Research Article An Improved Nondominated Sorting Genetic...

Research ArticleAn Improved Nondominated Sorting Genetic Algorithm forMultiobjective Problem

Ruihua Wang

Management Engineering Department of Zhengzhou University Zhengzhou Henan China

Correspondence should be addressed to Ruihua Wang wangrh0304zzueducn

Received 18 July 2016 Revised 10 October 2016 Accepted 27 November 2016

Academic Editor Ziran Wu

Copyright copy 2016 Ruihua WangThis 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

In this paper an improved NSGA2 algorithm is proposed which is used to solve the multiobjective problem For the originalNSGA2 algorithm the paper made one improvement joining the local search strategy into the NSGA2 algorithm After eachiteration calculation of the NSGA2 algorithm a kind of local search strategy is performed in the Pareto optimal set to search bettersolutions such that the NSGA2 algorithm can gain a better local search ability which is helpful to the optimization process Finallythe proposed modified NSGA2 algorithm (MNSGA2) is simulated in the two classic multiobjective problems which is called KURproblem and ZDT3 problemThe calculation results show the modified NSGA2 outperforms the original NSGA2 which indicatesthat the improvement strategy is helpful to improve the algorithm

1 Introduction

Themaximization of profit is always the pursuit of mankindno matter in which field especially the development of theenergy issues and the population problems in recent yearsFor many social and economic problems it is needed to findan optimal solution to reduce the input and increase the out-put As a result the optimization problem is more and moreconcerned by scholars In recent years many scholars haveput forward a lot of optimization problem solving methodsfrommany aspects Many of them are to find the value of thecontrol variables to minimize the single objective Howevermany real world decision problems not only need to considerjust one factor but a variety of factors to get a decision Inother words a lot of practical problems are themultiobjectiveprogramming problem Therefore in recent years scholarshave conducted in-depth research onmultiobjective planningand got a lot of meaningful research results which lay thefoundation for the future research of multiobjective problemIn recent years the heuristic search algorithm has a greatdevelopment Some researchers introduced and developedmany heuristic search algorithm according to the behaviorof the social animal in the nature and the characteristicof the optimization problems to be optimized like genetic

algorithm [1] particle swarm optimization algorithm [2] dif-ferential evolution algorithm [3] shuffle frog leap algorithm[4] artificial bee colony algorithm [5] and so on In all ofthese heuristic search algorithms the genetic algorithm isvery representative According to its good calculation perfor-mance and genetic idea the genetic algorithm is largely usedin many aspects [6ndash10] The first genetic algorithm is used todeal with the single objective optimization problem while asthe time passes by many researchers introduce a lot of multi-objective optimization algorithms according to the idea of thegenetic algorithm These multiobjective optimization algo-rithms contain vector evaluation genetic algorithm [11]multiobjective genetic algorithm [12 13] strength Paretoevolutionary algorithm [14 15] and nondominating sortinggenetic algorithm [16] After several years later in order tomodify each of these multiobjective genetic algorithms cal-culating performance a lot of researchers also introduce theimproved version of these multiobjective genetic algorithmslike strength Pareto evolutionary algorithm 2 [17] which is theimproved version of the strength Pareto evolutionary algo-rithm strength Pareto evolutionary algorithm [18] which isalso the improved version of the strength Pareto evolutionaryalgorithm nondominated sorting genetic algorithm 2 whichis the improved version of the nondominating sorting genetic

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1519542 7 pageshttpdxdoiorg10115520161519542

2 Mathematical Problems in Engineering

algorithm [19] and so on In this way many multiobjectivegenetic algorithms are proposed in these years whichmade agreat contribution to thesemultiobjective optimization prob-lems

Nondominated sorting genetic algorithm 2 (which wecall it as the NSGA2 algorithm in the rest of this paper) isthe improved version of the nondominating sorting geneticalgorithm which is first proposed by Deb in 2001 Duringthese years development and use the NSGA2 algorithmhas been proved as a very good multiobjective geneticalgorithm which has been successfully used in many multiobjective optimization problems [20ndash26] because the calcu-lation performance of one algorithm varies very much whendealing with different optimization problems Beside thisthe algorithmrsquos calculation performance also depends on theoptimizing strategy inside of the algorithm itself So a lot ofresearchers also proposed some improvement strategies intothe NSGA2 algorithm to deal with the shortcomings whenoptimizing some optimization problems In this paper inorder to overcome the poor local search ability problem inthe NSGA2 algorithm a local search strategy is joined intothe NSGA2 algorithm to enhance the local search abilityThe general idea of the proposed improvement strategy isto search locally around the optimization solution found sofar to find a better solution such that the algorithm can geta better local search ability and can make a more detailedsearch In order to see whether the local search ability canimprove the calculation performance of the original NSGA2algorithm or not this paper also introduce two classic multi-objective optimization problems to let the modified NSGA2algorithm to simulate At last the outcome of the simulationworks is compared to the original NSGA2 algorithm Thecomparison shows the local search ability can really improvethe calculation performance of the NSGA2 algorithm

The rest of this paper is organized as follows Section 2describes themathematicalmodel of themultiobjective prob-lem Section 3 shows the principles and steps of the NSGA2algorithm where we introduce three aspects to describethe original NSGA2 algorithm fast nondominate sortingcrowding distance function and environmental selectionbased on PCD At the end of Section 3 the basic stepsof the original NSGA2 algorithm are introduced to thereaders The simulation works and the results analysis ofdifferent multiobjective optimization problem are given inSection 4 where we introduced two classic multiobjectiveoptimization problems Section 5 draws the conclusion of thispaper

2 Multiobjective Problem

A multiobjective problem can be seen as a problem whichshould optimize at least two objectives while it satisfies a lotof constrains A multiobjective optimization problem can bemathematically described as

Min 119865 = 1198911 (119883) 1198912 (119883) 119891119896 (119883)St 119892119894 (119883) le 0 119894 = 1 2 119899

119883 = (1199091 1199092 119909119898) isin 119877119898(1)

where 119891119894(119909) represents the 119894-th objective function 119883 repre-sents the control variables vector 119899 represents the number ofconstrains and 119896 is the total number of the objective to beoptimized

But the purpose of the decision scheme of the multiob-jective optimization problem is not to make each objectiveto be optimized optimal but to provide a series of solutionsfor the decision makers In accordance with their owndecision intention the decision makers choose the decisionscheme to make decisions A series of optimal solution setof multiobjective optimization problems found in the endis called the Pareto optimal solution set The mathematicalexplanation of the Pareto optimal solution set can be shownas follows

119875 = 119909lowast isin Ω | notexist119909119896 isin Ω 119909119896 ≺ 119909lowast (2)

where the 119909lowast represents the Pareto optimal solutions whosemathematical expression can be shown as follows

notexist119909119895 isin Ω 119909lowast ≺ 119909119895 (3)

In (3) the expression 1199091 ≺ 1199092 represents the individual 1199091dominating the individual 1199092 For a minimization optimiza-tion problem a solution dominating another solution in anmultiobjective can be mathematically is expressed as follows

forall119894 isin 1 2 119873 119891119894 (1199091) le 119891119894 (1199092)exist119895 isin 1 2 119873 119891119895 (1199091) lt 119891119895 (1199092) (4)

From (4) we can conclude that if a solution 1199091rsquos eachobjective value in a multiobjective problem is not bigger thana1199092rsquos corresponding objective value and at least one objectivecan be found the 1199091rsquos value is smaller than the 1199092rsquos then wecan say that the solution 1199091 dominates 1199092

The multiobjective optimization problem is exactlyaccording to seeking the nondominating solution and usesome kind of evolutionary strategy to evolve the evolutionarypopulation to finally find the Pareto optimization set ofthe multiobjective optimization problem to be optimizedIn order to more intuitively demonstrate the value of eachobjective value in the Pareto optimal solution set and its dis-tribution in the solution space the Pareto front is introducedThe Pareto front is a line or a curved surface formed by theobjective values of each solution in the Pareto optimal setwhose mathematical expression is as follows

PF = 119865 (119909) = (1198911 (119909) 1198912 (119909) 119891119898 (119909)) | 119909 isin 119875 (5)

3 Improved NSGA2 Algorithm

31 NSGA2Algorithm TheNSGA2 algorithm is themodifiedversion of NSGA by Deb and other researchers after severalyears of development and use the NSGA2 algorithm is seenas a very good and mature multiobjective optimization algo-rithm at present In NSGA2 algorithm the crowding distancefunction is used to calculate each individualrsquos fitness value inthe algorithm According to the fast nondominating sortingidea and environmental selection of PCD the algorithm

Mathematical Problems in Engineering 3

uses the two strategies to evolve the evolutionary populationto optimize the corresponding problem In the rest of thispaper three important aspects of this algorithm is introducedto study the original NSGA2 fast nondominating sortingwhich is used to evaluate each individual in the NSGA2 algo-rithm crowding distance function which is used to calculateeach individualrsquos fitness value Environmental selection basedon PCD is used to select individuals to the next generation ofthe evolutionary population in the algorithm

311 Fast Nondominated Sorting (FNS) In the evolutionarymultiobjective optimization algorithm the evolution of thepopulation is to eliminate the inferior solution and select theexcellent solutions by a certain strategy such that the inferiorsolution is eliminated and the excellent solution is retainedin the next generation of the evolutionary population so as tofind the final excellent solution set through the step by stepevolution In NSGA2 the fast nondominated sorting strategyis used to evaluate each individualrsquos dominating status in theevolutionary population and according to the dominatingstatus of each individual the evolutionary population isdivided into different subgroups which is a necessary wayto calculate the fitness of each individual in the crowd andto evaluate the quality of each individual In the fast non-dominated sorting strategy the first step of this strategy is tocalculate each individualrsquos (eg individual 119894) correspondingdominating solutions number 119904119894 and the individuals number119899119894 who dominate the individual 119894 in the evolutionary popu-lation When an individualrsquos corresponding 119899119894 equals 0 thensend it to the first level of the nondominating individuals Inthe fast nondominated sorting strategy it declares that whenthe nondominating individualsrsquo dominating grade in a level isdetermined already the level of all individuals in the controlof other individuals is no longer considered Considering thisfor an individual 119894rsquos dominating individual set 119904119894 each of theindividualrsquos (let us say individual 119895) corresponding 119899119895 in the119894rsquos dominating individual set 119904119894 should subtract 1 After thesubtraction when 119899119895 = 0 the individual 119895will be put into thesecond nondominating levelThe rest of the individuals in theevolutionary population use the same way to find which non-dominating level they belong to After each of the individualsin the evolutionary population finds its own nondominatinglevel the lowest levelrsquos corresponding individuals are closeto the real Pareto front of the multiobjective optimizationproblem to be optimized So the lowest levelrsquos correspondingindividuals are more superior than the other individuals inthe evolutionary population of the NSGA2 algorithm Andthe NSGA2 algorithm exactly uses a larger probability toselect the lowest level compared to the high level into thenext generation of the evolutionary population such thatthe NSGA2 algorithm can evolve to the direction of the realPareto optimization set of the multiobjective optimizationproblem to be optimized

312 Crowding Distance Function The purpose of the mul-tiobjective optimization algorithm is not just only to find thePareto optimal set which is close to the real Pareto optimal setbut also to find the solutions in the Pareto optimal set have avery good distribution such that better and more reasonable

solutions can be provided to the decisionmakerThe purposeof themultiobjective optimization algorithm is to find a seriesof uniform distribution of the solution for decisionmakers tochoose In order to obtain the distribution of the solution setwell various evolutionary algorithms have used a lot of dif-ferent kinds of treatmentmeasures Most algorithms evaluatethe distribution of each individual by constructing a functionthat considers the distribution of individual distribution Inthe NSGA2 algorithm by introducing the crowding distancefunction the congestion degree of each individual and otherindividuals is calculated and the local aggregation of the solu-tion set is avoided by a certain strategyThe crowding distancefunction of the NSGA2 algorithm can be expressed as

PCD (119896) = 1119899119899sum119895=1

119891119894 (119896 + 1) minus 119891119894 (119896 minus 1)119891max119894 minus 119891min

119894

(6)

where PCD(119896) represents the 119896-th individualrsquos crowdingdistance value 119891119894(119896 + 1) represents the 119896 + 1-th individualrsquosthe 119894-th objective function value and the 119891max

119894 and the119891min119894 separately represents the 119894-th objective function valuersquos

maximum and minimum value in the solution set Themathematical expression of the crowding distance functioncan be seen that the PCD is the mean value of the Euclideandistance and each objective function of the individual

On calculating the PCD value of each individual we needto notice that the boundary pointrsquos corresponding crowdingdistance function value of each objective function is set as1 From the PCD mathematical expression we can see thatthe value of each individualrsquos PCD value ranges from 0 to1 The purpose of the introduction of PCD concept into theNSGA2 algorithm is to evaluate the congestion of each otherin the same nondominated level A bigger PCD value can tellthe NSGA2 algorithm its corresponding individual is moresparse with the surrounding individualsThe individuals whohave a small PCD value have a very good distribution in theevolutionary population in the NSGA2 algorithm Accordingto the introduction of the PCD function into the NSGA2algorithm the individuals in the Pareto front will have agreat distribution in the Pareto optimal set such that it canguarantee the evenness and diversity of the evolutionarypopulation in the NSGA2 algorithm

313 Environmental Selection Based on PCD In multiobjec-tive evolution algorithm the individuals evolve towards theoptimal solution set depending on a certain evolution strat-egy In the NSGA2 algorithm the population evolutionarystrategy can be concluded as according to fast nondominatedsorting strategy to determine each individualrsquos nondominatedlevel By comparing the PCD value and each individualrsquosnondominated level to determine the elite individuals fromthe current population and former population and save theelite individuals to the next generation and then the evolu-tion of the dominating individuals in the NSGA2 algorithmis completed In the NSGA2 algorithm the selection of theindividuals to the next generation for evolution follows thefollowing two rules (1) if the nondominating level of anindividual 119901 is higher than 119902 then the former individual ischosen to the next generation (2) if the nondominating level


of an individual119901 is equal to 119902 in this situation the PCDvalueof these two individuals should be compared In the NSGA2algorithm the one with a smaller PCD value will be selectedto the next generation evolution population In the NSGA2algorithm 119875 ≺ 119876 is defined as the following mathematicalform to express the environmental selection strategy

119901 ≺ 119902

119901rank lt 119902rankor

119901rank lt 119902rank PCD (119901) gt PCD (119902) (7)

Through the NSGA2 algorithmrsquos environment selectionstrategy we can see when two individualsrsquo nondominatinglevel is different in this situation the individual with smallernondominating level such that the NSGA2 algorithm canensure the elite population on the contrary when twoindividualsrsquo nondominating level is the same choosing theindividual with larger crowding distance value can ensure thediversity of population such that the evolution speed of theNSGA2 algorithm can be accelerated

314 NSGA2 Algorithm Calculation Steps

Step 1 Randomly initialize the initial population and eachparameterrsquos value in the NSGA2 algorithm and then set theiteration number as 1 then start the algorithmrsquos iterationcalculation

Step 2 Determine if the iteration number right now attainsthe biggest iteration times set at Step 1 if the former is biggerthan the latter then end the iteration calculation and outputthe Pareto optimal set if the former is not bigger than thelatter then continue the iteration calculation until the twonumbers are equal to each other

Step 3 Use the fast nondominated sorting strategies toperform the evolutionary population And then perform thegenetic operation like selection crossover and mutation tothe evolutionary population to form the new generationwhich is called 119876 in this paper

Step 4 Combine the 119875 population and the 119876 population asthe new mixture population which is called 119877 and performthe fast nondominated sorting strategy to each individual inthe119877population then calculate the crowding distance of eachindividual in the 119877 population and sort them

Step 5 Retain the elite individuals as the new generation ofthe evolutionary population which we call it as 119875119905+1 in ourpaper

Step 6 Add 1 to the iteration number 119894 = 119894 + 1 and oneiteration calculation is done then go to Step 2 and continuethe rest of the iteration calculation

32 Improvement Strategy Although the NSGA2 algorithmis largely and successfully used in many fields as an excellentalgorithm it still has many drawbacks to be settled In recentyears with the heuristic search algorithm being largely used

in many aspects many researchers have found many evolu-tionary algorithms having poor local search ability problemwhich is very important for the algorithmrsquos calculationperformance The algorithm with a better local search abilitycan use a more precise way to seek the optimal solution of theoptimization problem to be settled A better individual canbe found by searching around a nondominated individualSo searching around the nondominated individuals for betterindividual is needed The studies show that the local searchability of the original NSGA2 algorithm is poor especiallywhen it is close to the Pareto frontier at the end of the evolu-tionary process the evolutionary efficiency is greatly reducedTo enhance the local search ability the local search strategy isadded into the NSGA2 algorithm after the next evolutionarypopulation is generatedThe local search strategy is exploringan individual of the evolutionary population to determinewhether a better performing solution can be foundThere aretwo types of thementioned solutionwith better performance(1) the new individual dominates the original (2) the newand the original individuals do not dominate each other butthe fitness of the new individual is better than the originalone So for an individual 119909 search for a new individual 1199091015840 inits neighborhood if 1199091015840 which is a new individual searchedout from individual 1199091015840s neighborhood dominates 119909 or itscorresponding fitness is better replace the latter otherwiseproceed with the local search

The specific steps of the local search strategy is as follows

Step 1 Set the local search iteration time 119894 = 1 and start localsearch calculating

Step 2 For each solution1199091015840 in the Pareto optimal set generatea new feasible solution 119909 around it and calculate its fitness

Step 3 If 1199091015840 dominates 119909 or fitness(1199091015840) lt fitness(119909) thenreplace 119909 with 1199091015840 if not maintain the 119909 unchangedStep 4 119894 = 119894 + 1Step 5 If 119894 = 119873119888 stop local search calculation and output thePareto optimal set to the NSGA2 algorithm if not continuethe local search calculation

4 Simulation Works

41 Parameter Selection There are a lot of parameters inNSGA2 algorithms such as population size iteration timescross probability and mutation probability The value ofthese parameters has a great influence on the NSGA2 algo-rithmrsquos calculation performance So when we use the NSGA2algorithm the parameter value selection becomes a veryimportant problem For some complicated multiobjectiveproblem a small population size cannot help NSGA2 algo-rithm achieve a very ideal Pareto optimal set A very largepopulation size can slow down the calculation speed of theNSGA2 algorithm although it can get a good Pareto optimalset So how to select the population size value according tothe complexity of the multiobjective problem is a problem tobe further researched Right now the population size value isselected by the experience of the usersThe researcher adjusts


the population value step by step until the Pareto optimalset is stable The NSGA2 algorithm is a variation of the GAalgorithm they have many things in common although thefirst one is multiobjective algorithm while the second oneis single objective algorithm So the NSGA2 is also uses aiterative way to calculate the proposed problem Then a veryimportant parameter must be valued first which is iterationtimes A very large iteration time can help the NSGA2algorithm gain a very good Pareto optimal set but it can alsoenlarge the calculation time which is bad to the algorithmusers While a small iteration times cannot guarantee thatthe NSGA2 algorithm achieves an ideal Pareto optimal setit can help the NSGA2 algorithm calculate very fast Manyresearchers now also use a step by step method to adjust theiteration timersquos value to select it If the Pareto optimal setwill not change a lot in the NSGA2 algorithm the iterationvalue can be used as the final value of this parameter of theNSGA2 algorithm The genetic manipulation includes selec-tion crossover and mutation operations [6 7] which is veryimportant in the NSGA2 algorithm In the NSGA2 algorithmthe genetic operations are the main way of evolution whichdirectly determine the outcome performance In the geneticoperations there are a lot of parameters which can change thegenetic evolution performance of the NSGA2 algorithm Sothe genetic parameters is also very important and is key to theNSGA2 algorithmrsquos calculation performance In the NSGA2algorithm the tournament selection is used in the selectionprocess whose parameter is called tournament value Thetournament value determines the selection pressure thebigger the value is the bigger the selection pressure is Inthe NSGA2 algorithm the tournament value is suggestedto set 2 The NSGA2 algorithm crossover process uses akind of simulated binary crossover method to cross-overtwo individuals There is also a parameter in the crossoverprocess which is called crossover distribution parameterThevalue of the crossover distribution parameter determines theprobability of the likelihood of offspring and father indi-vidual The bigger the crossover distribution parameter thebigger the probability of the likelihood of offspring and fatherindividual This paper sets the value of the crossover distri-bution parameter as 20 In the NSGA2 algorithm a kind ofpolynomial mutation is used in the mutation process whoseparameter inside is called mutation distribution parameterThemutation distribution parameter is more or less the sameas the crossover distribution parameter In our paper thevalue of the mutation distribution parameter is set as 20 Asthe local search strategy is added in theNSGA2 algorithm thelocal search iteration time should be valued In this paper thevalue of the local search iteration time is set as 10

42 Simulation In order to research the improvement effectto NSGA2 this paper selects some classic multiobjectiveproblem for simulation The first problem is called KURwhich is MISA problem The mathematical formulation is asfollows

1198911 = minus10 exp (minus02 lowast sqrt (1199092 + 1199102))minus 10 exp (minus02 lowast sqrt (1199102 + 1199112))

M-NSGA2NSGA2

minus12

minus10

minus8

minus6

minus4

minus2

0

2

f1

minus19 minus18 minus17 minus16 minus15 minus14minus20f2

Figure 1 Two Pareto fronts of KUR problem

1198912 = |119909|08 + 5 lowast sin (1199093) + 1003816100381610038161003816119910100381610038161003816100381608 + 5 lowast sin (1199103)+ |119911|08 + 5 lowast sin (1199113)

(8)

where 119909 119910 and 119911 are the control variables and their domainsare all [minus5 5]

Figure 1 shows the NSGA2rsquos and the I-NSGA2rsquos Paretofronts of this problem FromFigure 1we can see the differenceof the two algorithms Pareto fronts the distribution of I-NSGA2rsquos Pareto fronts is more uniform than the NSGA2algorithm Furthermore the I-NSGA2rsquos Pareto fronts is closerto the coordinate origin So the I-NSGA2 can get a betterPareto optimal solution than the original one

To further study and explain the difference of the twooutcomes we can conclude that just because of the localsearch strategy when the NSGA2 algorithm finds a Paretoset solution in each iterative calculation the local search canbegin a deeper and more accurate search for better solutionand this is why theM-NSGA2rsquos Pareto set is more close to thecoordinate origin point than the original algorithm

The second simulation case is also a classic multiobjectiveproblem called ZDT3 problem The ZDT3 problem can bemathematically expressed as follows

min 1198911 (119909) = 1199091min 1198912 (119909) = 119892[1 minus radic1198911119892 minus 1198911119892 sin (101205871198911)]

119892 = 1 + 9 119899sum119894=2

119909119894119899 minus 1 (9)

where 0 lt 119909119894 lt 1 and 119899 = 30The comparison of the two different NSGA2 algorithms

in the ZDT3 problems is shown in Figure 2 From Figure 2we can clearly see the difference of the two Pareto fronts


M-NSGA2NSGA2

minus08

minus04

00

04

08

12

f2

02 04 06 08 1000f1

Figure 2 Two Pareto fronts of ZDT3 problem

The Pareto front of the M-NSGA2 algorithm can concludethe second one which means the first Pareto front is closerto the real Pareto front of the ZDT3 problem By comparingthis casersquos results with the first case we also can see themore complex the problem is the more the superiority of theM-NSGA2 will be obvious Seeing this comparison we canconclude that the modified strategy in M-NSGA2 algorithmcan improve the optimization effect of the NSGA2 algorithm

5 Conclusion

This paper introduced a new improvement strategy toNSGA2 algorithm which uses a local search way to searcha better solution around the optimized solution of NSGA2algorithm In this paper we firstly researched the basicNSGA2 algorithm from three different but important partsthe fast nondominating sorting the crowding distance func-tion and the environmental selection based on PCD Afterthat we introduced the NSGA2 algorithm calculation stepsfor the multiobjective problem After the introduction ofNSGA2 algorithm we also introduced the local searchimprovement for the basic NSGA2 problem and formed theM-NSGA2 algorithm In the end in order to test effect ofthe improvement in NSGA2 algorithm we use two mul-tiobjective problems to simulate the two algorithms Fromthe comparison figures we can see the difference of the twoalgorithms calculation results

Disclosure

Ruihua Wang is a PhD candidate in School of Traffic andTransportation of Beijing Jiaotong University and a lecturerinManagement EngineeringDepartment of ZhengzhouUni-versity

Competing Interests

The author declares that they have no competing interests

References

[1] C Razvan G Lucian and M Ioan ldquoAn OOP MATLAB exten-sible framework for the implementation of genetic algorithmsPart I the frameworkrdquo in Proceedings of the 8th InternationalConference Interdisciplinarity in Engineering (INTER-ENG rsquo14)vol 19 pp 193ndash200 Targu Mures Romania 2015

[2] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer New York NY USA2011

[3] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[4] M Eusuff K Lansey and F Pasha ldquoShuffled frog-leaping algo-rithm a memetic meta-heuristic for discrete optimizationrdquoEngineering Optimization vol 38 no 2 pp 129ndash154 2006

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

[6] Y-B Park J-S Yoo and H-S Park ldquoA genetic algorithm forthe vendor-managed inventory routing problemwith lost salesrdquoExpert SystemswithApplications vol 53 no 1 pp 149ndash159 2016

[7] P Ghosh M Mitchell J A Tanyi and A Y Hung ldquoIncorpo-rating priors for medical image segmentation using a geneticalgorithmrdquo Neurocomputing vol 195 pp 181ndash194 2016

[8] Y Liu H Dong N Lohse and S Petrovic ldquoA multi-objectivegenetic algorithm for optimisation of energy consumption andshop floor production performancerdquo International Journal ofProduction Economics vol 179 pp 259ndash272 2016

[9] M A Salido J Escamilla A Giret and F Barber ldquoA geneticalgorithm for energy-efficiency in job-shop schedulingrdquo TheInternational Journal of Advanced Manufacturing Technologyvol 85 no 5 pp 1304ndash1314 2016

[10] O Abdelkhalek S Krichen and A Guitouni ldquoA genetic algo-rithm based decision support system for the multi-objectivenode placement problem in next wireless generation networkrdquoApplied Soft Computing Journal vol 33 no 8 pp 278ndash291 2015

[11] M Salami and T Hendtlass ldquoA fitness estimation strategy forgenetic algorithmsrdquo Lecture Notes in Computer Science vol2358 pp 502ndash513 2002

[12] E E Korkmaz ldquoMulti-objective genetic algorithms for group-ing problemsrdquo Applied Intelligence vol 33 no 2 pp 179ndash1922010

[13] M Kaya ldquoAutonomous classifiers with understandable ruleusing multi-objective genetic algorithmsrdquo Expert Systems withApplications vol 37 no 4 pp 3489ndash3494 2010

[14] M A Abido ldquoMultiobjective optimal VAR dispatch usingStrength Pareto Evolutionary Algorithmrdquo in Proceedings of theIEEECongress on Evolutionary Computation pp 730ndash736 2002

[15] M Eghbal N Yorino E E El-Araby andY Zoka ldquoOptimal VArexpansion considering ATC using strength pareto evolutionaryalgorithmrdquo in Proceedings of the 40th North American PowerSymposium (NAPS rsquo08) pp 550ndash555 IEEE Calgary Canada2008

[16] 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


[17] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improving thestrength Pareto evolutionary algorithmrdquo Eurogen vol 3242 no103 pp 95ndash100 2001

[18] M Kim T Hiroyasu M Miki et al ldquoSPEA2+ improving theperformance of the strength Pareto evolutionary algorithm 2rdquoin Parallel Problem Solving from NaturemdashPPSN VIII 8th Inter-national Conference Birmingham UK September 18ndash22 2004Proceedings vol 3242 pp 742ndash751 Springer Berlin Germany2004

[19] KDeb S Agrawal A Pratap et al ldquoA fast elitist non-dominatedsorting genetic algorithm for multi-objective optimizationNSGA-IIrdquo in Parallel Problem Solving fromNature PPSNVI 6thInternational Conference Paris France September 18ndash20 2000Proceedings vol 1917 of Lecture Notes in Computer Science pp849ndash858 Springer Berlin Germany 2000

[20] B Milosevic and M Begovic ldquoNondominated sorting geneticalgorithm for optimal phasor measurement placementrdquo IEEETransactions on Power Systems vol 18 no 1 pp 69ndash75 2003

[21] S Kuriakose andM S Shunmugam ldquoMulti-objective optimiza-tion of wire-electro discharge machining process by non-domi-nated sorting genetic algorithmrdquo Journal of Materials ProcessingTechnology vol 170 no 1-2 pp 133ndash141 2005

[22] D Kanagarajan R Karthikeyan K Palanikumar and J PDavim ldquoOptimization of electrical discharge machining char-acteristics of WCCo composites using non-dominated sort-ing genetic algorithm (NSGA-II)rdquo International Journal ofAdvancedManufacturing Technology vol 36 no 11-12 pp 1124ndash1132 2008

[23] S Panda ldquoMulti-objective PID controller tuning for a FACTS-based damping stabilizer using non-dominated sorting geneticalgorithm-IIrdquo International Journal of Electrical Poweramp EnergySystems vol 33 no 7 pp 1296ndash1308 2011

[24] S Panda and N K Yegireddy ldquoAutomatic generation control ofmulti-area power system using multi-objective non-dominatedsorting genetic algorithm-IIrdquo International Journal of ElectricalPower amp Energy Systems vol 53 no 1 pp 54ndash63 2013

[25] S R M Yandamuri K Srinivasan and S Murty BhallamudildquoMultiobjective optimal waste load allocation models for riversusing nondominated sorting genetic algorithm-IIrdquo Journal ofWater Resources Planning and Management vol 132 no 3 pp133ndash143 2006

[26] L Wang T-G Wang and Y Luo ldquoImproved non-dominatedsorting genetic algorithm (NSGA)-II in multi-objective opti-mization studies of wind turbine bladesrdquo Applied Mathematicsand Mechanics vol 32 no 6 pp 739ndash748 2011

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Stochastic AnalysisInternational Journal of


algorithm [19] and so on In this way many multiobjectivegenetic algorithms are proposed in these years whichmade agreat contribution to thesemultiobjective optimization prob-lems

Nondominated sorting genetic algorithm 2 (which wecall it as the NSGA2 algorithm in the rest of this paper) isthe improved version of the nondominating sorting geneticalgorithm which is first proposed by Deb in 2001 Duringthese years development and use the NSGA2 algorithmhas been proved as a very good multiobjective geneticalgorithm which has been successfully used in many multiobjective optimization problems [20ndash26] because the calcu-lation performance of one algorithm varies very much whendealing with different optimization problems Beside thisthe algorithmrsquos calculation performance also depends on theoptimizing strategy inside of the algorithm itself So a lot ofresearchers also proposed some improvement strategies intothe NSGA2 algorithm to deal with the shortcomings whenoptimizing some optimization problems In this paper inorder to overcome the poor local search ability problem inthe NSGA2 algorithm a local search strategy is joined intothe NSGA2 algorithm to enhance the local search abilityThe general idea of the proposed improvement strategy isto search locally around the optimization solution found sofar to find a better solution such that the algorithm can geta better local search ability and can make a more detailedsearch In order to see whether the local search ability canimprove the calculation performance of the original NSGA2algorithm or not this paper also introduce two classic multi-objective optimization problems to let the modified NSGA2algorithm to simulate At last the outcome of the simulationworks is compared to the original NSGA2 algorithm Thecomparison shows the local search ability can really improvethe calculation performance of the NSGA2 algorithm

The rest of this paper is organized as follows Section 2describes themathematicalmodel of themultiobjective prob-lem Section 3 shows the principles and steps of the NSGA2algorithm where we introduce three aspects to describethe original NSGA2 algorithm fast nondominate sortingcrowding distance function and environmental selectionbased on PCD At the end of Section 3 the basic stepsof the original NSGA2 algorithm are introduced to thereaders The simulation works and the results analysis ofdifferent multiobjective optimization problem are given inSection 4 where we introduced two classic multiobjectiveoptimization problems Section 5 draws the conclusion of thispaper

2 Multiobjective Problem

A multiobjective problem can be seen as a problem whichshould optimize at least two objectives while it satisfies a lotof constrains A multiobjective optimization problem can bemathematically described as

Min 119865 = 1198911 (119883) 1198912 (119883) 119891119896 (119883)St 119892119894 (119883) le 0 119894 = 1 2 119899

119883 = (1199091 1199092 119909119898) isin 119877119898(1)

where 119891119894(119909) represents the 119894-th objective function 119883 repre-sents the control variables vector 119899 represents the number ofconstrains and 119896 is the total number of the objective to beoptimized

But the purpose of the decision scheme of the multiob-jective optimization problem is not to make each objectiveto be optimized optimal but to provide a series of solutionsfor the decision makers In accordance with their owndecision intention the decision makers choose the decisionscheme to make decisions A series of optimal solution setof multiobjective optimization problems found in the endis called the Pareto optimal solution set The mathematicalexplanation of the Pareto optimal solution set can be shownas follows

119875 = 119909lowast isin Ω | notexist119909119896 isin Ω 119909119896 ≺ 119909lowast (2)

where the 119909lowast represents the Pareto optimal solutions whosemathematical expression can be shown as follows

notexist119909119895 isin Ω 119909lowast ≺ 119909119895 (3)

In (3) the expression 1199091 ≺ 1199092 represents the individual 1199091dominating the individual 1199092 For a minimization optimiza-tion problem a solution dominating another solution in anmultiobjective can be mathematically is expressed as follows

forall119894 isin 1 2 119873 119891119894 (1199091) le 119891119894 (1199092)exist119895 isin 1 2 119873 119891119895 (1199091) lt 119891119895 (1199092) (4)

From (4) we can conclude that if a solution 1199091rsquos eachobjective value in a multiobjective problem is not bigger thana1199092rsquos corresponding objective value and at least one objectivecan be found the 1199091rsquos value is smaller than the 1199092rsquos then wecan say that the solution 1199091 dominates 1199092

The multiobjective optimization problem is exactlyaccording to seeking the nondominating solution and usesome kind of evolutionary strategy to evolve the evolutionarypopulation to finally find the Pareto optimization set ofthe multiobjective optimization problem to be optimizedIn order to more intuitively demonstrate the value of eachobjective value in the Pareto optimal solution set and its dis-tribution in the solution space the Pareto front is introducedThe Pareto front is a line or a curved surface formed by theobjective values of each solution in the Pareto optimal setwhose mathematical expression is as follows

PF = 119865 (119909) = (1198911 (119909) 1198912 (119909) 119891119898 (119909)) | 119909 isin 119875 (5)

3 Improved NSGA2 Algorithm

31 NSGA2Algorithm TheNSGA2 algorithm is themodifiedversion of NSGA by Deb and other researchers after severalyears of development and use the NSGA2 algorithm is seenas a very good and mature multiobjective optimization algo-rithm at present In NSGA2 algorithm the crowding distancefunction is used to calculate each individualrsquos fitness value inthe algorithm According to the fast nondominating sortingidea and environmental selection of PCD the algorithm






PCD (119896) = 1119899119899sum119895=1


119894

(6)








119901 ≺ 119902

















4 Simulation Works






M-NSGA2NSGA2

minus12

minus10

minus8

minus6

minus4

minus2

0

2

f1




(8)






119892 = 1 + 9 119899sum119894=2

119909119894119899 minus 1 (9)




M-NSGA2NSGA2

minus08

minus04

00

04

08

12

f2

02 04 06 08 1000f1



5 Conclusion


Disclosure


Competing Interests


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














PCD (119896) = 1119899119899sum119895=1


119894

(6)








119901 ≺ 119902

















4 Simulation Works






M-NSGA2NSGA2

minus12

minus10

minus8

minus6

minus4

minus2

0

2

f1




(8)






119892 = 1 + 9 119899sum119894=2

119909119894119899 minus 1 (9)




M-NSGA2NSGA2

minus08

minus04

00

04

08

12

f2

02 04 06 08 1000f1



5 Conclusion


Disclosure


Competing Interests


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901 ≺ 119902

















4 Simulation Works






M-NSGA2NSGA2

minus12

minus10

minus8

minus6

minus4

minus2

0

2

f1




(8)






119892 = 1 + 9 119899sum119894=2

119909119894119899 minus 1 (9)




M-NSGA2NSGA2

minus08

minus04

00

04

08

12

f2

02 04 06 08 1000f1



5 Conclusion


Disclosure


Competing Interests


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













M-NSGA2NSGA2

minus12

minus10

minus8

minus6

minus4

minus2

0

2

f1




(8)






119892 = 1 + 9 119899sum119894=2

119909119894119899 minus 1 (9)




M-NSGA2NSGA2

minus08

minus04

00

04

08

12

f2

02 04 06 08 1000f1



5 Conclusion


Disclosure


Competing Interests


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










M-NSGA2NSGA2

minus08

minus04

00

04

08

12

f2

02 04 06 08 1000f1



5 Conclusion


Disclosure


Competing Interests


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article An Improved Nondominated Sorting Genetic...

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