Solving of Single-objective Problems based on a Modified Multiple-crossover Genetic Algorithm: Test Function Study

Transcription

1 Internatonal Conference on Systems, Sgnal Processng and Electroncs Engneerng (ICSSEE'0 December 6-7, 0 Duba (UAE Solvng of Sngle-objectve Problems based on a Modfed Multple-crossover Genetc Algorthm: Test Functon Study M. Andalb Sahnehsarae, M. J. Mahmoodabad, S. Entezar Sahnehsarae Abstract In ths paper, frst a modfed multple-crossover genetc algorthm s proposed. Used operators such as reproducton, crossover and mutaton are ntroduced completely. It s selected some test functons to challenge the ablty of ths algorthm. All test functons are sngle-objectve. Numercal results are presented. The obtaned numercal results are compared wth the results related to three other algorthms appled on the same test functons to valdate the proposed algorthm. Keywords genetc algorthm, multple-crossover, sngleobjectve optmzaton, test functons. I. INTRODUCTION ENETIC Algorthm (GA that belongs to the group of Gevolutonary algorthms s one of the most powerful tools to solve optmzaton problems. In two recent decades, many researchers have been nterested n usng of GA n dfferent felds []. Ths nterest s because of two man advantages avalable n GA n comparson wth other old and tradtonal optmzaton algorthms []. The advantages consst of applyng optmzaton varables n coded forms, and the ablty of GA n the use of a set of solutons, named populaton. After dentfyng the functon that must be optmzed and t s named cost or objectve functon, GA s started by specfyng a set of solutons, named ntal populaton. Each soluton n populaton s called a chromosome. Then, t s assgned a ftness value to each chromosome wth respect to the objectve functon. Now, GA by use of three operators named reproducton, crossover, and mutaton tres to produce a new and better populaton, of course. Ths loop s repeated to obtan optmal solutons. Hence, a GA can end after some certan teraton as the stoppng crteron. Each one of these three operators can be performed n several dfferent methods. Morteza Andalb Sahnehsarae was wth the Unversty of Gulan, Rasht, Iran. He s now wth the Natonal Iranan Gas Company (NIGC, Ramsar, Iran (correspondng author to provde phone: +8665; fax: ; e-mal: morteza_andalb@yahoo.com. Mohammad Javad Mahmoodabad, s wth Unversty of Gulan, Rasht, Iran (e-mal: mahmoodabad@gulan.ac.r. Samaneh Entezar Sahnehsarae s wth the Imam Sajjad Hosptal, Ramsar, Iran (e-mal: entezar.hosptal@yahoo.com. For example, the reproducton operator can be appled n some dfferent approaches that the famous ones are roulette wheel as well as tournament method. Every GA can use an especal combnaton of dfferent operators. But the truth of the matter s that an especal GA never has an absolute domnance relatve to other GAs. All known crossover operators use two chromosomes. In ths work, t has been consdered the ablty of a GA wth the crossover operator that uses three chromosomes. Frst, ths method has used by Lung Chen and Chang [, ]. They appled ths novel crossover operator n ther GA to solve an especal problem. The present paper s gong to challenge and consder the ablty of the proposed GA n solvng of dfferent sngle-objectve problems. Hence, one of the best procedures for ths consderaton s the use of some famous test functons. Here, well-known test functons that all belong to the class of sngle-objectve functons, are selected [5]. Some these functons are unmodal and the others are multmodal. A functon s unmodal f there be exactly one optmum pont. For a multmodal functon, there are some local optmum ponts n addton to one global optmum pont. Furthermore, t s presented three algorthms named, Fast Evolutonary Programmng (FEP, Classc Evolutonary Programmng (CEP [5] and multple-crossover GA [, ]. Each of these three algorthms s run for the same test functons. The obtaned numercal results are compared to evaluate the performance of the proposed method. II. A MODIFIED MULTIPLE-CROSSOVER GA The man dea of evolutonary algorthms was brought up by Rechenberg n 60 s. GA s a branch of these algorthms. John Holland mooted GA n 75 wth nspraton of Darwn s theory about the survval of fttest [6]. Hence, the structure of GA s based on genes and chromosomes. In general, GA s a method to optmze sngle-objectve or multobjectve and constraned or non-constraned problems. As sad n ntroducton, GA s started wth a set of solutons that s selected randomly. Ths set of solutons s called ntal populaton. In general, such set of solutons n every stage of GA s called a populaton. Each soluton n a populaton s a chromosome or an ndvdual. The number of chromosomes n a populaton state populaton sze. One of abltes of GA s to

2 Internatonal Conference on Systems, Sgnal Processng and Electroncs Engneerng (ICSSEE'0 December 6-7, 0 Duba (UAE work wth coded varables. There are several methods for encodng n GA that bnary encodng s the most common among them. In ths method, after producng the ntal populaton, GA must encode all parameters as bnary dgts. But n ths study, t has been used real-coded GA. Now the am of GA s to produce a better populaton n comparson wth before populaton. Ths procedure s done wth respect to ftness value assgned to each chromosome. Ftness value s usually obtaned from puttng each chromosome nto the objectve functon. An objectve functon s the functon that must be optmzed. GA wll succeed to produce a new generaton after performng three stages. Each of stages s done by an especal operator. They are reproducton, crossover, and mutaton. Fg. shows a flow dagram of a realcoded GA. It s descrbed the detals of three used genetc operators n ths study below. smpler the tournament method as well as beng more effcent n convergence than roulette wheel method, t s used tournament mechansm as a reproducton operator. In selecton by tournament method, two chromosomes are selected randomly to wn accordng to ther ftness values. The chromosome wth better ftness value wll be duplcated nto the populaton and the chromosome wth worse ftness value wll be dscarded from the populaton. Ths operaton s done for p r N chromosomes, where p and N are the probablty r of reproducton and populaton sze, respectvely. At the end of selecton operaton, although there wll be a new populaton of better chromosomes than prevous populaton, but t has not produced a new generaton yet. B. Modfed Multple-crossover A tradtonal crossover operator contans only two chromosomes. A novel crossover operator has been proposed n []. Ths novel crossover uses three chromosomes. It has been called multple-crossover. Here, t s presented a modfed multple-crossover wth some changes n ponted multple-crossover formula. Let Γ, Γ, and Γ represent three chromosomes that are selected from the parent populaton. If Γ has the smallest ftness value among ftness values of these three chromosomes the formula of the modfed multple-crossover proposed s stated below., Γ + ( Γ - Γ Γ ( Γ - Γ Γ ( Γ - Γ Γ Γ ρλ - Γ ρ λ - Γ ρλ - Γ +, (. Γ + Fg. Flow dagram of a real-coded GA A. Reproducton Reproducton that s called selecton as well, s usually the frst operaton that s appled on the populaton. Reproducton operaton selects the better ones from a populaton to form a new populaton named parents. There are several selecton mechansms. Two mechansms named roulette wheel and tournament methods are more famous. On account of beng Where ρ ρ,and [0, ] are random values and, ρ λ = Γ Γ, =,, p c. The sgn. denotes the absolute ( N value. It s appled chromosomes to the crossover operaton n ths study, where parameter p s referred to c probablty of crossover. After performng crossover operaton, t s created some new chromosomes or ndvduals named offsprng. But t has not been produced a new generaton yet. C. 7BNew Mutaton After reproducton and crossover, there s a new populaton wth the same amount of chromosomes. Some have been drectly coped and the others have produced by the new multple-crossover. In order to ensure that the chromosomes are not all exactly the same, t must be allowed for a small chance of mutaton. Snce the used GA n ths paper s a realcoded type, t s presented below formula to compute the mutaton for a randomly selected chromosome Γ. ( Γ Γ + s. Φ.

3 Internatonal Conference on Systems, Sgnal Processng and Electroncs Engneerng (ICSSEE'0 December 6-7, 0 Duba (UAE Where both s and Φ are vectors ( n, and n s the number of the dmensons of each chromosome. In the other hand, s s the chromosome that has the smallest ftness value among all chromosomes n the current populaton, and Φ s defned as rand (,n where returns an - by -n matrx contanng pseudorandom values drawn from the standard unform dstrbuton on the open nterval (0,. Fnally, the operator. multples the correspondng members from vectors s and Φ. Lke what was mentoned for reproducton and crossover, p m N chromosomes mutate randomly durng the mutaton procedure, where p m s the probablty of mutaton. After performng three operators, reproducton, crossover, and mutaton by the algorthm, t can be clamed to have a new generaton. Ths loop s so terated untl the algorthm stoppng crteron become satsfed. In ths work, t s wanted algorthm to be stopped after producng a certan number of generatons (G. III. OPTIMIZATION In the most general terms, optmzaton theory s a body of mathematcal results and numercal methods for fndng and dentfyng the best canddate from a collecton of alternatves wthout havng to explctly enumerate and evaluate all possble alternatves [7]. Optmal soluton of an optmzaton problem mnmzes or maxmzes the objectve functon(s. A. Sngle-objectve problems An optmzaton problem s dvded to two classes of sngleobjectve and mult objectve problems wth the pont of the number of objectve functons. Here, t s presented some famous test functons to challenge the ablty of proposed algorthm. All these test functons are sngle-objectve that must be mnmzed. Some essental nformaton about the test functons has been summarzed n Table I. IV. NUMERICAL RESULTS In ths secton, performance of the proposed GA s evaluated. Hence, the results of 50 ndependent runs of the proposed algorthm for test functons are presented. These results are compared wth the smlar results related to three algorthms FEP, CEP and multple- crossover GA. Table II shows ths comparson. The results have been llustrated under the ttles of Mean Best and Std Dev for each method. Mean Best ndcates the mean of the best solutons after 50 ndependent runs and Std Dev stands for the standard devaton. For ease n comparson affar, better results have become boldface for each functon separately. The results related to both FEP and CEP mentoned here are the same exactly comng n [5]. To perform multple-crossover GA and new multple-crossover GA, the proposed method, t has been supposed that p r = 0. 6, p c = 0., p m = 0., N = 00, and G = 00. In the most cases, an optmzaton problem s summarzed to a mnmzaton problem wthout losng the publcty of the problem. TABLE I SINGLE-OBJECTIVE TEST FUNCTIONS Name Comment Dmenson (n Generaton Intal & Search Range Sphere ( f Unmodal 500 [-00,00] n Schwefel. ( f Unmodal 000 [-0,0] n Schwefel. ( f Unmodal 5000 [-00,00] n Schwefel. ( f Unmodal 5000 [-00,00] n Rosenbrock ( f5 Unmodal 0000 [-,] n Step ( f6 Unmodal dscontnuous 500 [-00,00] n Quartc ( f 7 Unmodal nosy 00 [-.8,.8] n Rastrgn ( f8 Multmodal 5000 [-5.,5.] n Ackley ( f Multmodal 500 [-,] n Grewank ( f0 Multmodal 000 [-600,600] n Camel-Back ( f Multmodal 00 [-5,5] n Brann ( f Multmodal 00 [-5,0] [0,5] Golsten-Prce ( f Multmodal 00 [-,] n 5

4 Internatonal Conference on Systems, Sgnal Processng and Electroncs Engneerng (ICSSEE'0 December 6-7, 0 Duba (UAE As t s observed, the proposed method performs much better than three other methods on functons f - f, f and f 0. For step functon f 6, both FEP and proposed method present the best results. For three functons f 5, f and 7 f 8, FEP performs better than others. For functons f and f, there s not a sensble domnance among the algorthms except n ther standard devatons.fnally, for functon f the obtaned results show that three algorthms CEP, multplecrossover GA and modfed multple-crossover GA present the same results for the Mean Best and better than FEP, but CEP has better Std Dev. Accordng to presented results, the proposed GA performs much better than three other methods on all unmodal test functons except two. And t s very good on all multmodal test functons except one. Hence, the proposed modfed multple-crossover GA can be ntroduced as an acceptable method to solve many sngle-objectve optmzaton problems. Improvement n multple-crossover operator s the man reason of beng better the proposed method than multplecrossover GA, although t has been created a slght change n mutaton, too. Thus, the selected name, modfed multplecrossover GA, for ths algorthm can be proper. V. CONCLUSION Ths paper frst proposes a modfed multple-crossover GA that has been mproved on ts multple-crossover and mutaton operators relatve to multple-crossover GA. Then, the performance of ths proposed GA s evaluated on a number of test functons. The obtaned results are compared wth the results of three other algorthms. The comparson shows that the proposed algorthm has acceptable performance and performs much better on the most test functons than three other algorthms. TABLE II NUMERICAL RESULTS FEP CEP Multple Crossover Proposed Method Mean Best Std Dev Mean Best Std Dev Mean Best Std Dev Mean Best Std Dev f f f f f f f f f f f f f

5 Internatonal Conference on Systems, Sgnal Processng and Electroncs Engneerng (ICSSEE'0 December 6-7, 0 Duba (UAE APPENDIX BENCHMARK FUNCTIONS A. Sphere f( = x, -00 x 00 = mn( f = f(0,...,0 = 0. B. Schwefel s. f( = x + x, -0 x 0 = = mn( f = f(0,...,0 = 0. C. Schwefel s. f( = (, = x j x j = mn( f D. Schwefel s. f ( = max x,, E. Rosenbrock { } mn( f -00 x ( x - x + ( x - 00 f5( = 00 +, - x = mn( f5 = f5(,..., = 0. F. F. Step f ( = x + 0.5, -00 x 00 6 = G. Quartc ( mn( f f7 ( = x = H. Rastrgn f8( = ( x = mn( f 6 mn( f x 6 + random[0,, -.8 x cos(πx I. Ackley - f ( = -0 exp (-0. - exp ( = x, cos(πx e mn( f = x J. Grewank x f0( = - cos( 000 x + = = x 600 mn( f 0 0 K. Camel-Back 6 f( = x -.x + x + xx - x + x - 5 x 5 xmn = (0.088, , (0.088, mn( f = L. Brann 5. 5 f( = ( x - x + x - 6 π π + 0 (- cos x + 0 8π - 5 x 0, 0 x 5 xmn = (-.,.75, (.,.75, (.5,.5 mn( f = 0.8. M. Goldsten-Prce f ( = [+ ( x + x + ( -x + 6x x + x ] [ + (x + x - x -x (8 - x + x + 8x - 6xx + 7x ] - x, mn( f (0, =. REFERENCES [] Mtchell, M.,. An ntroducton to genetc algorthms. 5th ed., The MIT Press, Massachusetts. [] Huang, Y.P., and Huang, C.H., 7. Real-valued genetc algorthm for fuzzy grey predcton system. Fuzzy Sets and Systems, 87, pp [] Chang, W.D., 007. A multple-crossover genetc approach to multvarable PID controllers tunng. Expert Systams wth Applcatons,, pp [] Chen, J.L., and Chang, W.D., 00. Feedback lnearzaton control of a two-lnk robot usng a mult-crossover genetc algorthm. Expert Systems wth Applcatons, 6, pp [5] Yao, X., Lu, Y., and Ln, G., July,. Evolutonary programmng made faster. IEEE Transactons on Evolutonary Computaton, Vol., no., pp [6] Holland, J.,75. Adaptaton n Natural and Artfcal Systems. Unversty of Mchgan Press, Ann Arbor, Mchgan. 7

6 Internatonal Conference on Systems, Sgnal Processng and Electroncs Engneerng (ICSSEE'0 December 6-7, 0 Duba (UAE [7] Ravndran, A., Ragsdell, K.M., and Reklats, G.V., 006. Engneerng Optmzaton: Methods and Applcatons. nd ed., Wley, Hoboken, New Jersey. 8