OePC: A portable software for parallel geetic algorithms used i optimizatio problems Ioais G. Tsoulos, Departmet of Computer Egieerig, School of Applied Techology Techological educatioal istitute of Epirus, Greece Abstract OePC is a software etirely writte i ANSI C++ usig the QT library to be portable. The software implemets parallel geetic algorithms that are based to a ovel algorithm to tackle the global optimizatio problem. The proposed algorithm cotais a ehaced stoppig rule ad a periodical applicatio of a local search procedure. Also, the software cotais a script laguage to assist the programmers. The article itroduces the software ad the uderlyig algorithm as well as some experimetal results. Idex Terms : Geetic algorithms, Global optimizatio, Parallel Computig, Programmig tool I. INTRODUCTION The commo problem of estimatig the global miimum of a multi - dimesioal fuctio is give as x = arg mi f x x S wheres is a subset of R ad is defied by: S = a 1, b 1 a 2, b 2 a, b This problem appears i may scietific areas such as [1]-[2], chemistry [3]-[4], ecoomics [5] etc. Oe commo techique to tackle this is the aturally ispired method of geetic algorithms [6]. Geetic algorithms have bee used i a may fields [7]-[9] ad they have may advatages such as: Adaptatio i every problem Requiremet for the objective fuctio oly ad ot for gradiet fuctios They ca be parallelized easily The most frequet model for parallel geetic algorithms is the server - cliet model, such as the so called Islad model [10]-[11], where cliets ru a a local geetic algorithm ad the server collects iformatio from the cliets ad occasioally distributes iformatio to them, such as the best discovered miimum.the proposed software amed OePC implemets a cliet - server model for parallel geetic algorithms with some advaced features such as: 1. A modified stoppig rule. This termiatio rule has bee proposed also i a recet work [12] ad it is geeral eough to be adapted i ay geetic algorithm. 2. A periodical applicatio of a local search procedure. 3. Exchage of best chromosomes betwee cliets. The proposed software has bee writte etirely i ANSI C++ usig the Qt library freely available from http://qt.io i order to be portable i ay operatig system. The rest of this article has as follows: i sectio II a detailed descriptio of the method is give, i sectio III full documetatio of the software is provided, i sectio IV some experimets that outlie the usability of the software are listed ad fially i sectio V some coclusios are preseted accompaied with some guidelies for future research. II. METHOD DESCRIPTION A. Server algorithm The machie deoted as server is resposible to gather from cliet machies the correspodig best discovered values. The algorithm executed o server is outlied below. Iitalizatio Step 1. Setg m = 2. Set N, the umber of cliets Check Step 1. If all cliets have fiished the a. Reportg m as the global miimum b. Termiate 2. Edif Loop Step 1. For i=1...n Do a. Obtai the miimum g i from the cliet i b. If g i < g m the g m = g i 2. EdFor 3. GotoCheck Step 78
B. Cliet algorithm O each cliet a geetic algorithm with the termiatio rule described i [12] accompaied with a additioal local search operator is applied. The steps of the algorithm are give below: 1. Iitializatio Step a. Set iter=0, where iter is the curret umber of geeratios b. SetN c the umber of chromosomes c. Iitialize chromosomes X i, i = 1,, N c d. Set IMAX as the maximum umber of allowed geeratios e. Set p s as the selectio rate ad p m as the mutatio rate. Both rates are i rage [0,1]. f. Set f l = as the best discovered fitess. g. Set L I the umber of geeratios that should pass before the local search procedure is applied. h. Set L C the umber of chromosomes that will participate i local search procedure. 2. Termiatio Check. At every geeratio the variace σ (iter ) of f l is calculated. If there was o improvemet of the geetic algorithm for a umber of geeratios, the the algorithm should termiate. The stoppig rule has as follows: σ iter σ last 2 OR iter > IMAX (3) Where last deotes the geeratio umber where f l was produced iitially. If the above equatio is true thegoto Step 10. 3. Fitess calculatio. Calculate the fitess f i of every chromosome of the populatio. 4. Geetic operators. a. Selectio procedure. The chromosomes are sorted i descedig order accordig to their fitess value. The first 1 p s N c chromosomes are trasferred to the ext geeratio. The rest of the chromosomes are substituted by offsprigs created through crossover procedure: For every offsprig two chromosomes (parets) are selected from the old populatio usig touramet selectio. The procedure of touramet selectio has as follows:a set of N>1 radomly selected chromosomes is produced ad the chromosome with the best fitess value i this set is selected ad the others are discarded. Havig selected the parets, the offsprigs x ad y are produced accordig to the equatios x i = a i x i + 1 a i x i ad y i = a i y i + 1 a i y i where a i are radom umber i the rage [-0.5,1.5] as suggested i [13]. b. Mutatio procedure. Mutatethe offsprigs produced durig crossover with probabilityp m. Suppose that the elemet i of a give chromosome x is deoted as x i. The ew elemet x i is calculated with a equatio borrowed from the popular PSO optimizatio method [14]: x i = c 1 r 1 x i b x i + c 2 r 2 x i l x i All Rights Reserved 2017 IJARCSEE where c 1 ad c 2 are two positive costats (acceleratio coefficiets), r 1 ad r 2 are radom umbers i the rage [0,1], the vector x b is a copy of the best so far positio of chromosome x ad x l is the best discovered chromosome so far. c. Replacethe p s N c worst chromosomes i the populatio with the offsprigs created by the geetic operators. 5. Set iter=iter+1 6. Local Search Step a. If iters mod L I =0 the i. Select radomly L C chromosomes from the geetic populatio ad create the set L S from these chromosomes. ii. For every chromosome X i L S 1. Select radomly aother chromosome Y from the populatio 2. Create a offsprig of X i ad Y. Deote this chromosome as Z 3. Obtai the fitess f(z) of the chromosome Z 4. If f z < f x i the X i = Z iii. EdFor b. Edif 7. Obtaithe best value i the populatio, deoted asf l for the correspodig chromosome x l. 8. Sed x l, f l to the server machie. 9. Goto step 3. 10. Sed x l, f l to the server machie. 11. Termiate. III. SOFTWARE DOCUMENTATION I this sectio a detailed descriptio of the mai parts of the software is provided. More detailed iformatio about the software that icludes some screeshots ca be foud i the URL http://itsoulos.teiep.gr/oepcsite/idex.html. Also a last copy of the software is provided i https://github.com/itsoulos/oepc A. User provided fuctios The user should code the objective fuctio i C++. The C++ files should have the followig commad before ay fuctio i the file exter C ad the lie after them. The user should supply the followig fuctios: 1. getdimesio().it is a iteger fuctio which returs the dimesio of the objective fuctio. 2. getleftmargi(left).it is a iteger fuctio which returs the dimesio of the objective fuctio. 79
3. getrightmargi(right).it is a iteger fuctio which returs the dimesio of the objective fuctio. 4. fumi(x). It is a iteger fuctio which returs the dimesio of the objective fuctio. As a example cosider the Rastrigi fuctio: f x 1, x 2 = x 1 2 + x 2 2 cos 18x 1 cos 18x 2 The code for this objective fuctio is show i Fig. 1 Also the user may optioally provide the followig fuctios: 1. void iit(qjsoobject params). This fuctio iitializes the objective problem ad it is ot madatory. I params parameter user stores some useful parameters for the problem. The user should use the iclude directive #iclude <QJsoObject> at the begiig of the file. As a example cosider the iitializatio fuctio for the problem of eergy potetial show i Fig. 2, where some critical parameters of the objective fuctio are iitialized through the iit() fuctio. 2. double doe(double *x). This fuctio should be called at the ed of optimizatio. The double parameter x is the vector of global miimum discovered by the optimizatio process. The fuctio should retur the fial value of the objective fuctio. It is useful whe a local optimizatio algorithm is required at the ed of the optimizatio of i the eural etwork case, where the eural etwork should be applied to the test set. The objective problem is writte i ANSI C++ ad it should build as a shared library. The use ca use ay programmig tool to achieve that, but the best choice is to use Qt build eviromet for that. The project file is used i Qt programmig eviromet to build the associated project. I our case is used to build the optimizatio fuctio. A example for buildig the potetial optimizatio fuctio is illustrated i Fig 3. B. The program OePCServer Istallatio The user should issue the followig commads to build the program OePC Cliet (uder some Uix machie): 1. Dowload the code (OePcCliet.tar.gz) 2. guzip OePcCliet.tar.gz 3. tar xfv OePcCliet.tar 4. cd OePcCliet 5. qmake. 6. make User Iterface The graphical iterface of the program is writte etirely i Qt ad it cotais for mai tabs: 1. Ifo tab. The tab cotais iformatio about the machie ruig the server. The most importat iformatio from this tab is the IP of the server ad the port where it is ruig. This iformatio should be used by the cliets that wat to coect to geetic server. Also, the user ca load OeScript programs usig the butto LOAD SCRIPT. 2. Cliets tab. This is the tab where the user ca start the parallel geetic algorithm by pressig the butto Ru Experimet. This butto seds o every cliet the objective fuctio. The fuctio will be set i the correspodig format of every cliet depedig of the ruig operatig system. Afterwards each cliet starts the geetic algorithm. The Cliets tab displays also the followig iformatio for every cliet: a. Name of the cliet b. Operatig system of the cliet c. Status of the cliet (ruig, waitig, termiated, paused) d. Discovered global miimum from the cliet 3. Problems tab. The iformatio about the stored objective problems is outlied i this tab. This iformatio is stored i a sqlite3 database amed gaserver.db3 i the same folder with the ruig server. The user ca load compiled objective problems i shared library format from the hard disk ad he ca also chage or add parameters to the objective problem through the relevat dialog. 4. Messages tab. This tab displays debug iformatio for every cliet as well as about the status of the ruig server. C. The program OePCCliet Istallatio The user should issue the followig commads to build the program OePC Cliet (uder some Uix machie): 1. Dowload the code (OePcCliet.tar.gz) 2. guzip OePcCliet.tar.gz 3. tar xfv OePcCliet.tar 4. cd OePcCliet 5. qmake. 6. make User iterface The graphical applicatio is writte etirely i Qt i order to be portable. Nevertheless, the user ca start the cliet i commad lie mode by startig the executable with the commad lie gui=o. The graphical applicatio is orgaized i three tabs: 1. Coectio tab. I this tab the user should supply the ip of the server i the Host textfield as well as the ruig port. Without this iformatio the cliet caot coect to the server. Also, the user ca provide a ame for the cliet i order to distict them from other ruig cliets. 2. Ru tab. Uder Ru tab the user ca modify some of the most critical parameters of the geetic algorithm such as chromosomes, geeratios etc. Also, the user ca moitor the progress of the geetic algorithm durig the executio of the algorithm. 3. Messages tab. Some debug iformatio is displayed i this tab. Commad lie optios The user ca use commad lie argumets for the applicatio. 1. geeratios=g. The iteger value g specifies the maximum umber of geeratios allowed for the geetic algorithm. The default value is 200. 2. chromosomes=c. The iteger value c specifies the amout of chromosomes used i the geetic 80
populatio. The default value for this parameter is 200. 3. selectiorate=s. The double parameter s determies the selectio rate for the geetic algorithm. The default value is 0.90 (90%), which meas that 90% of the populatio will be copied itact to the ext geeratio. 4. mutatiorate=m. The double parameter m determies the mutatio rate used i geetic algorithm. The default value is 0.05 (5%). 5. serverip=s. The strig parameter s specifies the ip server of the OePC server. 6. serverport=p. The iteger parameter p specifies the port of the OePC server. 7. gui=b. The boolea value b ca specifies if gui will be used for the cliet. If b is yes or true the gui will be used otherwise it will ot. 8. seed=s. The iteger parameter s specifies the seed for the radom geerator. 9. machieame=ame. The strig parameter ame specifies the ame of the cliet that will be displayed uder tab Cliets i OePC server. 10. parallelchromosomes=c. The iteger parameter c specifies the umber of chromosomes that will be exchaged betwee this cliet ad the other cliets i the parallel populatio. The default value for this parameter is 10. 11. localsearchchromosomes=c. The iteger parameter c specifies the amout of chromosomes that will take part ito local search step of the geetic algorithm. The default value for this # iclude <math.h> exter "C" itgetdimesio() retur 2; void getleftmargi(double *x) x[0]=-1; x[1]=-1; void getrightmargi(double *x) x[0]=1; x[1]=1; double fumi(double *x) retur x[0]*x[0]+x[1]*x[1]-cos(18.0*x[0]) -cos(18.0*x[1]); Figure 1. The Rastrigi fuctio. All Rights Reserved 2017 IJARCSEE parameter is 20. 12. localsearchgeeratios=g. The iteger parameter g specifies the amout of geeratios that will be executed before the local search step of the geetic algorithm. The default value for this parameter is 50. D. The laguage OeScript The laguage OeScript ca be used to simplify the executio of a series of problems i OePC without the iterferece of the programmer. The commads ca be writte usig capital or lowercase letters. For the time beig the laguage has a simple set of commads which are: 1. SET PROBLEM ame. Sets the curret problem to ame. 2. SET PARAMETER paramname paramvalue. Chage the value of the parameter paramname to paramvalue. 3. RUN. Executes the curret problem usig the attached cliets. 4. PRINT FILE. Prit the curret global miimum to file FILE. As a example of a script i OeScript cosider the program listed i Fig. 4 used to set the objective problem to SINU. double potetialepsilo=1.0; double potetialsigma=1.0; itatoms=10; ittolmiiters=2001; QStrigoutputfile="potetial.txt"; double *xx=null; double *yy=null; double *zz=null; void iit(qjsoobjectobj) if(obj.cotais("atoms")) atoms= obj["atoms"].tostrig().toit(); if(obj.cotais("tolmiiters")) tolmiiters= obj["tolmiiters"].tostrig().toit(); if(obj.cotais("outputfile")) outputfile= obj["outputfile"].tostrig(); xx=ew double[atoms]; yy =ew double[atoms]; zz =ew double[atoms]; Figure 2. Iitializatio fuctio of Potetial problem. 81
QT -= gui TARGET = potetial TEMPLATE = lib DEFINES += POTENTIAL_LIBRARY SOURCES += potetial.cpp tolmi.cc HEADERS += potetial.hpotetial_global.h uix SET target.path PROBLEM = /usr/lib siu SET INSTALLS PARAMETER += target dimesio 16 RUN Figure 3. Project file to build the potetial PRINT /home/user/log_siu16.txt objective fuctio. Figure 4. Project file to build the potetial objective fuctio. IV. EXPERIMENTS The method was tested o some test fuctios from the relevat literature as well as o the Leard Joes potetial problem. The results are compared agaist the well-kow software for parallel computatio called GALib-mpi[18]. The experimets were performed 30 times usig differet seed for the radom geerator each time ad averages were take. The umber of chromosomes was set to 200 ad the maximum umber of allowed geeratios was set to 2000. A. Experimets o Test Fuctio The followig test fuctios were used i the experimets: 1. Gkls. f(x)=gkls(x,,w) is a fuctio with w local miima described i [19], x 1,1, [2,100]. I the coducted experimets the cases of =2,3,4 with 50 local miima were used. 2. Test2. The fuctio is give by 1 2 i=1 x 4 i 16x 2 i + 5x i with x [ 5,5]. The fuctio has 2 local miima i the specified rage. I the coducted experimets the cases of =5,6,7 was cosidered. 3. Siusoidal. The fuctio f(x) is give by 2.5 i=1 si x i z + i=1 si 5 x i z with 0 x i π ad z = π. The global miimum is 6-3.5 4. The Chemical Equilibrium.The problem is described i [15] ad it is described by the followig set of equatios: x 1 x 2 + x 1 3x 5 = 0 2x 1 x 2 + x 1 + x 2 x 2 3 + R 8 x 2 Rx 5 + 2R 10 x 2 2 + R 7 x 2 x 3 + R 9 x 2 x 4 = 0 2x 2 x 2 3 + 2R 5 x 2 3 8x 5 + R 6 x 3 + R 7 x 2 x 3 = 0 R 9 x 2 x 4 + 2x 2 4 + 4Rx 5 = 0 x 1 x 2 + 1 + R 10 x 2 2 + x 2 x 2 3 + R 8 x 2 + R 5 x 2 3 + x 2 4 1 + R 6 x 3 + R 7 x 2 x 3 + R 9 x 2 x 4 = 0 where R = 10 R 5 = 0.193 R 6 = 0.002596 R 7 = 0.00348 R 8 = 0.00001799 R 9 = 0.0002155 R 10 = 0.00003846 The correspodig objective fuctio is the summatio of the absolute values of all equatios i the system i.e. f i (x) i=1 The global miimum is 0. 5. The Kiematic Applicatio. This fuctio is described i [15] for the iverse positio problem for a six revolute joit problem i mechaics. The problem is provided as a system of equatios: x 2 2 i + x i+1 = 0 a 1i x 1 x 3 + a 2i x 1 x 4 + a 3i x 2 x 3 + a 4i x 2 x 4 + a 5i x 2 x 7 + a 6i x 5 x 8 + a 7i x 6 x 7 + a 8i x 6 x 8 + a 9i x 1 + a 10i x 2 + a 11i x 3 + a 12i x 4 + a 13i x 5 + a 14i x 6 + a 15i x 7 + a 16i x 8 + a 17i = 0 Where 1 i 4 ad the table a ij is a = 0.249150 0.125016 0.635550 1.489477 1.6091354 0.68660736 0.1157199 0.230623 0.27942343 0.11922812 0.6664 1.3281073 1.4348016 0.719947 0.110362 0.258645 0.0 0.43241927 0.2907020 1.165172 0.026384 0.0 1.2587767 0.26908 0.80052768 0.0 0.629388 0.53816 0.0 0.86483855 0.5814 0.582585 0.075 0.03715727 0.195946 0.208169 0.083050 0.0354368 1.228034 2.686832 0.38615961 0.085383482 0.0 0.699103 0.75526603 0.0 0.079034 0.3574441 0.50420168 0.039251967 0.026387 1.249911 1.0916287 0.0 0.057131 1.467736 0.0 0.43241927 1.1628081 1.165172 0.04920729 0.0 1.2587767 1.0763397 0.04920729 0.01387301 2.162575 0.696868 The correspodig objective fuctio is the summatio of the absolute values of all equatios i the system i.e. f i (x) i=1 The global miimum is 0. 6. The Combustio Applicatio. This problem is also described i [15] as a series of equatios: x 2 + 2x 6 + 2x 9 x 10 10 5 = 0 x 3 + x 8 3 10 5 = 0 x 1 + x 3 + 2x 5 + 2x 8 + x 9 + x 10 5 10 5 = 0 x 4 + 2x 7 10 5 = 0 0.51437 10 7 2 x 5 x 1 = 0 0.1006932 10 6 2 x 6 2x 2 = 0 0.7816278 10 15 2 x 7 x 4 = 0 0.1496236 10 6 x 8 x 1 x 3 = 0 0.619441 10 7 x 9 x 1 x 2 = 0 0.2089296 10 14 2 x 10 x 1 x 2 = 0 82
Agai, the correspodig objective fuctio is the summatio of the absolute values of all equatios i the system i.e. f i (x) i=1 The global miimum is 0. The results from the applicatios of OePC ad Galib to the problems above are preseted i TableI for two processors, i TableII for four processors ad i TableIII for eight processors. The cells deote average umber of geeratios ad the figures i paretheses deote the fractio of rus that located the global miimum ad were ot trapped i oe of the local miima. Absece of this fractio deotes 100% success i locatig the global miimum. I all tables the colum FUNCTION deotes the fuctio ame, the colum GAlib deotes the results from the applicatio of GALIB ad the colum ONEPC deotes the results from the applicatio of the proposed software. As we ca deduce from the experimetal results the proposed software seems to require lower umber of geeratios tha GAlib to discover the global miimum. Also the fractio of rus that discovered the global miimum is higher tha GAlib. B. The case of Leard Joes Potetial The molecular coformatio correspodig to the global miimum of the eergy of N atoms iteractig via the Leard-Joes potetial [16] is used as a test case here. The fuctio to be miimized is give by: V LJ r = 4ε σ σ r r The fuctio is miimized firstly with OePC ad afterwards the local search procedure Tolmi (a BFGS variat of Powell [17]) is used to ehace the detected miimum. Success rates of discoverig the global miimum by GAlib are preseted i TableIV ad success rates obtaied by the proposed method are preseted i Table V. Agai i most cases the fractio of rus that discovered the global miimum is higher tha GAlib. 12 V. CONCLUSIONS A portable software for global optimizatio was itroduced with the followig features: 1. It ca be istalled i most operatig systems. 2. There is o eed for parallel libraries such as OpeMPI[20]. 3. The system cotiues to work eve if some of the odes have lost coectio. 4. The cliets ca operate with or without GUI. 5. The user ca cotrol the iitializatio parameters of the objective problems through the iit() procedure. 6. The server ca use a script laguage to cotrol the optimizatio procedure Future research may iclude: 1. Additioal commads for the script laguage of the server. 2. More advaced stoppig rules. 6 Table IExperimetal results usig two processors for a series of optimizatio problems PROBLEM GaLib OePC GKLS350 310.50 205.18 GKLS450 370.20(0.30) 393.78(0.77) GKLS550 1548.70(0.93) 569.80(0.80) SINU16 1933.50 1633.63 SINU32 1933.57(0.90) 1644.18 SINU64 272.50(0.17) 155.68(0.17) TEST2N4 543.60 425.02 TEST2N5 857.63 521.22 TEST2N6 1037.37 786.27 TEST2N7 1207.90(0.93) 891.7 CHEMICAL 1917.25(0.93) 1415.31 KINEMATIC 1976.37 1601.81 COMBUSTION 1865.67 19.19 Table IIExperimetal results usig four processors for a series of optimizatio problems PROBLEM GaLib OePC GKLS350 429.93 255.58 GKLS450 424.87(0.33) 359.73(0.90) GKLS550 1322.43(0.90) 585.87 SINU16 1867.20(0.90) 1611.76 SINU32 17.63(0.90) 1657.10 SINU64 339.50(0.23) 72.29(0.30) TEST2N4 712.67 342.78 TEST2N5 984.57(0.97) 498.70 TEST2N6 983.47(0.90) 624.65 TEST2N7 1015.67(0.70) 957.80 CHEMICAL 1825.37 1319.69 KINEMATIC 1903.94 1644.89 COMBUSTION 1836.54 1241.43 Table IIIExperimetal results usig eight processors for a series of optimizatio problems PROBLEM GaLib OePC GKLS350 418.17 224.49 GKLS450 626.(0.) 373.13 GKLS550 1291.00(0.83) 592.63 SINU16 1933.26(0.93) 1642.59 SINU32 1933.63(0.87) 1678.78 SINU64 373.87(0.23) 55.38(0.67) TEST2N4 461.77 375.70 TEST2N5 947.70(0.97) 506.12 TEST2N6 1092.13(0.93) 629.29 TEST2N7 1097.43(0.73) 883.76 CHEMICAL 1908.33 1383.93 All Rights Reserved 2017 IJARCSEE 83
KIMEMATIC 1945.60 1636.10 COMBUSTION 1875.49 1252.35 Table IVSuccess rate results for the Potetial problem with GaLib NATOMS C2 C4 C8 5 100% 100% 100% 6 27% 37% 50% 7 70% 73% 97% 8 93% 100% 100% 9 63% 87% 93% 10 53% 73% 97% 12 63% 83% 93% Glover (eds.), New ideas i Optimizatio, McGraw-Hill, Cambridge, UK, pp. 11-32, 1999 [15] C. Grosa ad A. Abraham. A ew approach for solvig oliear equatios systems. IEEE Trasactios o Systems, Ma, ad Cyberetics Part A: Systems ad Humas vol.38, pp. 698 714, 2008. [16] J.E. Leard-Joes, O the Determiatio of Molecular Fields, Proc. R. Soc. Lod. A, vol. 106: pp. 463 477, 1924. [17] M.J.D Powell, A Tolerat Algorithm for Liearly Costraied Optimizatio Calculatios, Mathematical Programmig vol. 45, pp 547, 1989 [18] M. WALL GAlib: A C++ library of geetic algorithm compoets. Mechaical Egieerig Departmet, Massachusetts Istitute of Techology 87: pp.54, 1996. [19] M. Gaviao, D.E. Ksasov, D. Lera ad Y.D. Sergeyev, Software for geeratio of classes of test fuctios with kow local ad global miima for global optimizatio, ACM Tras. Math. Softw. Vol. 29, pp. 469-480, 2003. [20] R. L. Graham, T. S. Woodall ad J. M. Squyres, Ope MPI: A Flexible High Performace MPI, Parallel Processig ad Applied Mathematics Volume 3911 of the series Lecture Notes i Computer Sciece, pp 228-239, 2006. Table VSuccess rate results for the Potetial problem with OePC NATOMS C2 C4 C8 5 100% 100% 100% 6 100% 100% 100% 7 80% 100% 100% 8 60% 80% 100% 9 70% 100% 100% 10 % 80% 90% 12 100% 100% 100% REFERENCES [1] P. O. Yapo, H. V. Gupta ad S. Sorooshia, Multi-objective global optimizatio for hydrologic models, Joural of Hydrologyvol 204, pp. 83-97, 1998. [2] Q. Dua, S. Sorooshia ad V. Gupta, Effective ad efficiet global optimizatio for coceptual raifall-ruoff models, Water Resources Researchvol 28, pp. 1015-1031, 1992. [3] D. J. Wales ad H. A. Scheraga, Global Optimizatio of Clusters, Crystals, ad Biomolecules, Sciece vol27, pp. 1368-1372, 1999. [4] P.M. Pardalos, D. Shalloway ad G. Xue, Optimizatio methods for computig global miima of ocovex potetial eergy fuctios, Joural of Global Optimizatio vol. 4, pp. 117-133, 1994. [5] Zwe-Lee Gaig, Particle swarm optimizatio to solvig the ecoomic dispatch cosiderig the geerator costraits, IEEE Trasactios o Power Systems vol. 18, pp. 1187-1195, 2003. [6] D.E. Goldberg ad J. H. Hollad, Geetic Algorithms ad Machie Learig, Machie Learig vol. 3, pp. 95-99, 1988. [7] J.J. Grefestette, R. Gopal, B. J. Rosmaita ad D. Va Gucht, Geetic Algorithms for the Travelig Salesma Problem, I: Proceedigs of the 1st Iteratioal Coferece o Geetic Algorithms, pp. 160-168, Lawrece Erlbaum Associates, 1985. [8] P. Kaelo ad M.M. Ali, Itegrated crossover rules i real coded geetic algorithms, Europea Joural of Operatioal Research vol. 176, pp. 60-76, 2007. [9] T. Prasad ad N. Park, Multiobjective Geetic Algorithms for Desig of Water Distributio Networks, J. Water Resour. Pla. Maage. vol. 130, pp. 73-82, 2004. [10] A. L. Corcora ad R. L. Waiwright, A parallel islad model geetic algorithm for the multiprocessor schedulig problem, SAC '94 Proceedigs of the 1994 ACM symposium o Applied computig, pp. 483-487, 1994. [11] D. Whitley, S. Raa ad R. B. Heckedor, Islad model geetic algorithms ad liearly separable problems, Evolutioary Computig Volume 1305 of the series Lecture Notes i Computer Sciece, pp 109-125, 2005. [12] I.G. Tsoulos, Modificatios of real code geetic algorithm for global optimizatio, Applied Mathematics ad Computatio vol. 203, pp. 598-607, 2008. [13] Z. Michaelewicz, Geetic Algorithms + Data Structures = Evolutio Programs. Spriger - Verlag, 1996. [14] J. Keedy ad R.C. Eberhart, The particle swarm: social adaptatio i iformatio processig systems, i: D. Core, M. Dorigo ad F. 84