Memetic Algorithm: Hybridization of Hill Climbing with Selection Operator

Transcription

1 Iteratioal Joural of Soft Computig ad Egieerig (IJSCE) ISSN: , Volume-3, Issue-2, May 2013 Memetic Algorithm: Hybridizatio of Hill Climbig with Selectio Operator Rakesh Kumar, Sajay Tyagi, Maju Sharma Abstract Geetic Algorithms are the populatio based search ad optimizatio techique that mimic the process of atural evolutio. Premature Covergece ad geetic drift are the iheret characteristics of geetic algorithms that make them icapable of fidig global optimal solutio. A memetic algorithm is a extesio of geetic algorithm that icorporates the local search techiques withi geetic operatios so as to prevet the premature covergece ad improve performace i case of NP-hard problems. This paper proposes a ew memetic algorithm where hill climbig local search is applied to each idividual selected after selectio operatio. The experimets have bee coducted usig four differet bechmark fuctios ad implemetatio is carried out usig MATLAB. The fuctio s result shows that the proposed memetic algorithm performs better tha the geetic algorithm i terms of producig more optimal results ad maitais balace betwee exploitatio ad exploratio withi the search space. Idex Terms bechmark fuctios, hybrid geetic algorithms, hill climbig, memetic algorithms. I. INTRODUCTION Geetic algorithms are the search techique based o the evolutioary ideas of atural selectio ad geetics [1]. Geetic algorithms use the priciples ispired by atural populatio geetics to evolve solutios to problems. They follow the priciple of survival of fittest [2], for better adaptatio of species to their eviromet. For more tha four decades, they have bee applied o wide rage of optimizatio problems. The performace of geetic algorithms depeds o the balacig betwee the exploitatio ad exploratio techiques. Exploitatio meas to use the already available kowledge to fid out the better solutio ad Exploratio is to ivestigate ew ad ukow area i search space. The power of geetic algorithms comes from their ability to combie both exploratio ad exploitatio i a optimal way [3]. Geetic algorithms are ispired from biological geetics model ad most of its termiology has bee borrowed from geetics. Each allele has a uique positio o chromosome called locus. Geetic algorithm uses a iterative process to create a populatio. The algorithm stops, whe the populatio coverges towards the optimal solutio. It cosists of followig steps:- Mauscript received o May, Rakesh Kumar, DCSA, Kurukshetra Uiversity, Kurukshetra, Haryaa, Idia. Sajay Tyagi, DCSA, Kurukshetra Uiversity, Kurukshetra, Haryaa, Idia. Maju Sharma DCSA, Kurukshetra Uiversity, Kurukshetra, Haryaa, Idia. INITIALIZATION: Radomly geerate a populatio of N chromosomes. SELECTION: Idividuals are selected to create mate pool for reproductio accordig to selectio methods. REPRODUCTION: Crossover ad mutatio operators applied o the mate pool idividuals. REPLACEMENT: Idividuals from old populatio are replaced by ew oes accordig to replacemet strategies. I practice, the populatio size is fiite that iflueces the performace of geetic algorithm ad leads to the problem of geetic drift that occurs mostly i case of multimodal search space. Icorporatig a local search method withi the geetic operators ca itroduce ew gees tha ca overcome the problem of geetic drift ad accelerate the search towards global optima [4]. A combiatio of geetic algorithm ad a local search method is called as hybrid geetic algorithm or memetic algorithm. I hybrid geetic algorithms, kowledge ad local search ca be icorporated at ay stage like iitializatio, selectio, crossover ad mutatio. This paper icorporates hill climbig based local search after selectio step, the ew algorithm is proposed called as hybrid geetic ad hill climbig algorithm (HGHCA).The proposed HGHCA is compared with Geetic algorithm(ga) o stadard bechmark multimodal fuctios. The paper is orgaized i the followig sectios. I sectio 2, literature review is give o differet researches related to hybrid geetic algorithms. I sectio 3, memetic algorithm approach ad hill climbig search alog with their pseudo codes are discussed. I sectio 4, bechmark test fuctios cosidered for implemetatio are described. Implemetatio details ad computatioal results are specified i sectio 5 ad coclusio ad future work are give i sectio 6. II. RELATED WORK Hollad [3] ad David Goldberg [1] by usig k armed badit aalogy showed that both exploratio ad exploitatio are used by geetic algorithm at the same time. Due to certai parameters, it has bee observed that, stochastic errors occur i geetic algorithm that leads to geetic drift [5,6]. Rakesh Kumar et al. proposed a ovel crossover operator that uses the priciple of Tabu search. They compared the proposed crossover with PMX ad foud that the proposed crossover yielded better results tha PMX [7]. H.A. Sausi et al. ivestigated the performace of geetic algorithm ad memetic algorithm for costraied optimizatio kapsack problem. The aalysis results showed that memetic algorithm coverges faster tha geetic algorithm ad produces more optimal result [8]. A comparative aalysis of memetic algorithm based o hill climbig search ad geetic algorithm has bee performed for 140

2 Memetic Algorithm: Hybridizatio of Hill Climbig with Selectio Operator the cryptaalysis o simplified data ecryptio stadard problem by Pooam Garg [9].She cocluded that memetic algorithm is superior for fidig umber of keys tha geetic algorithms. Atariksha [10], proposed a hybrid geetic algorithm based o GA ad Artificial Immue etwork Algorithm (GAIN) for fidig optimal collisio free path i case of mobile robot movig i static eviromet filled with obstacles. She cocluded that GAIN is better for solvig such kid of problems. E.Burke et al. proposed a memetic algorithm that based o Tabu search techique to solve the maiteace schedulig problem. The proposed MA performs better ad ca be usefully applied to real problems [11]. Mali et al [12] proposed a memetic algorithm for feature selectio i volumetric data cotaiig spatially distributed clusters of iformative features i eurosciece applicatio. They cocluded that the proposed MA idetified a majority of relevat features as compared to geetic algorithm. III. METHODOLOGY A. Memetic Algorithm Approach Icorporatig problem specific iformatio i a geetic algorithm at ay level of geetic operatio form a hybrid geetic algorithm [13].The techique of hybridizatio of kowledge ad global geetic algorithm is memetic algorithm (MA). MA is motivated by Dawkis otatio of a meme. A meme is a uit of iformatio that reproduces itself as people exchage ideas [14]. MA bids the fuctioality of GA with several heuristic s search techiques like hill climbig, simulated aealig, Tabu search etc. A umber of issues should be carefully addressed whe a effective hybrid geetic algorithm is costructed.two popular ways of hybridizatio depeds o the cocepts of Baldwi effect [15] ad Lamarckism [16]. Accordig to Baldwiia search strategy, the local optimizatio ca iteracts ad allow the local search to chage the fitess of idividual but geotype itself remai uchaged. The disadvatage of Baldwiism is that it is slow. Accordig to Lamarckism, the characteristics acquired by idividual durig its lifetime may become heritable traits. Accordig to this approach both the fitess ad geotype of idividuals are chaged durig local optimizatio phase. Most of the MA is based o Lamarckism approach of hybridizatio. The proposed memetic algorithm icorporates hill climbig local search after selectio process i order to icrease exploitatio. I the proposed approach, members selected usig roulette wheel selectio has bee used as iitial poit to carry out hill climbig search. I this approach, each idividual is improved usig hill climbig before passig to reproductio phase. Algorithm 1: Proposed Hybrid Geetic Algorithm Procedure MA(fitfx, psize, Pc, Pm) // fitfx fitess fuctio to evaluate chromosome // psize size of populatio i each geeratio // Pc crossover probability // Pm mutatio probability // mxge maximum umber of geeratios ecode solutio space Iitialize populatio ge=1 while (ge <= mxge) evaluate (mi(fitfx)) for i = 1 to psize mate1, mate2=select(populatio) // apply local search to each selected idividual optmate1=hill climbig (mate1) optmate2=hill climbig (mate2) If ( rd(0,1)<=pc) child = crossover(optmate1, optmate2) ed If If ( rd(0,1)<=pm) mchild = mutatio(child) Ed If Ed for Add offsprig to ew geeratio ge=ge+1 Ed while retur best chromosomes B. Hill climbig local search Hill climbig is a optimizatio algorithm for sigle objective fuctio. I hill climbig algorithm a loop is performed i which the curretly kow best idividual produce oe offsprig. If the fitess of ew idividual is better tha paret it replaces it, else stop the loop. Algorithm 2: Hill Climbig Algorithm Procedure Hill climbig (paret) //paret curretly kow best solutio While (termiatio criteria is ot specified) do New_solutio < eighbors (paret) If (New_Solutio is better tha paret) Paret = New_solutio Ed If Ed While retur best solutio IV. TEST FUNCTIONS Differet researchers have used differet fuctio group to evaluate the geetic algorithm performace. I this paper, the authors examie four differet bechmark multimodal fuctios i order to study the performace of purposed memetic algorithm. Fuctio F1 F2 F3 F4 Table 1 Bechmark Test Fuctios Rastrigi s fuctio (F1) is highly multimodal ad has a complexity of O(.log()), where is the umber of fuctio parameters. It has several local miima [17]. Fuctio defiitio: 1(x) = 10* <=x i <=5.12 global miimum: f(x)=0, x i =0 Name Rastrigi s Fuctio Schwefel s Fuctio Ackley s Fuctio Griewagk s Fuctio [x i 2-10*cos(2. x i )] 141

3 Iteratioal Joural of Soft Computig ad Egieerig (IJSCE) ISSN: , Volume-3, Issue-2, May 2013 Griewagk s Fuctio (F4) is similar to Rastrigi ad has may wide spread regularly distributed local miima[18]. Fuctio defiitio: f4(x)= 1/ <=x i <=600 2 x i - global miimum: f(x)=0,x i =0 cos (x i / i ) + 1 Fig. 1 Fuctio Graph for F1 for =2 I Schwefel s fuctio (F2), the global miimum is geometrically distat over parameter spaces, from ext best local optima [18].Therefore search algorithms are proe to coverge i wrog directio. Fuctio defiitio: 2(x) = [-x.si( i x )] -500<=x i <=500 global miimum: x i = f(x)= Fig. 4 Fuctio Graph for F1 for =2 V. IMPLEMENTATION & OBSERVATIONS I this paper, MATLAB code has bee developed to fid the performace of proposed memetic algorithm ad geetic algorithm. The code cosiders the group of four bechmark multimodal fuctios ad uses the same iitial populatio, same crossover ad mutatio probability i both selectio cases to compare the performace of geetic algorithm (GA) with the proposed memetic algorithm (MA). Mi ad Average value of fuctios is computed for 100 & 50 geeratios ad plotted to compare the result of two approach. Fig. 2 Fuctio Graph for F2 for =2 Ackley s Fuctio (F3) is cotiuous multimodal fuctio obtaied by modulatig a expoetial fuctio with a cosie wave of moderate amplitude [17] Fuctio defiitio: 3(x) =-a. e(-b. 1/ x i 2 ) e(1/ <=x i <= a=20, b=0.2, c=2*p global miimum: f(x)=0, x i =0 cos( c. )) +a +e x i Fig. 3 Fuctio Graph for F3 for =2 Fig. 5 Compariso of Miimum fitess value of f(x) i F1 142

4 Memetic Algorithm: Hybridizatio of Hill Climbig with Selectio Operator This sectio cotais the result of code rus. Compariso of two algorithms is based o their respective fuctio values. Parameters used for implemetatio are- Populatio size (N): 5, 10, 15, 20 Number of geeratios (Ge): 50, 100 Ecodig: Value ecodig Selectio: Roulette wheel selectio Crossover operator: Arithmetic crossover operator Mutatio: Creep Mutatio Crossover probability (p c =0.7) Mutatio probability (p m =0.01) Fig. 8 Compariso of Average fitess value of f(x) i F2 Fig. 6 Compariso of Average fitess value of f(x) i F1 Fig. 9 Compariso of Miimum fitess value of f(x) i F3 Fig. 7 Compariso of Miimum fitess value of f(x) i F2 Fig. 11 Compariso of Miimum fitess value of f(x) i F4 143

5 Iteratioal Joural of Soft Computig ad Egieerig (IJSCE) ISSN: , Volume-3, Issue-2, May 2013 balace betwee exploitatio ad exploratio is maitaied. The local search operator geerates more meaigful buildig blocks that help the geetic algorithms i makig small moves withi a defied search area. It has bee foud that with icreasig problem size, problem did ot coverge prematurely ad the same tred of results is observed i each problem size. Fig. 10 Compariso of Average fitess value of f(x) i F3 Fig. 12 Compariso of Average fitess value of f(x) i F4 Results for Miimum as well as Average fitess value usig differet populatio size or geeratio i four bechmark multimodal fuctios are give i Table 2, Table 3, Table4 ad Table 5. Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 ad Figure 12 shows the performace curve of two algorithms for 100 geeratios. It has bee observed that the proposed memetic algorithm has outperformed geetic algorithm i terms of covergece ad optimal solutio. The proposed MA maitais more diversity i populatio ad prevet algorithm to stick i local optima ad geetic drift problem. By applyig the local search operatio after roulette wheel selectio, a good VI. CONCLUSION The paper compares two algorithms amely geetic algorithm ad proposed memetic algorithm o the stadard multimodal bechmark test fuctios. It was foud that the proposed memetic algorithm provides better results tha the geetic algorithm. The proposed algorithm uses the cocept of hill climbig local search after selectio operatio so as to allow the geetic algorithm to improve the exploitig ability of search without limitig its explorig ability. The proposed algorithm improves the performace i terms of covergece ad optimal solutio as well as maitais diversity i the populatio & solves the problem of premature covergece ad geetic drift. The Proposed algorithm ca prove to be better for differet NP Hard problems also. This algorithm ca be tested ad implemeted i differet combiatio of crossover ad iitializatio i future to substatiate its performace. Hybridizatio of selectio ad kowledge of icorporates hill climbig, has icreased the existig geetic algorithm techique ad amplified its search performace. REFERENCES [1] D. E. Goldberg, Geetic algorithm i search ad optimizatio ad machie learig, Addiso Wesley Logma, Ic., ISBN , [2] D. Fogel, Evolutioary computatio, IEEE Press, [3] J. Hollad, Adaptatio i atural ad artificial systems, Uiversity of Michigam Press, A Arbor, [4] W. E. Hart, Adaptive global optimizatio with local search, Doctoral diss., Sa Diego, Uiversity of Califoria, [5] D. E. Goldberg ad P. Segrest, Fiite Markov chai aalysis of geetic algorithms, Proceedigs of 2d Iteratioal Cof. o Geetic Algorithms, Lawrece Erlbaum Associates, 1987, pp 1-8. [6] L.Booker, Improvig search i geetic algorithm, geetic algorithm ad simulated aealig, Pitma, vol 5, 1987, pp [7] Rakesh kumar ad Jyotishree, Novel kowledge based tabu crossover i geetic algorithms, Iteratioal Joural of Advaced research i Computer sciece ad software Egieerig, vol 2, No. 8, Aug 2012, pp [8] H. A. Sasi, A. Zubair ad R. O. Oladele, Comparative assessmet of geetic ad memetic algorithms, Joural of Emergig Treds i Computig ad Iformatio Sciece, vol 2, No. 10, Oct 2011, pp [9] Pooam Garg, A compariso betwee memetic algorithm ad geetic algorithm for the cryptaalysis of simplified data ecryptio stadard algorithm, Iteratioal Joural of Network Security ad its Applicatios,vol 1, No. 1, April 2009, pp [10] Atariksha Bhaduri, A mobile robot path plaig usig geetic artificial immue etwork algorithm, Proceedigs of World Cogress o Nature ad biologically Ispired Computig, NaBIC, IEEE, 2009, pp [11] E. K. Burke ad A. J. Smith, A memetic algorithm for the maiteace schedulig problem, Proceedigs of Iteratioal Cof o Neural Iformatio Processig ad Itelliget Iformatio, Spriger 2010, pp [12] Mali Bjorsdotter ad Joha Wessberg, A memetic algorithm for selectio of 3D clustered featured with applicatios i eurosciece, Proceedigs of Iteratioal Coferece o Patter Recogitio, IEEE, 2010, pp

6 Memetic Algorithm: Hybridizatio of Hill Climbig with Selectio Operator [13] P. Mascato ad P. C. Cotta, A getle itroductio to memetic algorithms, Hadbook of Metaheuristics, 2003, pp [14] R. Dawkis, The selfish gee, Oxford Uiversity Press, Oxford, [15] K. Ku, M. Mak, Empherical aalysis of the factors that affect the Baldwi effect: Parallel problem solvig from ature, Proceedigs of 5 th Iteratioal Coferece o lecture otes i computer sciece, Berli, Spriger, Heidelberg, 1998, pp [16] G. M. Morris, D. S. Goodsell, R. S. Halleday, W. E. Hartad ad R. K. Belew, Automated dockig usig a Lamarckia geetic algorithm ad a empherical bedig free eergy fuctio, Joural of Computatioal Chemistry, vol 19, 1998, pp [17] J. G. Digalakis ad K. G. Margaritis, A experimetal study of Bechmarkig Fuctios for geetic algorithms, Iteratioal Joural of Computer Mathematics, vol 79, No. 4, 2002,pp [18] Marci Molgo ad Czeslaw Smaticki, Test fuctios for optimizatio eeds, kwietia, vol 3, Dr. Rakesh Kumar obtaied his B.Sc. Degree, Master s degree Gold Medalist (Master of Computer Applicatios) ad PhD (Computer Sciece & Applicatios) from Kurukshetra Uiversity, Kurukshetra. Curretly, he is Professor i the Departmet of Computer Sciece ad Applicatios, Kurukshetra Uiversity, Kurukshetra, Haryaa, Idia. His research iterests are i Geetic Algorithm, Software Testig, Artificial Itelligece ad Networkig. He is a seior member of Iteratioal Associatio of Computer Sciece ad Iformatio Techology (IACSIT). Dr. Sajay Tyagi obtaied his Master s degree (Master of Computer Applicatios) ad PhD (Computer Sciece & Applicatios) from Kurukshetra Uiversity, Kurukshetra. Curretly, he is Assistat Professor i the Departmet of Computer Sciece ad Applicatios, Kurukshetra Uiversity, Kurukshetra, Haryaa, Idia. His research iterests are i Geetic Algorithm ad Software Testig. He has preseted 26 papers i Natioal & Iteratioal Cofereces. Ms Maju Sharma obtaied her B. Tech (Computer Sciece ad Egg.) degree from Kurukshetra Uiversity. Curretly, she is pursuig M.Tech (Computer Sciece & Applicatios) from the Departmet of Computer Sciece ad Applicatios, Kurukshetra Uiversity, Kurukshetra, Haryaa, Idia. She has preseted papers i 3 Natioal Cofereces. Her research iterests are Geetic Algorithms, Soft Computig methods ad Software Egieerig. 145