Evolutionary Algorithm With Experimental Design Technique

Transcription

1 Evolutoary Algorthm Wth Expermetal Desg Techque Qgfu Zhag Departmet of Computer Scece Uversty of Essex Wvehoe Park Colchester, CO4 3SQ Uted Kgdom Abstract: - Major steps evolutoary algorthms volve samplg pot from a space. Most of the exstg evolutoary algorthms adopt radom samplg. Expermetal desg techque s a sophstcated samplg method. Ths paper troduces a crossover based o orthogoal array desg. The proposed crossover ca estmate the best pot the space determed by the parets. The expermetal results show that the algorthm wth ths crossover s effectve for optmzato problems. Keywords: - Evolutoary algorthms, expermetal desg techque, crossover. Itroducto Evolutoary algorthms mmc the evoluto of bologcal speces to fd optmal or close-tooptmal solutos to a hard search or optmzato problems wth large ad complex search space [,2,3]. Ulke tradtoal search ad optmzato techques, evolutoary algorthms mata a collecto of potetal solutos stead of just a sgle soluto. Usg bologcally spred operators such as selecto, crossover ad mutato, the populato s evolved ad mproved utl a stoppg codto s met. Evolutoary algorthms have bee successfully appled to varous optmzato ad search problems. However, the theoretcal foudatos of evolutoary algorthms are to some extet weak. There are oly few rules of thumbs for the desg of evolutoary algorthms. Major steps evolutoary algorthms volve samplg pots from a space. From example, crossovers sample pots from the space determed by the paret pots. Most of the exstg evolutoary algorthms adopt radom samplg to select ad check pots. However, radom samplg s ot equvalet to dverse samplg. Expermet desg techque s a brach of statstcs [4,5,6]. It ca be appled to reveal the behavor of a system whose cost depeds o cotrollable factors ad ucotrollable factors (e. g. ose. Expermet desg techques sample a small porto of expermetal pots accordg to small rules ad the aalyze the results of ths expermet. They ca detfy the most fluetal factors o the cost; fd the mportat teractos amog the factors, ad the dscover the best combato of factor level that mmzes the cost. The author ad hs co-workers have appled expermetal desg techques to the desg of evolutoary algorthms [7,8,9,0]. The ma dea s that the samplg problem evolutoary algorthms ca be cosdered as a expermetal desg problem. Therefore, the sophstcated expermetal desg techques ca be used to stregthe these expermets. Ths paper revews the two basc expermetal desg techques, ad troduces crossover operators based o these techques for real fucto optmzato problem. The proposed crossover ca be used to estmate the best pot the space determed by the parets. Expermetal results show that the algorthm wth our crossover s effectve for optmzato problems. 2 Expermetal Desg Techques I ths secto, we revew the orthogoal array desg methods [4]. Cosder a system whose cost y depeds o cotrollable factors x, x 2,, x p, each at several levels, ad other ucotrollable factors (e.g. oses. The orthogoal array desg method ca be used to guess the best level of each factor by testg a small porto of combatos of factor levels. It employs orthogoal array to determe what combato of factor levels to be tested. May orthogoal arrays have bee developed. The L 9 orthogoal array s show table.

2 L 9 orthogoal array ca arrage the expermet of 4 factors wth 3 levels. I L 9 orthogoal array, each row represets a ru, ths s, a combato of factor levels to be tested. The colum elemets represet three levels of the colum factors. The orthogoalty of a orthogoal array meas that (a for ay colum, each level of the factor occurs the same tmes, (b for ay two colums, each possble combato of levels of the factors occurs the same tmes. For a system wth four cotrollable factors wth three levels, we could estmate the best combato of the factor levels the followg way: ( Test e combatos of factor levels lsted L 9 ad compute ther correspodg cost f respectvely. (2 For k=,2,3,4, ad j=0,,2, compute F k, j, the average cost of the combatos where factor K s at level j, respectvely. For example: F, =(f + f 2 + f 3 /3 (3 For factor k (k=,2,3,4, ts best level best(k s estmated as: Best(k=arg m F k, j j= 0,,2 Although there s a possblty that the combato (Best(,,best(4 s ot the best amog all possble combatos of factor levels, t s the best estmato a sese. It s worth metog that the orthogoal array desg techque could also be used to estmate the effects of factors, ad to aalyze the effects of the teractos of factors. Table : Orthogoal Array L 9 Ru Factor Factor2 Factor3 Factor Aother very useful expermetal desg techque s the uform array desg [6]. Geerally speakg, the sze of the uform array s much smaller tha that of that of the orthogoal array. The ma shortcomg of the uform array s that t caot be used to aalyze the effects of the factors ad the teractos of the factors. 3. Crossovers Based o Expermetal Desg Techques. Expermetal desg techques ca be appled to ehace crossover ad mutato [7]. I ths secto, we take the crossover based o the orthogoal array L 9 (OCX9 as a example to show how to corporate the expermetal desg techques to crossovers. Cosder the followg global optmzato problem: m f(x x D where f(x s a -dmesoal real-valued fucto, ad D s a boud set o R. OCX9 ca be regarded as a mappg from D 2 to D 2,.e.: ( x, x 2 = OCX 9( x, where x, x 2 D, ofte called parets, are the put of OCX9, ad x, x 2 D, ofte called offsprg, are ts output. Let x = ( x, L, x ad = (, L,, OCX9 cossts of the followg steps: Step Choose a λ (0, ad let x0 = λx + ( λ Step 2 For each dex {,2, L}, radomly geerate mark ( {0,,2,3}. Step 3 For each factor level combato l = ( l, l2, l3, l4 the orthogoal array L 9, assocate l wth a soluto y the followg: y = x lmark( Step 4 Let o (=,2,, 9 be the solutos assocated wth the combatos L 9, ad evaluate f(o respectvely. Step 5 Accordg to the method Secto 2, estmate the best combato, ad let o 0 be ts correspodg soluto to the problem, ad evaluate f(o 0. Step 6 Fd two of the best solutos x, x 2 amog o 0,, o 0 to be the offsprg of the crossover. It s possble that we aalyze the teractos amog the varable the above procedure by usg the methods orthogoal desg techque. Obvously, we ca also apply the uform desg to the desg

3 of crossover the same way. However, the data aalyss may become mpossble. There s a possblty that the above OCX9 gves two offsprg wth hgher costs tha ther parets. To assure that offsprg are better tha ther parets, we may employ competto amog the parets ad the temporal offsprg Step 6 to produce two solutos for the ext geerato. Most of the crossovers the lterature are smply to geerate two offsprg some radom way. OCX9 assumes that f(x a small area has some learty, ad performs a reasoable data aalyss based o orthogoal array. OCX eeds more computatoal amout tha other crossovers. However, OCX ofte gves much better offsprg. 4. Evolutoary Algorthm wth OCX9 I ths secto, a evolutoary algorthm wth OCX9 for real value fucto mmzato s sketched as follows: Step Italzato Set k= ad choose the parameters such as P m ad mu_c. Geerate (ofte radomly 2N solutos: x x x, 2,..., 2 N D Whle (ot stoppg codto do beg Step 2 OCX9 If k s eve umber, the for each j {,2, L, N} x, x : OCX 9( x, x else ( 2 j 2 j = 2 j 2 j ( x, N : = OCX 9( x, N for each j {,2, L, N } x, x : OCX 9( x, x ( 2 j 2 j+ = 2 j 2 j+ Step 3 Mutato For each {,2, L,2N} mu _ c If P m >radom(0,, the x = x + N (0, k ad perform some ecessary repar o x such that x D, where radom(0, ad N(0, represet a uform radom umber ad a vector of depedet radom Gaussas, respectvely. Step 4 k:=k+ ed. The proposed algorthm starts wth a populato of 2N caddate solutos. These solutos are called dvduals. Regardless of ts cost, each dvdual has oe ad oly oe chace of beg a paret. The matg strategy s determstc ad very smple. A dvdual has two eghbors amog whch oe s ts Brother (. e., geerated from the same parets. It always mates wth aother eghbor. Such matg strategy largely reduces the commucato overhead ts mplemetato, partcularly a shared-memory mache. By the use of OCX9, each par of parets ca geerate two offsprg, Mutato s also used wth probablty the algorthm to help both escapg from local mmum (especally the earler stage of the algorthm ad makg fe local tug (especally the later stage of the algorthm. The proposed algorthm dffers from most of the exstg algorthm that t uses offsprg selecto stead of paret selecto. The reaso why we dscard global selecto s twofold. Frst, global selecto ad competto s a serous bottleeck parallelzg evolutoary algorthms. Secodly, the orthogoal array desg techque, whch s emboded OCX, has played a role gudg the search. A good global optmzato algorthm should have some ablty to make the best use of all the formato that has bee kow the prevous stage. By corporatg the expermetal desg techques to the desg of crossover, we mprove such ablty of the resultg algorthm. 5. Expermetal Results The algorthm descrbed the above secto s coceptual ad urefed. There s stll much room for mprovemets. I order to objectvely evaluate ts performace, however, we mplemeted the algorthms wthout ay trck. We have tested the algorthm wth examples wth dverse fucto shapes. Some of them are preseted ths secto. For the followg example, P m s always set at 0., λ at 0.5, ad mu_c at 5. Example Let the fucto f(x be defed as: 2 f ( x = x 4 cos( x 2 = = 5 x 5 =, L, Ths fucto s a corrupted verso of a smple covex quadratc fucto. Ths fucto has the

4 global mmum 4 at the org, ad a large umber of local mmum pots. We ow gve the expermetal results for three versos of the fucto wth =0, 50, 00, respectvely. ( =0. Te depedet rus were performed. I each ru, the populato s talzed at radom, ad the populato sze (. e. 2N s always set at 40. I Table 2, the mmal fucto values geerato 33 ad 00 for 0 depedet rus are gve, respectvely. Table 2 RUN Geerato 33 Geerato I [], Styblsk ad Tag appled both approxmato wth smoothg ad fast smulated aealg to solve ths verso of the problem. For te radom ru, the average mmal fucto value foud s wth the average umber of fucto evaluatos equal to 6620 for SAS, ad average mmum foud s wth the average umber of fucto evaluatos beg 50,000 for FSA. As to our algorthm, the average total umber of fucto evaluatos s 6072 over geerato 33. As see, our algorthm s slghtly weaker tha SAS the ablty to make local fg. However, our algorthm s much easer to ru parallel computers tha SAS. (2 =50. I ths case, the populato sze s set at 00. The results of te radom rus are gve table 3. Table 3 RUN Geerato 00 Geerato (4 =00 I ths case, the populato sze s set at 50. The results of te radom rus are gve table 4. Table 4 RUN Geerato 400 Geerato Example 2 f ( x = ( x 2 = 6x + 5x 5 x 5 =, L, The oe-dmesoal verso of the fucto s used by Szu.[2] Its 2-D ad 0-D versos are tested []. The total umber of ts local mma s 2. All the mma must have ther coordates equal to x = or (=,2,,, ad the global mmum s As poted out [], we ca use the umber of postve coordates to characterze the qualty of the solutos other tha the global, the smaller NPC s, the better the soluto s. If NPC of a soluto s zero, the from ths soluto, we ca apply some local optmzato methods to approxmate the global soluto wth very hgh accuracy. 4 2 Te radom rus were performed for 00-D verso of the fucto. We set the populato sze at 00 for each ru.

5 Table 5 gves mma ad NPCs of the mmal soluto geerato 00 ad 500 for each ru. Example 3 Grewak fucto [3] x f ( x = x cos( 4000 = = 600 x 600 =, L,0 The fucto has ts global mmum f=- at x=0 ad several thousads local mma. The radom rus were performed for ths fucto. We set the populato sze at 00. The mmal fucto values geerato 33 ad 00 for each ru are gve table 6. Table 5 RUN Geerato 00 Geerato 500 M. NPC M. NPC Table 6 RUN Geerato 33 Geerato The radom rus were performed for ths fucto. We set the populato sze at 00. The mmal fucto values geerato 33 ad 00 for each ru are gve table Coclusos The crossover based o orthogoal array desg for a real value fucto optmzato problem was descrbed ths paper. The proposed crossover s able to estmate the best pot the space determed by the paret. Recetly, a lot of efforts have bee made to develop evolutoary algorthms for the optmzato problem wth hgh olear teractos amog the varable. Sce orthogoal array desg ca detect the teracto amog the varables, we beleve that orthogoal array desg ca also be used to mprove the ablty of evolutoary algorthm for such hard optmzato problems. Refereces [] T. Back, U. Hammel, ad H. P. Evolutoary Computato: Commets o the Hstory ad Curret State, IEEE Trasactos o Evolutoary Computato. Vol., No., 997, pp. 3-7 [2] Mchalewcz, Z., Geetc Algorthms + Data Structures = Evoluto Programs, 2d, Exteded Edto, Sprger-Verlag, 994. [3] D. B. Fogel, Evolutoary Computato: Toward a New Phlosophy of Mache Itellgece, Pscataway, NJ: IEEE Press, 995 [4] G. M. Clarke ad R. E. Kempso, Itroducto to the Desg ad Expermets, Lodo, Arold. 997 [5] Hcks, C. R., Fudametal Cocepts the Desg of Expermets, 3rd Edto, CBS College Publshg, 982. [6] K. T. Fag, Uform Desg, Scece Press, Bejg, 994 [7] Q. Zhag, S. Wu, ad H. Che, Some Results o Evolutoary Computato wth Expermetal Desg Techques. Workg paper, Dept. of Computer, Chagsha Isttute of Techology, Jue 995. [8] Q. Zhag, W. Peg, ad H. Che, Orthogoal Evolutoary Algorthm for Global Optmzato Problems, Workg paper, Dept. of Computer, Chagsha Isttute of Techology, Aug. 995 [9] S. Wu, Q. Zhag, ad H. Che, A New Evolutoary Model Based o Famly Eugecs: the frst results: The proceedg of 996 IEEE Iteratoa Coferece o Evolutoary Computato (ICEC'96, 996, Nagoya, Japa. pp

6 [0] Q. Zhag ad Y.-Y. Leug, Orthogoal Geetc Algorthm for Multmeda Multcast Routg, IEEE Trasactos o Evolutoary Computato, Vol. 3, No., 999, pp53-62 []M.A.Styblsk ad T.S.Tag, Expermets Nocovex Optmzato: Stochastc Approxmato ad Fucto Smoothg ad Smulated Aealg, Neural Networks, Vol. 3., 990. pp [2]Szu, H.H., ad Hartley, R.L., Nocovex Optmzato by Fast Smulated Aealg, Proceedgs of the IEEE, Vol 75, No., Nov [3]A.Tor ad A.Zlskas, Global Optmzato, Lecture Notes Computer Scece, Vol. 350, Sprger-Verlag, New York,987.