] [phase]rdquo and the results of differentalgorithms on test problems are named as ldquocns [chaoticmap] [phase] [test problem]rdquo In addition ldquoirdquo represents the

phase for initial population ldquocrdquo represents the phase forcrossover operator and ldquomrdquo represents the phase formutationoperator For example the results through modified initialpopulation by logistic map solving ZDT1 problem are namedas ldquocns logistic i zdt1rdquo

Each combination of one chaotic map and one phaseneeds one experiment In this research 10 chaotic maps with3 different phases based on 2 metrics solving 5 test problemsneed 150 basic experiments and obtain 300 results Eachexperiment obtains a Pareto frontThe values of convergencemetric 120574 and the diversity metric Δ are also calculated

In order to compare with the results of original NSGA-IIalgorithm we focused on the difference of the 120574 and Δ valuesof the original NSGA-II algorithm and the new algorithmFor example the 120574 of results of ldquocns sinusoidal i zdt1rdquois named as ldquocns sinusoidal i zdt1 gamardquo and the 120574 ofresults of NSGA-II solving ZDT1 problem is namedas ldquons zdt1 gamardquo Then the difference is named asldquons zdt1 gamamdashcns sinusoidal i zdt1 gamardquo When theprocesses of algorithms get to convergence the difference isvery small The properties of convergence and diversity inthe process of iterations need to be taken into account sothe 120574 values of each generation in the iterations are recordedand the differences of 120574 of each generation are obtained Thisprocess also applies to Δ

Some main parameters in the process of NSGA-II algo-rithm are introduced in the following paragraphs Then theresults of experiments are shown and analyzed

71TheMainParameters Themainparameters in the processof NSGA-II algorithm are presented in this section Choosingan appropriate representation of a chromosome is veryimportant for solving problems Real numbers are chosento represent the genes One chromosome represents oneindividual The initial population has 100 individuals andeach chromosome has a certain number of genes whichare represented by a real number Each individual of theinitial population is generated randomly with the range

8 Mathematical Problems in Engineering

Table 2 Parameters in the process of algorithms

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6119899iter 250119899pop 100119899var 30 30 30 10 10119901119888

09119901119898

130 130 130 110 110

based on the test problems The iteration will not terminateuntil the number of iterations gets to 250 For the processof NSGA-II algorithm a parent population is selected bytournament selection depending on the nondominated rankand the crowed-comparison operator Then the new popula-tion is generated by crossover and mutation operators Thecrossover operation is executed with the probability of 119901

119888=

09 The probability of mutation 119901119898is equal to the reciprocal

of 119899var which is the dimension number of a chromosome thatis 119901119898= 1119899var

Those parameters are summarized in Table 2 In thetable 119899iter is the number of iterations 119899pop is the scaleof the population 119899var is the number of dimensions of achromosome and119901

119888and119901119898are the probabilities of crossover

and mutation operations

72 Convergence Performance It is known that the 120574 differ-ence is used to evaluate the performance of the chaotic mapsin different phases inmultiobjective evolutionary algorithmsAn example is chosen for further explanation in detail As inFigure 1 the graph shows the results of solving ZDT1 prob-lems with Bakerrsquos map in crossover operator in NSGA-IIThedifferences of 120574 between the experiment ldquocns baker c zdt1rdquoand the experiment ldquons zdt1rdquo in the 250 iterations are givenAs seen from the figure the black line is above the red linewhich represents 0 so the new algorithm ldquocns bakers crdquo isbetter thanNSGA-II algorithm in solvingZDT1 problemwithregard to the convergence metric

The 120574 results of all the experiments are given similar toFigure 1 Since it is difficult to show so many graphs in thispaper the results of three typical problems are chosen thatis ZDT1 which is a simple convex problem ZDT3 whosePareto front is piecewise and ZDT4 which has local optimaThe graphs in Figures 2 3 and 4 provide a comparison of theperformance of solving different MOPs with chaotic maps ininitial population ZDT4 is chosen to show the performanceof chaotic maps in different phases on solving the sameMOPas shown in Figures 4 5 and 6 Each subgraph is labeled withthe name of the chaotic map used

In order to quantify the effect of chaotic maps and phaseswith regard to the metric 120574 the average of 120574 difference in 250generations is calculated to represent the effect of the newalgorithms

Since the order of magnitude of 120574 is not the samethe comparison of these 120574 values is not convenient Thenormalized values are obtained by dividing the 120574 values bythemaximumof the absolute values of the 120574 based on one testproblem The results of normalization are shown in Table 3

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

Bakerrsquos06

05

04

03

02

01

0

minus01

minus02

minus03

minus04

Figure 1 Performance of Bakerrsquos maps in crossover operator insolving ZDT1

Table 3 can be presented in a more intuitive way If 120574 ge03 the numerical value of 120574 is replaced by ldquo++rdquo Similarlyldquo+rdquo represents 01 le 120574 lt 03 ldquo0rdquo represents minus01 le 120574 lt01 ldquominusrdquo represents minus03 le 120574 lt minus01 and ldquominusminusrdquo represents120574 lt minus03 Therefore ldquo++rdquo means that the effect of the newalgorithm with chaotic maps is much better whereas ldquominusminusrdquo ismuch worse Table 4 shows the results

As shown in Table 4 most of the combinations of chaoticmaps and phases have a positive effect on improving the per-formance of NSGA-II algorithm The effect of some chaoticmaps is very good especially in some particular phases Forexample Bakerrsquos map in crossover operator Gauss map incrossover operator and initial population ICMIC map ininitial population sinusoidal map in initial population tentmap in crossover operation and Zaslavskii map in initialpopulation have very good effect

Since ZDT4 problem has 219 or 794times1011 different localPareto-optimal fronts in the search space the solutions easilyget entrapped into local optimum As seen from Table 4chaotic maps used for crossover and mutation operator havesignificant improvement on evolutionary algorithms solvingZDT4 problem especially cat map has the best performancein tenmaps Circle map and cubicmap have less contributionin solving those MOPs The distribution of cat map isrelatively uniform It is probably the reason for the goodperformance in solving problems with local optima

The original NSGA-II algorithm is not good at solvingZDT3 and ZDT6 problems because Pareto-optimal front ofZDT3 is disconnected and solutions of ZDT6 are nonuni-formly spaced However it can be seen in Table 4 that chaoticmaps can improve NSGA-II especially in crossover operationand initial population in solving ZDT3 and ZDT6 problem

In order to eliminate the special effect of the NSGA-II algorithm the polynomial mutation operator in NSGA-II is changed by the Gauss mutation and Cauchy mutationoperators Four typical chaotic maps which include twochaotic maps with best performance and two chaotic maps

Mathematical Problems in Engineering 9

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

GenerationGeneration

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

Figure 2 Performance of chaotic maps in initial population in solving ZDT1

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma 06

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

Figure 3 Performance of chaotic maps in initial population in solving ZDT3

10 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 4 Performance of chaotic maps in initial population in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 5 Performance of chaotic maps in crossover operator in solving ZDT4

Mathematical Problems in Engineering 11

Table 3 The normalized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker 0617 0080 0049 0682 0149 0094 0668 0109 0003 0220 minus0040 0167 0567 0105 minus0090Cat minus0060 0076 0101 0013 0084 0024 minus0007 0090 0174 0191 0109 0158 minus0028 0022 minus0096Circle minus0142 minus0040 minus0016 0013 minus0121 minus0007 minus0153 0028 minus0016 0151 minus0069 0098 minus0176 minus0072 minus0002Cubic minus0544 0064 minus0025 minus0626 minus0041 0006 minus0334 minus0049 0077 0032 minus0781 0071 minus0288 0040 0008Gauss 0307 0513 0089 0306 0507 0070 0454 0585 0037 0159 0005 0191 0114 minus0010 0132ICMIC minus0415 0558 0132 minus0280 0609 0144 minus0295 0510 0004 0003 minus0380 0088 minus0162 0252 0163Logistic 0070 0242 0189 0072 0204 minus0031 0012 0158 0127 0017 minus0819 0183 0152 0129 0132Sinusoidal 0077 1 0616 0121 1 0742 0169 1 0734 minus0091 minus1 minus0138 0148 0688 1Tent 0655 0177 0062 0704 0103 minus0005 0731 0043 minus0088 0190 minus0008 minus0058 0569 0124 minus0003Zaslavskii minus0051 0462 0032 minus0150 0518 0064 minus0086 0499 0110 minus0103 minus0339 0108 minus0060 0174 0183

Table 4 The visualized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker ++ 0 0 ++ + 0 ++ + 0 + 0 + ++ + 0Cat 0 0 + 0 0 0 0 0 + + + + 0 0 0Circle minus 0 0 0 minus 0 minus 0 0 + 0 0 minus 0 0Cubic mdash 0 0 mdash 0 0 mdash 0 0 0 mdash 0 minus 0 0Gauss ++ ++ 0 ++ ++ 0 ++ ++ 0 + 0 + + 0 +ICMIC mdash ++ + minus ++ + minus ++ 0 0 mdash 0 minus + +Logistic 0 + + 0 + 0 0 + + 0 mdash + + + +Sinusoidal 0 ++ ++ + ++ ++ + ++ ++ 0 mdash minus + ++ ++Tent ++ + 0 ++ + 0 ++ 0 0 + 0 0 ++ + 0Zaslavskii 0 ++ 0 minus ++ 0 0 ++ + minus mdash + 0 + +

Table 5 Results of Gauss mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle minus04886 minus00170 19394 minus03130 00971 minus04370 1856683 750688 28693 minus05529Cubic minus24674 minus00589 minus62676 minus00630 minus25351 minus12051 491193 minus448985 minus23071 57609Sinusoidal 10277 60982 04343 94173 minus00396 45810 minus106768 minus525536 584566 285351Tent 35035 06784 59502 11086 18616 00623 minus272428 1014576 150061 112005

Table 6 Results of Cauchy mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle 13978 05012 13997 07723 07187 04043 1354511 minus136938 01105 minus26454Cubic minus26201 minus04932 minus42148 minus05317 minus24831 minus07487 minus771621 minus456939 minus57560 54533Sinusoidal 01470 46977 08327 82820 03179 45376 minus202995 minus468243 65462 238545Tent 27613 03380 52687 09699 26813 minus01916 minus283172 579606 127523 52806

with worst performance are chosen to be used in theexperiments These chaotic maps are circle map cubic mapsinusoidal map and tent map The values of 120574 differences areshown in Tables 5 and 6 As seen from Tables 5 and 6 theperformance of sinusoidal map and tent map is better thanthe performance of circlemap and cubicmap Sinusoidalmapin initial population is better than that in crossover operationand tent map in crossover operation is better than that ininitial population This means the rules of combinations of

chaotic maps and phases in solving MOPs are almost thesame as in the previous observations So the rules based onthe framework of NSGA-II algorithm are applicable to otherMOEAs

In general Bakerrsquos map with a phase for crossover oper-ator sinusoidal map with phases for initial population andmutation operator and tent map with a phase for crossoveroperator could be the best choice for improving evolutionaryalgorithms for MOPs without local optimum For problems

12 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 6 Performance of chaotic maps in mutation operator in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

Figure 7 Performance of chaotic maps in crossover operator in solving ZDT1 with metric Δ

Mathematical Problems in Engineering 13

Table7Th

eaverage

results

ofΔ

ZDT1

ZDT2

ZDT3

ZDT4

ZDT6

ci

mc

im

ci

mc

im

ci

mBa

ker

minus00337

minus00026

00015

00226

00310

00024

minus000

96000

69000

06minus006

0300343

minus00513

00125

00024

minus00015

Cat

000

0500017

000

4200234

minus01021

minus00145

minus00019

minus00052

000

0600277

minus000

90minus00148

minus000

01minus000

02minus000

68Circle

minus00325

minus00035

minus000

64minus02019

minus00632

minus01410

minus00052

minus000

6500010

minus00255

minus00259

minus00136

00031

minus00020

000

64Cu

bic

minus000

4700121

minus00022

minus00439

minus00148

00361

000

00minus000

0600031

minus00052

minus00279

00238

minus00170

00054

minus00017

Gauss

00121

00075

minus00180

minus01075

00329

minus00711

minus00024

minus00032

00019

00330

minus00214

00155

00180

000

03000

64ICMIC

00081

000

92000

02minus00734

minus00850

minus00917

minus00036

000

08minus00054

minus00356

minus00497

00238

00123

000

0100085

Logistic

minus00118

00076

00075

minus01935

00239

00509

minus00084

minus00014

00080

00275

minus00100

minus00232

minus000

6900020

00077

Sinu

soidal

00149

000

69minus00564

minus01750

minus000

67minus02902

minus00014

00029

minus00286

minus00390

minus00164

minus01945

00129

00088

minus004

88Tent

minus00350

000

9300016

00215

minus00173

minus00261

minus00059

minus00168

00071

minus00761

minus00139

00075

00125

00028

minus000

02Za

slavskii

002307

00056

minus00014

minus00292

minus00189

minus00013

000

99000

40minus00037

00201

minus00056

000

9000038

00027

00084

14 Mathematical Problems in Engineering

Table 8 Statistical data for combinations of chaotic maps and phases on different problems

Threshold ZDT1 ZDT2 ZDT3 ZDT4 ZDT60 18 9 15 10 200001 16 9 9 10 180002 13 9 7 10 180003 13 8 6 10 140004 13 8 5 10 120005 12 8 4 10 120006 11 8 4 10 11

0 50 100 150 200 250minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

Generation

Diff

eren

ce o

f delt

a

Bakerrsquos

Figure 8 Performance of Bakerrsquos maps in crossover operator insolving ZDT1 with metric Δ

with local optimum cat map has good performance onimproving evolutionary algorithms

73 Diversity Performance Similar to the convergencemetric120574 the differences of the diversity metricΔ between the resultsof the new algorithm with chaotic maps and the originalNSGA-II algorithm are used to measure the performanceFigure 7 shows the performance of chaotic maps in crossoveroperator in solving ZDT1 with metric Δ Each subgraphshows the effect of one chaotic map To be seen more clearlythe first subgraph in Figure 7 is shown in Figure 8

As seen fromFigures 7 and 8 theΔ difference is not stablein 250 generations The average values of Δ difference in 250generations are calculated to represent the effect of the newalgorithms For brevity the rest of results are not shown ingraphs but in Table 7

As seen from Table 7 the Δ values have little differenceWe count the number of combinations of chaotic maps andphases for solving one problem in different threshold valuesFor example there are 18 combinations whose values of Δ aregreater than zero Based on the number of the combinationsof chaotic maps and phases in different threshold values thevalues of diversity metric Δ are summarized in Table 8

In Table 8 the rank of the number of Δ in differentthreshold values is ZDT1 gt ZDT6 gt ZDT4 gt ZDT2 gt

ZDT3 especially for larger threshold ZDT1 problem whichis a convex function and has no local optima is a relativelyeasy problem Chaotic maps bring the biggest improve-ments on solving ZDT1 Though the solutions of ZDT6 arenonuniformly spaced chaotic maps can find better spread ofsolutions While ZDT4 problem is a complex problem andthe solutions are easily trapped into local optima chaoticmaps can improve the distribution of the solutions ZDT2problem is a convex function and the solutions sometimesfall into the local optimumThe effects of chaoticmaps can begeneralizedThe Pareto front of ZDT3 problem is segmentedso the Δ value of ZDT3 is larger and the ranking of ZDT3 islower It is our observation that Δ is not fit for evaluating thesolutions to problems which are disconnected

Based on the diversity metric Δ chaotic maps havethe best improvement on solving convex problems withoutlocal optima and have better effect on solving problemswhich have nonuniform solutions For problems with localminimum chaotic maps embedded algorithms can improvethe performance with regard to metric Δ

A short summary can be given according to the aboveexperiments First chaotic maps can improve the perfor-mance of MOEAs but the results showed that no one chaoticmap outperforms other maps for all of the problems Theresults in this paper give some guidance on how to choose achaotic map and a phase in MOEAs Second an interestingdiscovery is that cat map has best performance on solvingproblems with local optima Uniformity of cat map may beone of the reasons for the good performance of solving ZDT4

8 Conclusion

The focus of this paper is to explore the relationships ofchaotic maps and phases in MOEAs in solving MOPs Themain framework of algorithms in experiments is the NSGA-II algorithm The combinations of ten chaotic maps andthree phases are chosen in the experiments Two metricsconvergence metric 120574 and diversity metric Δ are used toevaluate the convergence and diversity properties of thealgorithms with chaotic maps The test problems are ZDTseries which were all MOPs The ergodicity and initial valuesensitivity of chaotic maps can help evolutionary algorithmsavoid solutions from falling into local optimal and get betterconvergence In the experimental results almost all chaoticmaps have good effects on improving the performance ofevolutionary algorithms to solveMOPs without local optima

Mathematical Problems in Engineering 15

Cat map has best performance on solving problems withlocal optimum This work gives insight on choosing chaoticmaps and phases for different problems Our future workwill perform further experiments with more chaotic maps onother MOEAs and formulate the theory analysis

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China under Grant no 61101153 and Prof Qiuis supported by NSF CNS 1359557

References

[1] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing vol 13 pp2790ndash2802 2013

[2] J A Adeyemo ldquoReservoir operation using multi-objective evo-lutionary algorithms-a reviewrdquo Asian Journal of Scientific Res-earch vol 4 no 1 pp 16ndash27 2011

[3] H Hu L Xu R Wei and B Zhu ldquoMulti-objective controloptimization for greenhouse environment using evolutionaryalgorithmsrdquo Sensors vol 11 no 6 pp 5792ndash5807 2011

[4] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[5] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[6] E Zitzler M Laumanns L Thiele et al SPEA2 Improving theStrength Pareto Evolutionary Algorithm Eidgenossische Techn-ische Hochschule Zurich (ETH) Institut fur Technische Infor-matik und Kommunikationsnetze (TIK) 2001

[7] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[8] B Alatas E Akin and A B Ozer ldquoChaos embedded particleswarm optimization algorithmsrdquo Chaos Solitons and Fractalsvol 40 no 4 pp 1715ndash1734 2009

[9] G ZilongW Sunrsquoan andZ Jian ldquoA novel immune evolutionaryalgorithm incorporating chaos optimizationrdquo Pattern Recogni-tion Letters vol 27 no 1 pp 2ndash8 2006

[10] M S Tavazoei and M Haeri ldquoComparison of different one-dimensional maps as chaotic search pattern in chaos optimiza-tion algorithmsrdquo Applied Mathematics and Computation vol187 no 2 pp 1076ndash1085 2007

[11] G Yu X Wang and P Li ldquoApplication of chaotic theory indifferential evolution algorithmsrdquo in Proceedings of the 6th Inte-rnational Conference on Natural Computation (ICNC rsquo10) pp3816ndash3820 August 2010

[12] B Alatas and E Akin ldquoMulti-objective rulemining using a cha-otic particle swarm optimization algorithmrdquo Knowledge-BasedSystems vol 22 no 6 pp 455ndash460 2009

[13] L D S Coelho and V C Mariani ldquoChaotic artificial immuneapproach applied to economic dispatch of electric energy usingthermal unitsrdquo Chaos Solitons and Fractals vol 40 no 5 pp2376ndash2383 2009

[14] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[15] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[16] L D S Coelho ldquoA quantum particle swarm optimizer with cha-otic mutation operatorrdquo Chaos Solitons and Fractals vol 37 no5 pp 1409ndash1418 2008

[17] L S Dos Coelho and P Alotto ldquoMultiobjective electromagneticoptimization based on a nondominated sorting genetic appr-oach with a chaotic crossover operatorrdquo IEEE Transactions onMagnetics vol 44 no 6 pp 1078ndash1081 2008

[18] H F Zhang J Z Zhou Y C Zhang N Fang and R ZhangldquoShort termhydrothermal scheduling usingmulti-objective dif-ferential evolution with three chaotic sequencesrdquo InternationalJournal of Electrical Power amp Energy Systems vol 47 pp 85ndash992013

[19] Y-J Wang and J-S Zhang ldquoGlobal optimization by an impr-oved differential evolutionary algorithmrdquo Applied Mathematicsand Computation vol 188 no 1 pp 669ndash680 2007

[20] R Caponetto L Fortuna S Fazzino and M G Xibilia ldquoCha-otic sequences to improve the performance of evolutionary alg-orithmsrdquo IEEE Transactions on Evolutionary Computation vol7 no 3 pp 289ndash304 2003

[21] M Ahmadi and H Mojallali ldquoChaotic invasive weed optimiz-ation algorithm with application to parameter estimation ofchaotic systemsrdquo Chaos Solitons amp Fractals vol 45 no 9-10pp 1108ndash1120 2012

[22] Z S Ma ldquoChaotic populations in genetic algorithmsrdquo AppliedSoft Computing vol 12 pp 2409ndash2424 2012

[23] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[24] W-M Zheng ldquoKneading plane of the circle maprdquo ChaosSolitons and Fractals vol 4 no 7 pp 1221ndash1233 1994

[25] T D Rogers and D C Whitley ldquoChaos in the cubic mappingrdquoMathematical Modelling vol 4 no 1 pp 9ndash25 1983

[26] M Bucolo R Caponetto L Fortuna M Frasca and A RizzoldquoDoes chaos work better than noiserdquo IEEE Circuits and SystemsMagazine vol 2 no 3 pp 4ndash19 2002

[27] D He C He L-G Jiang H-W Zhu and G-R Hu ldquoChaoticcharacteristics of a one-dimensional iterative map with infinitecollapsesrdquo IEEE Transactions on Circuits and Systems I vol 48no 7 pp 900ndash906 2001

[28] R May ldquoSimple mathematical models with very complicateddynamicsrdquo inTheTheory of Chaotic Attractors B Hunt T Y LiJ Kennedy and H Nusse Eds pp 85ndash93 Springer New YorkNY USA 2004

[29] R Barton ldquoChaos and fractalsrdquo The Mathematics Teacher vol83 pp 524ndash529 1990

[30] A Baykasoglu ldquoDesign optimization with chaos embeddedgreat deluge algorithmrdquoApplied Soft Computing Journal vol 12no 3 pp 1055ndash1067 2012

[31] J Fridrich ldquoSymmetric ciphers based on two-dimensional cha-otic mapsrdquo International Journal of Bifurcation and Chaos in

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl

Mathematical Problems in Engineering 3

of mutation operator in QPSO were replaced with chaoticsequences based on Zaslavskii mapThe results demonstratedthat it is a powerful strategy to diversify the population andimprove the performance in preventing premature conver-gence to local minima Dos Coelho and Alotto consideredthe chaotic crossover operator using the Zaslavskii map tosolve multiobjective optimization problems [17] Zhang et al[18] proposed three chaotic sequences based multiobjec-tive differential evolution (CS-MODE) to solve short-termhydrothermal optimal scheduling with economic emission(SHOSEE) In themodifiedmutation operator chaotic theoryis used to increase the population diversity and some adap-tive tuning parameters are produced by chaotic mappings tocontrol the evolution

The other strategy is to use the chaos optimization as anoperator For example Alatas [14] applied chaotic search incase that a solution does not obtain improvements in artificialbee colony (ABC) algorithm The results showed that thestrategy has better performance than that of ABC algorithmWang and Zhang [19] employed chaos analogously Whenthe value of objective function had no improvement incontinuous iterations one chaotic system was applied toreinitialize half of the population It replaced the worst halfpart of the population in order to jump out of the localoptimum whereas the best half part is kept unchanged

Since evolutionary algorithms have sensitive dependenceon their initial condition and parameters the improvementson these parameters can have a good effect That may be oneof the reasons that the first strategy is widely adopted In thefirst strategy it is necessary to consider the phases of replacingrandom sequences with chaotic sequences and the differentchaotic maps adopted

For the phases of the evolutionary algorithms Caponettoet al [20] introduced chaotic sequences instead of randomones during all the phases of the evolution process Resultsshowed that the behaviors of all operators were influenced bychaotic sequences Alatas [15] Ahmadi andMojallali [21] andMa [22] focused on random parameters in initial populationCoelho [16] and Zhang et al [18] did their research onmutation operator However which phase is the best choicewas not discussed

To study the performance of different chaotic mapssome researchers give the comparisons of different chaoticmaps solving both single objective optimization problemsand MOPs Talatahari et al [23] proposed a novel chaoticimproved imperialist competitive algorithm (CICA) forglobal optimization Seven chaotic maps were utilized toimprove the movement step of the algorithm and thelogistic and sinusoidal maps were found as the best choicesCaponetto et al [20] proposed an experimental analysison the convergence of evolutionary algorithms Six chaoticmaps four phases and single-objective statistical testsshowed an improvement of evolutionary algorithms whenchaotic sequences were used instead of random processes Luet al [1] proposed a chaotic nondominated sorting geneticalgorithm (CNSGA) to solve the automatic test task schedul-ing problem (TTSP) According to the different capabilitiesof the logistic and the cat chaotic operators the CNSGAapproach using the cat population initialization the cat or

logistic crossover operator and the logisticmutation operatorperforms well and is very suitable for solving the TTSP Thecomparisons of the performance of chaotic maps in theseresearches are based on solving one specific problem so theresults cannot be generalized to offer guidance on how tochoose a chaotic map for solving other problems Further-more most researches focus on single objective problems

In contrast this paper performs extensive experimentson genetic multiobjective evolutionary algorithms embed-ded with chaotic sequences It focuses on exploring therelationships of phases and chaotic maps on improvingmultiobjective evolutionary algorithms As mentioned aboveten chaoticmaps and three phases of evolutionary algorithmsare considered Five general benchmark problems are used todemonstrate that the conclusions can be generalized Finallythe guidance is presented to help researchers choose the suit-able chaotic map and phases in multiobjective evolutionaryalgorithms for different MOPs

3 Phases in Chaos EmbeddedEvolutionary Algorithms

With the ergodic property chaos is adopted to enrich thesearching behavior and to avoid solutions being trapped intolocal optimum in optimization problems In this sectionthree key phases in evolutionary algorithms initializationcrossover and mutation are chosen to be embedded withchaos Those three phases are described as follows

31 Initialization Initial population is the starting pointof iterations Ergodicity and diversity of initial populationare very important for making sure that the individuals inthe population spread in the search spaces uniformly asfar as possible In this case initial population is generatedby chaotic maps which can form a feasible solution spacewith good distribution by the properties of randomicity andergodicity of chaos Chaotic sequences can guarantee thediversity of the initial population speed up its convergenceand enhance global search capability

More specifically a chaotic map such as logistic mapor cat map is adopted instead of random population ini-tialization of evolutionary algorithms In the experiments ofmultiobjective evolutionary algorithms with chaos the initialpopulation is generated by chaos maps For example one ofthe individuals can be denoted by 119909

119904= 1199091

119904 1199092

119904 119909119894

119904 119909119899

119904

119904 = 1 2 119873 119894 = 1 2 119873 For the logistic mapinitialization 119909119894+1

119904= 4119909119894

119904(1 minus 119909

119894

119904)

32 Crossover Operator Crossover operator is most impor-tant for evolutionary algorithms Most of the offsprings aregenerated through the crossover operator It has a great influ-ence on the convergence speed A good crossover operatormay prevent premature convergence Ergodicity of chaoshelps search all the solutions avoid solutions from falling intolocal optimum and gain the global optimum

There are many different crossover operators such assimulated binary crossover operator [7] in NSGA-II algo-rithmandmultiparent arithmetic crossover operator Chaotic

4 Mathematical Problems in Engineering

sequences substitute random parameters in the crossoveroperators Chaotic sequences do not change the randomnessof the parameter but display better randomness and thereforeenhance the global performance of evolutionary algorithms

In this paper simulated binary crossover (SBX) opera-tor is adopted in the experiment According to SBX twochild individuals 119909

1198881= 119909

1

1198881 119909

119894

1198881 119909

119899

1198881 and 119909

1198882=

1199091

1198882 119909

119894

1198882 119909

119899

1198882 are generated by a pair of parents 119909

1199011=

1199091

1199011 119909

119894

1199011 119909

119899

1199011 and 119909

1199012= 1199091

1199012 119909

119894

1199012 119909

119899

1199012 as

follows

119909119894

1198881=1

2[(1 minus 120573) 119909

119894

1199011+ (1 + 120573) 119909

119894

1199012]

119909119894

1198882=1

2[(1 + 120573) 119909

119894

1199011+ (1 minus 120573) 119909

119894

1199012]

(1)

and 120573 is generated in the following manner

120573 =

(2119906)1(120578119888+1)

if 119906 le 05

(1

2 (1 minus 119906))

1(120578119888+1)

others(2)

where 119906 is a random number in the range [0 1] 120578119888is the

distribution index for the crossover operatorSince119906 is a randomnumber119906 can be generated by chaotic

maps For instance if the chaotic map is a logistic map and inthe 119894th iteration 119906 = 119906

119894 then in the (119894 + 1)th iteration 119906

119904=

119906119894+1= 4 times 119906

119894(1 minus 119906

119894)

33 Mutation Operator Mutation operator is indispensablein the process of evolutionary algorithms This mechanismavoids solutions from falling into local optimum and guar-antees more possibilities of obtaining global optimum Theproperties of chaos like randomness and sensitivity to initialconditions contribute to preventing solutions from beingtrapped into local optimum

Random parameters in mutation operators for instancepolynomial variation are replaced by chaotic sequences Fora solution 119909

119904 the polynomial mutation is described as

119909lowast

119904= 119909119904+ (119909119906

119904minus 119909119897

119904) times 120575119904 (3)

where 119909119906119904and 119909119897

119904are the upper and lower bounds of 119909

119904 and

120575119904= (2119906119904)1(120578119898+1)

minus 1 if 119906119904lt 05

1 minus (2 times (1 minus 119906119904))1(120578119898+1)

others(4)

where 119906119904is a random number ranging from 0 to 1 120578

119898is the

distribution index for the mutation operatorThe phase for mutation is that 119906

119904is calculated by chaotic

maps in iterations For example if the chaotic map is logisticmap and in the 119894th iteration 119906

119904= 119906119894 then in the (119894 + 1)th

iteration 119906119904= 119906119894+1= 4 times 119906

119894(1 minus 119906

119894)

As a representative of MOEAs the framework of NSGA-II algorithm is adopted in the experiments In order toeliminate the effect of NSGA-II algorithm other two differentmutation operators that is Gauss mutation and Cauchymutation are chosen to replace polynomial variation

331 Gauss Mutation If random variable 119883 has the proba-bility density function

119901 (119909) =1

radic2120587120590119890minus((119909minus120583)

2

21205902

)

minusinfin lt 119909 lt +infin (5)

then119883 obeys Gauss normal distribution with the parameters120583 120590 that is119883 sim 119873(120583 1205902)

Gaussmutationmeans that the randomnumbers obeyinggauss distribution substitute 120575

119904in polynomial mutation that

is 120575119904sim 119873(120583 120590

2

)

332 Cauchy Mutation The probability density function ofCauchy distribution concentrated near the origin It is definedas

119891 (119909) =1

120587

119905

1199052 + 1199092 minusinfin lt 119909 lt +infin 119905 gt 0 (6)

It is similar to Gauss probability density function Thedifference is that the value of Cauchy distribution is lowerthan the value of Gauss distribution in the vertical directionand Cauchy distribution is closer to the horizontal axis inthe horizontal direction Cauchy mutation means that therandom numbers obeying Cauchy distribution substitute 120575

119904

in polynomial mutation

4 Chaotic Maps

Chaotic maps generate chaotic sequences in the processof evolutionary algorithms Ten chaotic maps includingboth one-dimensional maps and two-dimensional maps areintroduced in this section They will be used to improve theperformance of MOP algorithms

41 One-Dimensional Maps

(1) Circle Map Circle map is a member of a family ofdynamical systems on the circle first defined by AndreyKolmogorov He proposed this family as a simplified modelfor driven mechanical rotors specifically a free-spinningwheel weakly coupled by a spring to a motor The circlemap equations also describe a simplified model of the phase-locked loop in electronics The circle map [24] is given byiterating the map

119909119896+1= 119909119896+ 119887 minus (

119886

2120587) sin (2120587119909

119896) mod (1) (7)

with 119886 = 05 and 119887 = 02 it generates chaotic sequence in(0 1)

(2) Cubic Map Cubic map is one of the most commonly usedmaps in generating chaotic sequences in various applicationsThis map is formally defined by the following equation [25]

119909119896+1= 120588119909119896(1 minus 119909

2

119896) 119909

119896isin (0 1) (8)

Cubic map generates chaotic sequences in (0 1) with 120588 =259

Mathematical Problems in Engineering 5

(3) Gauss Map Gauss map is also one of the well-known andcommonly employed maps in generating chaotic sequences[26] as follows

119909119896+1=

0 119909119896= 0

1

119909119896

mod (1) otherwise (9)

This map also generates chaotic sequences in (0 1)

(4) ICMIC Map The iterative chaotic map with infinitecollapses (ICMIC) [27] is defined by the following equation

119909119896+1= sin( 119886

119909119896

) 119886 isin (0infin) 119909119896isin (minus1 1) (10)

The parameter ldquo119886rdquo is an adjustable parameter This paperchooses 119886 = 2 Because the range of119909

119896is not (0 1) the chaotic

sequences need to be transformed to change the range

(5) LogisticMap As awell-known chaoticmap logisticmap isone of the simplest maps and was introduced by May in 2004[28] It is often cited as an example of how complex behaviorcan arise from a very simple nonlinear dynamical equationLogistic map generates chaotic sequences in (0 1) This mapis formally defined by the following equation

119909119896+1= 119886119909119896(1 minus 119909

119896) (11)

Parameter 119886 is set to 4 in the simulation

(6) Sinusoidal IteratorThe sinusoidal iterator [29] is formallydefined by the following equation

119909119896+1= 1198861199092

119896sin (120587119909

119896) 119909

119896isin (0 1) (12)

In this paper the simplified equation is used in the followingiteration

119909119896+1= sin (120587119909

119896) 119909

119896isin (0 1) (13)

(7) Tent Map Tent chaotic map is very similar to the logisticmap which displays specific chaotic effects [30] This map isdefined by the following equation

119909119896+1= 2119909119896 119909

119896lt 05

2 (1 minus 119909119896) 119909

119896ge 05

(14)

where 119909119896is ranging from 0 to 1

Tent map generates chaotic sequences in (0 1)

42 Two-Dimensional Maps

(1) Bakerrsquos Map The Baker map [31] is described by thefollowing formulas

119861 (119909 119910) =

(2119909 2119910) for 0 le 119909 lt 05(2 minus 2119909 1 minus

119910

2) for 05 le 119909 lt 1

(15)

In the following simulations one dimension of Bakerrsquosmap which is similar to tent map is adopted The equationis defined by

119909119896+1= 2119909119896 for 0 le 119909

119896lt 05

2 minus 2119909119896 for 05 le 119909

119896lt 1

(16)

This map generates chaotic sequences in (0 1)

(2) Arnoldrsquos Cat Map Arnoldrsquos cat map is named afterVladimir Arnold who demonstrated its effects in the 1960susing an image of a cat It is represented by [32]

119909119896+1= 119909119896+ 119910119896mod (1)

119910119896+1= 119909119896+ 2119910119896mod (1)

(17)

It is obvious that the sequences 119909119896isin (0 1) and 119910

119896isin (0 1)

(3) Zaslavskii Map Zaslavskii map [33] is an interestingdynamic system with chaotic behavior The discretized equa-tion is given by

119909119896+1= (119909119896+ V + 119886119910

119896+1) mod (1)

119910119896+1= cos (2120587119909

119896) + 119890minus119903

119910119896

(18)

The Zaslavskii map shows a strange attractor with thelargest Lyapunov exponent for V = 400 119903 = 3 and119886 = 126695 In this case it can be calculated that 119910

119896+1isin

[minus10512 10512] Only one dimension is chosen in thefollowing simulation Since the scale of 119910

119896+1is not [0 1] the

chaotic sequences generated need scale transformation

5 Chaotic Properties of Sequences Generatedby Scale Transformation

Asmentioned in the previous sections the scale of sequencesgenerated by chaotic maps is not always fit for the problemsto be solved Some sequences have to change their scale andsome sequences are generated by one dimension of a two-dimension chaoticmap Hence it is necessary to demonstratethe chaotic properties of sequences after these changes

Detecting the presence of chaos in a dynamical system isusually solved by measuring the largest Lyapunov exponentwhich describes quantitatively the speed of index divergenceor convergence between the adjacent phase space orbits Apositive largest Lyapunov exponent indicates chaos Sincethe chaotic sequences adopted in this paper are discrete theLyapunov exponent of discrete series can be calculated bysmall data sets arithmetic [34] This method makes full useof all the data obtains higher accuracy and has strongerrobustness for the amount of data the embedding dimensionand the time delay

51 Small Data Sets Arithmetic The reconstructed trajectory119883 can be expressed as a matrix where each row is a phase-space vector that is

119883 = (1198831 1198832 119883

119872)119879

(19)

6 Mathematical Problems in Engineering

where 119883119894is the state of the system at discrete time 119894 For an

119873-point time series 1199091 1199092 119909

119873 each119883

119894is given by

119883119894= (119909119894 119909119894+119869 119909

119894+(119898minus1)119869) (20)

where 119869 is the lag or reconstruction delay and 119898 is theembedding dimension Thus119883 is an119872times119898matrix and theconstants119898119872 119869 and119873 are related as

119872 = 119873 minus (119898 minus 1) 119869 (21)After reconstructing the dynamics the algorithm locates thenearest neighbor of each point on the trajectory The nearestneighbor 119883

119895 where 119895 isin 1 2 119872 is found by searching

for the point that minimizes the distance to the particularreference point119883

119895 This is expressed as

119889119895(0) = min

119883119895

10038171003817100381710038171003817119883119895minus 119883119895

10038171003817100381710038171003817 (22)

where 119889119895(0) is the initial distance from the 119895th point to its

nearest neighbor and denotes the Euclidean norm Weimpose an additional constraint that the nearest neighborshave a temporal separation greater than the mean period ofthe time series

10038161003816100381610038161003816119895 minus 11989510038161003816100381610038161003816gt 119901 (23)

where 119901 is the mean period of time series 119901 can be estimatedby the reciprocal of the mean frequency of the powerspectrum This allows us to consider each pair of neighborsas nearby initial conditions for different trajectories Thelargest Lyapunov exponent is estimated as the mean rate ofseparation of the nearest neighbors

For each reference point119883119895 119889119895(119894) is the distance between

the 119895th pair of nearest neighbors after 119894 discrete time

119889119895(119894) =

10038171003817100381710038171003817119883119895+119894minus 119883119895+119894

10038171003817100381710038171003817 119894 = 1 2 min (119872 minus 119895119872 minus 119895)

(24)Assume that reference point 119883

119895and its nearest neighbor

119883119895have index divergence rate 120582

1 then

119889119895(119894) = 119862

1198951198901205821(119894sdotΔ119905)

119862119895= 119889119895(0) (25)

where 119862119895is the initial separation By taking the logarithm of

both sides of (25) we getln 119889119895(119894) asymp ln119862

119895+ 1205821(119894 sdot Δ119905) (26)

Equation (26) represents a set of approximately parallel lines(for 119895 = 1 2 119872) each with a slope 119904 roughly proportionalto 1205821 The largest Lyapunov exponent is easily and accurately

calculated using a least square fit to the ldquoaveragerdquo line definedby

119910 (119894) =1

Δ119905⟨ln 119889119895(119894)⟩ (27)

where ⟨ ⟩ denotes the average over all values of 119895 So

119910 (119894) =1

119902Δ119905

119902

sum

119895=1

ln 119889119895(119894) (28)

where 119902 is the number of 119889119895(119894) with 119889

119895(119894) = 0

Choose a linear area of the curve 119910(119894) sim 119894 and apply theleast square method to get the regression straight line Thenthe slope of the regression straight line is the largest Lyapunovexponent 120582

1

52 The Lyapunov Exponent of Sequences In the calculationprocess the embedding dimension 119898 is calculated throughthe method of false nearest neighbors (FNN) For the timedelay 119869 a good approximation of 119869 is equal to the numberlagging where the autocorrelation function drops to 1 minus 1119890of its initial value

Since different test problems have different rangeschaotic sequences need to be changed to different scalesTwo kinds of sequences used in experiments need to beinvestigated sequences with scales changed and sequencesgenerated by one dimension of a two-dimension chaoticmap

521 Sequences with Scales Changed Since the sequence 1199091

to 119909100

generated by ICMIC is not in (0 1) the new sequence1199101to 119910100

has to be generated by the following function

119910119894=1

2(119909119894+ 1) 119894 isin [1 100] (29)

The sequence 1199101to 119910100

is in the range of (0 1) The Lya-punov exponent of the new sequence is calculated throughsmall data sets arithmetic The average Lyapunov exponentof 10 runs is 00744 Since it is a positive number the newsequence 119910

1to 119910100

conforms to the chaotic nature

522 Sequences Generated by One Dimension of a Two-Dimension ChaoticMap For the Zaslavskii map one dimen-sion119910

119896is chosen in the following simulationThe sequence119910

1

to 119910100

is generated by 100 iterations through Zaslavskii mapThe new sequence 119911

1to 119911100

is generated by the followingfunction

119911119894=(119910119894+ 10513)

21026 119894 isin [1 100] (30)

Then the sequence 1199111to 119911100

is in (0 1) By a similarprocessing with ICMIC the average Lyapunov exponent is000194 Then the new sequence 119911

1to 119911100

conforms to thechaotic nature

6 Test Problem and Performance Measures

61 Test Problems Two-objective optimization problems arechosen to test and measure the performance improvementof the evolutionary algorithms using chaotic maps in threephases We use well-defined benchmark functions as objec-tive functions Their properties are shown in Table 1

62 Performance Measures Two criteria are used to evaluatethe performance of multiobjective optimization (1) conver-gence to the Pareto-optimal set and (2) maintenance of diver-sity in solutions of the Pareto-optimal set [7] Twometrics areadopted to evaluate the effects of the combinations of phasesand chaotic maps

The first metric 120574 measures the extent of convergence toa known set of Pareto-optimal solutions It is defined as

120574 =1

119873

119873

sum

119894=1

119889119894 (31)

Mathematical Problems in Engineering 7

Table 1 Test problems

Problem 119899 Variable bounds Objective functions Optimal solutions

ZDT1 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909)]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT2 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus (119909

1119892(119909))

2

]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT3 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909) minus (119909

1119892(119909)) sin(10120587119909

1)]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT4 101199091isin [0 1]

119909119894isin [minus5 5]

119894 = 2 119899

1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909)]

119892(119909) = 1 + (10(119899 minus 1) + sum119899

119894=2[1199092

119894minus 10 cos(4120587119909

119894)])

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT6 10 [0 1]1198911(119909) = 1 minus exp(minus4119909

1)sin6(6120587119909

1)

1198912(119909) = 119892(119909)[1 minus (119891

1(119909)119892(119909))

2

]

119892(119909) = 1 + (9[(sum119899

119894=2119909119894) (119899 minus 1)]

025

)

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

where 119889119894is the minimum Euclidean distance of every

obtained solution to the Pareto-optimal front The smallerthe value of this metric is the nearer the convergence towardPareto-front is

The other metric Δ measures the extent of spreadachieved among the obtained solutions The metric Δ isdefined by

Δ =119889119891+ 119889119897+ sum119873minus1

119894=1

10038161003816100381610038161003816119889119894minus 11988910038161003816100381610038161003816

119889119891+ 119889119897+ (119873 minus 1) 119889

(32)

The parameter 119889119894is the Euclidean distance between consecu-

tive solutions in the obtained nondominated set of solutionsTheparameters119889

119897and119889119891are the Euclidean distances between

the extreme solutions and the boundary solutions of theobtained nondominated set The parameter 119889 is the averageof all distances 119889

119894 119894 = 1 2 119873 minus 1 assuming that there are

119873 solutions on the best nondominated front

7 Experiments and Results

To explore the relationship of phases and chaotic mapsto solve MOPs NSGA-II algorithm is chosen as the mainframeworkThe ten chaotic maps mentioned in Section 4 areembedded in three different phases in the original NSGA-II algorithm Each time only one parameter is modifiedFor example if initial population is generated by chaoticmap the crossover and mutation operator are not changedSimilarly if crossover operator is modified by a chaoticmap the initial population and mutation operator are notchanged The solutions generated by the chaos embeddedNSGA-II algorithm are evaluated by two metrics 120574 and ΔFor readerrsquos convenience the new algorithms with differentcombinations of chaotic maps and phases are named asldquocns [chaotic map] [phase]rdquo and the results of differentalgorithms on test problems are named as ldquocns [chaoticmap] [phase] [test problem]rdquo In addition ldquoirdquo represents the

phase for initial population ldquocrdquo represents the phase forcrossover operator and ldquomrdquo represents the phase formutationoperator For example the results through modified initialpopulation by logistic map solving ZDT1 problem are namedas ldquocns logistic i zdt1rdquo

Each combination of one chaotic map and one phaseneeds one experiment In this research 10 chaotic maps with3 different phases based on 2 metrics solving 5 test problemsneed 150 basic experiments and obtain 300 results Eachexperiment obtains a Pareto frontThe values of convergencemetric 120574 and the diversity metric Δ are also calculated

In order to compare with the results of original NSGA-IIalgorithm we focused on the difference of the 120574 and Δ valuesof the original NSGA-II algorithm and the new algorithmFor example the 120574 of results of ldquocns sinusoidal i zdt1rdquois named as ldquocns sinusoidal i zdt1 gamardquo and the 120574 ofresults of NSGA-II solving ZDT1 problem is namedas ldquons zdt1 gamardquo Then the difference is named asldquons zdt1 gamamdashcns sinusoidal i zdt1 gamardquo When theprocesses of algorithms get to convergence the difference isvery small The properties of convergence and diversity inthe process of iterations need to be taken into account sothe 120574 values of each generation in the iterations are recordedand the differences of 120574 of each generation are obtained Thisprocess also applies to Δ

Some main parameters in the process of NSGA-II algo-rithm are introduced in the following paragraphs Then theresults of experiments are shown and analyzed

71TheMainParameters Themainparameters in the processof NSGA-II algorithm are presented in this section Choosingan appropriate representation of a chromosome is veryimportant for solving problems Real numbers are chosento represent the genes One chromosome represents oneindividual The initial population has 100 individuals andeach chromosome has a certain number of genes whichare represented by a real number Each individual of theinitial population is generated randomly with the range

8 Mathematical Problems in Engineering

Table 2 Parameters in the process of algorithms

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6119899iter 250119899pop 100119899var 30 30 30 10 10119901119888

09119901119898

130 130 130 110 110

based on the test problems The iteration will not terminateuntil the number of iterations gets to 250 For the processof NSGA-II algorithm a parent population is selected bytournament selection depending on the nondominated rankand the crowed-comparison operator Then the new popula-tion is generated by crossover and mutation operators Thecrossover operation is executed with the probability of 119901

119888=

09 The probability of mutation 119901119898is equal to the reciprocal

of 119899var which is the dimension number of a chromosome thatis 119901119898= 1119899var

Those parameters are summarized in Table 2 In thetable 119899iter is the number of iterations 119899pop is the scaleof the population 119899var is the number of dimensions of achromosome and119901

119888and119901119898are the probabilities of crossover

and mutation operations

72 Convergence Performance It is known that the 120574 differ-ence is used to evaluate the performance of the chaotic mapsin different phases inmultiobjective evolutionary algorithmsAn example is chosen for further explanation in detail As inFigure 1 the graph shows the results of solving ZDT1 prob-lems with Bakerrsquos map in crossover operator in NSGA-IIThedifferences of 120574 between the experiment ldquocns baker c zdt1rdquoand the experiment ldquons zdt1rdquo in the 250 iterations are givenAs seen from the figure the black line is above the red linewhich represents 0 so the new algorithm ldquocns bakers crdquo isbetter thanNSGA-II algorithm in solvingZDT1 problemwithregard to the convergence metric

The 120574 results of all the experiments are given similar toFigure 1 Since it is difficult to show so many graphs in thispaper the results of three typical problems are chosen thatis ZDT1 which is a simple convex problem ZDT3 whosePareto front is piecewise and ZDT4 which has local optimaThe graphs in Figures 2 3 and 4 provide a comparison of theperformance of solving different MOPs with chaotic maps ininitial population ZDT4 is chosen to show the performanceof chaotic maps in different phases on solving the sameMOPas shown in Figures 4 5 and 6 Each subgraph is labeled withthe name of the chaotic map used

In order to quantify the effect of chaotic maps and phaseswith regard to the metric 120574 the average of 120574 difference in 250generations is calculated to represent the effect of the newalgorithms

Since the order of magnitude of 120574 is not the samethe comparison of these 120574 values is not convenient Thenormalized values are obtained by dividing the 120574 values bythemaximumof the absolute values of the 120574 based on one testproblem The results of normalization are shown in Table 3

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

Bakerrsquos06

05

04

03

02

01

0

minus01

minus02

minus03

minus04

Figure 1 Performance of Bakerrsquos maps in crossover operator insolving ZDT1

Table 3 can be presented in a more intuitive way If 120574 ge03 the numerical value of 120574 is replaced by ldquo++rdquo Similarlyldquo+rdquo represents 01 le 120574 lt 03 ldquo0rdquo represents minus01 le 120574 lt01 ldquominusrdquo represents minus03 le 120574 lt minus01 and ldquominusminusrdquo represents120574 lt minus03 Therefore ldquo++rdquo means that the effect of the newalgorithm with chaotic maps is much better whereas ldquominusminusrdquo ismuch worse Table 4 shows the results

As shown in Table 4 most of the combinations of chaoticmaps and phases have a positive effect on improving the per-formance of NSGA-II algorithm The effect of some chaoticmaps is very good especially in some particular phases Forexample Bakerrsquos map in crossover operator Gauss map incrossover operator and initial population ICMIC map ininitial population sinusoidal map in initial population tentmap in crossover operation and Zaslavskii map in initialpopulation have very good effect

Since ZDT4 problem has 219 or 794times1011 different localPareto-optimal fronts in the search space the solutions easilyget entrapped into local optimum As seen from Table 4chaotic maps used for crossover and mutation operator havesignificant improvement on evolutionary algorithms solvingZDT4 problem especially cat map has the best performancein tenmaps Circle map and cubicmap have less contributionin solving those MOPs The distribution of cat map isrelatively uniform It is probably the reason for the goodperformance in solving problems with local optima

The original NSGA-II algorithm is not good at solvingZDT3 and ZDT6 problems because Pareto-optimal front ofZDT3 is disconnected and solutions of ZDT6 are nonuni-formly spaced However it can be seen in Table 4 that chaoticmaps can improve NSGA-II especially in crossover operationand initial population in solving ZDT3 and ZDT6 problem

In order to eliminate the special effect of the NSGA-II algorithm the polynomial mutation operator in NSGA-II is changed by the Gauss mutation and Cauchy mutationoperators Four typical chaotic maps which include twochaotic maps with best performance and two chaotic maps

Mathematical Problems in Engineering 9

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

GenerationGeneration

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

Figure 2 Performance of chaotic maps in initial population in solving ZDT1

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma 06

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

Figure 3 Performance of chaotic maps in initial population in solving ZDT3

10 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 4 Performance of chaotic maps in initial population in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 5 Performance of chaotic maps in crossover operator in solving ZDT4

Mathematical Problems in Engineering 11

Table 3 The normalized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker 0617 0080 0049 0682 0149 0094 0668 0109 0003 0220 minus0040 0167 0567 0105 minus0090Cat minus0060 0076 0101 0013 0084 0024 minus0007 0090 0174 0191 0109 0158 minus0028 0022 minus0096Circle minus0142 minus0040 minus0016 0013 minus0121 minus0007 minus0153 0028 minus0016 0151 minus0069 0098 minus0176 minus0072 minus0002Cubic minus0544 0064 minus0025 minus0626 minus0041 0006 minus0334 minus0049 0077 0032 minus0781 0071 minus0288 0040 0008Gauss 0307 0513 0089 0306 0507 0070 0454 0585 0037 0159 0005 0191 0114 minus0010 0132ICMIC minus0415 0558 0132 minus0280 0609 0144 minus0295 0510 0004 0003 minus0380 0088 minus0162 0252 0163Logistic 0070 0242 0189 0072 0204 minus0031 0012 0158 0127 0017 minus0819 0183 0152 0129 0132Sinusoidal 0077 1 0616 0121 1 0742 0169 1 0734 minus0091 minus1 minus0138 0148 0688 1Tent 0655 0177 0062 0704 0103 minus0005 0731 0043 minus0088 0190 minus0008 minus0058 0569 0124 minus0003Zaslavskii minus0051 0462 0032 minus0150 0518 0064 minus0086 0499 0110 minus0103 minus0339 0108 minus0060 0174 0183

Table 4 The visualized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker ++ 0 0 ++ + 0 ++ + 0 + 0 + ++ + 0Cat 0 0 + 0 0 0 0 0 + + + + 0 0 0Circle minus 0 0 0 minus 0 minus 0 0 + 0 0 minus 0 0Cubic mdash 0 0 mdash 0 0 mdash 0 0 0 mdash 0 minus 0 0Gauss ++ ++ 0 ++ ++ 0 ++ ++ 0 + 0 + + 0 +ICMIC mdash ++ + minus ++ + minus ++ 0 0 mdash 0 minus + +Logistic 0 + + 0 + 0 0 + + 0 mdash + + + +Sinusoidal 0 ++ ++ + ++ ++ + ++ ++ 0 mdash minus + ++ ++Tent ++ + 0 ++ + 0 ++ 0 0 + 0 0 ++ + 0Zaslavskii 0 ++ 0 minus ++ 0 0 ++ + minus mdash + 0 + +

Table 5 Results of Gauss mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle minus04886 minus00170 19394 minus03130 00971 minus04370 1856683 750688 28693 minus05529Cubic minus24674 minus00589 minus62676 minus00630 minus25351 minus12051 491193 minus448985 minus23071 57609Sinusoidal 10277 60982 04343 94173 minus00396 45810 minus106768 minus525536 584566 285351Tent 35035 06784 59502 11086 18616 00623 minus272428 1014576 150061 112005

Table 6 Results of Cauchy mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle 13978 05012 13997 07723 07187 04043 1354511 minus136938 01105 minus26454Cubic minus26201 minus04932 minus42148 minus05317 minus24831 minus07487 minus771621 minus456939 minus57560 54533Sinusoidal 01470 46977 08327 82820 03179 45376 minus202995 minus468243 65462 238545Tent 27613 03380 52687 09699 26813 minus01916 minus283172 579606 127523 52806

with worst performance are chosen to be used in theexperiments These chaotic maps are circle map cubic mapsinusoidal map and tent map The values of 120574 differences areshown in Tables 5 and 6 As seen from Tables 5 and 6 theperformance of sinusoidal map and tent map is better thanthe performance of circlemap and cubicmap Sinusoidalmapin initial population is better than that in crossover operationand tent map in crossover operation is better than that ininitial population This means the rules of combinations of

chaotic maps and phases in solving MOPs are almost thesame as in the previous observations So the rules based onthe framework of NSGA-II algorithm are applicable to otherMOEAs

In general Bakerrsquos map with a phase for crossover oper-ator sinusoidal map with phases for initial population andmutation operator and tent map with a phase for crossoveroperator could be the best choice for improving evolutionaryalgorithms for MOPs without local optimum For problems

12 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 6 Performance of chaotic maps in mutation operator in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

Figure 7 Performance of chaotic maps in crossover operator in solving ZDT1 with metric Δ

Mathematical Problems in Engineering 13

Table7Th

eaverage

results

ofΔ

ZDT1

ZDT2

ZDT3

ZDT4

ZDT6

ci

mc

im

ci

mc

im

ci

mBa

ker

minus00337

minus00026

00015

00226

00310

00024

minus000

96000

69000

06minus006

0300343

minus00513

00125

00024

minus00015

Cat

000

0500017

000

4200234

minus01021

minus00145

minus00019

minus00052

000

0600277

minus000

90minus00148

minus000

01minus000

02minus000

68Circle

minus00325

minus00035

minus000

64minus02019

minus00632

minus01410

minus00052

minus000

6500010

minus00255

minus00259

minus00136

00031

minus00020

000

64Cu

bic

minus000

4700121

minus00022

minus00439

minus00148

00361

000

00minus000

0600031

minus00052

minus00279

00238

minus00170

00054

minus00017

Gauss

00121

00075

minus00180

minus01075

00329

minus00711

minus00024

minus00032

00019

00330

minus00214

00155

00180

000

03000

64ICMIC

00081

000

92000

02minus00734

minus00850

minus00917

minus00036

000

08minus00054

minus00356

minus00497

00238

00123

000

0100085

Logistic

minus00118

00076

00075

minus01935

00239

00509

minus00084

minus00014

00080

00275

minus00100

minus00232

minus000

6900020

00077

Sinu

soidal

00149

000

69minus00564

minus01750

minus000

67minus02902

minus00014

00029

minus00286

minus00390

minus00164

minus01945

00129

00088

minus004

88Tent

minus00350

000

9300016

00215

minus00173

minus00261

minus00059

minus00168

00071

minus00761

minus00139

00075

00125

00028

minus000

02Za

slavskii

002307

00056

minus00014

minus00292

minus00189

minus00013

000

99000

40minus00037

00201

minus00056

000

9000038

00027

00084

14 Mathematical Problems in Engineering

Table 8 Statistical data for combinations of chaotic maps and phases on different problems

Threshold ZDT1 ZDT2 ZDT3 ZDT4 ZDT60 18 9 15 10 200001 16 9 9 10 180002 13 9 7 10 180003 13 8 6 10 140004 13 8 5 10 120005 12 8 4 10 120006 11 8 4 10 11

0 50 100 150 200 250minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

Generation

Diff

eren

ce o

f delt

a

Bakerrsquos

Figure 8 Performance of Bakerrsquos maps in crossover operator insolving ZDT1 with metric Δ

with local optimum cat map has good performance onimproving evolutionary algorithms

73 Diversity Performance Similar to the convergencemetric120574 the differences of the diversity metricΔ between the resultsof the new algorithm with chaotic maps and the originalNSGA-II algorithm are used to measure the performanceFigure 7 shows the performance of chaotic maps in crossoveroperator in solving ZDT1 with metric Δ Each subgraphshows the effect of one chaotic map To be seen more clearlythe first subgraph in Figure 7 is shown in Figure 8

As seen fromFigures 7 and 8 theΔ difference is not stablein 250 generations The average values of Δ difference in 250generations are calculated to represent the effect of the newalgorithms For brevity the rest of results are not shown ingraphs but in Table 7

As seen from Table 7 the Δ values have little differenceWe count the number of combinations of chaotic maps andphases for solving one problem in different threshold valuesFor example there are 18 combinations whose values of Δ aregreater than zero Based on the number of the combinationsof chaotic maps and phases in different threshold values thevalues of diversity metric Δ are summarized in Table 8

In Table 8 the rank of the number of Δ in differentthreshold values is ZDT1 gt ZDT6 gt ZDT4 gt ZDT2 gt

ZDT3 especially for larger threshold ZDT1 problem whichis a convex function and has no local optima is a relativelyeasy problem Chaotic maps bring the biggest improve-ments on solving ZDT1 Though the solutions of ZDT6 arenonuniformly spaced chaotic maps can find better spread ofsolutions While ZDT4 problem is a complex problem andthe solutions are easily trapped into local optima chaoticmaps can improve the distribution of the solutions ZDT2problem is a convex function and the solutions sometimesfall into the local optimumThe effects of chaoticmaps can begeneralizedThe Pareto front of ZDT3 problem is segmentedso the Δ value of ZDT3 is larger and the ranking of ZDT3 islower It is our observation that Δ is not fit for evaluating thesolutions to problems which are disconnected

Based on the diversity metric Δ chaotic maps havethe best improvement on solving convex problems withoutlocal optima and have better effect on solving problemswhich have nonuniform solutions For problems with localminimum chaotic maps embedded algorithms can improvethe performance with regard to metric Δ

A short summary can be given according to the aboveexperiments First chaotic maps can improve the perfor-mance of MOEAs but the results showed that no one chaoticmap outperforms other maps for all of the problems Theresults in this paper give some guidance on how to choose achaotic map and a phase in MOEAs Second an interestingdiscovery is that cat map has best performance on solvingproblems with local optima Uniformity of cat map may beone of the reasons for the good performance of solving ZDT4

8 Conclusion

The focus of this paper is to explore the relationships ofchaotic maps and phases in MOEAs in solving MOPs Themain framework of algorithms in experiments is the NSGA-II algorithm The combinations of ten chaotic maps andthree phases are chosen in the experiments Two metricsconvergence metric 120574 and diversity metric Δ are used toevaluate the convergence and diversity properties of thealgorithms with chaotic maps The test problems are ZDTseries which were all MOPs The ergodicity and initial valuesensitivity of chaotic maps can help evolutionary algorithmsavoid solutions from falling into local optimal and get betterconvergence In the experimental results almost all chaoticmaps have good effects on improving the performance ofevolutionary algorithms to solveMOPs without local optima

Mathematical Problems in Engineering 15

Cat map has best performance on solving problems withlocal optimum This work gives insight on choosing chaoticmaps and phases for different problems Our future workwill perform further experiments with more chaotic maps onother MOEAs and formulate the theory analysis

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China under Grant no 61101153 and Prof Qiuis supported by NSF CNS 1359557

References

[1] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing vol 13 pp2790ndash2802 2013

[2] J A Adeyemo ldquoReservoir operation using multi-objective evo-lutionary algorithms-a reviewrdquo Asian Journal of Scientific Res-earch vol 4 no 1 pp 16ndash27 2011

[3] H Hu L Xu R Wei and B Zhu ldquoMulti-objective controloptimization for greenhouse environment using evolutionaryalgorithmsrdquo Sensors vol 11 no 6 pp 5792ndash5807 2011

[4] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[5] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[6] E Zitzler M Laumanns L Thiele et al SPEA2 Improving theStrength Pareto Evolutionary Algorithm Eidgenossische Techn-ische Hochschule Zurich (ETH) Institut fur Technische Infor-matik und Kommunikationsnetze (TIK) 2001

[7] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[8] B Alatas E Akin and A B Ozer ldquoChaos embedded particleswarm optimization algorithmsrdquo Chaos Solitons and Fractalsvol 40 no 4 pp 1715ndash1734 2009

[9] G ZilongW Sunrsquoan andZ Jian ldquoA novel immune evolutionaryalgorithm incorporating chaos optimizationrdquo Pattern Recogni-tion Letters vol 27 no 1 pp 2ndash8 2006

[10] M S Tavazoei and M Haeri ldquoComparison of different one-dimensional maps as chaotic search pattern in chaos optimiza-tion algorithmsrdquo Applied Mathematics and Computation vol187 no 2 pp 1076ndash1085 2007

[11] G Yu X Wang and P Li ldquoApplication of chaotic theory indifferential evolution algorithmsrdquo in Proceedings of the 6th Inte-rnational Conference on Natural Computation (ICNC rsquo10) pp3816ndash3820 August 2010

[12] B Alatas and E Akin ldquoMulti-objective rulemining using a cha-otic particle swarm optimization algorithmrdquo Knowledge-BasedSystems vol 22 no 6 pp 455ndash460 2009

[13] L D S Coelho and V C Mariani ldquoChaotic artificial immuneapproach applied to economic dispatch of electric energy usingthermal unitsrdquo Chaos Solitons and Fractals vol 40 no 5 pp2376ndash2383 2009

[14] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[15] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[16] L D S Coelho ldquoA quantum particle swarm optimizer with cha-otic mutation operatorrdquo Chaos Solitons and Fractals vol 37 no5 pp 1409ndash1418 2008

[17] L S Dos Coelho and P Alotto ldquoMultiobjective electromagneticoptimization based on a nondominated sorting genetic appr-oach with a chaotic crossover operatorrdquo IEEE Transactions onMagnetics vol 44 no 6 pp 1078ndash1081 2008

[18] H F Zhang J Z Zhou Y C Zhang N Fang and R ZhangldquoShort termhydrothermal scheduling usingmulti-objective dif-ferential evolution with three chaotic sequencesrdquo InternationalJournal of Electrical Power amp Energy Systems vol 47 pp 85ndash992013

[19] Y-J Wang and J-S Zhang ldquoGlobal optimization by an impr-oved differential evolutionary algorithmrdquo Applied Mathematicsand Computation vol 188 no 1 pp 669ndash680 2007

[20] R Caponetto L Fortuna S Fazzino and M G Xibilia ldquoCha-otic sequences to improve the performance of evolutionary alg-orithmsrdquo IEEE Transactions on Evolutionary Computation vol7 no 3 pp 289ndash304 2003

[21] M Ahmadi and H Mojallali ldquoChaotic invasive weed optimiz-ation algorithm with application to parameter estimation ofchaotic systemsrdquo Chaos Solitons amp Fractals vol 45 no 9-10pp 1108ndash1120 2012

[22] Z S Ma ldquoChaotic populations in genetic algorithmsrdquo AppliedSoft Computing vol 12 pp 2409ndash2424 2012

[23] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[24] W-M Zheng ldquoKneading plane of the circle maprdquo ChaosSolitons and Fractals vol 4 no 7 pp 1221ndash1233 1994

[25] T D Rogers and D C Whitley ldquoChaos in the cubic mappingrdquoMathematical Modelling vol 4 no 1 pp 9ndash25 1983

[26] M Bucolo R Caponetto L Fortuna M Frasca and A RizzoldquoDoes chaos work better than noiserdquo IEEE Circuits and SystemsMagazine vol 2 no 3 pp 4ndash19 2002

[27] D He C He L-G Jiang H-W Zhu and G-R Hu ldquoChaoticcharacteristics of a one-dimensional iterative map with infinitecollapsesrdquo IEEE Transactions on Circuits and Systems I vol 48no 7 pp 900ndash906 2001

[28] R May ldquoSimple mathematical models with very complicateddynamicsrdquo inTheTheory of Chaotic Attractors B Hunt T Y LiJ Kennedy and H Nusse Eds pp 85ndash93 Springer New YorkNY USA 2004

[29] R Barton ldquoChaos and fractalsrdquo The Mathematics Teacher vol83 pp 524ndash529 1990

[30] A Baykasoglu ldquoDesign optimization with chaos embeddedgreat deluge algorithmrdquoApplied Soft Computing Journal vol 12no 3 pp 1055ndash1067 2012

[31] J Fridrich ldquoSymmetric ciphers based on two-dimensional cha-otic mapsrdquo International Journal of Bifurcation and Chaos in

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014

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Stochastic AnalysisInternational Journal of

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl

4 Mathematical Problems in Engineering

sequences substitute random parameters in the crossoveroperators Chaotic sequences do not change the randomnessof the parameter but display better randomness and thereforeenhance the global performance of evolutionary algorithms

In this paper simulated binary crossover (SBX) opera-tor is adopted in the experiment According to SBX twochild individuals 119909

1198881= 119909

1

1198881 119909

119894

1198881 119909

119899

1198881 and 119909

1198882=

1199091

1198882 119909

119894

1198882 119909

119899

1198882 are generated by a pair of parents 119909

1199011=

1199091

1199011 119909

119894

1199011 119909

119899

1199011 and 119909

1199012= 1199091

1199012 119909

119894

1199012 119909

119899

1199012 as

follows

119909119894

1198881=1

2[(1 minus 120573) 119909

119894

1199011+ (1 + 120573) 119909

119894

1199012]

119909119894

1198882=1

2[(1 + 120573) 119909

119894

1199011+ (1 minus 120573) 119909

119894

1199012]

(1)

and 120573 is generated in the following manner

120573 =

(2119906)1(120578119888+1)

if 119906 le 05

(1

2 (1 minus 119906))

1(120578119888+1)

others(2)

where 119906 is a random number in the range [0 1] 120578119888is the

distribution index for the crossover operatorSince119906 is a randomnumber119906 can be generated by chaotic

maps For instance if the chaotic map is a logistic map and inthe 119894th iteration 119906 = 119906

119894 then in the (119894 + 1)th iteration 119906

119904=

119906119894+1= 4 times 119906

119894(1 minus 119906

119894)

33 Mutation Operator Mutation operator is indispensablein the process of evolutionary algorithms This mechanismavoids solutions from falling into local optimum and guar-antees more possibilities of obtaining global optimum Theproperties of chaos like randomness and sensitivity to initialconditions contribute to preventing solutions from beingtrapped into local optimum

Random parameters in mutation operators for instancepolynomial variation are replaced by chaotic sequences Fora solution 119909

119904 the polynomial mutation is described as

119909lowast

119904= 119909119904+ (119909119906

119904minus 119909119897

119904) times 120575119904 (3)

where 119909119906119904and 119909119897

119904are the upper and lower bounds of 119909

119904 and

120575119904= (2119906119904)1(120578119898+1)

minus 1 if 119906119904lt 05

1 minus (2 times (1 minus 119906119904))1(120578119898+1)

others(4)

where 119906119904is a random number ranging from 0 to 1 120578

119898is the

distribution index for the mutation operatorThe phase for mutation is that 119906

119904is calculated by chaotic

maps in iterations For example if the chaotic map is logisticmap and in the 119894th iteration 119906

119904= 119906119894 then in the (119894 + 1)th

iteration 119906119904= 119906119894+1= 4 times 119906

119894(1 minus 119906

119894)

As a representative of MOEAs the framework of NSGA-II algorithm is adopted in the experiments In order toeliminate the effect of NSGA-II algorithm other two differentmutation operators that is Gauss mutation and Cauchymutation are chosen to replace polynomial variation

331 Gauss Mutation If random variable 119883 has the proba-bility density function

119901 (119909) =1

radic2120587120590119890minus((119909minus120583)

2

21205902

)

minusinfin lt 119909 lt +infin (5)

then119883 obeys Gauss normal distribution with the parameters120583 120590 that is119883 sim 119873(120583 1205902)

Gaussmutationmeans that the randomnumbers obeyinggauss distribution substitute 120575

119904in polynomial mutation that

is 120575119904sim 119873(120583 120590

2

)

332 Cauchy Mutation The probability density function ofCauchy distribution concentrated near the origin It is definedas

119891 (119909) =1

120587

119905

1199052 + 1199092 minusinfin lt 119909 lt +infin 119905 gt 0 (6)

It is similar to Gauss probability density function Thedifference is that the value of Cauchy distribution is lowerthan the value of Gauss distribution in the vertical directionand Cauchy distribution is closer to the horizontal axis inthe horizontal direction Cauchy mutation means that therandom numbers obeying Cauchy distribution substitute 120575

119904

in polynomial mutation

4 Chaotic Maps

Chaotic maps generate chaotic sequences in the processof evolutionary algorithms Ten chaotic maps includingboth one-dimensional maps and two-dimensional maps areintroduced in this section They will be used to improve theperformance of MOP algorithms

41 One-Dimensional Maps

(1) Circle Map Circle map is a member of a family ofdynamical systems on the circle first defined by AndreyKolmogorov He proposed this family as a simplified modelfor driven mechanical rotors specifically a free-spinningwheel weakly coupled by a spring to a motor The circlemap equations also describe a simplified model of the phase-locked loop in electronics The circle map [24] is given byiterating the map

119909119896+1= 119909119896+ 119887 minus (

119886

2120587) sin (2120587119909

119896) mod (1) (7)

with 119886 = 05 and 119887 = 02 it generates chaotic sequence in(0 1)

(2) Cubic Map Cubic map is one of the most commonly usedmaps in generating chaotic sequences in various applicationsThis map is formally defined by the following equation [25]

119909119896+1= 120588119909119896(1 minus 119909

2

119896) 119909

119896isin (0 1) (8)

Cubic map generates chaotic sequences in (0 1) with 120588 =259

Mathematical Problems in Engineering 5

(3) Gauss Map Gauss map is also one of the well-known andcommonly employed maps in generating chaotic sequences[26] as follows

119909119896+1=

0 119909119896= 0

1

119909119896

mod (1) otherwise (9)

This map also generates chaotic sequences in (0 1)

(4) ICMIC Map The iterative chaotic map with infinitecollapses (ICMIC) [27] is defined by the following equation

119909119896+1= sin( 119886

119909119896

) 119886 isin (0infin) 119909119896isin (minus1 1) (10)

The parameter ldquo119886rdquo is an adjustable parameter This paperchooses 119886 = 2 Because the range of119909

119896is not (0 1) the chaotic

sequences need to be transformed to change the range

(5) LogisticMap As awell-known chaoticmap logisticmap isone of the simplest maps and was introduced by May in 2004[28] It is often cited as an example of how complex behaviorcan arise from a very simple nonlinear dynamical equationLogistic map generates chaotic sequences in (0 1) This mapis formally defined by the following equation

119909119896+1= 119886119909119896(1 minus 119909

119896) (11)

Parameter 119886 is set to 4 in the simulation

(6) Sinusoidal IteratorThe sinusoidal iterator [29] is formallydefined by the following equation

119909119896+1= 1198861199092

119896sin (120587119909

119896) 119909

119896isin (0 1) (12)

In this paper the simplified equation is used in the followingiteration

119909119896+1= sin (120587119909

119896) 119909

119896isin (0 1) (13)

(7) Tent Map Tent chaotic map is very similar to the logisticmap which displays specific chaotic effects [30] This map isdefined by the following equation

119909119896+1= 2119909119896 119909

119896lt 05

2 (1 minus 119909119896) 119909

119896ge 05

(14)

where 119909119896is ranging from 0 to 1

Tent map generates chaotic sequences in (0 1)

42 Two-Dimensional Maps

(1) Bakerrsquos Map The Baker map [31] is described by thefollowing formulas

119861 (119909 119910) =

(2119909 2119910) for 0 le 119909 lt 05(2 minus 2119909 1 minus

119910

2) for 05 le 119909 lt 1

(15)

In the following simulations one dimension of Bakerrsquosmap which is similar to tent map is adopted The equationis defined by

119909119896+1= 2119909119896 for 0 le 119909

119896lt 05

2 minus 2119909119896 for 05 le 119909

119896lt 1

(16)

This map generates chaotic sequences in (0 1)

(2) Arnoldrsquos Cat Map Arnoldrsquos cat map is named afterVladimir Arnold who demonstrated its effects in the 1960susing an image of a cat It is represented by [32]

119909119896+1= 119909119896+ 119910119896mod (1)

119910119896+1= 119909119896+ 2119910119896mod (1)

(17)

It is obvious that the sequences 119909119896isin (0 1) and 119910

119896isin (0 1)

(3) Zaslavskii Map Zaslavskii map [33] is an interestingdynamic system with chaotic behavior The discretized equa-tion is given by

119909119896+1= (119909119896+ V + 119886119910

119896+1) mod (1)

119910119896+1= cos (2120587119909

119896) + 119890minus119903

119910119896

(18)

The Zaslavskii map shows a strange attractor with thelargest Lyapunov exponent for V = 400 119903 = 3 and119886 = 126695 In this case it can be calculated that 119910

119896+1isin

[minus10512 10512] Only one dimension is chosen in thefollowing simulation Since the scale of 119910

119896+1is not [0 1] the

chaotic sequences generated need scale transformation

5 Chaotic Properties of Sequences Generatedby Scale Transformation

Asmentioned in the previous sections the scale of sequencesgenerated by chaotic maps is not always fit for the problemsto be solved Some sequences have to change their scale andsome sequences are generated by one dimension of a two-dimension chaoticmap Hence it is necessary to demonstratethe chaotic properties of sequences after these changes

Detecting the presence of chaos in a dynamical system isusually solved by measuring the largest Lyapunov exponentwhich describes quantitatively the speed of index divergenceor convergence between the adjacent phase space orbits Apositive largest Lyapunov exponent indicates chaos Sincethe chaotic sequences adopted in this paper are discrete theLyapunov exponent of discrete series can be calculated bysmall data sets arithmetic [34] This method makes full useof all the data obtains higher accuracy and has strongerrobustness for the amount of data the embedding dimensionand the time delay

51 Small Data Sets Arithmetic The reconstructed trajectory119883 can be expressed as a matrix where each row is a phase-space vector that is

119883 = (1198831 1198832 119883

119872)119879

(19)

6 Mathematical Problems in Engineering

where 119883119894is the state of the system at discrete time 119894 For an

119873-point time series 1199091 1199092 119909

119873 each119883

119894is given by

119883119894= (119909119894 119909119894+119869 119909

119894+(119898minus1)119869) (20)

where 119869 is the lag or reconstruction delay and 119898 is theembedding dimension Thus119883 is an119872times119898matrix and theconstants119898119872 119869 and119873 are related as

119872 = 119873 minus (119898 minus 1) 119869 (21)After reconstructing the dynamics the algorithm locates thenearest neighbor of each point on the trajectory The nearestneighbor 119883

119895 where 119895 isin 1 2 119872 is found by searching

for the point that minimizes the distance to the particularreference point119883

119895 This is expressed as

119889119895(0) = min

119883119895

10038171003817100381710038171003817119883119895minus 119883119895

10038171003817100381710038171003817 (22)

where 119889119895(0) is the initial distance from the 119895th point to its

nearest neighbor and denotes the Euclidean norm Weimpose an additional constraint that the nearest neighborshave a temporal separation greater than the mean period ofthe time series

10038161003816100381610038161003816119895 minus 11989510038161003816100381610038161003816gt 119901 (23)

where 119901 is the mean period of time series 119901 can be estimatedby the reciprocal of the mean frequency of the powerspectrum This allows us to consider each pair of neighborsas nearby initial conditions for different trajectories Thelargest Lyapunov exponent is estimated as the mean rate ofseparation of the nearest neighbors

For each reference point119883119895 119889119895(119894) is the distance between

the 119895th pair of nearest neighbors after 119894 discrete time

119889119895(119894) =

10038171003817100381710038171003817119883119895+119894minus 119883119895+119894

10038171003817100381710038171003817 119894 = 1 2 min (119872 minus 119895119872 minus 119895)

(24)Assume that reference point 119883

119895and its nearest neighbor

119883119895have index divergence rate 120582

1 then

119889119895(119894) = 119862

1198951198901205821(119894sdotΔ119905)

119862119895= 119889119895(0) (25)

where 119862119895is the initial separation By taking the logarithm of

both sides of (25) we getln 119889119895(119894) asymp ln119862

119895+ 1205821(119894 sdot Δ119905) (26)

Equation (26) represents a set of approximately parallel lines(for 119895 = 1 2 119872) each with a slope 119904 roughly proportionalto 1205821 The largest Lyapunov exponent is easily and accurately

calculated using a least square fit to the ldquoaveragerdquo line definedby

119910 (119894) =1

Δ119905⟨ln 119889119895(119894)⟩ (27)

where ⟨ ⟩ denotes the average over all values of 119895 So

119910 (119894) =1

119902Δ119905

119902

sum

119895=1

ln 119889119895(119894) (28)

where 119902 is the number of 119889119895(119894) with 119889

119895(119894) = 0

Choose a linear area of the curve 119910(119894) sim 119894 and apply theleast square method to get the regression straight line Thenthe slope of the regression straight line is the largest Lyapunovexponent 120582

1

52 The Lyapunov Exponent of Sequences In the calculationprocess the embedding dimension 119898 is calculated throughthe method of false nearest neighbors (FNN) For the timedelay 119869 a good approximation of 119869 is equal to the numberlagging where the autocorrelation function drops to 1 minus 1119890of its initial value

Since different test problems have different rangeschaotic sequences need to be changed to different scalesTwo kinds of sequences used in experiments need to beinvestigated sequences with scales changed and sequencesgenerated by one dimension of a two-dimension chaoticmap

521 Sequences with Scales Changed Since the sequence 1199091

to 119909100

generated by ICMIC is not in (0 1) the new sequence1199101to 119910100

has to be generated by the following function

119910119894=1

2(119909119894+ 1) 119894 isin [1 100] (29)

The sequence 1199101to 119910100

is in the range of (0 1) The Lya-punov exponent of the new sequence is calculated throughsmall data sets arithmetic The average Lyapunov exponentof 10 runs is 00744 Since it is a positive number the newsequence 119910

1to 119910100

conforms to the chaotic nature

522 Sequences Generated by One Dimension of a Two-Dimension ChaoticMap For the Zaslavskii map one dimen-sion119910

119896is chosen in the following simulationThe sequence119910

1

to 119910100

is generated by 100 iterations through Zaslavskii mapThe new sequence 119911

1to 119911100

is generated by the followingfunction

119911119894=(119910119894+ 10513)

21026 119894 isin [1 100] (30)

Then the sequence 1199111to 119911100

is in (0 1) By a similarprocessing with ICMIC the average Lyapunov exponent is000194 Then the new sequence 119911

1to 119911100

conforms to thechaotic nature

6 Test Problem and Performance Measures

61 Test Problems Two-objective optimization problems arechosen to test and measure the performance improvementof the evolutionary algorithms using chaotic maps in threephases We use well-defined benchmark functions as objec-tive functions Their properties are shown in Table 1

62 Performance Measures Two criteria are used to evaluatethe performance of multiobjective optimization (1) conver-gence to the Pareto-optimal set and (2) maintenance of diver-sity in solutions of the Pareto-optimal set [7] Twometrics areadopted to evaluate the effects of the combinations of phasesand chaotic maps

The first metric 120574 measures the extent of convergence toa known set of Pareto-optimal solutions It is defined as

120574 =1

119873

119873

sum

119894=1

119889119894 (31)

Mathematical Problems in Engineering 7

Table 1 Test problems

Problem 119899 Variable bounds Objective functions Optimal solutions

ZDT1 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909)]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT2 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus (119909

1119892(119909))

2

]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT3 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909) minus (119909

1119892(119909)) sin(10120587119909

1)]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT4 101199091isin [0 1]

119909119894isin [minus5 5]

119894 = 2 119899

1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909)]

119892(119909) = 1 + (10(119899 minus 1) + sum119899

119894=2[1199092

119894minus 10 cos(4120587119909

119894)])

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT6 10 [0 1]1198911(119909) = 1 minus exp(minus4119909

1)sin6(6120587119909

1)

1198912(119909) = 119892(119909)[1 minus (119891

1(119909)119892(119909))

2

]

119892(119909) = 1 + (9[(sum119899

119894=2119909119894) (119899 minus 1)]

025

)

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

where 119889119894is the minimum Euclidean distance of every

obtained solution to the Pareto-optimal front The smallerthe value of this metric is the nearer the convergence towardPareto-front is

The other metric Δ measures the extent of spreadachieved among the obtained solutions The metric Δ isdefined by

Δ =119889119891+ 119889119897+ sum119873minus1

119894=1

10038161003816100381610038161003816119889119894minus 11988910038161003816100381610038161003816

119889119891+ 119889119897+ (119873 minus 1) 119889

(32)

The parameter 119889119894is the Euclidean distance between consecu-

tive solutions in the obtained nondominated set of solutionsTheparameters119889

119897and119889119891are the Euclidean distances between

the extreme solutions and the boundary solutions of theobtained nondominated set The parameter 119889 is the averageof all distances 119889

119894 119894 = 1 2 119873 minus 1 assuming that there are

119873 solutions on the best nondominated front

7 Experiments and Results

To explore the relationship of phases and chaotic mapsto solve MOPs NSGA-II algorithm is chosen as the mainframeworkThe ten chaotic maps mentioned in Section 4 areembedded in three different phases in the original NSGA-II algorithm Each time only one parameter is modifiedFor example if initial population is generated by chaoticmap the crossover and mutation operator are not changedSimilarly if crossover operator is modified by a chaoticmap the initial population and mutation operator are notchanged The solutions generated by the chaos embeddedNSGA-II algorithm are evaluated by two metrics 120574 and ΔFor readerrsquos convenience the new algorithms with differentcombinations of chaotic maps and phases are named asldquocns [chaotic map] [phase]rdquo and the results of differentalgorithms on test problems are named as ldquocns [chaoticmap] [phase] [test problem]rdquo In addition ldquoirdquo represents the

phase for initial population ldquocrdquo represents the phase forcrossover operator and ldquomrdquo represents the phase formutationoperator For example the results through modified initialpopulation by logistic map solving ZDT1 problem are namedas ldquocns logistic i zdt1rdquo

Each combination of one chaotic map and one phaseneeds one experiment In this research 10 chaotic maps with3 different phases based on 2 metrics solving 5 test problemsneed 150 basic experiments and obtain 300 results Eachexperiment obtains a Pareto frontThe values of convergencemetric 120574 and the diversity metric Δ are also calculated

In order to compare with the results of original NSGA-IIalgorithm we focused on the difference of the 120574 and Δ valuesof the original NSGA-II algorithm and the new algorithmFor example the 120574 of results of ldquocns sinusoidal i zdt1rdquois named as ldquocns sinusoidal i zdt1 gamardquo and the 120574 ofresults of NSGA-II solving ZDT1 problem is namedas ldquons zdt1 gamardquo Then the difference is named asldquons zdt1 gamamdashcns sinusoidal i zdt1 gamardquo When theprocesses of algorithms get to convergence the difference isvery small The properties of convergence and diversity inthe process of iterations need to be taken into account sothe 120574 values of each generation in the iterations are recordedand the differences of 120574 of each generation are obtained Thisprocess also applies to Δ

Some main parameters in the process of NSGA-II algo-rithm are introduced in the following paragraphs Then theresults of experiments are shown and analyzed

71TheMainParameters Themainparameters in the processof NSGA-II algorithm are presented in this section Choosingan appropriate representation of a chromosome is veryimportant for solving problems Real numbers are chosento represent the genes One chromosome represents oneindividual The initial population has 100 individuals andeach chromosome has a certain number of genes whichare represented by a real number Each individual of theinitial population is generated randomly with the range

8 Mathematical Problems in Engineering

Table 2 Parameters in the process of algorithms

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6119899iter 250119899pop 100119899var 30 30 30 10 10119901119888

09119901119898

130 130 130 110 110

based on the test problems The iteration will not terminateuntil the number of iterations gets to 250 For the processof NSGA-II algorithm a parent population is selected bytournament selection depending on the nondominated rankand the crowed-comparison operator Then the new popula-tion is generated by crossover and mutation operators Thecrossover operation is executed with the probability of 119901

119888=

09 The probability of mutation 119901119898is equal to the reciprocal

of 119899var which is the dimension number of a chromosome thatis 119901119898= 1119899var

Those parameters are summarized in Table 2 In thetable 119899iter is the number of iterations 119899pop is the scaleof the population 119899var is the number of dimensions of achromosome and119901

119888and119901119898are the probabilities of crossover

and mutation operations

72 Convergence Performance It is known that the 120574 differ-ence is used to evaluate the performance of the chaotic mapsin different phases inmultiobjective evolutionary algorithmsAn example is chosen for further explanation in detail As inFigure 1 the graph shows the results of solving ZDT1 prob-lems with Bakerrsquos map in crossover operator in NSGA-IIThedifferences of 120574 between the experiment ldquocns baker c zdt1rdquoand the experiment ldquons zdt1rdquo in the 250 iterations are givenAs seen from the figure the black line is above the red linewhich represents 0 so the new algorithm ldquocns bakers crdquo isbetter thanNSGA-II algorithm in solvingZDT1 problemwithregard to the convergence metric

The 120574 results of all the experiments are given similar toFigure 1 Since it is difficult to show so many graphs in thispaper the results of three typical problems are chosen thatis ZDT1 which is a simple convex problem ZDT3 whosePareto front is piecewise and ZDT4 which has local optimaThe graphs in Figures 2 3 and 4 provide a comparison of theperformance of solving different MOPs with chaotic maps ininitial population ZDT4 is chosen to show the performanceof chaotic maps in different phases on solving the sameMOPas shown in Figures 4 5 and 6 Each subgraph is labeled withthe name of the chaotic map used

In order to quantify the effect of chaotic maps and phaseswith regard to the metric 120574 the average of 120574 difference in 250generations is calculated to represent the effect of the newalgorithms

Since the order of magnitude of 120574 is not the samethe comparison of these 120574 values is not convenient Thenormalized values are obtained by dividing the 120574 values bythemaximumof the absolute values of the 120574 based on one testproblem The results of normalization are shown in Table 3

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

Bakerrsquos06

05

04

03

02

01

0

minus01

minus02

minus03

minus04

Figure 1 Performance of Bakerrsquos maps in crossover operator insolving ZDT1

Table 3 can be presented in a more intuitive way If 120574 ge03 the numerical value of 120574 is replaced by ldquo++rdquo Similarlyldquo+rdquo represents 01 le 120574 lt 03 ldquo0rdquo represents minus01 le 120574 lt01 ldquominusrdquo represents minus03 le 120574 lt minus01 and ldquominusminusrdquo represents120574 lt minus03 Therefore ldquo++rdquo means that the effect of the newalgorithm with chaotic maps is much better whereas ldquominusminusrdquo ismuch worse Table 4 shows the results

As shown in Table 4 most of the combinations of chaoticmaps and phases have a positive effect on improving the per-formance of NSGA-II algorithm The effect of some chaoticmaps is very good especially in some particular phases Forexample Bakerrsquos map in crossover operator Gauss map incrossover operator and initial population ICMIC map ininitial population sinusoidal map in initial population tentmap in crossover operation and Zaslavskii map in initialpopulation have very good effect

Since ZDT4 problem has 219 or 794times1011 different localPareto-optimal fronts in the search space the solutions easilyget entrapped into local optimum As seen from Table 4chaotic maps used for crossover and mutation operator havesignificant improvement on evolutionary algorithms solvingZDT4 problem especially cat map has the best performancein tenmaps Circle map and cubicmap have less contributionin solving those MOPs The distribution of cat map isrelatively uniform It is probably the reason for the goodperformance in solving problems with local optima

The original NSGA-II algorithm is not good at solvingZDT3 and ZDT6 problems because Pareto-optimal front ofZDT3 is disconnected and solutions of ZDT6 are nonuni-formly spaced However it can be seen in Table 4 that chaoticmaps can improve NSGA-II especially in crossover operationand initial population in solving ZDT3 and ZDT6 problem

In order to eliminate the special effect of the NSGA-II algorithm the polynomial mutation operator in NSGA-II is changed by the Gauss mutation and Cauchy mutationoperators Four typical chaotic maps which include twochaotic maps with best performance and two chaotic maps

Mathematical Problems in Engineering 9

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

GenerationGeneration

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

Figure 2 Performance of chaotic maps in initial population in solving ZDT1

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma 06

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

Figure 3 Performance of chaotic maps in initial population in solving ZDT3

10 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 4 Performance of chaotic maps in initial population in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 5 Performance of chaotic maps in crossover operator in solving ZDT4

Mathematical Problems in Engineering 11

Table 3 The normalized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker 0617 0080 0049 0682 0149 0094 0668 0109 0003 0220 minus0040 0167 0567 0105 minus0090Cat minus0060 0076 0101 0013 0084 0024 minus0007 0090 0174 0191 0109 0158 minus0028 0022 minus0096Circle minus0142 minus0040 minus0016 0013 minus0121 minus0007 minus0153 0028 minus0016 0151 minus0069 0098 minus0176 minus0072 minus0002Cubic minus0544 0064 minus0025 minus0626 minus0041 0006 minus0334 minus0049 0077 0032 minus0781 0071 minus0288 0040 0008Gauss 0307 0513 0089 0306 0507 0070 0454 0585 0037 0159 0005 0191 0114 minus0010 0132ICMIC minus0415 0558 0132 minus0280 0609 0144 minus0295 0510 0004 0003 minus0380 0088 minus0162 0252 0163Logistic 0070 0242 0189 0072 0204 minus0031 0012 0158 0127 0017 minus0819 0183 0152 0129 0132Sinusoidal 0077 1 0616 0121 1 0742 0169 1 0734 minus0091 minus1 minus0138 0148 0688 1Tent 0655 0177 0062 0704 0103 minus0005 0731 0043 minus0088 0190 minus0008 minus0058 0569 0124 minus0003Zaslavskii minus0051 0462 0032 minus0150 0518 0064 minus0086 0499 0110 minus0103 minus0339 0108 minus0060 0174 0183

Table 4 The visualized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker ++ 0 0 ++ + 0 ++ + 0 + 0 + ++ + 0Cat 0 0 + 0 0 0 0 0 + + + + 0 0 0Circle minus 0 0 0 minus 0 minus 0 0 + 0 0 minus 0 0Cubic mdash 0 0 mdash 0 0 mdash 0 0 0 mdash 0 minus 0 0Gauss ++ ++ 0 ++ ++ 0 ++ ++ 0 + 0 + + 0 +ICMIC mdash ++ + minus ++ + minus ++ 0 0 mdash 0 minus + +Logistic 0 + + 0 + 0 0 + + 0 mdash + + + +Sinusoidal 0 ++ ++ + ++ ++ + ++ ++ 0 mdash minus + ++ ++Tent ++ + 0 ++ + 0 ++ 0 0 + 0 0 ++ + 0Zaslavskii 0 ++ 0 minus ++ 0 0 ++ + minus mdash + 0 + +

Table 5 Results of Gauss mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle minus04886 minus00170 19394 minus03130 00971 minus04370 1856683 750688 28693 minus05529Cubic minus24674 minus00589 minus62676 minus00630 minus25351 minus12051 491193 minus448985 minus23071 57609Sinusoidal 10277 60982 04343 94173 minus00396 45810 minus106768 minus525536 584566 285351Tent 35035 06784 59502 11086 18616 00623 minus272428 1014576 150061 112005

Table 6 Results of Cauchy mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle 13978 05012 13997 07723 07187 04043 1354511 minus136938 01105 minus26454Cubic minus26201 minus04932 minus42148 minus05317 minus24831 minus07487 minus771621 minus456939 minus57560 54533Sinusoidal 01470 46977 08327 82820 03179 45376 minus202995 minus468243 65462 238545Tent 27613 03380 52687 09699 26813 minus01916 minus283172 579606 127523 52806

with worst performance are chosen to be used in theexperiments These chaotic maps are circle map cubic mapsinusoidal map and tent map The values of 120574 differences areshown in Tables 5 and 6 As seen from Tables 5 and 6 theperformance of sinusoidal map and tent map is better thanthe performance of circlemap and cubicmap Sinusoidalmapin initial population is better than that in crossover operationand tent map in crossover operation is better than that ininitial population This means the rules of combinations of

chaotic maps and phases in solving MOPs are almost thesame as in the previous observations So the rules based onthe framework of NSGA-II algorithm are applicable to otherMOEAs

In general Bakerrsquos map with a phase for crossover oper-ator sinusoidal map with phases for initial population andmutation operator and tent map with a phase for crossoveroperator could be the best choice for improving evolutionaryalgorithms for MOPs without local optimum For problems

12 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 6 Performance of chaotic maps in mutation operator in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

Figure 7 Performance of chaotic maps in crossover operator in solving ZDT1 with metric Δ

Mathematical Problems in Engineering 13

Table7Th

eaverage

results

ofΔ

ZDT1

ZDT2

ZDT3

ZDT4

ZDT6

ci

mc

im

ci

mc

im

ci

mBa

ker

minus00337

minus00026

00015

00226

00310

00024

minus000

96000

69000

06minus006

0300343

minus00513

00125

00024

minus00015

Cat

000

0500017

000

4200234

minus01021

minus00145

minus00019

minus00052

000

0600277

minus000

90minus00148

minus000

01minus000

02minus000

68Circle

minus00325

minus00035

minus000

64minus02019

minus00632

minus01410

minus00052

minus000

6500010

minus00255

minus00259

minus00136

00031

minus00020

000

64Cu

bic

minus000

4700121

minus00022

minus00439

minus00148

00361

000

00minus000

0600031

minus00052

minus00279

00238

minus00170

00054

minus00017

Gauss

00121

00075

minus00180

minus01075

00329

minus00711

minus00024

minus00032

00019

00330

minus00214

00155

00180

000

03000

64ICMIC

00081

000

92000

02minus00734

minus00850

minus00917

minus00036

000

08minus00054

minus00356

minus00497

00238

00123

000

0100085

Logistic

minus00118

00076

00075

minus01935

00239

00509

minus00084

minus00014

00080

00275

minus00100

minus00232

minus000

6900020

00077

Sinu

soidal

00149

000

69minus00564

minus01750

minus000

67minus02902

minus00014

00029

minus00286

minus00390

minus00164

minus01945

00129

00088

minus004

88Tent

minus00350

000

9300016

00215

minus00173

minus00261

minus00059

minus00168

00071

minus00761

minus00139

00075

00125

00028

minus000

02Za

slavskii

002307

00056

minus00014

minus00292

minus00189

minus00013

000

99000

40minus00037

00201

minus00056

000

9000038

00027

00084

14 Mathematical Problems in Engineering

Table 8 Statistical data for combinations of chaotic maps and phases on different problems

Threshold ZDT1 ZDT2 ZDT3 ZDT4 ZDT60 18 9 15 10 200001 16 9 9 10 180002 13 9 7 10 180003 13 8 6 10 140004 13 8 5 10 120005 12 8 4 10 120006 11 8 4 10 11

0 50 100 150 200 250minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

Generation

Diff

eren

ce o

f delt

a

Bakerrsquos

Figure 8 Performance of Bakerrsquos maps in crossover operator insolving ZDT1 with metric Δ

with local optimum cat map has good performance onimproving evolutionary algorithms

73 Diversity Performance Similar to the convergencemetric120574 the differences of the diversity metricΔ between the resultsof the new algorithm with chaotic maps and the originalNSGA-II algorithm are used to measure the performanceFigure 7 shows the performance of chaotic maps in crossoveroperator in solving ZDT1 with metric Δ Each subgraphshows the effect of one chaotic map To be seen more clearlythe first subgraph in Figure 7 is shown in Figure 8

As seen fromFigures 7 and 8 theΔ difference is not stablein 250 generations The average values of Δ difference in 250generations are calculated to represent the effect of the newalgorithms For brevity the rest of results are not shown ingraphs but in Table 7

As seen from Table 7 the Δ values have little differenceWe count the number of combinations of chaotic maps andphases for solving one problem in different threshold valuesFor example there are 18 combinations whose values of Δ aregreater than zero Based on the number of the combinationsof chaotic maps and phases in different threshold values thevalues of diversity metric Δ are summarized in Table 8

In Table 8 the rank of the number of Δ in differentthreshold values is ZDT1 gt ZDT6 gt ZDT4 gt ZDT2 gt

ZDT3 especially for larger threshold ZDT1 problem whichis a convex function and has no local optima is a relativelyeasy problem Chaotic maps bring the biggest improve-ments on solving ZDT1 Though the solutions of ZDT6 arenonuniformly spaced chaotic maps can find better spread ofsolutions While ZDT4 problem is a complex problem andthe solutions are easily trapped into local optima chaoticmaps can improve the distribution of the solutions ZDT2problem is a convex function and the solutions sometimesfall into the local optimumThe effects of chaoticmaps can begeneralizedThe Pareto front of ZDT3 problem is segmentedso the Δ value of ZDT3 is larger and the ranking of ZDT3 islower It is our observation that Δ is not fit for evaluating thesolutions to problems which are disconnected

Based on the diversity metric Δ chaotic maps havethe best improvement on solving convex problems withoutlocal optima and have better effect on solving problemswhich have nonuniform solutions For problems with localminimum chaotic maps embedded algorithms can improvethe performance with regard to metric Δ

A short summary can be given according to the aboveexperiments First chaotic maps can improve the perfor-mance of MOEAs but the results showed that no one chaoticmap outperforms other maps for all of the problems Theresults in this paper give some guidance on how to choose achaotic map and a phase in MOEAs Second an interestingdiscovery is that cat map has best performance on solvingproblems with local optima Uniformity of cat map may beone of the reasons for the good performance of solving ZDT4

8 Conclusion

The focus of this paper is to explore the relationships ofchaotic maps and phases in MOEAs in solving MOPs Themain framework of algorithms in experiments is the NSGA-II algorithm The combinations of ten chaotic maps andthree phases are chosen in the experiments Two metricsconvergence metric 120574 and diversity metric Δ are used toevaluate the convergence and diversity properties of thealgorithms with chaotic maps The test problems are ZDTseries which were all MOPs The ergodicity and initial valuesensitivity of chaotic maps can help evolutionary algorithmsavoid solutions from falling into local optimal and get betterconvergence In the experimental results almost all chaoticmaps have good effects on improving the performance ofevolutionary algorithms to solveMOPs without local optima

Mathematical Problems in Engineering 15

Cat map has best performance on solving problems withlocal optimum This work gives insight on choosing chaoticmaps and phases for different problems Our future workwill perform further experiments with more chaotic maps onother MOEAs and formulate the theory analysis

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China under Grant no 61101153 and Prof Qiuis supported by NSF CNS 1359557

References

[1] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing vol 13 pp2790ndash2802 2013

[2] J A Adeyemo ldquoReservoir operation using multi-objective evo-lutionary algorithms-a reviewrdquo Asian Journal of Scientific Res-earch vol 4 no 1 pp 16ndash27 2011

[3] H Hu L Xu R Wei and B Zhu ldquoMulti-objective controloptimization for greenhouse environment using evolutionaryalgorithmsrdquo Sensors vol 11 no 6 pp 5792ndash5807 2011

[4] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[5] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[6] E Zitzler M Laumanns L Thiele et al SPEA2 Improving theStrength Pareto Evolutionary Algorithm Eidgenossische Techn-ische Hochschule Zurich (ETH) Institut fur Technische Infor-matik und Kommunikationsnetze (TIK) 2001

[7] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[8] B Alatas E Akin and A B Ozer ldquoChaos embedded particleswarm optimization algorithmsrdquo Chaos Solitons and Fractalsvol 40 no 4 pp 1715ndash1734 2009

[9] G ZilongW Sunrsquoan andZ Jian ldquoA novel immune evolutionaryalgorithm incorporating chaos optimizationrdquo Pattern Recogni-tion Letters vol 27 no 1 pp 2ndash8 2006

[10] M S Tavazoei and M Haeri ldquoComparison of different one-dimensional maps as chaotic search pattern in chaos optimiza-tion algorithmsrdquo Applied Mathematics and Computation vol187 no 2 pp 1076ndash1085 2007

[11] G Yu X Wang and P Li ldquoApplication of chaotic theory indifferential evolution algorithmsrdquo in Proceedings of the 6th Inte-rnational Conference on Natural Computation (ICNC rsquo10) pp3816ndash3820 August 2010

[12] B Alatas and E Akin ldquoMulti-objective rulemining using a cha-otic particle swarm optimization algorithmrdquo Knowledge-BasedSystems vol 22 no 6 pp 455ndash460 2009

[13] L D S Coelho and V C Mariani ldquoChaotic artificial immuneapproach applied to economic dispatch of electric energy usingthermal unitsrdquo Chaos Solitons and Fractals vol 40 no 5 pp2376ndash2383 2009

[14] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[15] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[16] L D S Coelho ldquoA quantum particle swarm optimizer with cha-otic mutation operatorrdquo Chaos Solitons and Fractals vol 37 no5 pp 1409ndash1418 2008

[17] L S Dos Coelho and P Alotto ldquoMultiobjective electromagneticoptimization based on a nondominated sorting genetic appr-oach with a chaotic crossover operatorrdquo IEEE Transactions onMagnetics vol 44 no 6 pp 1078ndash1081 2008

[18] H F Zhang J Z Zhou Y C Zhang N Fang and R ZhangldquoShort termhydrothermal scheduling usingmulti-objective dif-ferential evolution with three chaotic sequencesrdquo InternationalJournal of Electrical Power amp Energy Systems vol 47 pp 85ndash992013

[19] Y-J Wang and J-S Zhang ldquoGlobal optimization by an impr-oved differential evolutionary algorithmrdquo Applied Mathematicsand Computation vol 188 no 1 pp 669ndash680 2007

[20] R Caponetto L Fortuna S Fazzino and M G Xibilia ldquoCha-otic sequences to improve the performance of evolutionary alg-orithmsrdquo IEEE Transactions on Evolutionary Computation vol7 no 3 pp 289ndash304 2003

[21] M Ahmadi and H Mojallali ldquoChaotic invasive weed optimiz-ation algorithm with application to parameter estimation ofchaotic systemsrdquo Chaos Solitons amp Fractals vol 45 no 9-10pp 1108ndash1120 2012

[22] Z S Ma ldquoChaotic populations in genetic algorithmsrdquo AppliedSoft Computing vol 12 pp 2409ndash2424 2012

[23] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[24] W-M Zheng ldquoKneading plane of the circle maprdquo ChaosSolitons and Fractals vol 4 no 7 pp 1221ndash1233 1994

[25] T D Rogers and D C Whitley ldquoChaos in the cubic mappingrdquoMathematical Modelling vol 4 no 1 pp 9ndash25 1983

[26] M Bucolo R Caponetto L Fortuna M Frasca and A RizzoldquoDoes chaos work better than noiserdquo IEEE Circuits and SystemsMagazine vol 2 no 3 pp 4ndash19 2002

[27] D He C He L-G Jiang H-W Zhu and G-R Hu ldquoChaoticcharacteristics of a one-dimensional iterative map with infinitecollapsesrdquo IEEE Transactions on Circuits and Systems I vol 48no 7 pp 900ndash906 2001

[28] R May ldquoSimple mathematical models with very complicateddynamicsrdquo inTheTheory of Chaotic Attractors B Hunt T Y LiJ Kennedy and H Nusse Eds pp 85ndash93 Springer New YorkNY USA 2004

[29] R Barton ldquoChaos and fractalsrdquo The Mathematics Teacher vol83 pp 524ndash529 1990

[30] A Baykasoglu ldquoDesign optimization with chaos embeddedgreat deluge algorithmrdquoApplied Soft Computing Journal vol 12no 3 pp 1055ndash1067 2012

[31] J Fridrich ldquoSymmetric ciphers based on two-dimensional cha-otic mapsrdquo International Journal of Bifurcation and Chaos in

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

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Stochastic AnalysisInternational Journal of

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
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      • • tmp1401803913pdfPD4Sl

Mathematical Problems in Engineering 5

(3) Gauss Map Gauss map is also one of the well-known andcommonly employed maps in generating chaotic sequences[26] as follows

119909119896+1=

0 119909119896= 0

1

119909119896

mod (1) otherwise (9)

This map also generates chaotic sequences in (0 1)

(4) ICMIC Map The iterative chaotic map with infinitecollapses (ICMIC) [27] is defined by the following equation

119909119896+1= sin( 119886

119909119896

) 119886 isin (0infin) 119909119896isin (minus1 1) (10)

The parameter ldquo119886rdquo is an adjustable parameter This paperchooses 119886 = 2 Because the range of119909

119896is not (0 1) the chaotic

sequences need to be transformed to change the range

(5) LogisticMap As awell-known chaoticmap logisticmap isone of the simplest maps and was introduced by May in 2004[28] It is often cited as an example of how complex behaviorcan arise from a very simple nonlinear dynamical equationLogistic map generates chaotic sequences in (0 1) This mapis formally defined by the following equation

119909119896+1= 119886119909119896(1 minus 119909

119896) (11)

Parameter 119886 is set to 4 in the simulation

(6) Sinusoidal IteratorThe sinusoidal iterator [29] is formallydefined by the following equation

119909119896+1= 1198861199092

119896sin (120587119909

119896) 119909

119896isin (0 1) (12)

In this paper the simplified equation is used in the followingiteration

119909119896+1= sin (120587119909

119896) 119909

119896isin (0 1) (13)

(7) Tent Map Tent chaotic map is very similar to the logisticmap which displays specific chaotic effects [30] This map isdefined by the following equation

119909119896+1= 2119909119896 119909

119896lt 05

2 (1 minus 119909119896) 119909

119896ge 05

(14)

where 119909119896is ranging from 0 to 1

Tent map generates chaotic sequences in (0 1)

42 Two-Dimensional Maps

(1) Bakerrsquos Map The Baker map [31] is described by thefollowing formulas

119861 (119909 119910) =

(2119909 2119910) for 0 le 119909 lt 05(2 minus 2119909 1 minus

119910

2) for 05 le 119909 lt 1

(15)

In the following simulations one dimension of Bakerrsquosmap which is similar to tent map is adopted The equationis defined by

119909119896+1= 2119909119896 for 0 le 119909

119896lt 05

2 minus 2119909119896 for 05 le 119909

119896lt 1

(16)

This map generates chaotic sequences in (0 1)

(2) Arnoldrsquos Cat Map Arnoldrsquos cat map is named afterVladimir Arnold who demonstrated its effects in the 1960susing an image of a cat It is represented by [32]

119909119896+1= 119909119896+ 119910119896mod (1)

119910119896+1= 119909119896+ 2119910119896mod (1)

(17)

It is obvious that the sequences 119909119896isin (0 1) and 119910

119896isin (0 1)

(3) Zaslavskii Map Zaslavskii map [33] is an interestingdynamic system with chaotic behavior The discretized equa-tion is given by

119909119896+1= (119909119896+ V + 119886119910

119896+1) mod (1)

119910119896+1= cos (2120587119909

119896) + 119890minus119903

119910119896

(18)

The Zaslavskii map shows a strange attractor with thelargest Lyapunov exponent for V = 400 119903 = 3 and119886 = 126695 In this case it can be calculated that 119910

119896+1isin

[minus10512 10512] Only one dimension is chosen in thefollowing simulation Since the scale of 119910

119896+1is not [0 1] the

chaotic sequences generated need scale transformation

5 Chaotic Properties of Sequences Generatedby Scale Transformation

Asmentioned in the previous sections the scale of sequencesgenerated by chaotic maps is not always fit for the problemsto be solved Some sequences have to change their scale andsome sequences are generated by one dimension of a two-dimension chaoticmap Hence it is necessary to demonstratethe chaotic properties of sequences after these changes

Detecting the presence of chaos in a dynamical system isusually solved by measuring the largest Lyapunov exponentwhich describes quantitatively the speed of index divergenceor convergence between the adjacent phase space orbits Apositive largest Lyapunov exponent indicates chaos Sincethe chaotic sequences adopted in this paper are discrete theLyapunov exponent of discrete series can be calculated bysmall data sets arithmetic [34] This method makes full useof all the data obtains higher accuracy and has strongerrobustness for the amount of data the embedding dimensionand the time delay

51 Small Data Sets Arithmetic The reconstructed trajectory119883 can be expressed as a matrix where each row is a phase-space vector that is

119883 = (1198831 1198832 119883

119872)119879

(19)

6 Mathematical Problems in Engineering

where 119883119894is the state of the system at discrete time 119894 For an

119873-point time series 1199091 1199092 119909

119873 each119883

119894is given by

119883119894= (119909119894 119909119894+119869 119909

119894+(119898minus1)119869) (20)

where 119869 is the lag or reconstruction delay and 119898 is theembedding dimension Thus119883 is an119872times119898matrix and theconstants119898119872 119869 and119873 are related as

119872 = 119873 minus (119898 minus 1) 119869 (21)After reconstructing the dynamics the algorithm locates thenearest neighbor of each point on the trajectory The nearestneighbor 119883

119895 where 119895 isin 1 2 119872 is found by searching

for the point that minimizes the distance to the particularreference point119883

119895 This is expressed as

119889119895(0) = min

119883119895

10038171003817100381710038171003817119883119895minus 119883119895

10038171003817100381710038171003817 (22)

where 119889119895(0) is the initial distance from the 119895th point to its

nearest neighbor and denotes the Euclidean norm Weimpose an additional constraint that the nearest neighborshave a temporal separation greater than the mean period ofthe time series

10038161003816100381610038161003816119895 minus 11989510038161003816100381610038161003816gt 119901 (23)

where 119901 is the mean period of time series 119901 can be estimatedby the reciprocal of the mean frequency of the powerspectrum This allows us to consider each pair of neighborsas nearby initial conditions for different trajectories Thelargest Lyapunov exponent is estimated as the mean rate ofseparation of the nearest neighbors

For each reference point119883119895 119889119895(119894) is the distance between

the 119895th pair of nearest neighbors after 119894 discrete time

119889119895(119894) =

10038171003817100381710038171003817119883119895+119894minus 119883119895+119894

10038171003817100381710038171003817 119894 = 1 2 min (119872 minus 119895119872 minus 119895)

(24)Assume that reference point 119883

119895and its nearest neighbor

119883119895have index divergence rate 120582

1 then

119889119895(119894) = 119862

1198951198901205821(119894sdotΔ119905)

119862119895= 119889119895(0) (25)

where 119862119895is the initial separation By taking the logarithm of

both sides of (25) we getln 119889119895(119894) asymp ln119862

119895+ 1205821(119894 sdot Δ119905) (26)

Equation (26) represents a set of approximately parallel lines(for 119895 = 1 2 119872) each with a slope 119904 roughly proportionalto 1205821 The largest Lyapunov exponent is easily and accurately

calculated using a least square fit to the ldquoaveragerdquo line definedby

119910 (119894) =1

Δ119905⟨ln 119889119895(119894)⟩ (27)

where ⟨ ⟩ denotes the average over all values of 119895 So

119910 (119894) =1

119902Δ119905

119902

sum

119895=1

ln 119889119895(119894) (28)

where 119902 is the number of 119889119895(119894) with 119889

119895(119894) = 0

Choose a linear area of the curve 119910(119894) sim 119894 and apply theleast square method to get the regression straight line Thenthe slope of the regression straight line is the largest Lyapunovexponent 120582

1

52 The Lyapunov Exponent of Sequences In the calculationprocess the embedding dimension 119898 is calculated throughthe method of false nearest neighbors (FNN) For the timedelay 119869 a good approximation of 119869 is equal to the numberlagging where the autocorrelation function drops to 1 minus 1119890of its initial value

Since different test problems have different rangeschaotic sequences need to be changed to different scalesTwo kinds of sequences used in experiments need to beinvestigated sequences with scales changed and sequencesgenerated by one dimension of a two-dimension chaoticmap

521 Sequences with Scales Changed Since the sequence 1199091

to 119909100

generated by ICMIC is not in (0 1) the new sequence1199101to 119910100

has to be generated by the following function

119910119894=1

2(119909119894+ 1) 119894 isin [1 100] (29)

The sequence 1199101to 119910100

is in the range of (0 1) The Lya-punov exponent of the new sequence is calculated throughsmall data sets arithmetic The average Lyapunov exponentof 10 runs is 00744 Since it is a positive number the newsequence 119910

1to 119910100

conforms to the chaotic nature

522 Sequences Generated by One Dimension of a Two-Dimension ChaoticMap For the Zaslavskii map one dimen-sion119910

119896is chosen in the following simulationThe sequence119910

1

to 119910100

is generated by 100 iterations through Zaslavskii mapThe new sequence 119911

1to 119911100

is generated by the followingfunction

119911119894=(119910119894+ 10513)

21026 119894 isin [1 100] (30)

Then the sequence 1199111to 119911100

is in (0 1) By a similarprocessing with ICMIC the average Lyapunov exponent is000194 Then the new sequence 119911

1to 119911100

conforms to thechaotic nature

6 Test Problem and Performance Measures

61 Test Problems Two-objective optimization problems arechosen to test and measure the performance improvementof the evolutionary algorithms using chaotic maps in threephases We use well-defined benchmark functions as objec-tive functions Their properties are shown in Table 1

62 Performance Measures Two criteria are used to evaluatethe performance of multiobjective optimization (1) conver-gence to the Pareto-optimal set and (2) maintenance of diver-sity in solutions of the Pareto-optimal set [7] Twometrics areadopted to evaluate the effects of the combinations of phasesand chaotic maps

The first metric 120574 measures the extent of convergence toa known set of Pareto-optimal solutions It is defined as

120574 =1

119873

119873

sum

119894=1

119889119894 (31)

Mathematical Problems in Engineering 7

Table 1 Test problems

Problem 119899 Variable bounds Objective functions Optimal solutions

ZDT1 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909)]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT2 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus (119909

1119892(119909))

2

]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT3 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909) minus (119909

1119892(119909)) sin(10120587119909

1)]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT4 101199091isin [0 1]

119909119894isin [minus5 5]

119894 = 2 119899

1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909)]

119892(119909) = 1 + (10(119899 minus 1) + sum119899

119894=2[1199092

119894minus 10 cos(4120587119909

119894)])

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT6 10 [0 1]1198911(119909) = 1 minus exp(minus4119909

1)sin6(6120587119909

1)

1198912(119909) = 119892(119909)[1 minus (119891

1(119909)119892(119909))

2

]

119892(119909) = 1 + (9[(sum119899

119894=2119909119894) (119899 minus 1)]

025

)

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

where 119889119894is the minimum Euclidean distance of every

obtained solution to the Pareto-optimal front The smallerthe value of this metric is the nearer the convergence towardPareto-front is

The other metric Δ measures the extent of spreadachieved among the obtained solutions The metric Δ isdefined by

Δ =119889119891+ 119889119897+ sum119873minus1

119894=1

10038161003816100381610038161003816119889119894minus 11988910038161003816100381610038161003816

119889119891+ 119889119897+ (119873 minus 1) 119889

(32)

The parameter 119889119894is the Euclidean distance between consecu-

tive solutions in the obtained nondominated set of solutionsTheparameters119889

119897and119889119891are the Euclidean distances between

the extreme solutions and the boundary solutions of theobtained nondominated set The parameter 119889 is the averageof all distances 119889

119894 119894 = 1 2 119873 minus 1 assuming that there are

119873 solutions on the best nondominated front

7 Experiments and Results

To explore the relationship of phases and chaotic mapsto solve MOPs NSGA-II algorithm is chosen as the mainframeworkThe ten chaotic maps mentioned in Section 4 areembedded in three different phases in the original NSGA-II algorithm Each time only one parameter is modifiedFor example if initial population is generated by chaoticmap the crossover and mutation operator are not changedSimilarly if crossover operator is modified by a chaoticmap the initial population and mutation operator are notchanged The solutions generated by the chaos embeddedNSGA-II algorithm are evaluated by two metrics 120574 and ΔFor readerrsquos convenience the new algorithms with differentcombinations of chaotic maps and phases are named asldquocns [chaotic map] [phase]rdquo and the results of differentalgorithms on test problems are named as ldquocns [chaoticmap] [phase] [test problem]rdquo In addition ldquoirdquo represents the

phase for initial population ldquocrdquo represents the phase forcrossover operator and ldquomrdquo represents the phase formutationoperator For example the results through modified initialpopulation by logistic map solving ZDT1 problem are namedas ldquocns logistic i zdt1rdquo

Each combination of one chaotic map and one phaseneeds one experiment In this research 10 chaotic maps with3 different phases based on 2 metrics solving 5 test problemsneed 150 basic experiments and obtain 300 results Eachexperiment obtains a Pareto frontThe values of convergencemetric 120574 and the diversity metric Δ are also calculated

In order to compare with the results of original NSGA-IIalgorithm we focused on the difference of the 120574 and Δ valuesof the original NSGA-II algorithm and the new algorithmFor example the 120574 of results of ldquocns sinusoidal i zdt1rdquois named as ldquocns sinusoidal i zdt1 gamardquo and the 120574 ofresults of NSGA-II solving ZDT1 problem is namedas ldquons zdt1 gamardquo Then the difference is named asldquons zdt1 gamamdashcns sinusoidal i zdt1 gamardquo When theprocesses of algorithms get to convergence the difference isvery small The properties of convergence and diversity inthe process of iterations need to be taken into account sothe 120574 values of each generation in the iterations are recordedand the differences of 120574 of each generation are obtained Thisprocess also applies to Δ

Some main parameters in the process of NSGA-II algo-rithm are introduced in the following paragraphs Then theresults of experiments are shown and analyzed

71TheMainParameters Themainparameters in the processof NSGA-II algorithm are presented in this section Choosingan appropriate representation of a chromosome is veryimportant for solving problems Real numbers are chosento represent the genes One chromosome represents oneindividual The initial population has 100 individuals andeach chromosome has a certain number of genes whichare represented by a real number Each individual of theinitial population is generated randomly with the range

8 Mathematical Problems in Engineering

Table 2 Parameters in the process of algorithms

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6119899iter 250119899pop 100119899var 30 30 30 10 10119901119888

09119901119898

130 130 130 110 110

based on the test problems The iteration will not terminateuntil the number of iterations gets to 250 For the processof NSGA-II algorithm a parent population is selected bytournament selection depending on the nondominated rankand the crowed-comparison operator Then the new popula-tion is generated by crossover and mutation operators Thecrossover operation is executed with the probability of 119901

119888=

09 The probability of mutation 119901119898is equal to the reciprocal

of 119899var which is the dimension number of a chromosome thatis 119901119898= 1119899var

Those parameters are summarized in Table 2 In thetable 119899iter is the number of iterations 119899pop is the scaleof the population 119899var is the number of dimensions of achromosome and119901

119888and119901119898are the probabilities of crossover

and mutation operations

72 Convergence Performance It is known that the 120574 differ-ence is used to evaluate the performance of the chaotic mapsin different phases inmultiobjective evolutionary algorithmsAn example is chosen for further explanation in detail As inFigure 1 the graph shows the results of solving ZDT1 prob-lems with Bakerrsquos map in crossover operator in NSGA-IIThedifferences of 120574 between the experiment ldquocns baker c zdt1rdquoand the experiment ldquons zdt1rdquo in the 250 iterations are givenAs seen from the figure the black line is above the red linewhich represents 0 so the new algorithm ldquocns bakers crdquo isbetter thanNSGA-II algorithm in solvingZDT1 problemwithregard to the convergence metric

The 120574 results of all the experiments are given similar toFigure 1 Since it is difficult to show so many graphs in thispaper the results of three typical problems are chosen thatis ZDT1 which is a simple convex problem ZDT3 whosePareto front is piecewise and ZDT4 which has local optimaThe graphs in Figures 2 3 and 4 provide a comparison of theperformance of solving different MOPs with chaotic maps ininitial population ZDT4 is chosen to show the performanceof chaotic maps in different phases on solving the sameMOPas shown in Figures 4 5 and 6 Each subgraph is labeled withthe name of the chaotic map used

In order to quantify the effect of chaotic maps and phaseswith regard to the metric 120574 the average of 120574 difference in 250generations is calculated to represent the effect of the newalgorithms

Since the order of magnitude of 120574 is not the samethe comparison of these 120574 values is not convenient Thenormalized values are obtained by dividing the 120574 values bythemaximumof the absolute values of the 120574 based on one testproblem The results of normalization are shown in Table 3

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

Bakerrsquos06

05

04

03

02

01

0

minus01

minus02

minus03

minus04

Figure 1 Performance of Bakerrsquos maps in crossover operator insolving ZDT1

Table 3 can be presented in a more intuitive way If 120574 ge03 the numerical value of 120574 is replaced by ldquo++rdquo Similarlyldquo+rdquo represents 01 le 120574 lt 03 ldquo0rdquo represents minus01 le 120574 lt01 ldquominusrdquo represents minus03 le 120574 lt minus01 and ldquominusminusrdquo represents120574 lt minus03 Therefore ldquo++rdquo means that the effect of the newalgorithm with chaotic maps is much better whereas ldquominusminusrdquo ismuch worse Table 4 shows the results

As shown in Table 4 most of the combinations of chaoticmaps and phases have a positive effect on improving the per-formance of NSGA-II algorithm The effect of some chaoticmaps is very good especially in some particular phases Forexample Bakerrsquos map in crossover operator Gauss map incrossover operator and initial population ICMIC map ininitial population sinusoidal map in initial population tentmap in crossover operation and Zaslavskii map in initialpopulation have very good effect

Since ZDT4 problem has 219 or 794times1011 different localPareto-optimal fronts in the search space the solutions easilyget entrapped into local optimum As seen from Table 4chaotic maps used for crossover and mutation operator havesignificant improvement on evolutionary algorithms solvingZDT4 problem especially cat map has the best performancein tenmaps Circle map and cubicmap have less contributionin solving those MOPs The distribution of cat map isrelatively uniform It is probably the reason for the goodperformance in solving problems with local optima

The original NSGA-II algorithm is not good at solvingZDT3 and ZDT6 problems because Pareto-optimal front ofZDT3 is disconnected and solutions of ZDT6 are nonuni-formly spaced However it can be seen in Table 4 that chaoticmaps can improve NSGA-II especially in crossover operationand initial population in solving ZDT3 and ZDT6 problem

In order to eliminate the special effect of the NSGA-II algorithm the polynomial mutation operator in NSGA-II is changed by the Gauss mutation and Cauchy mutationoperators Four typical chaotic maps which include twochaotic maps with best performance and two chaotic maps

Mathematical Problems in Engineering 9

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

GenerationGeneration

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

Figure 2 Performance of chaotic maps in initial population in solving ZDT1

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma 06

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

Figure 3 Performance of chaotic maps in initial population in solving ZDT3

10 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 4 Performance of chaotic maps in initial population in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 5 Performance of chaotic maps in crossover operator in solving ZDT4

Mathematical Problems in Engineering 11

Table 3 The normalized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker 0617 0080 0049 0682 0149 0094 0668 0109 0003 0220 minus0040 0167 0567 0105 minus0090Cat minus0060 0076 0101 0013 0084 0024 minus0007 0090 0174 0191 0109 0158 minus0028 0022 minus0096Circle minus0142 minus0040 minus0016 0013 minus0121 minus0007 minus0153 0028 minus0016 0151 minus0069 0098 minus0176 minus0072 minus0002Cubic minus0544 0064 minus0025 minus0626 minus0041 0006 minus0334 minus0049 0077 0032 minus0781 0071 minus0288 0040 0008Gauss 0307 0513 0089 0306 0507 0070 0454 0585 0037 0159 0005 0191 0114 minus0010 0132ICMIC minus0415 0558 0132 minus0280 0609 0144 minus0295 0510 0004 0003 minus0380 0088 minus0162 0252 0163Logistic 0070 0242 0189 0072 0204 minus0031 0012 0158 0127 0017 minus0819 0183 0152 0129 0132Sinusoidal 0077 1 0616 0121 1 0742 0169 1 0734 minus0091 minus1 minus0138 0148 0688 1Tent 0655 0177 0062 0704 0103 minus0005 0731 0043 minus0088 0190 minus0008 minus0058 0569 0124 minus0003Zaslavskii minus0051 0462 0032 minus0150 0518 0064 minus0086 0499 0110 minus0103 minus0339 0108 minus0060 0174 0183

Table 4 The visualized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker ++ 0 0 ++ + 0 ++ + 0 + 0 + ++ + 0Cat 0 0 + 0 0 0 0 0 + + + + 0 0 0Circle minus 0 0 0 minus 0 minus 0 0 + 0 0 minus 0 0Cubic mdash 0 0 mdash 0 0 mdash 0 0 0 mdash 0 minus 0 0Gauss ++ ++ 0 ++ ++ 0 ++ ++ 0 + 0 + + 0 +ICMIC mdash ++ + minus ++ + minus ++ 0 0 mdash 0 minus + +Logistic 0 + + 0 + 0 0 + + 0 mdash + + + +Sinusoidal 0 ++ ++ + ++ ++ + ++ ++ 0 mdash minus + ++ ++Tent ++ + 0 ++ + 0 ++ 0 0 + 0 0 ++ + 0Zaslavskii 0 ++ 0 minus ++ 0 0 ++ + minus mdash + 0 + +

Table 5 Results of Gauss mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle minus04886 minus00170 19394 minus03130 00971 minus04370 1856683 750688 28693 minus05529Cubic minus24674 minus00589 minus62676 minus00630 minus25351 minus12051 491193 minus448985 minus23071 57609Sinusoidal 10277 60982 04343 94173 minus00396 45810 minus106768 minus525536 584566 285351Tent 35035 06784 59502 11086 18616 00623 minus272428 1014576 150061 112005

Table 6 Results of Cauchy mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle 13978 05012 13997 07723 07187 04043 1354511 minus136938 01105 minus26454Cubic minus26201 minus04932 minus42148 minus05317 minus24831 minus07487 minus771621 minus456939 minus57560 54533Sinusoidal 01470 46977 08327 82820 03179 45376 minus202995 minus468243 65462 238545Tent 27613 03380 52687 09699 26813 minus01916 minus283172 579606 127523 52806

with worst performance are chosen to be used in theexperiments These chaotic maps are circle map cubic mapsinusoidal map and tent map The values of 120574 differences areshown in Tables 5 and 6 As seen from Tables 5 and 6 theperformance of sinusoidal map and tent map is better thanthe performance of circlemap and cubicmap Sinusoidalmapin initial population is better than that in crossover operationand tent map in crossover operation is better than that ininitial population This means the rules of combinations of

chaotic maps and phases in solving MOPs are almost thesame as in the previous observations So the rules based onthe framework of NSGA-II algorithm are applicable to otherMOEAs

In general Bakerrsquos map with a phase for crossover oper-ator sinusoidal map with phases for initial population andmutation operator and tent map with a phase for crossoveroperator could be the best choice for improving evolutionaryalgorithms for MOPs without local optimum For problems

12 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 6 Performance of chaotic maps in mutation operator in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

Figure 7 Performance of chaotic maps in crossover operator in solving ZDT1 with metric Δ

Mathematical Problems in Engineering 13

Table7Th

eaverage

results

ofΔ

ZDT1

ZDT2

ZDT3

ZDT4

ZDT6

ci

mc

im

ci

mc

im

ci

mBa

ker

minus00337

minus00026

00015

00226

00310

00024

minus000

96000

69000

06minus006

0300343

minus00513

00125

00024

minus00015

Cat

000

0500017

000

4200234

minus01021

minus00145

minus00019

minus00052

000

0600277

minus000

90minus00148

minus000

01minus000

02minus000

68Circle

minus00325

minus00035

minus000

64minus02019

minus00632

minus01410

minus00052

minus000

6500010

minus00255

minus00259

minus00136

00031

minus00020

000

64Cu

bic

minus000

4700121

minus00022

minus00439

minus00148

00361

000

00minus000

0600031

minus00052

minus00279

00238

minus00170

00054

minus00017

Gauss

00121

00075

minus00180

minus01075

00329

minus00711

minus00024

minus00032

00019

00330

minus00214

00155

00180

000

03000

64ICMIC

00081

000

92000

02minus00734

minus00850

minus00917

minus00036

000

08minus00054

minus00356

minus00497

00238

00123

000

0100085

Logistic

minus00118

00076

00075

minus01935

00239

00509

minus00084

minus00014

00080

00275

minus00100

minus00232

minus000

6900020

00077

Sinu

soidal

00149

000

69minus00564

minus01750

minus000

67minus02902

minus00014

00029

minus00286

minus00390

minus00164

minus01945

00129

00088

minus004

88Tent

minus00350

000

9300016

00215

minus00173

minus00261

minus00059

minus00168

00071

minus00761

minus00139

00075

00125

00028

minus000

02Za

slavskii

002307

00056

minus00014

minus00292

minus00189

minus00013

000

99000

40minus00037

00201

minus00056

000

9000038

00027

00084

14 Mathematical Problems in Engineering

Table 8 Statistical data for combinations of chaotic maps and phases on different problems

Threshold ZDT1 ZDT2 ZDT3 ZDT4 ZDT60 18 9 15 10 200001 16 9 9 10 180002 13 9 7 10 180003 13 8 6 10 140004 13 8 5 10 120005 12 8 4 10 120006 11 8 4 10 11

0 50 100 150 200 250minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

Generation

Diff

eren

ce o

f delt

a

Bakerrsquos

Figure 8 Performance of Bakerrsquos maps in crossover operator insolving ZDT1 with metric Δ

with local optimum cat map has good performance onimproving evolutionary algorithms

73 Diversity Performance Similar to the convergencemetric120574 the differences of the diversity metricΔ between the resultsof the new algorithm with chaotic maps and the originalNSGA-II algorithm are used to measure the performanceFigure 7 shows the performance of chaotic maps in crossoveroperator in solving ZDT1 with metric Δ Each subgraphshows the effect of one chaotic map To be seen more clearlythe first subgraph in Figure 7 is shown in Figure 8

As seen fromFigures 7 and 8 theΔ difference is not stablein 250 generations The average values of Δ difference in 250generations are calculated to represent the effect of the newalgorithms For brevity the rest of results are not shown ingraphs but in Table 7

As seen from Table 7 the Δ values have little differenceWe count the number of combinations of chaotic maps andphases for solving one problem in different threshold valuesFor example there are 18 combinations whose values of Δ aregreater than zero Based on the number of the combinationsof chaotic maps and phases in different threshold values thevalues of diversity metric Δ are summarized in Table 8

In Table 8 the rank of the number of Δ in differentthreshold values is ZDT1 gt ZDT6 gt ZDT4 gt ZDT2 gt

ZDT3 especially for larger threshold ZDT1 problem whichis a convex function and has no local optima is a relativelyeasy problem Chaotic maps bring the biggest improve-ments on solving ZDT1 Though the solutions of ZDT6 arenonuniformly spaced chaotic maps can find better spread ofsolutions While ZDT4 problem is a complex problem andthe solutions are easily trapped into local optima chaoticmaps can improve the distribution of the solutions ZDT2problem is a convex function and the solutions sometimesfall into the local optimumThe effects of chaoticmaps can begeneralizedThe Pareto front of ZDT3 problem is segmentedso the Δ value of ZDT3 is larger and the ranking of ZDT3 islower It is our observation that Δ is not fit for evaluating thesolutions to problems which are disconnected

Based on the diversity metric Δ chaotic maps havethe best improvement on solving convex problems withoutlocal optima and have better effect on solving problemswhich have nonuniform solutions For problems with localminimum chaotic maps embedded algorithms can improvethe performance with regard to metric Δ

A short summary can be given according to the aboveexperiments First chaotic maps can improve the perfor-mance of MOEAs but the results showed that no one chaoticmap outperforms other maps for all of the problems Theresults in this paper give some guidance on how to choose achaotic map and a phase in MOEAs Second an interestingdiscovery is that cat map has best performance on solvingproblems with local optima Uniformity of cat map may beone of the reasons for the good performance of solving ZDT4

8 Conclusion

The focus of this paper is to explore the relationships ofchaotic maps and phases in MOEAs in solving MOPs Themain framework of algorithms in experiments is the NSGA-II algorithm The combinations of ten chaotic maps andthree phases are chosen in the experiments Two metricsconvergence metric 120574 and diversity metric Δ are used toevaluate the convergence and diversity properties of thealgorithms with chaotic maps The test problems are ZDTseries which were all MOPs The ergodicity and initial valuesensitivity of chaotic maps can help evolutionary algorithmsavoid solutions from falling into local optimal and get betterconvergence In the experimental results almost all chaoticmaps have good effects on improving the performance ofevolutionary algorithms to solveMOPs without local optima

Mathematical Problems in Engineering 15

Cat map has best performance on solving problems withlocal optimum This work gives insight on choosing chaoticmaps and phases for different problems Our future workwill perform further experiments with more chaotic maps onother MOEAs and formulate the theory analysis

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China under Grant no 61101153 and Prof Qiuis supported by NSF CNS 1359557

References

[1] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing vol 13 pp2790ndash2802 2013

[2] J A Adeyemo ldquoReservoir operation using multi-objective evo-lutionary algorithms-a reviewrdquo Asian Journal of Scientific Res-earch vol 4 no 1 pp 16ndash27 2011

[3] H Hu L Xu R Wei and B Zhu ldquoMulti-objective controloptimization for greenhouse environment using evolutionaryalgorithmsrdquo Sensors vol 11 no 6 pp 5792ndash5807 2011

[4] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[5] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[6] E Zitzler M Laumanns L Thiele et al SPEA2 Improving theStrength Pareto Evolutionary Algorithm Eidgenossische Techn-ische Hochschule Zurich (ETH) Institut fur Technische Infor-matik und Kommunikationsnetze (TIK) 2001

[7] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[8] B Alatas E Akin and A B Ozer ldquoChaos embedded particleswarm optimization algorithmsrdquo Chaos Solitons and Fractalsvol 40 no 4 pp 1715ndash1734 2009

[9] G ZilongW Sunrsquoan andZ Jian ldquoA novel immune evolutionaryalgorithm incorporating chaos optimizationrdquo Pattern Recogni-tion Letters vol 27 no 1 pp 2ndash8 2006

[10] M S Tavazoei and M Haeri ldquoComparison of different one-dimensional maps as chaotic search pattern in chaos optimiza-tion algorithmsrdquo Applied Mathematics and Computation vol187 no 2 pp 1076ndash1085 2007

[11] G Yu X Wang and P Li ldquoApplication of chaotic theory indifferential evolution algorithmsrdquo in Proceedings of the 6th Inte-rnational Conference on Natural Computation (ICNC rsquo10) pp3816ndash3820 August 2010

[12] B Alatas and E Akin ldquoMulti-objective rulemining using a cha-otic particle swarm optimization algorithmrdquo Knowledge-BasedSystems vol 22 no 6 pp 455ndash460 2009

[13] L D S Coelho and V C Mariani ldquoChaotic artificial immuneapproach applied to economic dispatch of electric energy usingthermal unitsrdquo Chaos Solitons and Fractals vol 40 no 5 pp2376ndash2383 2009

[14] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[15] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[16] L D S Coelho ldquoA quantum particle swarm optimizer with cha-otic mutation operatorrdquo Chaos Solitons and Fractals vol 37 no5 pp 1409ndash1418 2008

[17] L S Dos Coelho and P Alotto ldquoMultiobjective electromagneticoptimization based on a nondominated sorting genetic appr-oach with a chaotic crossover operatorrdquo IEEE Transactions onMagnetics vol 44 no 6 pp 1078ndash1081 2008

[18] H F Zhang J Z Zhou Y C Zhang N Fang and R ZhangldquoShort termhydrothermal scheduling usingmulti-objective dif-ferential evolution with three chaotic sequencesrdquo InternationalJournal of Electrical Power amp Energy Systems vol 47 pp 85ndash992013

[19] Y-J Wang and J-S Zhang ldquoGlobal optimization by an impr-oved differential evolutionary algorithmrdquo Applied Mathematicsand Computation vol 188 no 1 pp 669ndash680 2007

[20] R Caponetto L Fortuna S Fazzino and M G Xibilia ldquoCha-otic sequences to improve the performance of evolutionary alg-orithmsrdquo IEEE Transactions on Evolutionary Computation vol7 no 3 pp 289ndash304 2003

[21] M Ahmadi and H Mojallali ldquoChaotic invasive weed optimiz-ation algorithm with application to parameter estimation ofchaotic systemsrdquo Chaos Solitons amp Fractals vol 45 no 9-10pp 1108ndash1120 2012

[22] Z S Ma ldquoChaotic populations in genetic algorithmsrdquo AppliedSoft Computing vol 12 pp 2409ndash2424 2012

[23] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[24] W-M Zheng ldquoKneading plane of the circle maprdquo ChaosSolitons and Fractals vol 4 no 7 pp 1221ndash1233 1994

[25] T D Rogers and D C Whitley ldquoChaos in the cubic mappingrdquoMathematical Modelling vol 4 no 1 pp 9ndash25 1983

[26] M Bucolo R Caponetto L Fortuna M Frasca and A RizzoldquoDoes chaos work better than noiserdquo IEEE Circuits and SystemsMagazine vol 2 no 3 pp 4ndash19 2002

[27] D He C He L-G Jiang H-W Zhu and G-R Hu ldquoChaoticcharacteristics of a one-dimensional iterative map with infinitecollapsesrdquo IEEE Transactions on Circuits and Systems I vol 48no 7 pp 900ndash906 2001

[28] R May ldquoSimple mathematical models with very complicateddynamicsrdquo inTheTheory of Chaotic Attractors B Hunt T Y LiJ Kennedy and H Nusse Eds pp 85ndash93 Springer New YorkNY USA 2004

[29] R Barton ldquoChaos and fractalsrdquo The Mathematics Teacher vol83 pp 524ndash529 1990

[30] A Baykasoglu ldquoDesign optimization with chaos embeddedgreat deluge algorithmrdquoApplied Soft Computing Journal vol 12no 3 pp 1055ndash1067 2012

[31] J Fridrich ldquoSymmetric ciphers based on two-dimensional cha-otic mapsrdquo International Journal of Bifurcation and Chaos in

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Volume 2014

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Stochastic AnalysisInternational Journal of

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl

6 Mathematical Problems in Engineering

where 119883119894is the state of the system at discrete time 119894 For an

119873-point time series 1199091 1199092 119909

119873 each119883

119894is given by

119883119894= (119909119894 119909119894+119869 119909

119894+(119898minus1)119869) (20)

where 119869 is the lag or reconstruction delay and 119898 is theembedding dimension Thus119883 is an119872times119898matrix and theconstants119898119872 119869 and119873 are related as

119872 = 119873 minus (119898 minus 1) 119869 (21)After reconstructing the dynamics the algorithm locates thenearest neighbor of each point on the trajectory The nearestneighbor 119883

119895 where 119895 isin 1 2 119872 is found by searching

for the point that minimizes the distance to the particularreference point119883

119895 This is expressed as

119889119895(0) = min

119883119895

10038171003817100381710038171003817119883119895minus 119883119895

10038171003817100381710038171003817 (22)

where 119889119895(0) is the initial distance from the 119895th point to its

nearest neighbor and denotes the Euclidean norm Weimpose an additional constraint that the nearest neighborshave a temporal separation greater than the mean period ofthe time series

10038161003816100381610038161003816119895 minus 11989510038161003816100381610038161003816gt 119901 (23)

where 119901 is the mean period of time series 119901 can be estimatedby the reciprocal of the mean frequency of the powerspectrum This allows us to consider each pair of neighborsas nearby initial conditions for different trajectories Thelargest Lyapunov exponent is estimated as the mean rate ofseparation of the nearest neighbors

For each reference point119883119895 119889119895(119894) is the distance between

the 119895th pair of nearest neighbors after 119894 discrete time

119889119895(119894) =

10038171003817100381710038171003817119883119895+119894minus 119883119895+119894

10038171003817100381710038171003817 119894 = 1 2 min (119872 minus 119895119872 minus 119895)

(24)Assume that reference point 119883

119895and its nearest neighbor

119883119895have index divergence rate 120582

1 then

119889119895(119894) = 119862

1198951198901205821(119894sdotΔ119905)

119862119895= 119889119895(0) (25)

where 119862119895is the initial separation By taking the logarithm of

both sides of (25) we getln 119889119895(119894) asymp ln119862

119895+ 1205821(119894 sdot Δ119905) (26)

Equation (26) represents a set of approximately parallel lines(for 119895 = 1 2 119872) each with a slope 119904 roughly proportionalto 1205821 The largest Lyapunov exponent is easily and accurately

calculated using a least square fit to the ldquoaveragerdquo line definedby

119910 (119894) =1

Δ119905⟨ln 119889119895(119894)⟩ (27)

where ⟨ ⟩ denotes the average over all values of 119895 So

119910 (119894) =1

119902Δ119905

119902

sum

119895=1

ln 119889119895(119894) (28)

where 119902 is the number of 119889119895(119894) with 119889

119895(119894) = 0

Choose a linear area of the curve 119910(119894) sim 119894 and apply theleast square method to get the regression straight line Thenthe slope of the regression straight line is the largest Lyapunovexponent 120582

1

52 The Lyapunov Exponent of Sequences In the calculationprocess the embedding dimension 119898 is calculated throughthe method of false nearest neighbors (FNN) For the timedelay 119869 a good approximation of 119869 is equal to the numberlagging where the autocorrelation function drops to 1 minus 1119890of its initial value

Since different test problems have different rangeschaotic sequences need to be changed to different scalesTwo kinds of sequences used in experiments need to beinvestigated sequences with scales changed and sequencesgenerated by one dimension of a two-dimension chaoticmap

521 Sequences with Scales Changed Since the sequence 1199091

to 119909100

generated by ICMIC is not in (0 1) the new sequence1199101to 119910100

has to be generated by the following function

119910119894=1

2(119909119894+ 1) 119894 isin [1 100] (29)

The sequence 1199101to 119910100

is in the range of (0 1) The Lya-punov exponent of the new sequence is calculated throughsmall data sets arithmetic The average Lyapunov exponentof 10 runs is 00744 Since it is a positive number the newsequence 119910

1to 119910100

conforms to the chaotic nature

522 Sequences Generated by One Dimension of a Two-Dimension ChaoticMap For the Zaslavskii map one dimen-sion119910

119896is chosen in the following simulationThe sequence119910

1

to 119910100

is generated by 100 iterations through Zaslavskii mapThe new sequence 119911

1to 119911100

is generated by the followingfunction

119911119894=(119910119894+ 10513)

21026 119894 isin [1 100] (30)

Then the sequence 1199111to 119911100

is in (0 1) By a similarprocessing with ICMIC the average Lyapunov exponent is000194 Then the new sequence 119911

1to 119911100

conforms to thechaotic nature

6 Test Problem and Performance Measures

61 Test Problems Two-objective optimization problems arechosen to test and measure the performance improvementof the evolutionary algorithms using chaotic maps in threephases We use well-defined benchmark functions as objec-tive functions Their properties are shown in Table 1

62 Performance Measures Two criteria are used to evaluatethe performance of multiobjective optimization (1) conver-gence to the Pareto-optimal set and (2) maintenance of diver-sity in solutions of the Pareto-optimal set [7] Twometrics areadopted to evaluate the effects of the combinations of phasesand chaotic maps

The first metric 120574 measures the extent of convergence toa known set of Pareto-optimal solutions It is defined as

120574 =1

119873

119873

sum

119894=1

119889119894 (31)

Mathematical Problems in Engineering 7

Table 1 Test problems

Problem 119899 Variable bounds Objective functions Optimal solutions

ZDT1 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909)]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT2 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus (119909

1119892(119909))

2

]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT3 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909) minus (119909

1119892(119909)) sin(10120587119909

1)]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT4 101199091isin [0 1]

119909119894isin [minus5 5]

119894 = 2 119899

1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909)]

119892(119909) = 1 + (10(119899 minus 1) + sum119899

119894=2[1199092

119894minus 10 cos(4120587119909

119894)])

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT6 10 [0 1]1198911(119909) = 1 minus exp(minus4119909

1)sin6(6120587119909

1)

1198912(119909) = 119892(119909)[1 minus (119891

1(119909)119892(119909))

2

]

119892(119909) = 1 + (9[(sum119899

119894=2119909119894) (119899 minus 1)]

025

)

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

where 119889119894is the minimum Euclidean distance of every

obtained solution to the Pareto-optimal front The smallerthe value of this metric is the nearer the convergence towardPareto-front is

The other metric Δ measures the extent of spreadachieved among the obtained solutions The metric Δ isdefined by

Δ =119889119891+ 119889119897+ sum119873minus1

119894=1

10038161003816100381610038161003816119889119894minus 11988910038161003816100381610038161003816

119889119891+ 119889119897+ (119873 minus 1) 119889

(32)

The parameter 119889119894is the Euclidean distance between consecu-

tive solutions in the obtained nondominated set of solutionsTheparameters119889

119897and119889119891are the Euclidean distances between

the extreme solutions and the boundary solutions of theobtained nondominated set The parameter 119889 is the averageof all distances 119889

119894 119894 = 1 2 119873 minus 1 assuming that there are

119873 solutions on the best nondominated front

7 Experiments and Results

To explore the relationship of phases and chaotic mapsto solve MOPs NSGA-II algorithm is chosen as the mainframeworkThe ten chaotic maps mentioned in Section 4 areembedded in three different phases in the original NSGA-II algorithm Each time only one parameter is modifiedFor example if initial population is generated by chaoticmap the crossover and mutation operator are not changedSimilarly if crossover operator is modified by a chaoticmap the initial population and mutation operator are notchanged The solutions generated by the chaos embeddedNSGA-II algorithm are evaluated by two metrics 120574 and ΔFor readerrsquos convenience the new algorithms with differentcombinations of chaotic maps and phases are named asldquocns [chaotic map] [phase]rdquo and the results of differentalgorithms on test problems are named as ldquocns [chaoticmap] [phase] [test problem]rdquo In addition ldquoirdquo represents the

phase for initial population ldquocrdquo represents the phase forcrossover operator and ldquomrdquo represents the phase formutationoperator For example the results through modified initialpopulation by logistic map solving ZDT1 problem are namedas ldquocns logistic i zdt1rdquo

Each combination of one chaotic map and one phaseneeds one experiment In this research 10 chaotic maps with3 different phases based on 2 metrics solving 5 test problemsneed 150 basic experiments and obtain 300 results Eachexperiment obtains a Pareto frontThe values of convergencemetric 120574 and the diversity metric Δ are also calculated

In order to compare with the results of original NSGA-IIalgorithm we focused on the difference of the 120574 and Δ valuesof the original NSGA-II algorithm and the new algorithmFor example the 120574 of results of ldquocns sinusoidal i zdt1rdquois named as ldquocns sinusoidal i zdt1 gamardquo and the 120574 ofresults of NSGA-II solving ZDT1 problem is namedas ldquons zdt1 gamardquo Then the difference is named asldquons zdt1 gamamdashcns sinusoidal i zdt1 gamardquo When theprocesses of algorithms get to convergence the difference isvery small The properties of convergence and diversity inthe process of iterations need to be taken into account sothe 120574 values of each generation in the iterations are recordedand the differences of 120574 of each generation are obtained Thisprocess also applies to Δ

Some main parameters in the process of NSGA-II algo-rithm are introduced in the following paragraphs Then theresults of experiments are shown and analyzed

71TheMainParameters Themainparameters in the processof NSGA-II algorithm are presented in this section Choosingan appropriate representation of a chromosome is veryimportant for solving problems Real numbers are chosento represent the genes One chromosome represents oneindividual The initial population has 100 individuals andeach chromosome has a certain number of genes whichare represented by a real number Each individual of theinitial population is generated randomly with the range

8 Mathematical Problems in Engineering

Table 2 Parameters in the process of algorithms

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6119899iter 250119899pop 100119899var 30 30 30 10 10119901119888

09119901119898

130 130 130 110 110

based on the test problems The iteration will not terminateuntil the number of iterations gets to 250 For the processof NSGA-II algorithm a parent population is selected bytournament selection depending on the nondominated rankand the crowed-comparison operator Then the new popula-tion is generated by crossover and mutation operators Thecrossover operation is executed with the probability of 119901

119888=

09 The probability of mutation 119901119898is equal to the reciprocal

of 119899var which is the dimension number of a chromosome thatis 119901119898= 1119899var

Those parameters are summarized in Table 2 In thetable 119899iter is the number of iterations 119899pop is the scaleof the population 119899var is the number of dimensions of achromosome and119901

119888and119901119898are the probabilities of crossover

and mutation operations

72 Convergence Performance It is known that the 120574 differ-ence is used to evaluate the performance of the chaotic mapsin different phases inmultiobjective evolutionary algorithmsAn example is chosen for further explanation in detail As inFigure 1 the graph shows the results of solving ZDT1 prob-lems with Bakerrsquos map in crossover operator in NSGA-IIThedifferences of 120574 between the experiment ldquocns baker c zdt1rdquoand the experiment ldquons zdt1rdquo in the 250 iterations are givenAs seen from the figure the black line is above the red linewhich represents 0 so the new algorithm ldquocns bakers crdquo isbetter thanNSGA-II algorithm in solvingZDT1 problemwithregard to the convergence metric

The 120574 results of all the experiments are given similar toFigure 1 Since it is difficult to show so many graphs in thispaper the results of three typical problems are chosen thatis ZDT1 which is a simple convex problem ZDT3 whosePareto front is piecewise and ZDT4 which has local optimaThe graphs in Figures 2 3 and 4 provide a comparison of theperformance of solving different MOPs with chaotic maps ininitial population ZDT4 is chosen to show the performanceof chaotic maps in different phases on solving the sameMOPas shown in Figures 4 5 and 6 Each subgraph is labeled withthe name of the chaotic map used

In order to quantify the effect of chaotic maps and phaseswith regard to the metric 120574 the average of 120574 difference in 250generations is calculated to represent the effect of the newalgorithms

Since the order of magnitude of 120574 is not the samethe comparison of these 120574 values is not convenient Thenormalized values are obtained by dividing the 120574 values bythemaximumof the absolute values of the 120574 based on one testproblem The results of normalization are shown in Table 3

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

Bakerrsquos06

05

04

03

02

01

0

minus01

minus02

minus03

minus04

Figure 1 Performance of Bakerrsquos maps in crossover operator insolving ZDT1

Table 3 can be presented in a more intuitive way If 120574 ge03 the numerical value of 120574 is replaced by ldquo++rdquo Similarlyldquo+rdquo represents 01 le 120574 lt 03 ldquo0rdquo represents minus01 le 120574 lt01 ldquominusrdquo represents minus03 le 120574 lt minus01 and ldquominusminusrdquo represents120574 lt minus03 Therefore ldquo++rdquo means that the effect of the newalgorithm with chaotic maps is much better whereas ldquominusminusrdquo ismuch worse Table 4 shows the results

As shown in Table 4 most of the combinations of chaoticmaps and phases have a positive effect on improving the per-formance of NSGA-II algorithm The effect of some chaoticmaps is very good especially in some particular phases Forexample Bakerrsquos map in crossover operator Gauss map incrossover operator and initial population ICMIC map ininitial population sinusoidal map in initial population tentmap in crossover operation and Zaslavskii map in initialpopulation have very good effect

Since ZDT4 problem has 219 or 794times1011 different localPareto-optimal fronts in the search space the solutions easilyget entrapped into local optimum As seen from Table 4chaotic maps used for crossover and mutation operator havesignificant improvement on evolutionary algorithms solvingZDT4 problem especially cat map has the best performancein tenmaps Circle map and cubicmap have less contributionin solving those MOPs The distribution of cat map isrelatively uniform It is probably the reason for the goodperformance in solving problems with local optima

The original NSGA-II algorithm is not good at solvingZDT3 and ZDT6 problems because Pareto-optimal front ofZDT3 is disconnected and solutions of ZDT6 are nonuni-formly spaced However it can be seen in Table 4 that chaoticmaps can improve NSGA-II especially in crossover operationand initial population in solving ZDT3 and ZDT6 problem

In order to eliminate the special effect of the NSGA-II algorithm the polynomial mutation operator in NSGA-II is changed by the Gauss mutation and Cauchy mutationoperators Four typical chaotic maps which include twochaotic maps with best performance and two chaotic maps

Mathematical Problems in Engineering 9

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

GenerationGeneration

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

Figure 2 Performance of chaotic maps in initial population in solving ZDT1

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma 06

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

Figure 3 Performance of chaotic maps in initial population in solving ZDT3

10 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 4 Performance of chaotic maps in initial population in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 5 Performance of chaotic maps in crossover operator in solving ZDT4

Mathematical Problems in Engineering 11

Table 3 The normalized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker 0617 0080 0049 0682 0149 0094 0668 0109 0003 0220 minus0040 0167 0567 0105 minus0090Cat minus0060 0076 0101 0013 0084 0024 minus0007 0090 0174 0191 0109 0158 minus0028 0022 minus0096Circle minus0142 minus0040 minus0016 0013 minus0121 minus0007 minus0153 0028 minus0016 0151 minus0069 0098 minus0176 minus0072 minus0002Cubic minus0544 0064 minus0025 minus0626 minus0041 0006 minus0334 minus0049 0077 0032 minus0781 0071 minus0288 0040 0008Gauss 0307 0513 0089 0306 0507 0070 0454 0585 0037 0159 0005 0191 0114 minus0010 0132ICMIC minus0415 0558 0132 minus0280 0609 0144 minus0295 0510 0004 0003 minus0380 0088 minus0162 0252 0163Logistic 0070 0242 0189 0072 0204 minus0031 0012 0158 0127 0017 minus0819 0183 0152 0129 0132Sinusoidal 0077 1 0616 0121 1 0742 0169 1 0734 minus0091 minus1 minus0138 0148 0688 1Tent 0655 0177 0062 0704 0103 minus0005 0731 0043 minus0088 0190 minus0008 minus0058 0569 0124 minus0003Zaslavskii minus0051 0462 0032 minus0150 0518 0064 minus0086 0499 0110 minus0103 minus0339 0108 minus0060 0174 0183

Table 4 The visualized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker ++ 0 0 ++ + 0 ++ + 0 + 0 + ++ + 0Cat 0 0 + 0 0 0 0 0 + + + + 0 0 0Circle minus 0 0 0 minus 0 minus 0 0 + 0 0 minus 0 0Cubic mdash 0 0 mdash 0 0 mdash 0 0 0 mdash 0 minus 0 0Gauss ++ ++ 0 ++ ++ 0 ++ ++ 0 + 0 + + 0 +ICMIC mdash ++ + minus ++ + minus ++ 0 0 mdash 0 minus + +Logistic 0 + + 0 + 0 0 + + 0 mdash + + + +Sinusoidal 0 ++ ++ + ++ ++ + ++ ++ 0 mdash minus + ++ ++Tent ++ + 0 ++ + 0 ++ 0 0 + 0 0 ++ + 0Zaslavskii 0 ++ 0 minus ++ 0 0 ++ + minus mdash + 0 + +

Table 5 Results of Gauss mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle minus04886 minus00170 19394 minus03130 00971 minus04370 1856683 750688 28693 minus05529Cubic minus24674 minus00589 minus62676 minus00630 minus25351 minus12051 491193 minus448985 minus23071 57609Sinusoidal 10277 60982 04343 94173 minus00396 45810 minus106768 minus525536 584566 285351Tent 35035 06784 59502 11086 18616 00623 minus272428 1014576 150061 112005

Table 6 Results of Cauchy mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle 13978 05012 13997 07723 07187 04043 1354511 minus136938 01105 minus26454Cubic minus26201 minus04932 minus42148 minus05317 minus24831 minus07487 minus771621 minus456939 minus57560 54533Sinusoidal 01470 46977 08327 82820 03179 45376 minus202995 minus468243 65462 238545Tent 27613 03380 52687 09699 26813 minus01916 minus283172 579606 127523 52806

with worst performance are chosen to be used in theexperiments These chaotic maps are circle map cubic mapsinusoidal map and tent map The values of 120574 differences areshown in Tables 5 and 6 As seen from Tables 5 and 6 theperformance of sinusoidal map and tent map is better thanthe performance of circlemap and cubicmap Sinusoidalmapin initial population is better than that in crossover operationand tent map in crossover operation is better than that ininitial population This means the rules of combinations of

chaotic maps and phases in solving MOPs are almost thesame as in the previous observations So the rules based onthe framework of NSGA-II algorithm are applicable to otherMOEAs

In general Bakerrsquos map with a phase for crossover oper-ator sinusoidal map with phases for initial population andmutation operator and tent map with a phase for crossoveroperator could be the best choice for improving evolutionaryalgorithms for MOPs without local optimum For problems

12 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 6 Performance of chaotic maps in mutation operator in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

Figure 7 Performance of chaotic maps in crossover operator in solving ZDT1 with metric Δ

Mathematical Problems in Engineering 13

Table7Th

eaverage

results

ofΔ

ZDT1

ZDT2

ZDT3

ZDT4

ZDT6

ci

mc

im

ci

mc

im

ci

mBa

ker

minus00337

minus00026

00015

00226

00310

00024

minus000

96000

69000

06minus006

0300343

minus00513

00125

00024

minus00015

Cat

000

0500017

000

4200234

minus01021

minus00145

minus00019

minus00052

000

0600277

minus000

90minus00148

minus000

01minus000

02minus000

68Circle

minus00325

minus00035

minus000

64minus02019

minus00632

minus01410

minus00052

minus000

6500010

minus00255

minus00259

minus00136

00031

minus00020

000

64Cu

bic

minus000

4700121

minus00022

minus00439

minus00148

00361

000

00minus000

0600031

minus00052

minus00279

00238

minus00170

00054

minus00017

Gauss

00121

00075

minus00180

minus01075

00329

minus00711

minus00024

minus00032

00019

00330

minus00214

00155

00180

000

03000

64ICMIC

00081

000

92000

02minus00734

minus00850

minus00917

minus00036

000

08minus00054

minus00356

minus00497

00238

00123

000

0100085

Logistic

minus00118

00076

00075

minus01935

00239

00509

minus00084

minus00014

00080

00275

minus00100

minus00232

minus000

6900020

00077

Sinu

soidal

00149

000

69minus00564

minus01750

minus000

67minus02902

minus00014

00029

minus00286

minus00390

minus00164

minus01945

00129

00088

minus004

88Tent

minus00350

000

9300016

00215

minus00173

minus00261

minus00059

minus00168

00071

minus00761

minus00139

00075

00125

00028

minus000

02Za

slavskii

002307

00056

minus00014

minus00292

minus00189

minus00013

000

99000

40minus00037

00201

minus00056

000

9000038

00027

00084

14 Mathematical Problems in Engineering

Table 8 Statistical data for combinations of chaotic maps and phases on different problems

Threshold ZDT1 ZDT2 ZDT3 ZDT4 ZDT60 18 9 15 10 200001 16 9 9 10 180002 13 9 7 10 180003 13 8 6 10 140004 13 8 5 10 120005 12 8 4 10 120006 11 8 4 10 11

0 50 100 150 200 250minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

Generation

Diff

eren

ce o

f delt

a

Bakerrsquos

Figure 8 Performance of Bakerrsquos maps in crossover operator insolving ZDT1 with metric Δ

with local optimum cat map has good performance onimproving evolutionary algorithms

73 Diversity Performance Similar to the convergencemetric120574 the differences of the diversity metricΔ between the resultsof the new algorithm with chaotic maps and the originalNSGA-II algorithm are used to measure the performanceFigure 7 shows the performance of chaotic maps in crossoveroperator in solving ZDT1 with metric Δ Each subgraphshows the effect of one chaotic map To be seen more clearlythe first subgraph in Figure 7 is shown in Figure 8

As seen fromFigures 7 and 8 theΔ difference is not stablein 250 generations The average values of Δ difference in 250generations are calculated to represent the effect of the newalgorithms For brevity the rest of results are not shown ingraphs but in Table 7

As seen from Table 7 the Δ values have little differenceWe count the number of combinations of chaotic maps andphases for solving one problem in different threshold valuesFor example there are 18 combinations whose values of Δ aregreater than zero Based on the number of the combinationsof chaotic maps and phases in different threshold values thevalues of diversity metric Δ are summarized in Table 8

In Table 8 the rank of the number of Δ in differentthreshold values is ZDT1 gt ZDT6 gt ZDT4 gt ZDT2 gt

ZDT3 especially for larger threshold ZDT1 problem whichis a convex function and has no local optima is a relativelyeasy problem Chaotic maps bring the biggest improve-ments on solving ZDT1 Though the solutions of ZDT6 arenonuniformly spaced chaotic maps can find better spread ofsolutions While ZDT4 problem is a complex problem andthe solutions are easily trapped into local optima chaoticmaps can improve the distribution of the solutions ZDT2problem is a convex function and the solutions sometimesfall into the local optimumThe effects of chaoticmaps can begeneralizedThe Pareto front of ZDT3 problem is segmentedso the Δ value of ZDT3 is larger and the ranking of ZDT3 islower It is our observation that Δ is not fit for evaluating thesolutions to problems which are disconnected

Based on the diversity metric Δ chaotic maps havethe best improvement on solving convex problems withoutlocal optima and have better effect on solving problemswhich have nonuniform solutions For problems with localminimum chaotic maps embedded algorithms can improvethe performance with regard to metric Δ

A short summary can be given according to the aboveexperiments First chaotic maps can improve the perfor-mance of MOEAs but the results showed that no one chaoticmap outperforms other maps for all of the problems Theresults in this paper give some guidance on how to choose achaotic map and a phase in MOEAs Second an interestingdiscovery is that cat map has best performance on solvingproblems with local optima Uniformity of cat map may beone of the reasons for the good performance of solving ZDT4

8 Conclusion

The focus of this paper is to explore the relationships ofchaotic maps and phases in MOEAs in solving MOPs Themain framework of algorithms in experiments is the NSGA-II algorithm The combinations of ten chaotic maps andthree phases are chosen in the experiments Two metricsconvergence metric 120574 and diversity metric Δ are used toevaluate the convergence and diversity properties of thealgorithms with chaotic maps The test problems are ZDTseries which were all MOPs The ergodicity and initial valuesensitivity of chaotic maps can help evolutionary algorithmsavoid solutions from falling into local optimal and get betterconvergence In the experimental results almost all chaoticmaps have good effects on improving the performance ofevolutionary algorithms to solveMOPs without local optima

Mathematical Problems in Engineering 15

Cat map has best performance on solving problems withlocal optimum This work gives insight on choosing chaoticmaps and phases for different problems Our future workwill perform further experiments with more chaotic maps onother MOEAs and formulate the theory analysis

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China under Grant no 61101153 and Prof Qiuis supported by NSF CNS 1359557

References

[1] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing vol 13 pp2790ndash2802 2013

[2] J A Adeyemo ldquoReservoir operation using multi-objective evo-lutionary algorithms-a reviewrdquo Asian Journal of Scientific Res-earch vol 4 no 1 pp 16ndash27 2011

[3] H Hu L Xu R Wei and B Zhu ldquoMulti-objective controloptimization for greenhouse environment using evolutionaryalgorithmsrdquo Sensors vol 11 no 6 pp 5792ndash5807 2011

[4] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[5] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[6] E Zitzler M Laumanns L Thiele et al SPEA2 Improving theStrength Pareto Evolutionary Algorithm Eidgenossische Techn-ische Hochschule Zurich (ETH) Institut fur Technische Infor-matik und Kommunikationsnetze (TIK) 2001

[7] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[8] B Alatas E Akin and A B Ozer ldquoChaos embedded particleswarm optimization algorithmsrdquo Chaos Solitons and Fractalsvol 40 no 4 pp 1715ndash1734 2009

[9] G ZilongW Sunrsquoan andZ Jian ldquoA novel immune evolutionaryalgorithm incorporating chaos optimizationrdquo Pattern Recogni-tion Letters vol 27 no 1 pp 2ndash8 2006

[10] M S Tavazoei and M Haeri ldquoComparison of different one-dimensional maps as chaotic search pattern in chaos optimiza-tion algorithmsrdquo Applied Mathematics and Computation vol187 no 2 pp 1076ndash1085 2007

[11] G Yu X Wang and P Li ldquoApplication of chaotic theory indifferential evolution algorithmsrdquo in Proceedings of the 6th Inte-rnational Conference on Natural Computation (ICNC rsquo10) pp3816ndash3820 August 2010

[12] B Alatas and E Akin ldquoMulti-objective rulemining using a cha-otic particle swarm optimization algorithmrdquo Knowledge-BasedSystems vol 22 no 6 pp 455ndash460 2009

[13] L D S Coelho and V C Mariani ldquoChaotic artificial immuneapproach applied to economic dispatch of electric energy usingthermal unitsrdquo Chaos Solitons and Fractals vol 40 no 5 pp2376ndash2383 2009

[14] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[15] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[16] L D S Coelho ldquoA quantum particle swarm optimizer with cha-otic mutation operatorrdquo Chaos Solitons and Fractals vol 37 no5 pp 1409ndash1418 2008

[17] L S Dos Coelho and P Alotto ldquoMultiobjective electromagneticoptimization based on a nondominated sorting genetic appr-oach with a chaotic crossover operatorrdquo IEEE Transactions onMagnetics vol 44 no 6 pp 1078ndash1081 2008

[18] H F Zhang J Z Zhou Y C Zhang N Fang and R ZhangldquoShort termhydrothermal scheduling usingmulti-objective dif-ferential evolution with three chaotic sequencesrdquo InternationalJournal of Electrical Power amp Energy Systems vol 47 pp 85ndash992013

[19] Y-J Wang and J-S Zhang ldquoGlobal optimization by an impr-oved differential evolutionary algorithmrdquo Applied Mathematicsand Computation vol 188 no 1 pp 669ndash680 2007

[20] R Caponetto L Fortuna S Fazzino and M G Xibilia ldquoCha-otic sequences to improve the performance of evolutionary alg-orithmsrdquo IEEE Transactions on Evolutionary Computation vol7 no 3 pp 289ndash304 2003

[21] M Ahmadi and H Mojallali ldquoChaotic invasive weed optimiz-ation algorithm with application to parameter estimation ofchaotic systemsrdquo Chaos Solitons amp Fractals vol 45 no 9-10pp 1108ndash1120 2012

[22] Z S Ma ldquoChaotic populations in genetic algorithmsrdquo AppliedSoft Computing vol 12 pp 2409ndash2424 2012

[23] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[24] W-M Zheng ldquoKneading plane of the circle maprdquo ChaosSolitons and Fractals vol 4 no 7 pp 1221ndash1233 1994

[25] T D Rogers and D C Whitley ldquoChaos in the cubic mappingrdquoMathematical Modelling vol 4 no 1 pp 9ndash25 1983

[26] M Bucolo R Caponetto L Fortuna M Frasca and A RizzoldquoDoes chaos work better than noiserdquo IEEE Circuits and SystemsMagazine vol 2 no 3 pp 4ndash19 2002

[27] D He C He L-G Jiang H-W Zhu and G-R Hu ldquoChaoticcharacteristics of a one-dimensional iterative map with infinitecollapsesrdquo IEEE Transactions on Circuits and Systems I vol 48no 7 pp 900ndash906 2001

[28] R May ldquoSimple mathematical models with very complicateddynamicsrdquo inTheTheory of Chaotic Attractors B Hunt T Y LiJ Kennedy and H Nusse Eds pp 85ndash93 Springer New YorkNY USA 2004

[29] R Barton ldquoChaos and fractalsrdquo The Mathematics Teacher vol83 pp 524ndash529 1990

[30] A Baykasoglu ldquoDesign optimization with chaos embeddedgreat deluge algorithmrdquoApplied Soft Computing Journal vol 12no 3 pp 1055ndash1067 2012

[31] J Fridrich ldquoSymmetric ciphers based on two-dimensional cha-otic mapsrdquo International Journal of Bifurcation and Chaos in

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

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Mathematical Problems in Engineering

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Volume 2014

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Stochastic AnalysisInternational Journal of

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl

Mathematical Problems in Engineering 7

Table 1 Test problems

Problem 119899 Variable bounds Objective functions Optimal solutions

ZDT1 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909)]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT2 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus (119909

1119892(119909))

2

]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT3 30 [0 1]1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909) minus (119909

1119892(119909)) sin(10120587119909

1)]

119892(119909) = 1 + (9 (sum119899

119894=2119909119894) (119899 minus 1))

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT4 101199091isin [0 1]

119909119894isin [minus5 5]

119894 = 2 119899

1198911(119909) = 119909

1

1198912(119909) = 119892(119909)[1 minus radic119909

1119892(119909)]

119892(119909) = 1 + (10(119899 minus 1) + sum119899

119894=2[1199092

119894minus 10 cos(4120587119909

119894)])

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

ZDT6 10 [0 1]1198911(119909) = 1 minus exp(minus4119909

1)sin6(6120587119909

1)

1198912(119909) = 119892(119909)[1 minus (119891

1(119909)119892(119909))

2

]

119892(119909) = 1 + (9[(sum119899

119894=2119909119894) (119899 minus 1)]

025

)

1199091isin [0 1]

119909119894= 0

119894 = 2 119899

where 119889119894is the minimum Euclidean distance of every

obtained solution to the Pareto-optimal front The smallerthe value of this metric is the nearer the convergence towardPareto-front is

The other metric Δ measures the extent of spreadachieved among the obtained solutions The metric Δ isdefined by

Δ =119889119891+ 119889119897+ sum119873minus1

119894=1

10038161003816100381610038161003816119889119894minus 11988910038161003816100381610038161003816

119889119891+ 119889119897+ (119873 minus 1) 119889

(32)

The parameter 119889119894is the Euclidean distance between consecu-

tive solutions in the obtained nondominated set of solutionsTheparameters119889

119897and119889119891are the Euclidean distances between

the extreme solutions and the boundary solutions of theobtained nondominated set The parameter 119889 is the averageof all distances 119889

119894 119894 = 1 2 119873 minus 1 assuming that there are

119873 solutions on the best nondominated front

7 Experiments and Results

To explore the relationship of phases and chaotic mapsto solve MOPs NSGA-II algorithm is chosen as the mainframeworkThe ten chaotic maps mentioned in Section 4 areembedded in three different phases in the original NSGA-II algorithm Each time only one parameter is modifiedFor example if initial population is generated by chaoticmap the crossover and mutation operator are not changedSimilarly if crossover operator is modified by a chaoticmap the initial population and mutation operator are notchanged The solutions generated by the chaos embeddedNSGA-II algorithm are evaluated by two metrics 120574 and ΔFor readerrsquos convenience the new algorithms with differentcombinations of chaotic maps and phases are named asldquocns [chaotic map] [phase]rdquo and the results of differentalgorithms on test problems are named as ldquocns [chaoticmap] [phase] [test problem]rdquo In addition ldquoirdquo represents the

phase for initial population ldquocrdquo represents the phase forcrossover operator and ldquomrdquo represents the phase formutationoperator For example the results through modified initialpopulation by logistic map solving ZDT1 problem are namedas ldquocns logistic i zdt1rdquo

Each combination of one chaotic map and one phaseneeds one experiment In this research 10 chaotic maps with3 different phases based on 2 metrics solving 5 test problemsneed 150 basic experiments and obtain 300 results Eachexperiment obtains a Pareto frontThe values of convergencemetric 120574 and the diversity metric Δ are also calculated

In order to compare with the results of original NSGA-IIalgorithm we focused on the difference of the 120574 and Δ valuesof the original NSGA-II algorithm and the new algorithmFor example the 120574 of results of ldquocns sinusoidal i zdt1rdquois named as ldquocns sinusoidal i zdt1 gamardquo and the 120574 ofresults of NSGA-II solving ZDT1 problem is namedas ldquons zdt1 gamardquo Then the difference is named asldquons zdt1 gamamdashcns sinusoidal i zdt1 gamardquo When theprocesses of algorithms get to convergence the difference isvery small The properties of convergence and diversity inthe process of iterations need to be taken into account sothe 120574 values of each generation in the iterations are recordedand the differences of 120574 of each generation are obtained Thisprocess also applies to Δ

Some main parameters in the process of NSGA-II algo-rithm are introduced in the following paragraphs Then theresults of experiments are shown and analyzed

71TheMainParameters Themainparameters in the processof NSGA-II algorithm are presented in this section Choosingan appropriate representation of a chromosome is veryimportant for solving problems Real numbers are chosento represent the genes One chromosome represents oneindividual The initial population has 100 individuals andeach chromosome has a certain number of genes whichare represented by a real number Each individual of theinitial population is generated randomly with the range

8 Mathematical Problems in Engineering

Table 2 Parameters in the process of algorithms

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6119899iter 250119899pop 100119899var 30 30 30 10 10119901119888

09119901119898

130 130 130 110 110

based on the test problems The iteration will not terminateuntil the number of iterations gets to 250 For the processof NSGA-II algorithm a parent population is selected bytournament selection depending on the nondominated rankand the crowed-comparison operator Then the new popula-tion is generated by crossover and mutation operators Thecrossover operation is executed with the probability of 119901

119888=

09 The probability of mutation 119901119898is equal to the reciprocal

of 119899var which is the dimension number of a chromosome thatis 119901119898= 1119899var

Those parameters are summarized in Table 2 In thetable 119899iter is the number of iterations 119899pop is the scaleof the population 119899var is the number of dimensions of achromosome and119901

119888and119901119898are the probabilities of crossover

and mutation operations

72 Convergence Performance It is known that the 120574 differ-ence is used to evaluate the performance of the chaotic mapsin different phases inmultiobjective evolutionary algorithmsAn example is chosen for further explanation in detail As inFigure 1 the graph shows the results of solving ZDT1 prob-lems with Bakerrsquos map in crossover operator in NSGA-IIThedifferences of 120574 between the experiment ldquocns baker c zdt1rdquoand the experiment ldquons zdt1rdquo in the 250 iterations are givenAs seen from the figure the black line is above the red linewhich represents 0 so the new algorithm ldquocns bakers crdquo isbetter thanNSGA-II algorithm in solvingZDT1 problemwithregard to the convergence metric

The 120574 results of all the experiments are given similar toFigure 1 Since it is difficult to show so many graphs in thispaper the results of three typical problems are chosen thatis ZDT1 which is a simple convex problem ZDT3 whosePareto front is piecewise and ZDT4 which has local optimaThe graphs in Figures 2 3 and 4 provide a comparison of theperformance of solving different MOPs with chaotic maps ininitial population ZDT4 is chosen to show the performanceof chaotic maps in different phases on solving the sameMOPas shown in Figures 4 5 and 6 Each subgraph is labeled withthe name of the chaotic map used

In order to quantify the effect of chaotic maps and phaseswith regard to the metric 120574 the average of 120574 difference in 250generations is calculated to represent the effect of the newalgorithms

Since the order of magnitude of 120574 is not the samethe comparison of these 120574 values is not convenient Thenormalized values are obtained by dividing the 120574 values bythemaximumof the absolute values of the 120574 based on one testproblem The results of normalization are shown in Table 3

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

Bakerrsquos06

05

04

03

02

01

0

minus01

minus02

minus03

minus04

Figure 1 Performance of Bakerrsquos maps in crossover operator insolving ZDT1

Table 3 can be presented in a more intuitive way If 120574 ge03 the numerical value of 120574 is replaced by ldquo++rdquo Similarlyldquo+rdquo represents 01 le 120574 lt 03 ldquo0rdquo represents minus01 le 120574 lt01 ldquominusrdquo represents minus03 le 120574 lt minus01 and ldquominusminusrdquo represents120574 lt minus03 Therefore ldquo++rdquo means that the effect of the newalgorithm with chaotic maps is much better whereas ldquominusminusrdquo ismuch worse Table 4 shows the results

As shown in Table 4 most of the combinations of chaoticmaps and phases have a positive effect on improving the per-formance of NSGA-II algorithm The effect of some chaoticmaps is very good especially in some particular phases Forexample Bakerrsquos map in crossover operator Gauss map incrossover operator and initial population ICMIC map ininitial population sinusoidal map in initial population tentmap in crossover operation and Zaslavskii map in initialpopulation have very good effect

Since ZDT4 problem has 219 or 794times1011 different localPareto-optimal fronts in the search space the solutions easilyget entrapped into local optimum As seen from Table 4chaotic maps used for crossover and mutation operator havesignificant improvement on evolutionary algorithms solvingZDT4 problem especially cat map has the best performancein tenmaps Circle map and cubicmap have less contributionin solving those MOPs The distribution of cat map isrelatively uniform It is probably the reason for the goodperformance in solving problems with local optima

The original NSGA-II algorithm is not good at solvingZDT3 and ZDT6 problems because Pareto-optimal front ofZDT3 is disconnected and solutions of ZDT6 are nonuni-formly spaced However it can be seen in Table 4 that chaoticmaps can improve NSGA-II especially in crossover operationand initial population in solving ZDT3 and ZDT6 problem

In order to eliminate the special effect of the NSGA-II algorithm the polynomial mutation operator in NSGA-II is changed by the Gauss mutation and Cauchy mutationoperators Four typical chaotic maps which include twochaotic maps with best performance and two chaotic maps

Mathematical Problems in Engineering 9

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

GenerationGeneration

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

Figure 2 Performance of chaotic maps in initial population in solving ZDT1

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma 06

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

Figure 3 Performance of chaotic maps in initial population in solving ZDT3

10 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 4 Performance of chaotic maps in initial population in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 5 Performance of chaotic maps in crossover operator in solving ZDT4

Mathematical Problems in Engineering 11

Table 3 The normalized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker 0617 0080 0049 0682 0149 0094 0668 0109 0003 0220 minus0040 0167 0567 0105 minus0090Cat minus0060 0076 0101 0013 0084 0024 minus0007 0090 0174 0191 0109 0158 minus0028 0022 minus0096Circle minus0142 minus0040 minus0016 0013 minus0121 minus0007 minus0153 0028 minus0016 0151 minus0069 0098 minus0176 minus0072 minus0002Cubic minus0544 0064 minus0025 minus0626 minus0041 0006 minus0334 minus0049 0077 0032 minus0781 0071 minus0288 0040 0008Gauss 0307 0513 0089 0306 0507 0070 0454 0585 0037 0159 0005 0191 0114 minus0010 0132ICMIC minus0415 0558 0132 minus0280 0609 0144 minus0295 0510 0004 0003 minus0380 0088 minus0162 0252 0163Logistic 0070 0242 0189 0072 0204 minus0031 0012 0158 0127 0017 minus0819 0183 0152 0129 0132Sinusoidal 0077 1 0616 0121 1 0742 0169 1 0734 minus0091 minus1 minus0138 0148 0688 1Tent 0655 0177 0062 0704 0103 minus0005 0731 0043 minus0088 0190 minus0008 minus0058 0569 0124 minus0003Zaslavskii minus0051 0462 0032 minus0150 0518 0064 minus0086 0499 0110 minus0103 minus0339 0108 minus0060 0174 0183

Table 4 The visualized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker ++ 0 0 ++ + 0 ++ + 0 + 0 + ++ + 0Cat 0 0 + 0 0 0 0 0 + + + + 0 0 0Circle minus 0 0 0 minus 0 minus 0 0 + 0 0 minus 0 0Cubic mdash 0 0 mdash 0 0 mdash 0 0 0 mdash 0 minus 0 0Gauss ++ ++ 0 ++ ++ 0 ++ ++ 0 + 0 + + 0 +ICMIC mdash ++ + minus ++ + minus ++ 0 0 mdash 0 minus + +Logistic 0 + + 0 + 0 0 + + 0 mdash + + + +Sinusoidal 0 ++ ++ + ++ ++ + ++ ++ 0 mdash minus + ++ ++Tent ++ + 0 ++ + 0 ++ 0 0 + 0 0 ++ + 0Zaslavskii 0 ++ 0 minus ++ 0 0 ++ + minus mdash + 0 + +

Table 5 Results of Gauss mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle minus04886 minus00170 19394 minus03130 00971 minus04370 1856683 750688 28693 minus05529Cubic minus24674 minus00589 minus62676 minus00630 minus25351 minus12051 491193 minus448985 minus23071 57609Sinusoidal 10277 60982 04343 94173 minus00396 45810 minus106768 minus525536 584566 285351Tent 35035 06784 59502 11086 18616 00623 minus272428 1014576 150061 112005

Table 6 Results of Cauchy mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle 13978 05012 13997 07723 07187 04043 1354511 minus136938 01105 minus26454Cubic minus26201 minus04932 minus42148 minus05317 minus24831 minus07487 minus771621 minus456939 minus57560 54533Sinusoidal 01470 46977 08327 82820 03179 45376 minus202995 minus468243 65462 238545Tent 27613 03380 52687 09699 26813 minus01916 minus283172 579606 127523 52806

with worst performance are chosen to be used in theexperiments These chaotic maps are circle map cubic mapsinusoidal map and tent map The values of 120574 differences areshown in Tables 5 and 6 As seen from Tables 5 and 6 theperformance of sinusoidal map and tent map is better thanthe performance of circlemap and cubicmap Sinusoidalmapin initial population is better than that in crossover operationand tent map in crossover operation is better than that ininitial population This means the rules of combinations of

chaotic maps and phases in solving MOPs are almost thesame as in the previous observations So the rules based onthe framework of NSGA-II algorithm are applicable to otherMOEAs

In general Bakerrsquos map with a phase for crossover oper-ator sinusoidal map with phases for initial population andmutation operator and tent map with a phase for crossoveroperator could be the best choice for improving evolutionaryalgorithms for MOPs without local optimum For problems

12 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 6 Performance of chaotic maps in mutation operator in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

Figure 7 Performance of chaotic maps in crossover operator in solving ZDT1 with metric Δ

Mathematical Problems in Engineering 13

Table7Th

eaverage

results

ofΔ

ZDT1

ZDT2

ZDT3

ZDT4

ZDT6

ci

mc

im

ci

mc

im

ci

mBa

ker

minus00337

minus00026

00015

00226

00310

00024

minus000

96000

69000

06minus006

0300343

minus00513

00125

00024

minus00015

Cat

000

0500017

000

4200234

minus01021

minus00145

minus00019

minus00052

000

0600277

minus000

90minus00148

minus000

01minus000

02minus000

68Circle

minus00325

minus00035

minus000

64minus02019

minus00632

minus01410

minus00052

minus000

6500010

minus00255

minus00259

minus00136

00031

minus00020

000

64Cu

bic

minus000

4700121

minus00022

minus00439

minus00148

00361

000

00minus000

0600031

minus00052

minus00279

00238

minus00170

00054

minus00017

Gauss

00121

00075

minus00180

minus01075

00329

minus00711

minus00024

minus00032

00019

00330

minus00214

00155

00180

000

03000

64ICMIC

00081

000

92000

02minus00734

minus00850

minus00917

minus00036

000

08minus00054

minus00356

minus00497

00238

00123

000

0100085

Logistic

minus00118

00076

00075

minus01935

00239

00509

minus00084

minus00014

00080

00275

minus00100

minus00232

minus000

6900020

00077

Sinu

soidal

00149

000

69minus00564

minus01750

minus000

67minus02902

minus00014

00029

minus00286

minus00390

minus00164

minus01945

00129

00088

minus004

88Tent

minus00350

000

9300016

00215

minus00173

minus00261

minus00059

minus00168

00071

minus00761

minus00139

00075

00125

00028

minus000

02Za

slavskii

002307

00056

minus00014

minus00292

minus00189

minus00013

000

99000

40minus00037

00201

minus00056

000

9000038

00027

00084

14 Mathematical Problems in Engineering

Table 8 Statistical data for combinations of chaotic maps and phases on different problems

Threshold ZDT1 ZDT2 ZDT3 ZDT4 ZDT60 18 9 15 10 200001 16 9 9 10 180002 13 9 7 10 180003 13 8 6 10 140004 13 8 5 10 120005 12 8 4 10 120006 11 8 4 10 11

0 50 100 150 200 250minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

Generation

Diff

eren

ce o

f delt

a

Bakerrsquos

Figure 8 Performance of Bakerrsquos maps in crossover operator insolving ZDT1 with metric Δ

with local optimum cat map has good performance onimproving evolutionary algorithms

73 Diversity Performance Similar to the convergencemetric120574 the differences of the diversity metricΔ between the resultsof the new algorithm with chaotic maps and the originalNSGA-II algorithm are used to measure the performanceFigure 7 shows the performance of chaotic maps in crossoveroperator in solving ZDT1 with metric Δ Each subgraphshows the effect of one chaotic map To be seen more clearlythe first subgraph in Figure 7 is shown in Figure 8

As seen fromFigures 7 and 8 theΔ difference is not stablein 250 generations The average values of Δ difference in 250generations are calculated to represent the effect of the newalgorithms For brevity the rest of results are not shown ingraphs but in Table 7

As seen from Table 7 the Δ values have little differenceWe count the number of combinations of chaotic maps andphases for solving one problem in different threshold valuesFor example there are 18 combinations whose values of Δ aregreater than zero Based on the number of the combinationsof chaotic maps and phases in different threshold values thevalues of diversity metric Δ are summarized in Table 8

In Table 8 the rank of the number of Δ in differentthreshold values is ZDT1 gt ZDT6 gt ZDT4 gt ZDT2 gt

ZDT3 especially for larger threshold ZDT1 problem whichis a convex function and has no local optima is a relativelyeasy problem Chaotic maps bring the biggest improve-ments on solving ZDT1 Though the solutions of ZDT6 arenonuniformly spaced chaotic maps can find better spread ofsolutions While ZDT4 problem is a complex problem andthe solutions are easily trapped into local optima chaoticmaps can improve the distribution of the solutions ZDT2problem is a convex function and the solutions sometimesfall into the local optimumThe effects of chaoticmaps can begeneralizedThe Pareto front of ZDT3 problem is segmentedso the Δ value of ZDT3 is larger and the ranking of ZDT3 islower It is our observation that Δ is not fit for evaluating thesolutions to problems which are disconnected

Based on the diversity metric Δ chaotic maps havethe best improvement on solving convex problems withoutlocal optima and have better effect on solving problemswhich have nonuniform solutions For problems with localminimum chaotic maps embedded algorithms can improvethe performance with regard to metric Δ

A short summary can be given according to the aboveexperiments First chaotic maps can improve the perfor-mance of MOEAs but the results showed that no one chaoticmap outperforms other maps for all of the problems Theresults in this paper give some guidance on how to choose achaotic map and a phase in MOEAs Second an interestingdiscovery is that cat map has best performance on solvingproblems with local optima Uniformity of cat map may beone of the reasons for the good performance of solving ZDT4

8 Conclusion

The focus of this paper is to explore the relationships ofchaotic maps and phases in MOEAs in solving MOPs Themain framework of algorithms in experiments is the NSGA-II algorithm The combinations of ten chaotic maps andthree phases are chosen in the experiments Two metricsconvergence metric 120574 and diversity metric Δ are used toevaluate the convergence and diversity properties of thealgorithms with chaotic maps The test problems are ZDTseries which were all MOPs The ergodicity and initial valuesensitivity of chaotic maps can help evolutionary algorithmsavoid solutions from falling into local optimal and get betterconvergence In the experimental results almost all chaoticmaps have good effects on improving the performance ofevolutionary algorithms to solveMOPs without local optima

Mathematical Problems in Engineering 15

Cat map has best performance on solving problems withlocal optimum This work gives insight on choosing chaoticmaps and phases for different problems Our future workwill perform further experiments with more chaotic maps onother MOEAs and formulate the theory analysis

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China under Grant no 61101153 and Prof Qiuis supported by NSF CNS 1359557

References

[1] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing vol 13 pp2790ndash2802 2013

[2] J A Adeyemo ldquoReservoir operation using multi-objective evo-lutionary algorithms-a reviewrdquo Asian Journal of Scientific Res-earch vol 4 no 1 pp 16ndash27 2011

[3] H Hu L Xu R Wei and B Zhu ldquoMulti-objective controloptimization for greenhouse environment using evolutionaryalgorithmsrdquo Sensors vol 11 no 6 pp 5792ndash5807 2011

[4] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[5] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[6] E Zitzler M Laumanns L Thiele et al SPEA2 Improving theStrength Pareto Evolutionary Algorithm Eidgenossische Techn-ische Hochschule Zurich (ETH) Institut fur Technische Infor-matik und Kommunikationsnetze (TIK) 2001

[7] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[8] B Alatas E Akin and A B Ozer ldquoChaos embedded particleswarm optimization algorithmsrdquo Chaos Solitons and Fractalsvol 40 no 4 pp 1715ndash1734 2009

[9] G ZilongW Sunrsquoan andZ Jian ldquoA novel immune evolutionaryalgorithm incorporating chaos optimizationrdquo Pattern Recogni-tion Letters vol 27 no 1 pp 2ndash8 2006

[10] M S Tavazoei and M Haeri ldquoComparison of different one-dimensional maps as chaotic search pattern in chaos optimiza-tion algorithmsrdquo Applied Mathematics and Computation vol187 no 2 pp 1076ndash1085 2007

[11] G Yu X Wang and P Li ldquoApplication of chaotic theory indifferential evolution algorithmsrdquo in Proceedings of the 6th Inte-rnational Conference on Natural Computation (ICNC rsquo10) pp3816ndash3820 August 2010

[12] B Alatas and E Akin ldquoMulti-objective rulemining using a cha-otic particle swarm optimization algorithmrdquo Knowledge-BasedSystems vol 22 no 6 pp 455ndash460 2009

[13] L D S Coelho and V C Mariani ldquoChaotic artificial immuneapproach applied to economic dispatch of electric energy usingthermal unitsrdquo Chaos Solitons and Fractals vol 40 no 5 pp2376ndash2383 2009

[14] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[15] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[16] L D S Coelho ldquoA quantum particle swarm optimizer with cha-otic mutation operatorrdquo Chaos Solitons and Fractals vol 37 no5 pp 1409ndash1418 2008

[17] L S Dos Coelho and P Alotto ldquoMultiobjective electromagneticoptimization based on a nondominated sorting genetic appr-oach with a chaotic crossover operatorrdquo IEEE Transactions onMagnetics vol 44 no 6 pp 1078ndash1081 2008

[18] H F Zhang J Z Zhou Y C Zhang N Fang and R ZhangldquoShort termhydrothermal scheduling usingmulti-objective dif-ferential evolution with three chaotic sequencesrdquo InternationalJournal of Electrical Power amp Energy Systems vol 47 pp 85ndash992013

[19] Y-J Wang and J-S Zhang ldquoGlobal optimization by an impr-oved differential evolutionary algorithmrdquo Applied Mathematicsand Computation vol 188 no 1 pp 669ndash680 2007

[20] R Caponetto L Fortuna S Fazzino and M G Xibilia ldquoCha-otic sequences to improve the performance of evolutionary alg-orithmsrdquo IEEE Transactions on Evolutionary Computation vol7 no 3 pp 289ndash304 2003

[21] M Ahmadi and H Mojallali ldquoChaotic invasive weed optimiz-ation algorithm with application to parameter estimation ofchaotic systemsrdquo Chaos Solitons amp Fractals vol 45 no 9-10pp 1108ndash1120 2012

[22] Z S Ma ldquoChaotic populations in genetic algorithmsrdquo AppliedSoft Computing vol 12 pp 2409ndash2424 2012

[23] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[24] W-M Zheng ldquoKneading plane of the circle maprdquo ChaosSolitons and Fractals vol 4 no 7 pp 1221ndash1233 1994

[25] T D Rogers and D C Whitley ldquoChaos in the cubic mappingrdquoMathematical Modelling vol 4 no 1 pp 9ndash25 1983

[26] M Bucolo R Caponetto L Fortuna M Frasca and A RizzoldquoDoes chaos work better than noiserdquo IEEE Circuits and SystemsMagazine vol 2 no 3 pp 4ndash19 2002

[27] D He C He L-G Jiang H-W Zhu and G-R Hu ldquoChaoticcharacteristics of a one-dimensional iterative map with infinitecollapsesrdquo IEEE Transactions on Circuits and Systems I vol 48no 7 pp 900ndash906 2001

[28] R May ldquoSimple mathematical models with very complicateddynamicsrdquo inTheTheory of Chaotic Attractors B Hunt T Y LiJ Kennedy and H Nusse Eds pp 85ndash93 Springer New YorkNY USA 2004

[29] R Barton ldquoChaos and fractalsrdquo The Mathematics Teacher vol83 pp 524ndash529 1990

[30] A Baykasoglu ldquoDesign optimization with chaos embeddedgreat deluge algorithmrdquoApplied Soft Computing Journal vol 12no 3 pp 1055ndash1067 2012

[31] J Fridrich ldquoSymmetric ciphers based on two-dimensional cha-otic mapsrdquo International Journal of Bifurcation and Chaos in

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl

8 Mathematical Problems in Engineering

Table 2 Parameters in the process of algorithms

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6119899iter 250119899pop 100119899var 30 30 30 10 10119901119888

09119901119898

130 130 130 110 110

based on the test problems The iteration will not terminateuntil the number of iterations gets to 250 For the processof NSGA-II algorithm a parent population is selected bytournament selection depending on the nondominated rankand the crowed-comparison operator Then the new popula-tion is generated by crossover and mutation operators Thecrossover operation is executed with the probability of 119901

119888=

09 The probability of mutation 119901119898is equal to the reciprocal

of 119899var which is the dimension number of a chromosome thatis 119901119898= 1119899var

Those parameters are summarized in Table 2 In thetable 119899iter is the number of iterations 119899pop is the scaleof the population 119899var is the number of dimensions of achromosome and119901

119888and119901119898are the probabilities of crossover

and mutation operations

72 Convergence Performance It is known that the 120574 differ-ence is used to evaluate the performance of the chaotic mapsin different phases inmultiobjective evolutionary algorithmsAn example is chosen for further explanation in detail As inFigure 1 the graph shows the results of solving ZDT1 prob-lems with Bakerrsquos map in crossover operator in NSGA-IIThedifferences of 120574 between the experiment ldquocns baker c zdt1rdquoand the experiment ldquons zdt1rdquo in the 250 iterations are givenAs seen from the figure the black line is above the red linewhich represents 0 so the new algorithm ldquocns bakers crdquo isbetter thanNSGA-II algorithm in solvingZDT1 problemwithregard to the convergence metric

The 120574 results of all the experiments are given similar toFigure 1 Since it is difficult to show so many graphs in thispaper the results of three typical problems are chosen thatis ZDT1 which is a simple convex problem ZDT3 whosePareto front is piecewise and ZDT4 which has local optimaThe graphs in Figures 2 3 and 4 provide a comparison of theperformance of solving different MOPs with chaotic maps ininitial population ZDT4 is chosen to show the performanceof chaotic maps in different phases on solving the sameMOPas shown in Figures 4 5 and 6 Each subgraph is labeled withthe name of the chaotic map used

In order to quantify the effect of chaotic maps and phaseswith regard to the metric 120574 the average of 120574 difference in 250generations is calculated to represent the effect of the newalgorithms

Since the order of magnitude of 120574 is not the samethe comparison of these 120574 values is not convenient Thenormalized values are obtained by dividing the 120574 values bythemaximumof the absolute values of the 120574 based on one testproblem The results of normalization are shown in Table 3

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

Bakerrsquos06

05

04

03

02

01

0

minus01

minus02

minus03

minus04

Figure 1 Performance of Bakerrsquos maps in crossover operator insolving ZDT1

Table 3 can be presented in a more intuitive way If 120574 ge03 the numerical value of 120574 is replaced by ldquo++rdquo Similarlyldquo+rdquo represents 01 le 120574 lt 03 ldquo0rdquo represents minus01 le 120574 lt01 ldquominusrdquo represents minus03 le 120574 lt minus01 and ldquominusminusrdquo represents120574 lt minus03 Therefore ldquo++rdquo means that the effect of the newalgorithm with chaotic maps is much better whereas ldquominusminusrdquo ismuch worse Table 4 shows the results

As shown in Table 4 most of the combinations of chaoticmaps and phases have a positive effect on improving the per-formance of NSGA-II algorithm The effect of some chaoticmaps is very good especially in some particular phases Forexample Bakerrsquos map in crossover operator Gauss map incrossover operator and initial population ICMIC map ininitial population sinusoidal map in initial population tentmap in crossover operation and Zaslavskii map in initialpopulation have very good effect

Since ZDT4 problem has 219 or 794times1011 different localPareto-optimal fronts in the search space the solutions easilyget entrapped into local optimum As seen from Table 4chaotic maps used for crossover and mutation operator havesignificant improvement on evolutionary algorithms solvingZDT4 problem especially cat map has the best performancein tenmaps Circle map and cubicmap have less contributionin solving those MOPs The distribution of cat map isrelatively uniform It is probably the reason for the goodperformance in solving problems with local optima

The original NSGA-II algorithm is not good at solvingZDT3 and ZDT6 problems because Pareto-optimal front ofZDT3 is disconnected and solutions of ZDT6 are nonuni-formly spaced However it can be seen in Table 4 that chaoticmaps can improve NSGA-II especially in crossover operationand initial population in solving ZDT3 and ZDT6 problem

In order to eliminate the special effect of the NSGA-II algorithm the polynomial mutation operator in NSGA-II is changed by the Gauss mutation and Cauchy mutationoperators Four typical chaotic maps which include twochaotic maps with best performance and two chaotic maps

Mathematical Problems in Engineering 9

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

GenerationGeneration

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

Figure 2 Performance of chaotic maps in initial population in solving ZDT1

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma 06

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

Figure 3 Performance of chaotic maps in initial population in solving ZDT3

10 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 4 Performance of chaotic maps in initial population in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 5 Performance of chaotic maps in crossover operator in solving ZDT4

Mathematical Problems in Engineering 11

Table 3 The normalized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker 0617 0080 0049 0682 0149 0094 0668 0109 0003 0220 minus0040 0167 0567 0105 minus0090Cat minus0060 0076 0101 0013 0084 0024 minus0007 0090 0174 0191 0109 0158 minus0028 0022 minus0096Circle minus0142 minus0040 minus0016 0013 minus0121 minus0007 minus0153 0028 minus0016 0151 minus0069 0098 minus0176 minus0072 minus0002Cubic minus0544 0064 minus0025 minus0626 minus0041 0006 minus0334 minus0049 0077 0032 minus0781 0071 minus0288 0040 0008Gauss 0307 0513 0089 0306 0507 0070 0454 0585 0037 0159 0005 0191 0114 minus0010 0132ICMIC minus0415 0558 0132 minus0280 0609 0144 minus0295 0510 0004 0003 minus0380 0088 minus0162 0252 0163Logistic 0070 0242 0189 0072 0204 minus0031 0012 0158 0127 0017 minus0819 0183 0152 0129 0132Sinusoidal 0077 1 0616 0121 1 0742 0169 1 0734 minus0091 minus1 minus0138 0148 0688 1Tent 0655 0177 0062 0704 0103 minus0005 0731 0043 minus0088 0190 minus0008 minus0058 0569 0124 minus0003Zaslavskii minus0051 0462 0032 minus0150 0518 0064 minus0086 0499 0110 minus0103 minus0339 0108 minus0060 0174 0183

Table 4 The visualized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker ++ 0 0 ++ + 0 ++ + 0 + 0 + ++ + 0Cat 0 0 + 0 0 0 0 0 + + + + 0 0 0Circle minus 0 0 0 minus 0 minus 0 0 + 0 0 minus 0 0Cubic mdash 0 0 mdash 0 0 mdash 0 0 0 mdash 0 minus 0 0Gauss ++ ++ 0 ++ ++ 0 ++ ++ 0 + 0 + + 0 +ICMIC mdash ++ + minus ++ + minus ++ 0 0 mdash 0 minus + +Logistic 0 + + 0 + 0 0 + + 0 mdash + + + +Sinusoidal 0 ++ ++ + ++ ++ + ++ ++ 0 mdash minus + ++ ++Tent ++ + 0 ++ + 0 ++ 0 0 + 0 0 ++ + 0Zaslavskii 0 ++ 0 minus ++ 0 0 ++ + minus mdash + 0 + +

Table 5 Results of Gauss mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle minus04886 minus00170 19394 minus03130 00971 minus04370 1856683 750688 28693 minus05529Cubic minus24674 minus00589 minus62676 minus00630 minus25351 minus12051 491193 minus448985 minus23071 57609Sinusoidal 10277 60982 04343 94173 minus00396 45810 minus106768 minus525536 584566 285351Tent 35035 06784 59502 11086 18616 00623 minus272428 1014576 150061 112005

Table 6 Results of Cauchy mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle 13978 05012 13997 07723 07187 04043 1354511 minus136938 01105 minus26454Cubic minus26201 minus04932 minus42148 minus05317 minus24831 minus07487 minus771621 minus456939 minus57560 54533Sinusoidal 01470 46977 08327 82820 03179 45376 minus202995 minus468243 65462 238545Tent 27613 03380 52687 09699 26813 minus01916 minus283172 579606 127523 52806

with worst performance are chosen to be used in theexperiments These chaotic maps are circle map cubic mapsinusoidal map and tent map The values of 120574 differences areshown in Tables 5 and 6 As seen from Tables 5 and 6 theperformance of sinusoidal map and tent map is better thanthe performance of circlemap and cubicmap Sinusoidalmapin initial population is better than that in crossover operationand tent map in crossover operation is better than that ininitial population This means the rules of combinations of

chaotic maps and phases in solving MOPs are almost thesame as in the previous observations So the rules based onthe framework of NSGA-II algorithm are applicable to otherMOEAs

In general Bakerrsquos map with a phase for crossover oper-ator sinusoidal map with phases for initial population andmutation operator and tent map with a phase for crossoveroperator could be the best choice for improving evolutionaryalgorithms for MOPs without local optimum For problems

12 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 6 Performance of chaotic maps in mutation operator in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

Figure 7 Performance of chaotic maps in crossover operator in solving ZDT1 with metric Δ

Mathematical Problems in Engineering 13

Table7Th

eaverage

results

ofΔ

ZDT1

ZDT2

ZDT3

ZDT4

ZDT6

ci

mc

im

ci

mc

im

ci

mBa

ker

minus00337

minus00026

00015

00226

00310

00024

minus000

96000

69000

06minus006

0300343

minus00513

00125

00024

minus00015

Cat

000

0500017

000

4200234

minus01021

minus00145

minus00019

minus00052

000

0600277

minus000

90minus00148

minus000

01minus000

02minus000

68Circle

minus00325

minus00035

minus000

64minus02019

minus00632

minus01410

minus00052

minus000

6500010

minus00255

minus00259

minus00136

00031

minus00020

000

64Cu

bic

minus000

4700121

minus00022

minus00439

minus00148

00361

000

00minus000

0600031

minus00052

minus00279

00238

minus00170

00054

minus00017

Gauss

00121

00075

minus00180

minus01075

00329

minus00711

minus00024

minus00032

00019

00330

minus00214

00155

00180

000

03000

64ICMIC

00081

000

92000

02minus00734

minus00850

minus00917

minus00036

000

08minus00054

minus00356

minus00497

00238

00123

000

0100085

Logistic

minus00118

00076

00075

minus01935

00239

00509

minus00084

minus00014

00080

00275

minus00100

minus00232

minus000

6900020

00077

Sinu

soidal

00149

000

69minus00564

minus01750

minus000

67minus02902

minus00014

00029

minus00286

minus00390

minus00164

minus01945

00129

00088

minus004

88Tent

minus00350

000

9300016

00215

minus00173

minus00261

minus00059

minus00168

00071

minus00761

minus00139

00075

00125

00028

minus000

02Za

slavskii

002307

00056

minus00014

minus00292

minus00189

minus00013

000

99000

40minus00037

00201

minus00056

000

9000038

00027

00084

14 Mathematical Problems in Engineering

Table 8 Statistical data for combinations of chaotic maps and phases on different problems

Threshold ZDT1 ZDT2 ZDT3 ZDT4 ZDT60 18 9 15 10 200001 16 9 9 10 180002 13 9 7 10 180003 13 8 6 10 140004 13 8 5 10 120005 12 8 4 10 120006 11 8 4 10 11

0 50 100 150 200 250minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

Generation

Diff

eren

ce o

f delt

a

Bakerrsquos

Figure 8 Performance of Bakerrsquos maps in crossover operator insolving ZDT1 with metric Δ

with local optimum cat map has good performance onimproving evolutionary algorithms

73 Diversity Performance Similar to the convergencemetric120574 the differences of the diversity metricΔ between the resultsof the new algorithm with chaotic maps and the originalNSGA-II algorithm are used to measure the performanceFigure 7 shows the performance of chaotic maps in crossoveroperator in solving ZDT1 with metric Δ Each subgraphshows the effect of one chaotic map To be seen more clearlythe first subgraph in Figure 7 is shown in Figure 8

As seen fromFigures 7 and 8 theΔ difference is not stablein 250 generations The average values of Δ difference in 250generations are calculated to represent the effect of the newalgorithms For brevity the rest of results are not shown ingraphs but in Table 7

As seen from Table 7 the Δ values have little differenceWe count the number of combinations of chaotic maps andphases for solving one problem in different threshold valuesFor example there are 18 combinations whose values of Δ aregreater than zero Based on the number of the combinationsof chaotic maps and phases in different threshold values thevalues of diversity metric Δ are summarized in Table 8

In Table 8 the rank of the number of Δ in differentthreshold values is ZDT1 gt ZDT6 gt ZDT4 gt ZDT2 gt

ZDT3 especially for larger threshold ZDT1 problem whichis a convex function and has no local optima is a relativelyeasy problem Chaotic maps bring the biggest improve-ments on solving ZDT1 Though the solutions of ZDT6 arenonuniformly spaced chaotic maps can find better spread ofsolutions While ZDT4 problem is a complex problem andthe solutions are easily trapped into local optima chaoticmaps can improve the distribution of the solutions ZDT2problem is a convex function and the solutions sometimesfall into the local optimumThe effects of chaoticmaps can begeneralizedThe Pareto front of ZDT3 problem is segmentedso the Δ value of ZDT3 is larger and the ranking of ZDT3 islower It is our observation that Δ is not fit for evaluating thesolutions to problems which are disconnected

Based on the diversity metric Δ chaotic maps havethe best improvement on solving convex problems withoutlocal optima and have better effect on solving problemswhich have nonuniform solutions For problems with localminimum chaotic maps embedded algorithms can improvethe performance with regard to metric Δ

A short summary can be given according to the aboveexperiments First chaotic maps can improve the perfor-mance of MOEAs but the results showed that no one chaoticmap outperforms other maps for all of the problems Theresults in this paper give some guidance on how to choose achaotic map and a phase in MOEAs Second an interestingdiscovery is that cat map has best performance on solvingproblems with local optima Uniformity of cat map may beone of the reasons for the good performance of solving ZDT4

8 Conclusion

The focus of this paper is to explore the relationships ofchaotic maps and phases in MOEAs in solving MOPs Themain framework of algorithms in experiments is the NSGA-II algorithm The combinations of ten chaotic maps andthree phases are chosen in the experiments Two metricsconvergence metric 120574 and diversity metric Δ are used toevaluate the convergence and diversity properties of thealgorithms with chaotic maps The test problems are ZDTseries which were all MOPs The ergodicity and initial valuesensitivity of chaotic maps can help evolutionary algorithmsavoid solutions from falling into local optimal and get betterconvergence In the experimental results almost all chaoticmaps have good effects on improving the performance ofevolutionary algorithms to solveMOPs without local optima

Mathematical Problems in Engineering 15

Cat map has best performance on solving problems withlocal optimum This work gives insight on choosing chaoticmaps and phases for different problems Our future workwill perform further experiments with more chaotic maps onother MOEAs and formulate the theory analysis

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China under Grant no 61101153 and Prof Qiuis supported by NSF CNS 1359557

References

[1] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing vol 13 pp2790ndash2802 2013

[2] J A Adeyemo ldquoReservoir operation using multi-objective evo-lutionary algorithms-a reviewrdquo Asian Journal of Scientific Res-earch vol 4 no 1 pp 16ndash27 2011

[3] H Hu L Xu R Wei and B Zhu ldquoMulti-objective controloptimization for greenhouse environment using evolutionaryalgorithmsrdquo Sensors vol 11 no 6 pp 5792ndash5807 2011

[4] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[5] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[6] E Zitzler M Laumanns L Thiele et al SPEA2 Improving theStrength Pareto Evolutionary Algorithm Eidgenossische Techn-ische Hochschule Zurich (ETH) Institut fur Technische Infor-matik und Kommunikationsnetze (TIK) 2001

[7] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[8] B Alatas E Akin and A B Ozer ldquoChaos embedded particleswarm optimization algorithmsrdquo Chaos Solitons and Fractalsvol 40 no 4 pp 1715ndash1734 2009

[9] G ZilongW Sunrsquoan andZ Jian ldquoA novel immune evolutionaryalgorithm incorporating chaos optimizationrdquo Pattern Recogni-tion Letters vol 27 no 1 pp 2ndash8 2006

[10] M S Tavazoei and M Haeri ldquoComparison of different one-dimensional maps as chaotic search pattern in chaos optimiza-tion algorithmsrdquo Applied Mathematics and Computation vol187 no 2 pp 1076ndash1085 2007

[11] G Yu X Wang and P Li ldquoApplication of chaotic theory indifferential evolution algorithmsrdquo in Proceedings of the 6th Inte-rnational Conference on Natural Computation (ICNC rsquo10) pp3816ndash3820 August 2010

[12] B Alatas and E Akin ldquoMulti-objective rulemining using a cha-otic particle swarm optimization algorithmrdquo Knowledge-BasedSystems vol 22 no 6 pp 455ndash460 2009

[13] L D S Coelho and V C Mariani ldquoChaotic artificial immuneapproach applied to economic dispatch of electric energy usingthermal unitsrdquo Chaos Solitons and Fractals vol 40 no 5 pp2376ndash2383 2009

[14] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[15] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[16] L D S Coelho ldquoA quantum particle swarm optimizer with cha-otic mutation operatorrdquo Chaos Solitons and Fractals vol 37 no5 pp 1409ndash1418 2008

[17] L S Dos Coelho and P Alotto ldquoMultiobjective electromagneticoptimization based on a nondominated sorting genetic appr-oach with a chaotic crossover operatorrdquo IEEE Transactions onMagnetics vol 44 no 6 pp 1078ndash1081 2008

[18] H F Zhang J Z Zhou Y C Zhang N Fang and R ZhangldquoShort termhydrothermal scheduling usingmulti-objective dif-ferential evolution with three chaotic sequencesrdquo InternationalJournal of Electrical Power amp Energy Systems vol 47 pp 85ndash992013

[19] Y-J Wang and J-S Zhang ldquoGlobal optimization by an impr-oved differential evolutionary algorithmrdquo Applied Mathematicsand Computation vol 188 no 1 pp 669ndash680 2007

[20] R Caponetto L Fortuna S Fazzino and M G Xibilia ldquoCha-otic sequences to improve the performance of evolutionary alg-orithmsrdquo IEEE Transactions on Evolutionary Computation vol7 no 3 pp 289ndash304 2003

[21] M Ahmadi and H Mojallali ldquoChaotic invasive weed optimiz-ation algorithm with application to parameter estimation ofchaotic systemsrdquo Chaos Solitons amp Fractals vol 45 no 9-10pp 1108ndash1120 2012

[22] Z S Ma ldquoChaotic populations in genetic algorithmsrdquo AppliedSoft Computing vol 12 pp 2409ndash2424 2012

[23] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[24] W-M Zheng ldquoKneading plane of the circle maprdquo ChaosSolitons and Fractals vol 4 no 7 pp 1221ndash1233 1994

[25] T D Rogers and D C Whitley ldquoChaos in the cubic mappingrdquoMathematical Modelling vol 4 no 1 pp 9ndash25 1983

[26] M Bucolo R Caponetto L Fortuna M Frasca and A RizzoldquoDoes chaos work better than noiserdquo IEEE Circuits and SystemsMagazine vol 2 no 3 pp 4ndash19 2002

[27] D He C He L-G Jiang H-W Zhu and G-R Hu ldquoChaoticcharacteristics of a one-dimensional iterative map with infinitecollapsesrdquo IEEE Transactions on Circuits and Systems I vol 48no 7 pp 900ndash906 2001

[28] R May ldquoSimple mathematical models with very complicateddynamicsrdquo inTheTheory of Chaotic Attractors B Hunt T Y LiJ Kennedy and H Nusse Eds pp 85ndash93 Springer New YorkNY USA 2004

[29] R Barton ldquoChaos and fractalsrdquo The Mathematics Teacher vol83 pp 524ndash529 1990

[30] A Baykasoglu ldquoDesign optimization with chaos embeddedgreat deluge algorithmrdquoApplied Soft Computing Journal vol 12no 3 pp 1055ndash1067 2012

[31] J Fridrich ldquoSymmetric ciphers based on two-dimensional cha-otic mapsrdquo International Journal of Bifurcation and Chaos in

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl

Mathematical Problems in Engineering 9

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

GenerationGeneration

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

06

04

02

0

minus02

minus04

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

06

04

02

0

minus02

minus04

Figure 2 Performance of chaotic maps in initial population in solving ZDT1

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

06

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma 06

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

0 50 100 150 200 250Generation

04

02

0

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f gam

ma

04

02

0

minus02

Figure 3 Performance of chaotic maps in initial population in solving ZDT3

10 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 4 Performance of chaotic maps in initial population in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 5 Performance of chaotic maps in crossover operator in solving ZDT4

Mathematical Problems in Engineering 11

Table 3 The normalized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker 0617 0080 0049 0682 0149 0094 0668 0109 0003 0220 minus0040 0167 0567 0105 minus0090Cat minus0060 0076 0101 0013 0084 0024 minus0007 0090 0174 0191 0109 0158 minus0028 0022 minus0096Circle minus0142 minus0040 minus0016 0013 minus0121 minus0007 minus0153 0028 minus0016 0151 minus0069 0098 minus0176 minus0072 minus0002Cubic minus0544 0064 minus0025 minus0626 minus0041 0006 minus0334 minus0049 0077 0032 minus0781 0071 minus0288 0040 0008Gauss 0307 0513 0089 0306 0507 0070 0454 0585 0037 0159 0005 0191 0114 minus0010 0132ICMIC minus0415 0558 0132 minus0280 0609 0144 minus0295 0510 0004 0003 minus0380 0088 minus0162 0252 0163Logistic 0070 0242 0189 0072 0204 minus0031 0012 0158 0127 0017 minus0819 0183 0152 0129 0132Sinusoidal 0077 1 0616 0121 1 0742 0169 1 0734 minus0091 minus1 minus0138 0148 0688 1Tent 0655 0177 0062 0704 0103 minus0005 0731 0043 minus0088 0190 minus0008 minus0058 0569 0124 minus0003Zaslavskii minus0051 0462 0032 minus0150 0518 0064 minus0086 0499 0110 minus0103 minus0339 0108 minus0060 0174 0183

Table 4 The visualized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker ++ 0 0 ++ + 0 ++ + 0 + 0 + ++ + 0Cat 0 0 + 0 0 0 0 0 + + + + 0 0 0Circle minus 0 0 0 minus 0 minus 0 0 + 0 0 minus 0 0Cubic mdash 0 0 mdash 0 0 mdash 0 0 0 mdash 0 minus 0 0Gauss ++ ++ 0 ++ ++ 0 ++ ++ 0 + 0 + + 0 +ICMIC mdash ++ + minus ++ + minus ++ 0 0 mdash 0 minus + +Logistic 0 + + 0 + 0 0 + + 0 mdash + + + +Sinusoidal 0 ++ ++ + ++ ++ + ++ ++ 0 mdash minus + ++ ++Tent ++ + 0 ++ + 0 ++ 0 0 + 0 0 ++ + 0Zaslavskii 0 ++ 0 minus ++ 0 0 ++ + minus mdash + 0 + +

Table 5 Results of Gauss mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle minus04886 minus00170 19394 minus03130 00971 minus04370 1856683 750688 28693 minus05529Cubic minus24674 minus00589 minus62676 minus00630 minus25351 minus12051 491193 minus448985 minus23071 57609Sinusoidal 10277 60982 04343 94173 minus00396 45810 minus106768 minus525536 584566 285351Tent 35035 06784 59502 11086 18616 00623 minus272428 1014576 150061 112005

Table 6 Results of Cauchy mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle 13978 05012 13997 07723 07187 04043 1354511 minus136938 01105 minus26454Cubic minus26201 minus04932 minus42148 minus05317 minus24831 minus07487 minus771621 minus456939 minus57560 54533Sinusoidal 01470 46977 08327 82820 03179 45376 minus202995 minus468243 65462 238545Tent 27613 03380 52687 09699 26813 minus01916 minus283172 579606 127523 52806

with worst performance are chosen to be used in theexperiments These chaotic maps are circle map cubic mapsinusoidal map and tent map The values of 120574 differences areshown in Tables 5 and 6 As seen from Tables 5 and 6 theperformance of sinusoidal map and tent map is better thanthe performance of circlemap and cubicmap Sinusoidalmapin initial population is better than that in crossover operationand tent map in crossover operation is better than that ininitial population This means the rules of combinations of

chaotic maps and phases in solving MOPs are almost thesame as in the previous observations So the rules based onthe framework of NSGA-II algorithm are applicable to otherMOEAs

In general Bakerrsquos map with a phase for crossover oper-ator sinusoidal map with phases for initial population andmutation operator and tent map with a phase for crossoveroperator could be the best choice for improving evolutionaryalgorithms for MOPs without local optimum For problems

12 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 6 Performance of chaotic maps in mutation operator in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

Figure 7 Performance of chaotic maps in crossover operator in solving ZDT1 with metric Δ

Mathematical Problems in Engineering 13

Table7Th

eaverage

results

ofΔ

ZDT1

ZDT2

ZDT3

ZDT4

ZDT6

ci

mc

im

ci

mc

im

ci

mBa

ker

minus00337

minus00026

00015

00226

00310

00024

minus000

96000

69000

06minus006

0300343

minus00513

00125

00024

minus00015

Cat

000

0500017

000

4200234

minus01021

minus00145

minus00019

minus00052

000

0600277

minus000

90minus00148

minus000

01minus000

02minus000

68Circle

minus00325

minus00035

minus000

64minus02019

minus00632

minus01410

minus00052

minus000

6500010

minus00255

minus00259

minus00136

00031

minus00020

000

64Cu

bic

minus000

4700121

minus00022

minus00439

minus00148

00361

000

00minus000

0600031

minus00052

minus00279

00238

minus00170

00054

minus00017

Gauss

00121

00075

minus00180

minus01075

00329

minus00711

minus00024

minus00032

00019

00330

minus00214

00155

00180

000

03000

64ICMIC

00081

000

92000

02minus00734

minus00850

minus00917

minus00036

000

08minus00054

minus00356

minus00497

00238

00123

000

0100085

Logistic

minus00118

00076

00075

minus01935

00239

00509

minus00084

minus00014

00080

00275

minus00100

minus00232

minus000

6900020

00077

Sinu

soidal

00149

000

69minus00564

minus01750

minus000

67minus02902

minus00014

00029

minus00286

minus00390

minus00164

minus01945

00129

00088

minus004

88Tent

minus00350

000

9300016

00215

minus00173

minus00261

minus00059

minus00168

00071

minus00761

minus00139

00075

00125

00028

minus000

02Za

slavskii

002307

00056

minus00014

minus00292

minus00189

minus00013

000

99000

40minus00037

00201

minus00056

000

9000038

00027

00084

14 Mathematical Problems in Engineering

Table 8 Statistical data for combinations of chaotic maps and phases on different problems

Threshold ZDT1 ZDT2 ZDT3 ZDT4 ZDT60 18 9 15 10 200001 16 9 9 10 180002 13 9 7 10 180003 13 8 6 10 140004 13 8 5 10 120005 12 8 4 10 120006 11 8 4 10 11

0 50 100 150 200 250minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

Generation

Diff

eren

ce o

f delt

a

Bakerrsquos

Figure 8 Performance of Bakerrsquos maps in crossover operator insolving ZDT1 with metric Δ

with local optimum cat map has good performance onimproving evolutionary algorithms

73 Diversity Performance Similar to the convergencemetric120574 the differences of the diversity metricΔ between the resultsof the new algorithm with chaotic maps and the originalNSGA-II algorithm are used to measure the performanceFigure 7 shows the performance of chaotic maps in crossoveroperator in solving ZDT1 with metric Δ Each subgraphshows the effect of one chaotic map To be seen more clearlythe first subgraph in Figure 7 is shown in Figure 8

As seen fromFigures 7 and 8 theΔ difference is not stablein 250 generations The average values of Δ difference in 250generations are calculated to represent the effect of the newalgorithms For brevity the rest of results are not shown ingraphs but in Table 7

As seen from Table 7 the Δ values have little differenceWe count the number of combinations of chaotic maps andphases for solving one problem in different threshold valuesFor example there are 18 combinations whose values of Δ aregreater than zero Based on the number of the combinationsof chaotic maps and phases in different threshold values thevalues of diversity metric Δ are summarized in Table 8

In Table 8 the rank of the number of Δ in differentthreshold values is ZDT1 gt ZDT6 gt ZDT4 gt ZDT2 gt

ZDT3 especially for larger threshold ZDT1 problem whichis a convex function and has no local optima is a relativelyeasy problem Chaotic maps bring the biggest improve-ments on solving ZDT1 Though the solutions of ZDT6 arenonuniformly spaced chaotic maps can find better spread ofsolutions While ZDT4 problem is a complex problem andthe solutions are easily trapped into local optima chaoticmaps can improve the distribution of the solutions ZDT2problem is a convex function and the solutions sometimesfall into the local optimumThe effects of chaoticmaps can begeneralizedThe Pareto front of ZDT3 problem is segmentedso the Δ value of ZDT3 is larger and the ranking of ZDT3 islower It is our observation that Δ is not fit for evaluating thesolutions to problems which are disconnected

Based on the diversity metric Δ chaotic maps havethe best improvement on solving convex problems withoutlocal optima and have better effect on solving problemswhich have nonuniform solutions For problems with localminimum chaotic maps embedded algorithms can improvethe performance with regard to metric Δ

A short summary can be given according to the aboveexperiments First chaotic maps can improve the perfor-mance of MOEAs but the results showed that no one chaoticmap outperforms other maps for all of the problems Theresults in this paper give some guidance on how to choose achaotic map and a phase in MOEAs Second an interestingdiscovery is that cat map has best performance on solvingproblems with local optima Uniformity of cat map may beone of the reasons for the good performance of solving ZDT4

8 Conclusion

The focus of this paper is to explore the relationships ofchaotic maps and phases in MOEAs in solving MOPs Themain framework of algorithms in experiments is the NSGA-II algorithm The combinations of ten chaotic maps andthree phases are chosen in the experiments Two metricsconvergence metric 120574 and diversity metric Δ are used toevaluate the convergence and diversity properties of thealgorithms with chaotic maps The test problems are ZDTseries which were all MOPs The ergodicity and initial valuesensitivity of chaotic maps can help evolutionary algorithmsavoid solutions from falling into local optimal and get betterconvergence In the experimental results almost all chaoticmaps have good effects on improving the performance ofevolutionary algorithms to solveMOPs without local optima

Mathematical Problems in Engineering 15

Cat map has best performance on solving problems withlocal optimum This work gives insight on choosing chaoticmaps and phases for different problems Our future workwill perform further experiments with more chaotic maps onother MOEAs and formulate the theory analysis

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China under Grant no 61101153 and Prof Qiuis supported by NSF CNS 1359557

References

[1] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing vol 13 pp2790ndash2802 2013

[2] J A Adeyemo ldquoReservoir operation using multi-objective evo-lutionary algorithms-a reviewrdquo Asian Journal of Scientific Res-earch vol 4 no 1 pp 16ndash27 2011

[3] H Hu L Xu R Wei and B Zhu ldquoMulti-objective controloptimization for greenhouse environment using evolutionaryalgorithmsrdquo Sensors vol 11 no 6 pp 5792ndash5807 2011

[4] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[5] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[6] E Zitzler M Laumanns L Thiele et al SPEA2 Improving theStrength Pareto Evolutionary Algorithm Eidgenossische Techn-ische Hochschule Zurich (ETH) Institut fur Technische Infor-matik und Kommunikationsnetze (TIK) 2001

[7] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[8] B Alatas E Akin and A B Ozer ldquoChaos embedded particleswarm optimization algorithmsrdquo Chaos Solitons and Fractalsvol 40 no 4 pp 1715ndash1734 2009

[9] G ZilongW Sunrsquoan andZ Jian ldquoA novel immune evolutionaryalgorithm incorporating chaos optimizationrdquo Pattern Recogni-tion Letters vol 27 no 1 pp 2ndash8 2006

[10] M S Tavazoei and M Haeri ldquoComparison of different one-dimensional maps as chaotic search pattern in chaos optimiza-tion algorithmsrdquo Applied Mathematics and Computation vol187 no 2 pp 1076ndash1085 2007

[11] G Yu X Wang and P Li ldquoApplication of chaotic theory indifferential evolution algorithmsrdquo in Proceedings of the 6th Inte-rnational Conference on Natural Computation (ICNC rsquo10) pp3816ndash3820 August 2010

[12] B Alatas and E Akin ldquoMulti-objective rulemining using a cha-otic particle swarm optimization algorithmrdquo Knowledge-BasedSystems vol 22 no 6 pp 455ndash460 2009

[13] L D S Coelho and V C Mariani ldquoChaotic artificial immuneapproach applied to economic dispatch of electric energy usingthermal unitsrdquo Chaos Solitons and Fractals vol 40 no 5 pp2376ndash2383 2009

[14] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[15] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[16] L D S Coelho ldquoA quantum particle swarm optimizer with cha-otic mutation operatorrdquo Chaos Solitons and Fractals vol 37 no5 pp 1409ndash1418 2008

[17] L S Dos Coelho and P Alotto ldquoMultiobjective electromagneticoptimization based on a nondominated sorting genetic appr-oach with a chaotic crossover operatorrdquo IEEE Transactions onMagnetics vol 44 no 6 pp 1078ndash1081 2008

[18] H F Zhang J Z Zhou Y C Zhang N Fang and R ZhangldquoShort termhydrothermal scheduling usingmulti-objective dif-ferential evolution with three chaotic sequencesrdquo InternationalJournal of Electrical Power amp Energy Systems vol 47 pp 85ndash992013

[19] Y-J Wang and J-S Zhang ldquoGlobal optimization by an impr-oved differential evolutionary algorithmrdquo Applied Mathematicsand Computation vol 188 no 1 pp 669ndash680 2007

[20] R Caponetto L Fortuna S Fazzino and M G Xibilia ldquoCha-otic sequences to improve the performance of evolutionary alg-orithmsrdquo IEEE Transactions on Evolutionary Computation vol7 no 3 pp 289ndash304 2003

[21] M Ahmadi and H Mojallali ldquoChaotic invasive weed optimiz-ation algorithm with application to parameter estimation ofchaotic systemsrdquo Chaos Solitons amp Fractals vol 45 no 9-10pp 1108ndash1120 2012

[22] Z S Ma ldquoChaotic populations in genetic algorithmsrdquo AppliedSoft Computing vol 12 pp 2409ndash2424 2012

[23] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[24] W-M Zheng ldquoKneading plane of the circle maprdquo ChaosSolitons and Fractals vol 4 no 7 pp 1221ndash1233 1994

[25] T D Rogers and D C Whitley ldquoChaos in the cubic mappingrdquoMathematical Modelling vol 4 no 1 pp 9ndash25 1983

[26] M Bucolo R Caponetto L Fortuna M Frasca and A RizzoldquoDoes chaos work better than noiserdquo IEEE Circuits and SystemsMagazine vol 2 no 3 pp 4ndash19 2002

[27] D He C He L-G Jiang H-W Zhu and G-R Hu ldquoChaoticcharacteristics of a one-dimensional iterative map with infinitecollapsesrdquo IEEE Transactions on Circuits and Systems I vol 48no 7 pp 900ndash906 2001

[28] R May ldquoSimple mathematical models with very complicateddynamicsrdquo inTheTheory of Chaotic Attractors B Hunt T Y LiJ Kennedy and H Nusse Eds pp 85ndash93 Springer New YorkNY USA 2004

[29] R Barton ldquoChaos and fractalsrdquo The Mathematics Teacher vol83 pp 524ndash529 1990

[30] A Baykasoglu ldquoDesign optimization with chaos embeddedgreat deluge algorithmrdquoApplied Soft Computing Journal vol 12no 3 pp 1055ndash1067 2012

[31] J Fridrich ldquoSymmetric ciphers based on two-dimensional cha-otic mapsrdquo International Journal of Bifurcation and Chaos in

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Volume 2014

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Stochastic AnalysisInternational Journal of

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl

10 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 4 Performance of chaotic maps in initial population in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 5 Performance of chaotic maps in crossover operator in solving ZDT4

Mathematical Problems in Engineering 11

Table 3 The normalized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker 0617 0080 0049 0682 0149 0094 0668 0109 0003 0220 minus0040 0167 0567 0105 minus0090Cat minus0060 0076 0101 0013 0084 0024 minus0007 0090 0174 0191 0109 0158 minus0028 0022 minus0096Circle minus0142 minus0040 minus0016 0013 minus0121 minus0007 minus0153 0028 minus0016 0151 minus0069 0098 minus0176 minus0072 minus0002Cubic minus0544 0064 minus0025 minus0626 minus0041 0006 minus0334 minus0049 0077 0032 minus0781 0071 minus0288 0040 0008Gauss 0307 0513 0089 0306 0507 0070 0454 0585 0037 0159 0005 0191 0114 minus0010 0132ICMIC minus0415 0558 0132 minus0280 0609 0144 minus0295 0510 0004 0003 minus0380 0088 minus0162 0252 0163Logistic 0070 0242 0189 0072 0204 minus0031 0012 0158 0127 0017 minus0819 0183 0152 0129 0132Sinusoidal 0077 1 0616 0121 1 0742 0169 1 0734 minus0091 minus1 minus0138 0148 0688 1Tent 0655 0177 0062 0704 0103 minus0005 0731 0043 minus0088 0190 minus0008 minus0058 0569 0124 minus0003Zaslavskii minus0051 0462 0032 minus0150 0518 0064 minus0086 0499 0110 minus0103 minus0339 0108 minus0060 0174 0183

Table 4 The visualized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker ++ 0 0 ++ + 0 ++ + 0 + 0 + ++ + 0Cat 0 0 + 0 0 0 0 0 + + + + 0 0 0Circle minus 0 0 0 minus 0 minus 0 0 + 0 0 minus 0 0Cubic mdash 0 0 mdash 0 0 mdash 0 0 0 mdash 0 minus 0 0Gauss ++ ++ 0 ++ ++ 0 ++ ++ 0 + 0 + + 0 +ICMIC mdash ++ + minus ++ + minus ++ 0 0 mdash 0 minus + +Logistic 0 + + 0 + 0 0 + + 0 mdash + + + +Sinusoidal 0 ++ ++ + ++ ++ + ++ ++ 0 mdash minus + ++ ++Tent ++ + 0 ++ + 0 ++ 0 0 + 0 0 ++ + 0Zaslavskii 0 ++ 0 minus ++ 0 0 ++ + minus mdash + 0 + +

Table 5 Results of Gauss mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle minus04886 minus00170 19394 minus03130 00971 minus04370 1856683 750688 28693 minus05529Cubic minus24674 minus00589 minus62676 minus00630 minus25351 minus12051 491193 minus448985 minus23071 57609Sinusoidal 10277 60982 04343 94173 minus00396 45810 minus106768 minus525536 584566 285351Tent 35035 06784 59502 11086 18616 00623 minus272428 1014576 150061 112005

Table 6 Results of Cauchy mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle 13978 05012 13997 07723 07187 04043 1354511 minus136938 01105 minus26454Cubic minus26201 minus04932 minus42148 minus05317 minus24831 minus07487 minus771621 minus456939 minus57560 54533Sinusoidal 01470 46977 08327 82820 03179 45376 minus202995 minus468243 65462 238545Tent 27613 03380 52687 09699 26813 minus01916 minus283172 579606 127523 52806

with worst performance are chosen to be used in theexperiments These chaotic maps are circle map cubic mapsinusoidal map and tent map The values of 120574 differences areshown in Tables 5 and 6 As seen from Tables 5 and 6 theperformance of sinusoidal map and tent map is better thanthe performance of circlemap and cubicmap Sinusoidalmapin initial population is better than that in crossover operationand tent map in crossover operation is better than that ininitial population This means the rules of combinations of

chaotic maps and phases in solving MOPs are almost thesame as in the previous observations So the rules based onthe framework of NSGA-II algorithm are applicable to otherMOEAs

In general Bakerrsquos map with a phase for crossover oper-ator sinusoidal map with phases for initial population andmutation operator and tent map with a phase for crossoveroperator could be the best choice for improving evolutionaryalgorithms for MOPs without local optimum For problems

12 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 6 Performance of chaotic maps in mutation operator in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

Figure 7 Performance of chaotic maps in crossover operator in solving ZDT1 with metric Δ

Mathematical Problems in Engineering 13

Table7Th

eaverage

results

ofΔ

ZDT1

ZDT2

ZDT3

ZDT4

ZDT6

ci

mc

im

ci

mc

im

ci

mBa

ker

minus00337

minus00026

00015

00226

00310

00024

minus000

96000

69000

06minus006

0300343

minus00513

00125

00024

minus00015

Cat

000

0500017

000

4200234

minus01021

minus00145

minus00019

minus00052

000

0600277

minus000

90minus00148

minus000

01minus000

02minus000

68Circle

minus00325

minus00035

minus000

64minus02019

minus00632

minus01410

minus00052

minus000

6500010

minus00255

minus00259

minus00136

00031

minus00020

000

64Cu

bic

minus000

4700121

minus00022

minus00439

minus00148

00361

000

00minus000

0600031

minus00052

minus00279

00238

minus00170

00054

minus00017

Gauss

00121

00075

minus00180

minus01075

00329

minus00711

minus00024

minus00032

00019

00330

minus00214

00155

00180

000

03000

64ICMIC

00081

000

92000

02minus00734

minus00850

minus00917

minus00036

000

08minus00054

minus00356

minus00497

00238

00123

000

0100085

Logistic

minus00118

00076

00075

minus01935

00239

00509

minus00084

minus00014

00080

00275

minus00100

minus00232

minus000

6900020

00077

Sinu

soidal

00149

000

69minus00564

minus01750

minus000

67minus02902

minus00014

00029

minus00286

minus00390

minus00164

minus01945

00129

00088

minus004

88Tent

minus00350

000

9300016

00215

minus00173

minus00261

minus00059

minus00168

00071

minus00761

minus00139

00075

00125

00028

minus000

02Za

slavskii

002307

00056

minus00014

minus00292

minus00189

minus00013

000

99000

40minus00037

00201

minus00056

000

9000038

00027

00084

14 Mathematical Problems in Engineering

Table 8 Statistical data for combinations of chaotic maps and phases on different problems

Threshold ZDT1 ZDT2 ZDT3 ZDT4 ZDT60 18 9 15 10 200001 16 9 9 10 180002 13 9 7 10 180003 13 8 6 10 140004 13 8 5 10 120005 12 8 4 10 120006 11 8 4 10 11

0 50 100 150 200 250minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

Generation

Diff

eren

ce o

f delt

a

Bakerrsquos

Figure 8 Performance of Bakerrsquos maps in crossover operator insolving ZDT1 with metric Δ

with local optimum cat map has good performance onimproving evolutionary algorithms

73 Diversity Performance Similar to the convergencemetric120574 the differences of the diversity metricΔ between the resultsof the new algorithm with chaotic maps and the originalNSGA-II algorithm are used to measure the performanceFigure 7 shows the performance of chaotic maps in crossoveroperator in solving ZDT1 with metric Δ Each subgraphshows the effect of one chaotic map To be seen more clearlythe first subgraph in Figure 7 is shown in Figure 8

As seen fromFigures 7 and 8 theΔ difference is not stablein 250 generations The average values of Δ difference in 250generations are calculated to represent the effect of the newalgorithms For brevity the rest of results are not shown ingraphs but in Table 7

As seen from Table 7 the Δ values have little differenceWe count the number of combinations of chaotic maps andphases for solving one problem in different threshold valuesFor example there are 18 combinations whose values of Δ aregreater than zero Based on the number of the combinationsof chaotic maps and phases in different threshold values thevalues of diversity metric Δ are summarized in Table 8

In Table 8 the rank of the number of Δ in differentthreshold values is ZDT1 gt ZDT6 gt ZDT4 gt ZDT2 gt

ZDT3 especially for larger threshold ZDT1 problem whichis a convex function and has no local optima is a relativelyeasy problem Chaotic maps bring the biggest improve-ments on solving ZDT1 Though the solutions of ZDT6 arenonuniformly spaced chaotic maps can find better spread ofsolutions While ZDT4 problem is a complex problem andthe solutions are easily trapped into local optima chaoticmaps can improve the distribution of the solutions ZDT2problem is a convex function and the solutions sometimesfall into the local optimumThe effects of chaoticmaps can begeneralizedThe Pareto front of ZDT3 problem is segmentedso the Δ value of ZDT3 is larger and the ranking of ZDT3 islower It is our observation that Δ is not fit for evaluating thesolutions to problems which are disconnected

Based on the diversity metric Δ chaotic maps havethe best improvement on solving convex problems withoutlocal optima and have better effect on solving problemswhich have nonuniform solutions For problems with localminimum chaotic maps embedded algorithms can improvethe performance with regard to metric Δ

A short summary can be given according to the aboveexperiments First chaotic maps can improve the perfor-mance of MOEAs but the results showed that no one chaoticmap outperforms other maps for all of the problems Theresults in this paper give some guidance on how to choose achaotic map and a phase in MOEAs Second an interestingdiscovery is that cat map has best performance on solvingproblems with local optima Uniformity of cat map may beone of the reasons for the good performance of solving ZDT4

8 Conclusion

The focus of this paper is to explore the relationships ofchaotic maps and phases in MOEAs in solving MOPs Themain framework of algorithms in experiments is the NSGA-II algorithm The combinations of ten chaotic maps andthree phases are chosen in the experiments Two metricsconvergence metric 120574 and diversity metric Δ are used toevaluate the convergence and diversity properties of thealgorithms with chaotic maps The test problems are ZDTseries which were all MOPs The ergodicity and initial valuesensitivity of chaotic maps can help evolutionary algorithmsavoid solutions from falling into local optimal and get betterconvergence In the experimental results almost all chaoticmaps have good effects on improving the performance ofevolutionary algorithms to solveMOPs without local optima

Mathematical Problems in Engineering 15

Cat map has best performance on solving problems withlocal optimum This work gives insight on choosing chaoticmaps and phases for different problems Our future workwill perform further experiments with more chaotic maps onother MOEAs and formulate the theory analysis

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China under Grant no 61101153 and Prof Qiuis supported by NSF CNS 1359557

References

[1] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing vol 13 pp2790ndash2802 2013

[2] J A Adeyemo ldquoReservoir operation using multi-objective evo-lutionary algorithms-a reviewrdquo Asian Journal of Scientific Res-earch vol 4 no 1 pp 16ndash27 2011

[3] H Hu L Xu R Wei and B Zhu ldquoMulti-objective controloptimization for greenhouse environment using evolutionaryalgorithmsrdquo Sensors vol 11 no 6 pp 5792ndash5807 2011

[4] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[5] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[6] E Zitzler M Laumanns L Thiele et al SPEA2 Improving theStrength Pareto Evolutionary Algorithm Eidgenossische Techn-ische Hochschule Zurich (ETH) Institut fur Technische Infor-matik und Kommunikationsnetze (TIK) 2001

[7] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[8] B Alatas E Akin and A B Ozer ldquoChaos embedded particleswarm optimization algorithmsrdquo Chaos Solitons and Fractalsvol 40 no 4 pp 1715ndash1734 2009

[9] G ZilongW Sunrsquoan andZ Jian ldquoA novel immune evolutionaryalgorithm incorporating chaos optimizationrdquo Pattern Recogni-tion Letters vol 27 no 1 pp 2ndash8 2006

[10] M S Tavazoei and M Haeri ldquoComparison of different one-dimensional maps as chaotic search pattern in chaos optimiza-tion algorithmsrdquo Applied Mathematics and Computation vol187 no 2 pp 1076ndash1085 2007

[11] G Yu X Wang and P Li ldquoApplication of chaotic theory indifferential evolution algorithmsrdquo in Proceedings of the 6th Inte-rnational Conference on Natural Computation (ICNC rsquo10) pp3816ndash3820 August 2010

[12] B Alatas and E Akin ldquoMulti-objective rulemining using a cha-otic particle swarm optimization algorithmrdquo Knowledge-BasedSystems vol 22 no 6 pp 455ndash460 2009

[13] L D S Coelho and V C Mariani ldquoChaotic artificial immuneapproach applied to economic dispatch of electric energy usingthermal unitsrdquo Chaos Solitons and Fractals vol 40 no 5 pp2376ndash2383 2009

[14] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[15] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[16] L D S Coelho ldquoA quantum particle swarm optimizer with cha-otic mutation operatorrdquo Chaos Solitons and Fractals vol 37 no5 pp 1409ndash1418 2008

[17] L S Dos Coelho and P Alotto ldquoMultiobjective electromagneticoptimization based on a nondominated sorting genetic appr-oach with a chaotic crossover operatorrdquo IEEE Transactions onMagnetics vol 44 no 6 pp 1078ndash1081 2008

[18] H F Zhang J Z Zhou Y C Zhang N Fang and R ZhangldquoShort termhydrothermal scheduling usingmulti-objective dif-ferential evolution with three chaotic sequencesrdquo InternationalJournal of Electrical Power amp Energy Systems vol 47 pp 85ndash992013

[19] Y-J Wang and J-S Zhang ldquoGlobal optimization by an impr-oved differential evolutionary algorithmrdquo Applied Mathematicsand Computation vol 188 no 1 pp 669ndash680 2007

[20] R Caponetto L Fortuna S Fazzino and M G Xibilia ldquoCha-otic sequences to improve the performance of evolutionary alg-orithmsrdquo IEEE Transactions on Evolutionary Computation vol7 no 3 pp 289ndash304 2003

[21] M Ahmadi and H Mojallali ldquoChaotic invasive weed optimiz-ation algorithm with application to parameter estimation ofchaotic systemsrdquo Chaos Solitons amp Fractals vol 45 no 9-10pp 1108ndash1120 2012

[22] Z S Ma ldquoChaotic populations in genetic algorithmsrdquo AppliedSoft Computing vol 12 pp 2409ndash2424 2012

[23] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[24] W-M Zheng ldquoKneading plane of the circle maprdquo ChaosSolitons and Fractals vol 4 no 7 pp 1221ndash1233 1994

[25] T D Rogers and D C Whitley ldquoChaos in the cubic mappingrdquoMathematical Modelling vol 4 no 1 pp 9ndash25 1983

[26] M Bucolo R Caponetto L Fortuna M Frasca and A RizzoldquoDoes chaos work better than noiserdquo IEEE Circuits and SystemsMagazine vol 2 no 3 pp 4ndash19 2002

[27] D He C He L-G Jiang H-W Zhu and G-R Hu ldquoChaoticcharacteristics of a one-dimensional iterative map with infinitecollapsesrdquo IEEE Transactions on Circuits and Systems I vol 48no 7 pp 900ndash906 2001

[28] R May ldquoSimple mathematical models with very complicateddynamicsrdquo inTheTheory of Chaotic Attractors B Hunt T Y LiJ Kennedy and H Nusse Eds pp 85ndash93 Springer New YorkNY USA 2004

[29] R Barton ldquoChaos and fractalsrdquo The Mathematics Teacher vol83 pp 524ndash529 1990

[30] A Baykasoglu ldquoDesign optimization with chaos embeddedgreat deluge algorithmrdquoApplied Soft Computing Journal vol 12no 3 pp 1055ndash1067 2012

[31] J Fridrich ldquoSymmetric ciphers based on two-dimensional cha-otic mapsrdquo International Journal of Bifurcation and Chaos in

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

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

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl

Mathematical Problems in Engineering 11

Table 3 The normalized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker 0617 0080 0049 0682 0149 0094 0668 0109 0003 0220 minus0040 0167 0567 0105 minus0090Cat minus0060 0076 0101 0013 0084 0024 minus0007 0090 0174 0191 0109 0158 minus0028 0022 minus0096Circle minus0142 minus0040 minus0016 0013 minus0121 minus0007 minus0153 0028 minus0016 0151 minus0069 0098 minus0176 minus0072 minus0002Cubic minus0544 0064 minus0025 minus0626 minus0041 0006 minus0334 minus0049 0077 0032 minus0781 0071 minus0288 0040 0008Gauss 0307 0513 0089 0306 0507 0070 0454 0585 0037 0159 0005 0191 0114 minus0010 0132ICMIC minus0415 0558 0132 minus0280 0609 0144 minus0295 0510 0004 0003 minus0380 0088 minus0162 0252 0163Logistic 0070 0242 0189 0072 0204 minus0031 0012 0158 0127 0017 minus0819 0183 0152 0129 0132Sinusoidal 0077 1 0616 0121 1 0742 0169 1 0734 minus0091 minus1 minus0138 0148 0688 1Tent 0655 0177 0062 0704 0103 minus0005 0731 0043 minus0088 0190 minus0008 minus0058 0569 0124 minus0003Zaslavskii minus0051 0462 0032 minus0150 0518 0064 minus0086 0499 0110 minus0103 minus0339 0108 minus0060 0174 0183

Table 4 The visualized results of 120574

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i m c i m c i m c i m c i m

Baker ++ 0 0 ++ + 0 ++ + 0 + 0 + ++ + 0Cat 0 0 + 0 0 0 0 0 + + + + 0 0 0Circle minus 0 0 0 minus 0 minus 0 0 + 0 0 minus 0 0Cubic mdash 0 0 mdash 0 0 mdash 0 0 0 mdash 0 minus 0 0Gauss ++ ++ 0 ++ ++ 0 ++ ++ 0 + 0 + + 0 +ICMIC mdash ++ + minus ++ + minus ++ 0 0 mdash 0 minus + +Logistic 0 + + 0 + 0 0 + + 0 mdash + + + +Sinusoidal 0 ++ ++ + ++ ++ + ++ ++ 0 mdash minus + ++ ++Tent ++ + 0 ++ + 0 ++ 0 0 + 0 0 ++ + 0Zaslavskii 0 ++ 0 minus ++ 0 0 ++ + minus mdash + 0 + +

Table 5 Results of Gauss mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle minus04886 minus00170 19394 minus03130 00971 minus04370 1856683 750688 28693 minus05529Cubic minus24674 minus00589 minus62676 minus00630 minus25351 minus12051 491193 minus448985 minus23071 57609Sinusoidal 10277 60982 04343 94173 minus00396 45810 minus106768 minus525536 584566 285351Tent 35035 06784 59502 11086 18616 00623 minus272428 1014576 150061 112005

Table 6 Results of Cauchy mutation

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6c i c i c i c i c i

Circle 13978 05012 13997 07723 07187 04043 1354511 minus136938 01105 minus26454Cubic minus26201 minus04932 minus42148 minus05317 minus24831 minus07487 minus771621 minus456939 minus57560 54533Sinusoidal 01470 46977 08327 82820 03179 45376 minus202995 minus468243 65462 238545Tent 27613 03380 52687 09699 26813 minus01916 minus283172 579606 127523 52806

with worst performance are chosen to be used in theexperiments These chaotic maps are circle map cubic mapsinusoidal map and tent map The values of 120574 differences areshown in Tables 5 and 6 As seen from Tables 5 and 6 theperformance of sinusoidal map and tent map is better thanthe performance of circlemap and cubicmap Sinusoidalmapin initial population is better than that in crossover operationand tent map in crossover operation is better than that ininitial population This means the rules of combinations of

chaotic maps and phases in solving MOPs are almost thesame as in the previous observations So the rules based onthe framework of NSGA-II algorithm are applicable to otherMOEAs

In general Bakerrsquos map with a phase for crossover oper-ator sinusoidal map with phases for initial population andmutation operator and tent map with a phase for crossoveroperator could be the best choice for improving evolutionaryalgorithms for MOPs without local optimum For problems

12 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 6 Performance of chaotic maps in mutation operator in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

Figure 7 Performance of chaotic maps in crossover operator in solving ZDT1 with metric Δ

Mathematical Problems in Engineering 13

Table7Th

eaverage

results

ofΔ

ZDT1

ZDT2

ZDT3

ZDT4

ZDT6

ci

mc

im

ci

mc

im

ci

mBa

ker

minus00337

minus00026

00015

00226

00310

00024

minus000

96000

69000

06minus006

0300343

minus00513

00125

00024

minus00015

Cat

000

0500017

000

4200234

minus01021

minus00145

minus00019

minus00052

000

0600277

minus000

90minus00148

minus000

01minus000

02minus000

68Circle

minus00325

minus00035

minus000

64minus02019

minus00632

minus01410

minus00052

minus000

6500010

minus00255

minus00259

minus00136

00031

minus00020

000

64Cu

bic

minus000

4700121

minus00022

minus00439

minus00148

00361

000

00minus000

0600031

minus00052

minus00279

00238

minus00170

00054

minus00017

Gauss

00121

00075

minus00180

minus01075

00329

minus00711

minus00024

minus00032

00019

00330

minus00214

00155

00180

000

03000

64ICMIC

00081

000

92000

02minus00734

minus00850

minus00917

minus00036

000

08minus00054

minus00356

minus00497

00238

00123

000

0100085

Logistic

minus00118

00076

00075

minus01935

00239

00509

minus00084

minus00014

00080

00275

minus00100

minus00232

minus000

6900020

00077

Sinu

soidal

00149

000

69minus00564

minus01750

minus000

67minus02902

minus00014

00029

minus00286

minus00390

minus00164

minus01945

00129

00088

minus004

88Tent

minus00350

000

9300016

00215

minus00173

minus00261

minus00059

minus00168

00071

minus00761

minus00139

00075

00125

00028

minus000

02Za

slavskii

002307

00056

minus00014

minus00292

minus00189

minus00013

000

99000

40minus00037

00201

minus00056

000

9000038

00027

00084

14 Mathematical Problems in Engineering

Table 8 Statistical data for combinations of chaotic maps and phases on different problems

Threshold ZDT1 ZDT2 ZDT3 ZDT4 ZDT60 18 9 15 10 200001 16 9 9 10 180002 13 9 7 10 180003 13 8 6 10 140004 13 8 5 10 120005 12 8 4 10 120006 11 8 4 10 11

0 50 100 150 200 250minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

Generation

Diff

eren

ce o

f delt

a

Bakerrsquos

Figure 8 Performance of Bakerrsquos maps in crossover operator insolving ZDT1 with metric Δ

with local optimum cat map has good performance onimproving evolutionary algorithms

73 Diversity Performance Similar to the convergencemetric120574 the differences of the diversity metricΔ between the resultsof the new algorithm with chaotic maps and the originalNSGA-II algorithm are used to measure the performanceFigure 7 shows the performance of chaotic maps in crossoveroperator in solving ZDT1 with metric Δ Each subgraphshows the effect of one chaotic map To be seen more clearlythe first subgraph in Figure 7 is shown in Figure 8

As seen fromFigures 7 and 8 theΔ difference is not stablein 250 generations The average values of Δ difference in 250generations are calculated to represent the effect of the newalgorithms For brevity the rest of results are not shown ingraphs but in Table 7

As seen from Table 7 the Δ values have little differenceWe count the number of combinations of chaotic maps andphases for solving one problem in different threshold valuesFor example there are 18 combinations whose values of Δ aregreater than zero Based on the number of the combinationsof chaotic maps and phases in different threshold values thevalues of diversity metric Δ are summarized in Table 8

In Table 8 the rank of the number of Δ in differentthreshold values is ZDT1 gt ZDT6 gt ZDT4 gt ZDT2 gt

ZDT3 especially for larger threshold ZDT1 problem whichis a convex function and has no local optima is a relativelyeasy problem Chaotic maps bring the biggest improve-ments on solving ZDT1 Though the solutions of ZDT6 arenonuniformly spaced chaotic maps can find better spread ofsolutions While ZDT4 problem is a complex problem andthe solutions are easily trapped into local optima chaoticmaps can improve the distribution of the solutions ZDT2problem is a convex function and the solutions sometimesfall into the local optimumThe effects of chaoticmaps can begeneralizedThe Pareto front of ZDT3 problem is segmentedso the Δ value of ZDT3 is larger and the ranking of ZDT3 islower It is our observation that Δ is not fit for evaluating thesolutions to problems which are disconnected

Based on the diversity metric Δ chaotic maps havethe best improvement on solving convex problems withoutlocal optima and have better effect on solving problemswhich have nonuniform solutions For problems with localminimum chaotic maps embedded algorithms can improvethe performance with regard to metric Δ

A short summary can be given according to the aboveexperiments First chaotic maps can improve the perfor-mance of MOEAs but the results showed that no one chaoticmap outperforms other maps for all of the problems Theresults in this paper give some guidance on how to choose achaotic map and a phase in MOEAs Second an interestingdiscovery is that cat map has best performance on solvingproblems with local optima Uniformity of cat map may beone of the reasons for the good performance of solving ZDT4

8 Conclusion

The focus of this paper is to explore the relationships ofchaotic maps and phases in MOEAs in solving MOPs Themain framework of algorithms in experiments is the NSGA-II algorithm The combinations of ten chaotic maps andthree phases are chosen in the experiments Two metricsconvergence metric 120574 and diversity metric Δ are used toevaluate the convergence and diversity properties of thealgorithms with chaotic maps The test problems are ZDTseries which were all MOPs The ergodicity and initial valuesensitivity of chaotic maps can help evolutionary algorithmsavoid solutions from falling into local optimal and get betterconvergence In the experimental results almost all chaoticmaps have good effects on improving the performance ofevolutionary algorithms to solveMOPs without local optima

Mathematical Problems in Engineering 15

Cat map has best performance on solving problems withlocal optimum This work gives insight on choosing chaoticmaps and phases for different problems Our future workwill perform further experiments with more chaotic maps onother MOEAs and formulate the theory analysis

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China under Grant no 61101153 and Prof Qiuis supported by NSF CNS 1359557

References

[1] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing vol 13 pp2790ndash2802 2013

[2] J A Adeyemo ldquoReservoir operation using multi-objective evo-lutionary algorithms-a reviewrdquo Asian Journal of Scientific Res-earch vol 4 no 1 pp 16ndash27 2011

[3] H Hu L Xu R Wei and B Zhu ldquoMulti-objective controloptimization for greenhouse environment using evolutionaryalgorithmsrdquo Sensors vol 11 no 6 pp 5792ndash5807 2011

[4] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[5] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[6] E Zitzler M Laumanns L Thiele et al SPEA2 Improving theStrength Pareto Evolutionary Algorithm Eidgenossische Techn-ische Hochschule Zurich (ETH) Institut fur Technische Infor-matik und Kommunikationsnetze (TIK) 2001

[7] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[8] B Alatas E Akin and A B Ozer ldquoChaos embedded particleswarm optimization algorithmsrdquo Chaos Solitons and Fractalsvol 40 no 4 pp 1715ndash1734 2009

[9] G ZilongW Sunrsquoan andZ Jian ldquoA novel immune evolutionaryalgorithm incorporating chaos optimizationrdquo Pattern Recogni-tion Letters vol 27 no 1 pp 2ndash8 2006

[10] M S Tavazoei and M Haeri ldquoComparison of different one-dimensional maps as chaotic search pattern in chaos optimiza-tion algorithmsrdquo Applied Mathematics and Computation vol187 no 2 pp 1076ndash1085 2007

[11] G Yu X Wang and P Li ldquoApplication of chaotic theory indifferential evolution algorithmsrdquo in Proceedings of the 6th Inte-rnational Conference on Natural Computation (ICNC rsquo10) pp3816ndash3820 August 2010

[12] B Alatas and E Akin ldquoMulti-objective rulemining using a cha-otic particle swarm optimization algorithmrdquo Knowledge-BasedSystems vol 22 no 6 pp 455ndash460 2009

[13] L D S Coelho and V C Mariani ldquoChaotic artificial immuneapproach applied to economic dispatch of electric energy usingthermal unitsrdquo Chaos Solitons and Fractals vol 40 no 5 pp2376ndash2383 2009

[14] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[15] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[16] L D S Coelho ldquoA quantum particle swarm optimizer with cha-otic mutation operatorrdquo Chaos Solitons and Fractals vol 37 no5 pp 1409ndash1418 2008

[17] L S Dos Coelho and P Alotto ldquoMultiobjective electromagneticoptimization based on a nondominated sorting genetic appr-oach with a chaotic crossover operatorrdquo IEEE Transactions onMagnetics vol 44 no 6 pp 1078ndash1081 2008

[18] H F Zhang J Z Zhou Y C Zhang N Fang and R ZhangldquoShort termhydrothermal scheduling usingmulti-objective dif-ferential evolution with three chaotic sequencesrdquo InternationalJournal of Electrical Power amp Energy Systems vol 47 pp 85ndash992013

[19] Y-J Wang and J-S Zhang ldquoGlobal optimization by an impr-oved differential evolutionary algorithmrdquo Applied Mathematicsand Computation vol 188 no 1 pp 669ndash680 2007

[20] R Caponetto L Fortuna S Fazzino and M G Xibilia ldquoCha-otic sequences to improve the performance of evolutionary alg-orithmsrdquo IEEE Transactions on Evolutionary Computation vol7 no 3 pp 289ndash304 2003

[21] M Ahmadi and H Mojallali ldquoChaotic invasive weed optimiz-ation algorithm with application to parameter estimation ofchaotic systemsrdquo Chaos Solitons amp Fractals vol 45 no 9-10pp 1108ndash1120 2012

[22] Z S Ma ldquoChaotic populations in genetic algorithmsrdquo AppliedSoft Computing vol 12 pp 2409ndash2424 2012

[23] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[24] W-M Zheng ldquoKneading plane of the circle maprdquo ChaosSolitons and Fractals vol 4 no 7 pp 1221ndash1233 1994

[25] T D Rogers and D C Whitley ldquoChaos in the cubic mappingrdquoMathematical Modelling vol 4 no 1 pp 9ndash25 1983

[26] M Bucolo R Caponetto L Fortuna M Frasca and A RizzoldquoDoes chaos work better than noiserdquo IEEE Circuits and SystemsMagazine vol 2 no 3 pp 4ndash19 2002

[27] D He C He L-G Jiang H-W Zhu and G-R Hu ldquoChaoticcharacteristics of a one-dimensional iterative map with infinitecollapsesrdquo IEEE Transactions on Circuits and Systems I vol 48no 7 pp 900ndash906 2001

[28] R May ldquoSimple mathematical models with very complicateddynamicsrdquo inTheTheory of Chaotic Attractors B Hunt T Y LiJ Kennedy and H Nusse Eds pp 85ndash93 Springer New YorkNY USA 2004

[29] R Barton ldquoChaos and fractalsrdquo The Mathematics Teacher vol83 pp 524ndash529 1990

[30] A Baykasoglu ldquoDesign optimization with chaos embeddedgreat deluge algorithmrdquoApplied Soft Computing Journal vol 12no 3 pp 1055ndash1067 2012

[31] J Fridrich ldquoSymmetric ciphers based on two-dimensional cha-otic mapsrdquo International Journal of Bifurcation and Chaos in

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

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

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl

12 Mathematical Problems in Engineering

Bakerrsquos Cat Circle

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

0 50 100 150 200 250

minus40

minus20

0

20

Generation0 50 100 150 200 250

minus40

minus20

0

20

Generation

Diff

eren

ce o

f gam

ma

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

Figure 6 Performance of chaotic maps in mutation operator in solving ZDT4

Bakerrsquos Cat Circle

0 50 100 150 200 250Generation

Cubic

Gauss ICMIC Logistic Sinusoidal

Tent Zaslavskii

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

0

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

0 50 100 150 200 250Generation

03

02

01

00

minus01

minus02

0 50 100 150 200 250Generation

Diff

eren

ce o

f delt

a 03

02

01

minus01

minus02

Figure 7 Performance of chaotic maps in crossover operator in solving ZDT1 with metric Δ

Mathematical Problems in Engineering 13

Table7Th

eaverage

results

ofΔ

ZDT1

ZDT2

ZDT3

ZDT4

ZDT6

ci

mc

im

ci

mc

im

ci

mBa

ker

minus00337

minus00026

00015

00226

00310

00024

minus000

96000

69000

06minus006

0300343

minus00513

00125

00024

minus00015

Cat

000

0500017

000

4200234

minus01021

minus00145

minus00019

minus00052

000

0600277

minus000

90minus00148

minus000

01minus000

02minus000

68Circle

minus00325

minus00035

minus000

64minus02019

minus00632

minus01410

minus00052

minus000

6500010

minus00255

minus00259

minus00136

00031

minus00020

000

64Cu

bic

minus000

4700121

minus00022

minus00439

minus00148

00361

000

00minus000

0600031

minus00052

minus00279

00238

minus00170

00054

minus00017

Gauss

00121

00075

minus00180

minus01075

00329

minus00711

minus00024

minus00032

00019

00330

minus00214

00155

00180

000

03000

64ICMIC

00081

000

92000

02minus00734

minus00850

minus00917

minus00036

000

08minus00054

minus00356

minus00497

00238

00123

000

0100085

Logistic

minus00118

00076

00075

minus01935

00239

00509

minus00084

minus00014

00080

00275

minus00100

minus00232

minus000

6900020

00077

Sinu

soidal

00149

000

69minus00564

minus01750

minus000

67minus02902

minus00014

00029

minus00286

minus00390

minus00164

minus01945

00129

00088

minus004

88Tent

minus00350

000

9300016

00215

minus00173

minus00261

minus00059

minus00168

00071

minus00761

minus00139

00075

00125

00028

minus000

02Za

slavskii

002307

00056

minus00014

minus00292

minus00189

minus00013

000

99000

40minus00037

00201

minus00056

000

9000038

00027

00084

14 Mathematical Problems in Engineering

Table 8 Statistical data for combinations of chaotic maps and phases on different problems

Threshold ZDT1 ZDT2 ZDT3 ZDT4 ZDT60 18 9 15 10 200001 16 9 9 10 180002 13 9 7 10 180003 13 8 6 10 140004 13 8 5 10 120005 12 8 4 10 120006 11 8 4 10 11

0 50 100 150 200 250minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

Generation

Diff

eren

ce o

f delt

a

Bakerrsquos

Figure 8 Performance of Bakerrsquos maps in crossover operator insolving ZDT1 with metric Δ

with local optimum cat map has good performance onimproving evolutionary algorithms

73 Diversity Performance Similar to the convergencemetric120574 the differences of the diversity metricΔ between the resultsof the new algorithm with chaotic maps and the originalNSGA-II algorithm are used to measure the performanceFigure 7 shows the performance of chaotic maps in crossoveroperator in solving ZDT1 with metric Δ Each subgraphshows the effect of one chaotic map To be seen more clearlythe first subgraph in Figure 7 is shown in Figure 8

As seen fromFigures 7 and 8 theΔ difference is not stablein 250 generations The average values of Δ difference in 250generations are calculated to represent the effect of the newalgorithms For brevity the rest of results are not shown ingraphs but in Table 7

As seen from Table 7 the Δ values have little differenceWe count the number of combinations of chaotic maps andphases for solving one problem in different threshold valuesFor example there are 18 combinations whose values of Δ aregreater than zero Based on the number of the combinationsof chaotic maps and phases in different threshold values thevalues of diversity metric Δ are summarized in Table 8

In Table 8 the rank of the number of Δ in differentthreshold values is ZDT1 gt ZDT6 gt ZDT4 gt ZDT2 gt

ZDT3 especially for larger threshold ZDT1 problem whichis a convex function and has no local optima is a relativelyeasy problem Chaotic maps bring the biggest improve-ments on solving ZDT1 Though the solutions of ZDT6 arenonuniformly spaced chaotic maps can find better spread ofsolutions While ZDT4 problem is a complex problem andthe solutions are easily trapped into local optima chaoticmaps can improve the distribution of the solutions ZDT2problem is a convex function and the solutions sometimesfall into the local optimumThe effects of chaoticmaps can begeneralizedThe Pareto front of ZDT3 problem is segmentedso the Δ value of ZDT3 is larger and the ranking of ZDT3 islower It is our observation that Δ is not fit for evaluating thesolutions to problems which are disconnected

Based on the diversity metric Δ chaotic maps havethe best improvement on solving convex problems withoutlocal optima and have better effect on solving problemswhich have nonuniform solutions For problems with localminimum chaotic maps embedded algorithms can improvethe performance with regard to metric Δ

A short summary can be given according to the aboveexperiments First chaotic maps can improve the perfor-mance of MOEAs but the results showed that no one chaoticmap outperforms other maps for all of the problems Theresults in this paper give some guidance on how to choose achaotic map and a phase in MOEAs Second an interestingdiscovery is that cat map has best performance on solvingproblems with local optima Uniformity of cat map may beone of the reasons for the good performance of solving ZDT4

8 Conclusion

The focus of this paper is to explore the relationships ofchaotic maps and phases in MOEAs in solving MOPs Themain framework of algorithms in experiments is the NSGA-II algorithm The combinations of ten chaotic maps andthree phases are chosen in the experiments Two metricsconvergence metric 120574 and diversity metric Δ are used toevaluate the convergence and diversity properties of thealgorithms with chaotic maps The test problems are ZDTseries which were all MOPs The ergodicity and initial valuesensitivity of chaotic maps can help evolutionary algorithmsavoid solutions from falling into local optimal and get betterconvergence In the experimental results almost all chaoticmaps have good effects on improving the performance ofevolutionary algorithms to solveMOPs without local optima

Mathematical Problems in Engineering 15

Cat map has best performance on solving problems withlocal optimum This work gives insight on choosing chaoticmaps and phases for different problems Our future workwill perform further experiments with more chaotic maps onother MOEAs and formulate the theory analysis

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China under Grant no 61101153 and Prof Qiuis supported by NSF CNS 1359557

References

[1] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing vol 13 pp2790ndash2802 2013

[2] J A Adeyemo ldquoReservoir operation using multi-objective evo-lutionary algorithms-a reviewrdquo Asian Journal of Scientific Res-earch vol 4 no 1 pp 16ndash27 2011

[3] H Hu L Xu R Wei and B Zhu ldquoMulti-objective controloptimization for greenhouse environment using evolutionaryalgorithmsrdquo Sensors vol 11 no 6 pp 5792ndash5807 2011

[4] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[5] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[6] E Zitzler M Laumanns L Thiele et al SPEA2 Improving theStrength Pareto Evolutionary Algorithm Eidgenossische Techn-ische Hochschule Zurich (ETH) Institut fur Technische Infor-matik und Kommunikationsnetze (TIK) 2001

[7] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[8] B Alatas E Akin and A B Ozer ldquoChaos embedded particleswarm optimization algorithmsrdquo Chaos Solitons and Fractalsvol 40 no 4 pp 1715ndash1734 2009

[9] G ZilongW Sunrsquoan andZ Jian ldquoA novel immune evolutionaryalgorithm incorporating chaos optimizationrdquo Pattern Recogni-tion Letters vol 27 no 1 pp 2ndash8 2006

[10] M S Tavazoei and M Haeri ldquoComparison of different one-dimensional maps as chaotic search pattern in chaos optimiza-tion algorithmsrdquo Applied Mathematics and Computation vol187 no 2 pp 1076ndash1085 2007

[11] G Yu X Wang and P Li ldquoApplication of chaotic theory indifferential evolution algorithmsrdquo in Proceedings of the 6th Inte-rnational Conference on Natural Computation (ICNC rsquo10) pp3816ndash3820 August 2010

[12] B Alatas and E Akin ldquoMulti-objective rulemining using a cha-otic particle swarm optimization algorithmrdquo Knowledge-BasedSystems vol 22 no 6 pp 455ndash460 2009

[13] L D S Coelho and V C Mariani ldquoChaotic artificial immuneapproach applied to economic dispatch of electric energy usingthermal unitsrdquo Chaos Solitons and Fractals vol 40 no 5 pp2376ndash2383 2009

[14] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[15] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[16] L D S Coelho ldquoA quantum particle swarm optimizer with cha-otic mutation operatorrdquo Chaos Solitons and Fractals vol 37 no5 pp 1409ndash1418 2008

[17] L S Dos Coelho and P Alotto ldquoMultiobjective electromagneticoptimization based on a nondominated sorting genetic appr-oach with a chaotic crossover operatorrdquo IEEE Transactions onMagnetics vol 44 no 6 pp 1078ndash1081 2008

[18] H F Zhang J Z Zhou Y C Zhang N Fang and R ZhangldquoShort termhydrothermal scheduling usingmulti-objective dif-ferential evolution with three chaotic sequencesrdquo InternationalJournal of Electrical Power amp Energy Systems vol 47 pp 85ndash992013

[19] Y-J Wang and J-S Zhang ldquoGlobal optimization by an impr-oved differential evolutionary algorithmrdquo Applied Mathematicsand Computation vol 188 no 1 pp 669ndash680 2007

[20] R Caponetto L Fortuna S Fazzino and M G Xibilia ldquoCha-otic sequences to improve the performance of evolutionary alg-orithmsrdquo IEEE Transactions on Evolutionary Computation vol7 no 3 pp 289ndash304 2003

[21] M Ahmadi and H Mojallali ldquoChaotic invasive weed optimiz-ation algorithm with application to parameter estimation ofchaotic systemsrdquo Chaos Solitons amp Fractals vol 45 no 9-10pp 1108ndash1120 2012

[22] Z S Ma ldquoChaotic populations in genetic algorithmsrdquo AppliedSoft Computing vol 12 pp 2409ndash2424 2012

[23] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[24] W-M Zheng ldquoKneading plane of the circle maprdquo ChaosSolitons and Fractals vol 4 no 7 pp 1221ndash1233 1994

[25] T D Rogers and D C Whitley ldquoChaos in the cubic mappingrdquoMathematical Modelling vol 4 no 1 pp 9ndash25 1983

[26] M Bucolo R Caponetto L Fortuna M Frasca and A RizzoldquoDoes chaos work better than noiserdquo IEEE Circuits and SystemsMagazine vol 2 no 3 pp 4ndash19 2002

[27] D He C He L-G Jiang H-W Zhu and G-R Hu ldquoChaoticcharacteristics of a one-dimensional iterative map with infinitecollapsesrdquo IEEE Transactions on Circuits and Systems I vol 48no 7 pp 900ndash906 2001

[28] R May ldquoSimple mathematical models with very complicateddynamicsrdquo inTheTheory of Chaotic Attractors B Hunt T Y LiJ Kennedy and H Nusse Eds pp 85ndash93 Springer New YorkNY USA 2004

[29] R Barton ldquoChaos and fractalsrdquo The Mathematics Teacher vol83 pp 524ndash529 1990

[30] A Baykasoglu ldquoDesign optimization with chaos embeddedgreat deluge algorithmrdquoApplied Soft Computing Journal vol 12no 3 pp 1055ndash1067 2012

[31] J Fridrich ldquoSymmetric ciphers based on two-dimensional cha-otic mapsrdquo International Journal of Bifurcation and Chaos in

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl

Mathematical Problems in Engineering 13

Table7Th

eaverage

results

ofΔ

ZDT1

ZDT2

ZDT3

ZDT4

ZDT6

ci

mc

im

ci

mc

im

ci

mBa

ker

minus00337

minus00026

00015

00226

00310

00024

minus000

96000

69000

06minus006

0300343

minus00513

00125

00024

minus00015

Cat

000

0500017

000

4200234

minus01021

minus00145

minus00019

minus00052

000

0600277

minus000

90minus00148

minus000

01minus000

02minus000

68Circle

minus00325

minus00035

minus000

64minus02019

minus00632

minus01410

minus00052

minus000

6500010

minus00255

minus00259

minus00136

00031

minus00020

000

64Cu

bic

minus000

4700121

minus00022

minus00439

minus00148

00361

000

00minus000

0600031

minus00052

minus00279

00238

minus00170

00054

minus00017

Gauss

00121

00075

minus00180

minus01075

00329

minus00711

minus00024

minus00032

00019

00330

minus00214

00155

00180

000

03000

64ICMIC

00081

000

92000

02minus00734

minus00850

minus00917

minus00036

000

08minus00054

minus00356

minus00497

00238

00123

000

0100085

Logistic

minus00118

00076

00075

minus01935

00239

00509

minus00084

minus00014

00080

00275

minus00100

minus00232

minus000

6900020

00077

Sinu

soidal

00149

000

69minus00564

minus01750

minus000

67minus02902

minus00014

00029

minus00286

minus00390

minus00164

minus01945

00129

00088

minus004

88Tent

minus00350

000

9300016

00215

minus00173

minus00261

minus00059

minus00168

00071

minus00761

minus00139

00075

00125

00028

minus000

02Za

slavskii

002307

00056

minus00014

minus00292

minus00189

minus00013

000

99000

40minus00037

00201

minus00056

000

9000038

00027

00084

14 Mathematical Problems in Engineering

Table 8 Statistical data for combinations of chaotic maps and phases on different problems

Threshold ZDT1 ZDT2 ZDT3 ZDT4 ZDT60 18 9 15 10 200001 16 9 9 10 180002 13 9 7 10 180003 13 8 6 10 140004 13 8 5 10 120005 12 8 4 10 120006 11 8 4 10 11

0 50 100 150 200 250minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

Generation

Diff

eren

ce o

f delt

a

Bakerrsquos

Figure 8 Performance of Bakerrsquos maps in crossover operator insolving ZDT1 with metric Δ

with local optimum cat map has good performance onimproving evolutionary algorithms

73 Diversity Performance Similar to the convergencemetric120574 the differences of the diversity metricΔ between the resultsof the new algorithm with chaotic maps and the originalNSGA-II algorithm are used to measure the performanceFigure 7 shows the performance of chaotic maps in crossoveroperator in solving ZDT1 with metric Δ Each subgraphshows the effect of one chaotic map To be seen more clearlythe first subgraph in Figure 7 is shown in Figure 8

As seen fromFigures 7 and 8 theΔ difference is not stablein 250 generations The average values of Δ difference in 250generations are calculated to represent the effect of the newalgorithms For brevity the rest of results are not shown ingraphs but in Table 7

As seen from Table 7 the Δ values have little differenceWe count the number of combinations of chaotic maps andphases for solving one problem in different threshold valuesFor example there are 18 combinations whose values of Δ aregreater than zero Based on the number of the combinationsof chaotic maps and phases in different threshold values thevalues of diversity metric Δ are summarized in Table 8

In Table 8 the rank of the number of Δ in differentthreshold values is ZDT1 gt ZDT6 gt ZDT4 gt ZDT2 gt

ZDT3 especially for larger threshold ZDT1 problem whichis a convex function and has no local optima is a relativelyeasy problem Chaotic maps bring the biggest improve-ments on solving ZDT1 Though the solutions of ZDT6 arenonuniformly spaced chaotic maps can find better spread ofsolutions While ZDT4 problem is a complex problem andthe solutions are easily trapped into local optima chaoticmaps can improve the distribution of the solutions ZDT2problem is a convex function and the solutions sometimesfall into the local optimumThe effects of chaoticmaps can begeneralizedThe Pareto front of ZDT3 problem is segmentedso the Δ value of ZDT3 is larger and the ranking of ZDT3 islower It is our observation that Δ is not fit for evaluating thesolutions to problems which are disconnected

Based on the diversity metric Δ chaotic maps havethe best improvement on solving convex problems withoutlocal optima and have better effect on solving problemswhich have nonuniform solutions For problems with localminimum chaotic maps embedded algorithms can improvethe performance with regard to metric Δ

A short summary can be given according to the aboveexperiments First chaotic maps can improve the perfor-mance of MOEAs but the results showed that no one chaoticmap outperforms other maps for all of the problems Theresults in this paper give some guidance on how to choose achaotic map and a phase in MOEAs Second an interestingdiscovery is that cat map has best performance on solvingproblems with local optima Uniformity of cat map may beone of the reasons for the good performance of solving ZDT4

8 Conclusion

The focus of this paper is to explore the relationships ofchaotic maps and phases in MOEAs in solving MOPs Themain framework of algorithms in experiments is the NSGA-II algorithm The combinations of ten chaotic maps andthree phases are chosen in the experiments Two metricsconvergence metric 120574 and diversity metric Δ are used toevaluate the convergence and diversity properties of thealgorithms with chaotic maps The test problems are ZDTseries which were all MOPs The ergodicity and initial valuesensitivity of chaotic maps can help evolutionary algorithmsavoid solutions from falling into local optimal and get betterconvergence In the experimental results almost all chaoticmaps have good effects on improving the performance ofevolutionary algorithms to solveMOPs without local optima

Mathematical Problems in Engineering 15

Cat map has best performance on solving problems withlocal optimum This work gives insight on choosing chaoticmaps and phases for different problems Our future workwill perform further experiments with more chaotic maps onother MOEAs and formulate the theory analysis

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China under Grant no 61101153 and Prof Qiuis supported by NSF CNS 1359557

References

[1] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing vol 13 pp2790ndash2802 2013

[2] J A Adeyemo ldquoReservoir operation using multi-objective evo-lutionary algorithms-a reviewrdquo Asian Journal of Scientific Res-earch vol 4 no 1 pp 16ndash27 2011

[3] H Hu L Xu R Wei and B Zhu ldquoMulti-objective controloptimization for greenhouse environment using evolutionaryalgorithmsrdquo Sensors vol 11 no 6 pp 5792ndash5807 2011

[4] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[5] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[6] E Zitzler M Laumanns L Thiele et al SPEA2 Improving theStrength Pareto Evolutionary Algorithm Eidgenossische Techn-ische Hochschule Zurich (ETH) Institut fur Technische Infor-matik und Kommunikationsnetze (TIK) 2001

[7] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[8] B Alatas E Akin and A B Ozer ldquoChaos embedded particleswarm optimization algorithmsrdquo Chaos Solitons and Fractalsvol 40 no 4 pp 1715ndash1734 2009

[9] G ZilongW Sunrsquoan andZ Jian ldquoA novel immune evolutionaryalgorithm incorporating chaos optimizationrdquo Pattern Recogni-tion Letters vol 27 no 1 pp 2ndash8 2006

[10] M S Tavazoei and M Haeri ldquoComparison of different one-dimensional maps as chaotic search pattern in chaos optimiza-tion algorithmsrdquo Applied Mathematics and Computation vol187 no 2 pp 1076ndash1085 2007

[11] G Yu X Wang and P Li ldquoApplication of chaotic theory indifferential evolution algorithmsrdquo in Proceedings of the 6th Inte-rnational Conference on Natural Computation (ICNC rsquo10) pp3816ndash3820 August 2010

[12] B Alatas and E Akin ldquoMulti-objective rulemining using a cha-otic particle swarm optimization algorithmrdquo Knowledge-BasedSystems vol 22 no 6 pp 455ndash460 2009

[13] L D S Coelho and V C Mariani ldquoChaotic artificial immuneapproach applied to economic dispatch of electric energy usingthermal unitsrdquo Chaos Solitons and Fractals vol 40 no 5 pp2376ndash2383 2009

[14] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[15] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[16] L D S Coelho ldquoA quantum particle swarm optimizer with cha-otic mutation operatorrdquo Chaos Solitons and Fractals vol 37 no5 pp 1409ndash1418 2008

[17] L S Dos Coelho and P Alotto ldquoMultiobjective electromagneticoptimization based on a nondominated sorting genetic appr-oach with a chaotic crossover operatorrdquo IEEE Transactions onMagnetics vol 44 no 6 pp 1078ndash1081 2008

[18] H F Zhang J Z Zhou Y C Zhang N Fang and R ZhangldquoShort termhydrothermal scheduling usingmulti-objective dif-ferential evolution with three chaotic sequencesrdquo InternationalJournal of Electrical Power amp Energy Systems vol 47 pp 85ndash992013

[19] Y-J Wang and J-S Zhang ldquoGlobal optimization by an impr-oved differential evolutionary algorithmrdquo Applied Mathematicsand Computation vol 188 no 1 pp 669ndash680 2007

[20] R Caponetto L Fortuna S Fazzino and M G Xibilia ldquoCha-otic sequences to improve the performance of evolutionary alg-orithmsrdquo IEEE Transactions on Evolutionary Computation vol7 no 3 pp 289ndash304 2003

[21] M Ahmadi and H Mojallali ldquoChaotic invasive weed optimiz-ation algorithm with application to parameter estimation ofchaotic systemsrdquo Chaos Solitons amp Fractals vol 45 no 9-10pp 1108ndash1120 2012

[22] Z S Ma ldquoChaotic populations in genetic algorithmsrdquo AppliedSoft Computing vol 12 pp 2409ndash2424 2012

[23] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[24] W-M Zheng ldquoKneading plane of the circle maprdquo ChaosSolitons and Fractals vol 4 no 7 pp 1221ndash1233 1994

[25] T D Rogers and D C Whitley ldquoChaos in the cubic mappingrdquoMathematical Modelling vol 4 no 1 pp 9ndash25 1983

[26] M Bucolo R Caponetto L Fortuna M Frasca and A RizzoldquoDoes chaos work better than noiserdquo IEEE Circuits and SystemsMagazine vol 2 no 3 pp 4ndash19 2002

[27] D He C He L-G Jiang H-W Zhu and G-R Hu ldquoChaoticcharacteristics of a one-dimensional iterative map with infinitecollapsesrdquo IEEE Transactions on Circuits and Systems I vol 48no 7 pp 900ndash906 2001

[28] R May ldquoSimple mathematical models with very complicateddynamicsrdquo inTheTheory of Chaotic Attractors B Hunt T Y LiJ Kennedy and H Nusse Eds pp 85ndash93 Springer New YorkNY USA 2004

[29] R Barton ldquoChaos and fractalsrdquo The Mathematics Teacher vol83 pp 524ndash529 1990

[30] A Baykasoglu ldquoDesign optimization with chaos embeddedgreat deluge algorithmrdquoApplied Soft Computing Journal vol 12no 3 pp 1055ndash1067 2012

[31] J Fridrich ldquoSymmetric ciphers based on two-dimensional cha-otic mapsrdquo International Journal of Bifurcation and Chaos in

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

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

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl

14 Mathematical Problems in Engineering

Table 8 Statistical data for combinations of chaotic maps and phases on different problems

Threshold ZDT1 ZDT2 ZDT3 ZDT4 ZDT60 18 9 15 10 200001 16 9 9 10 180002 13 9 7 10 180003 13 8 6 10 140004 13 8 5 10 120005 12 8 4 10 120006 11 8 4 10 11

0 50 100 150 200 250minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

Generation

Diff

eren

ce o

f delt

a

Bakerrsquos

Figure 8 Performance of Bakerrsquos maps in crossover operator insolving ZDT1 with metric Δ

with local optimum cat map has good performance onimproving evolutionary algorithms

73 Diversity Performance Similar to the convergencemetric120574 the differences of the diversity metricΔ between the resultsof the new algorithm with chaotic maps and the originalNSGA-II algorithm are used to measure the performanceFigure 7 shows the performance of chaotic maps in crossoveroperator in solving ZDT1 with metric Δ Each subgraphshows the effect of one chaotic map To be seen more clearlythe first subgraph in Figure 7 is shown in Figure 8

As seen fromFigures 7 and 8 theΔ difference is not stablein 250 generations The average values of Δ difference in 250generations are calculated to represent the effect of the newalgorithms For brevity the rest of results are not shown ingraphs but in Table 7

As seen from Table 7 the Δ values have little differenceWe count the number of combinations of chaotic maps andphases for solving one problem in different threshold valuesFor example there are 18 combinations whose values of Δ aregreater than zero Based on the number of the combinationsof chaotic maps and phases in different threshold values thevalues of diversity metric Δ are summarized in Table 8

In Table 8 the rank of the number of Δ in differentthreshold values is ZDT1 gt ZDT6 gt ZDT4 gt ZDT2 gt

ZDT3 especially for larger threshold ZDT1 problem whichis a convex function and has no local optima is a relativelyeasy problem Chaotic maps bring the biggest improve-ments on solving ZDT1 Though the solutions of ZDT6 arenonuniformly spaced chaotic maps can find better spread ofsolutions While ZDT4 problem is a complex problem andthe solutions are easily trapped into local optima chaoticmaps can improve the distribution of the solutions ZDT2problem is a convex function and the solutions sometimesfall into the local optimumThe effects of chaoticmaps can begeneralizedThe Pareto front of ZDT3 problem is segmentedso the Δ value of ZDT3 is larger and the ranking of ZDT3 islower It is our observation that Δ is not fit for evaluating thesolutions to problems which are disconnected

Based on the diversity metric Δ chaotic maps havethe best improvement on solving convex problems withoutlocal optima and have better effect on solving problemswhich have nonuniform solutions For problems with localminimum chaotic maps embedded algorithms can improvethe performance with regard to metric Δ

A short summary can be given according to the aboveexperiments First chaotic maps can improve the perfor-mance of MOEAs but the results showed that no one chaoticmap outperforms other maps for all of the problems Theresults in this paper give some guidance on how to choose achaotic map and a phase in MOEAs Second an interestingdiscovery is that cat map has best performance on solvingproblems with local optima Uniformity of cat map may beone of the reasons for the good performance of solving ZDT4

8 Conclusion

The focus of this paper is to explore the relationships ofchaotic maps and phases in MOEAs in solving MOPs Themain framework of algorithms in experiments is the NSGA-II algorithm The combinations of ten chaotic maps andthree phases are chosen in the experiments Two metricsconvergence metric 120574 and diversity metric Δ are used toevaluate the convergence and diversity properties of thealgorithms with chaotic maps The test problems are ZDTseries which were all MOPs The ergodicity and initial valuesensitivity of chaotic maps can help evolutionary algorithmsavoid solutions from falling into local optimal and get betterconvergence In the experimental results almost all chaoticmaps have good effects on improving the performance ofevolutionary algorithms to solveMOPs without local optima

Mathematical Problems in Engineering 15

Cat map has best performance on solving problems withlocal optimum This work gives insight on choosing chaoticmaps and phases for different problems Our future workwill perform further experiments with more chaotic maps onother MOEAs and formulate the theory analysis

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China under Grant no 61101153 and Prof Qiuis supported by NSF CNS 1359557

References

[1] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing vol 13 pp2790ndash2802 2013

[2] J A Adeyemo ldquoReservoir operation using multi-objective evo-lutionary algorithms-a reviewrdquo Asian Journal of Scientific Res-earch vol 4 no 1 pp 16ndash27 2011

[3] H Hu L Xu R Wei and B Zhu ldquoMulti-objective controloptimization for greenhouse environment using evolutionaryalgorithmsrdquo Sensors vol 11 no 6 pp 5792ndash5807 2011

[4] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[5] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[6] E Zitzler M Laumanns L Thiele et al SPEA2 Improving theStrength Pareto Evolutionary Algorithm Eidgenossische Techn-ische Hochschule Zurich (ETH) Institut fur Technische Infor-matik und Kommunikationsnetze (TIK) 2001

[7] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[8] B Alatas E Akin and A B Ozer ldquoChaos embedded particleswarm optimization algorithmsrdquo Chaos Solitons and Fractalsvol 40 no 4 pp 1715ndash1734 2009

[9] G ZilongW Sunrsquoan andZ Jian ldquoA novel immune evolutionaryalgorithm incorporating chaos optimizationrdquo Pattern Recogni-tion Letters vol 27 no 1 pp 2ndash8 2006

[10] M S Tavazoei and M Haeri ldquoComparison of different one-dimensional maps as chaotic search pattern in chaos optimiza-tion algorithmsrdquo Applied Mathematics and Computation vol187 no 2 pp 1076ndash1085 2007

[11] G Yu X Wang and P Li ldquoApplication of chaotic theory indifferential evolution algorithmsrdquo in Proceedings of the 6th Inte-rnational Conference on Natural Computation (ICNC rsquo10) pp3816ndash3820 August 2010

[12] B Alatas and E Akin ldquoMulti-objective rulemining using a cha-otic particle swarm optimization algorithmrdquo Knowledge-BasedSystems vol 22 no 6 pp 455ndash460 2009

[13] L D S Coelho and V C Mariani ldquoChaotic artificial immuneapproach applied to economic dispatch of electric energy usingthermal unitsrdquo Chaos Solitons and Fractals vol 40 no 5 pp2376ndash2383 2009

[14] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[15] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[16] L D S Coelho ldquoA quantum particle swarm optimizer with cha-otic mutation operatorrdquo Chaos Solitons and Fractals vol 37 no5 pp 1409ndash1418 2008

[17] L S Dos Coelho and P Alotto ldquoMultiobjective electromagneticoptimization based on a nondominated sorting genetic appr-oach with a chaotic crossover operatorrdquo IEEE Transactions onMagnetics vol 44 no 6 pp 1078ndash1081 2008

[18] H F Zhang J Z Zhou Y C Zhang N Fang and R ZhangldquoShort termhydrothermal scheduling usingmulti-objective dif-ferential evolution with three chaotic sequencesrdquo InternationalJournal of Electrical Power amp Energy Systems vol 47 pp 85ndash992013

[19] Y-J Wang and J-S Zhang ldquoGlobal optimization by an impr-oved differential evolutionary algorithmrdquo Applied Mathematicsand Computation vol 188 no 1 pp 669ndash680 2007

[20] R Caponetto L Fortuna S Fazzino and M G Xibilia ldquoCha-otic sequences to improve the performance of evolutionary alg-orithmsrdquo IEEE Transactions on Evolutionary Computation vol7 no 3 pp 289ndash304 2003

[21] M Ahmadi and H Mojallali ldquoChaotic invasive weed optimiz-ation algorithm with application to parameter estimation ofchaotic systemsrdquo Chaos Solitons amp Fractals vol 45 no 9-10pp 1108ndash1120 2012

[22] Z S Ma ldquoChaotic populations in genetic algorithmsrdquo AppliedSoft Computing vol 12 pp 2409ndash2424 2012

[23] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[24] W-M Zheng ldquoKneading plane of the circle maprdquo ChaosSolitons and Fractals vol 4 no 7 pp 1221ndash1233 1994

[25] T D Rogers and D C Whitley ldquoChaos in the cubic mappingrdquoMathematical Modelling vol 4 no 1 pp 9ndash25 1983

[26] M Bucolo R Caponetto L Fortuna M Frasca and A RizzoldquoDoes chaos work better than noiserdquo IEEE Circuits and SystemsMagazine vol 2 no 3 pp 4ndash19 2002

[27] D He C He L-G Jiang H-W Zhu and G-R Hu ldquoChaoticcharacteristics of a one-dimensional iterative map with infinitecollapsesrdquo IEEE Transactions on Circuits and Systems I vol 48no 7 pp 900ndash906 2001

[28] R May ldquoSimple mathematical models with very complicateddynamicsrdquo inTheTheory of Chaotic Attractors B Hunt T Y LiJ Kennedy and H Nusse Eds pp 85ndash93 Springer New YorkNY USA 2004

[29] R Barton ldquoChaos and fractalsrdquo The Mathematics Teacher vol83 pp 524ndash529 1990

[30] A Baykasoglu ldquoDesign optimization with chaos embeddedgreat deluge algorithmrdquoApplied Soft Computing Journal vol 12no 3 pp 1055ndash1067 2012

[31] J Fridrich ldquoSymmetric ciphers based on two-dimensional cha-otic mapsrdquo International Journal of Bifurcation and Chaos in

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

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

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl

Mathematical Problems in Engineering 15

Cat map has best performance on solving problems withlocal optimum This work gives insight on choosing chaoticmaps and phases for different problems Our future workwill perform further experiments with more chaotic maps onother MOEAs and formulate the theory analysis

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China under Grant no 61101153 and Prof Qiuis supported by NSF CNS 1359557

References

[1] H Lu R Y Niu J Liu and Z Zhu ldquoA chaotic non-dominatedsorting genetic algorithm for the multi-objective automatic testtask scheduling problemrdquo Applied Soft Computing vol 13 pp2790ndash2802 2013

[2] J A Adeyemo ldquoReservoir operation using multi-objective evo-lutionary algorithms-a reviewrdquo Asian Journal of Scientific Res-earch vol 4 no 1 pp 16ndash27 2011

[3] H Hu L Xu R Wei and B Zhu ldquoMulti-objective controloptimization for greenhouse environment using evolutionaryalgorithmsrdquo Sensors vol 11 no 6 pp 5792ndash5807 2011

[4] T Niknam ldquoAn efficient hybrid evolutionary algorithm basedon PSO andHBMOalgorithms formulti-objectiveDistributionFeeder Reconfigurationrdquo Energy Conversion and Managementvol 50 no 8 pp 2074ndash2082 2009

[5] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[6] E Zitzler M Laumanns L Thiele et al SPEA2 Improving theStrength Pareto Evolutionary Algorithm Eidgenossische Techn-ische Hochschule Zurich (ETH) Institut fur Technische Infor-matik und Kommunikationsnetze (TIK) 2001

[7] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[8] B Alatas E Akin and A B Ozer ldquoChaos embedded particleswarm optimization algorithmsrdquo Chaos Solitons and Fractalsvol 40 no 4 pp 1715ndash1734 2009

[9] G ZilongW Sunrsquoan andZ Jian ldquoA novel immune evolutionaryalgorithm incorporating chaos optimizationrdquo Pattern Recogni-tion Letters vol 27 no 1 pp 2ndash8 2006

[10] M S Tavazoei and M Haeri ldquoComparison of different one-dimensional maps as chaotic search pattern in chaos optimiza-tion algorithmsrdquo Applied Mathematics and Computation vol187 no 2 pp 1076ndash1085 2007

[11] G Yu X Wang and P Li ldquoApplication of chaotic theory indifferential evolution algorithmsrdquo in Proceedings of the 6th Inte-rnational Conference on Natural Computation (ICNC rsquo10) pp3816ndash3820 August 2010

[12] B Alatas and E Akin ldquoMulti-objective rulemining using a cha-otic particle swarm optimization algorithmrdquo Knowledge-BasedSystems vol 22 no 6 pp 455ndash460 2009

[13] L D S Coelho and V C Mariani ldquoChaotic artificial immuneapproach applied to economic dispatch of electric energy usingthermal unitsrdquo Chaos Solitons and Fractals vol 40 no 5 pp2376ndash2383 2009

[14] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[15] B Alatas ldquoChaotic harmony search algorithmsrdquo Applied Math-ematics and Computation vol 216 no 9 pp 2687ndash2699 2010

[16] L D S Coelho ldquoA quantum particle swarm optimizer with cha-otic mutation operatorrdquo Chaos Solitons and Fractals vol 37 no5 pp 1409ndash1418 2008

[17] L S Dos Coelho and P Alotto ldquoMultiobjective electromagneticoptimization based on a nondominated sorting genetic appr-oach with a chaotic crossover operatorrdquo IEEE Transactions onMagnetics vol 44 no 6 pp 1078ndash1081 2008

[18] H F Zhang J Z Zhou Y C Zhang N Fang and R ZhangldquoShort termhydrothermal scheduling usingmulti-objective dif-ferential evolution with three chaotic sequencesrdquo InternationalJournal of Electrical Power amp Energy Systems vol 47 pp 85ndash992013

[19] Y-J Wang and J-S Zhang ldquoGlobal optimization by an impr-oved differential evolutionary algorithmrdquo Applied Mathematicsand Computation vol 188 no 1 pp 669ndash680 2007

[20] R Caponetto L Fortuna S Fazzino and M G Xibilia ldquoCha-otic sequences to improve the performance of evolutionary alg-orithmsrdquo IEEE Transactions on Evolutionary Computation vol7 no 3 pp 289ndash304 2003

[21] M Ahmadi and H Mojallali ldquoChaotic invasive weed optimiz-ation algorithm with application to parameter estimation ofchaotic systemsrdquo Chaos Solitons amp Fractals vol 45 no 9-10pp 1108ndash1120 2012

[22] Z S Ma ldquoChaotic populations in genetic algorithmsrdquo AppliedSoft Computing vol 12 pp 2409ndash2424 2012

[23] S Talatahari B Farahmand Azar R Sheikholeslami and A HGandomi ldquoImperialist competitive algorithm combined withchaos for global optimizationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 3 pp 1312ndash13192012

[24] W-M Zheng ldquoKneading plane of the circle maprdquo ChaosSolitons and Fractals vol 4 no 7 pp 1221ndash1233 1994

[25] T D Rogers and D C Whitley ldquoChaos in the cubic mappingrdquoMathematical Modelling vol 4 no 1 pp 9ndash25 1983

[26] M Bucolo R Caponetto L Fortuna M Frasca and A RizzoldquoDoes chaos work better than noiserdquo IEEE Circuits and SystemsMagazine vol 2 no 3 pp 4ndash19 2002

[27] D He C He L-G Jiang H-W Zhu and G-R Hu ldquoChaoticcharacteristics of a one-dimensional iterative map with infinitecollapsesrdquo IEEE Transactions on Circuits and Systems I vol 48no 7 pp 900ndash906 2001

[28] R May ldquoSimple mathematical models with very complicateddynamicsrdquo inTheTheory of Chaotic Attractors B Hunt T Y LiJ Kennedy and H Nusse Eds pp 85ndash93 Springer New YorkNY USA 2004

[29] R Barton ldquoChaos and fractalsrdquo The Mathematics Teacher vol83 pp 524ndash529 1990

[30] A Baykasoglu ldquoDesign optimization with chaos embeddedgreat deluge algorithmrdquoApplied Soft Computing Journal vol 12no 3 pp 1055ndash1067 2012

[31] J Fridrich ldquoSymmetric ciphers based on two-dimensional cha-otic mapsrdquo International Journal of Bifurcation and Chaos in

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

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

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl

16 Mathematical Problems in Engineering

Applied Sciences and Engineering vol 8 no 6 pp 1259ndash12841998

[32] V I Arnold and A Avez Problemes ergodiques de la mecaniqueclassique Gauthier-Villars Paris France 1967

[33] G M Zaslavsky ldquoThe simplest case of a strange attractorrdquo Phy-sics Letters A vol 69 no 3 pp 145ndash147 197879

[34] M T Rosenstein J J Collins and C J De Luca ldquoA practicalmethod for calculating largest Lyapunov exponents from smalldata setsrdquo Physica D vol 65 no 1-2 pp 117ndash134 1993

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

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl

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

• The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
• • Repository Citation
  • • The Effects of Using Chaotic Map on Improving the Performance of Multiobjective Evolutionary Algorithms
    • • Digital Object Identifier (DOI)
      • NotesCitation Information
      • • tmp1401803913pdfPD4Sl