PhD THESES KAPOSVAR UNIVERSITY. FACULTY OF ECONOMIC SCIENCES Department of Information Technology. Head of Doctoral School: PROF. DR.


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1 PhD THESES KAPOSVAR UNIVERSITY FACULTY OF ECONOMIC SCIENCES Deparmen of Informaion Technology Head of Docoral School: PROF. DR. GÁBOR UDOVECZ Docor of Hungarian Academy of Sciences Supervisor: DR. HABIL. BÉLA CSUKÁS CSc Associae Professor DEVELOPMENT AND APPLICATION OF GENETIC ALGORITM FOR MULTICRITERIA ECONOMIC DECISION MAKING Auhor: Sándor Balogh KAPOSVÁR 2009
2 1 BACKGROUND AND OBJECTIVES In he convenional opimizaion mehodologies he model of he invesigaed problem used o be simplified o he formalism of a mahemaical consruc ha makes possible he deerminaion of he exac opimum. Considering he imporance of he deails in he engineering problem solving, in he pas decades increasing effor has been made for he possibly mos deailed model based opimizaion. The dynamic simulaion ools for he various processes developed rapidly, while he opimizaion of more and more complex hybrid (coninuous/discree) models became inracable in he developmen of inparallel opimizaion mehods. Anoher acual challenge is he opimal soluion of he large scale, long erm processes of changing srucure wih increasing complexiy. Having recognized hese difficulies, he inexac, heurisic and/or evoluionary mehods of he arificial and compuaional inelligence became more and more imporan. In he case of an inexac approach here is no guaranee for he deerminaion of he absolue opimum. This disadvanage is compensaed by he fac, ha good enough soluions can be deermined on he basis of he necessarily mos deailed models. One of he heurisic opimizaion mehods of he Compuaional Inelligence is he Geneic Algorihm. I me hese mehods in my MSc hesis in he early 90h, and have been dealing wih i in my work since ha ime. This work is characerized and moivaed by he demand on opimize deailed and/or large scale hybrid problems, coming from various field of applicaion, which could no be solved wih he available ools. In addiion, in he case of an economic objecive we had o opimize according o a number of naural crieria, or we had o combine he economic and naural objecives. The inparallel developing generic simulaion mehod more and more oleraes he arbirary discree and coninuous changes. In addiion, in accordance wih he general endency coninuously increased he compuaional demand for he simulaion of he possible soluions. The fundamenal objecive of his work is he comparison of he published mehods wih my resuls came from he coninuous developmen of he geneic algorihm, moivaed by he soluion of he various pracical problems in he 1
3 pas decades. Based on his comparison, a combined geneic algorihm has o be developed for supporing he mulicrieria economic decisions. The deailed objecives of my research are he followings: 1. Considering he pracical needs, a complex geneic coding is o be elaboraed ha makes possible he unified descripion of he opionally hierarchical implemenaion of genes, consising of coninuous/discree and permuaional subsequences. Moreover he coding has o suppor he declaraion of he incompaible genes by means of he so called srucure laice. The planed complex coding has o describe he convenional kins of code as special cases o avoid he applicaion of special specific mehods. 2. In accordance wih his unified coding, uilizing he se of available geneic operaors and opionally inroducing modified operaors a general framework is o be developed for he auomaic selecion of he respecive operaors o he acual coding. 3. Anoher imporan objecive is o elaborae a paramerizable mehod wih he respecive paramerizable geneic operaors ha suppor he execuion evoluionary process wih small generaion number and populaion size. Accordingly, aking ino accoun he experiences from he lieraure and from he successfully solved pracical problems a new, so called grid mehod is o be developed. 4. The muli crieria problem solving is o be improved by an algorihm, calculaing he combined Pareodominance wih he respecive paramerizable geneic operaors. 5. Finally, I am o implemen he elaboraed algorihms for he evoluionary opimizaion of less number of variance wih grid compuaional demand. Accordingly, I ry o uilize he possibiliy o sore all of he generaed geneic codes, on he one hand. In addiion, my objecive is he auomaically conrolled parallel simulaion based evaluaion, uilizing 2
4 he possibiliies of he macrogranularly parallelizable (e.g. PC cluser) archiecures. The applicaion of he developed mehods for he soluion of decision making, economic opimizaion and logisical problem solving will be illusraed by simple examples, followed by a couple of es problems elaboraed for he evaluaion of geneic algorihms. 2 MATERIALS AND METHODS In he presen research and developmen work many open source code sofware developmen ools, as well as he collaboraing generic simulaor, developed by our research eam were applied. For he realizaion of he macro granularly parallel evoluionary developmen, a compuer cluser was buil and configured. Accordingly, he mehods applied for he developmen and esing of he elaboraed geneic algorihm were he followings: sofware ools, applied for he developmen of he geneic algorihm; he generic simulaor, collaboraing wih he geneic algorihm; hardware and sofware ools, applied for he realizaion of he macro granularly parallel operaion. The mos imporan open source code sofware ools, used for he developmen of he geneic algorihm were he followings: fox oolki: (hp://www.foxoolki.org); plplo: (hp://www.plplo.org); clap: (hp://clap.sourceforge.ne/); c++ compilers: gcc mingw32. For he macro granularly parallel execuion of he evoluionary simulaions, a PC cluser, conaining 16 unis was buil. The operaion of he cluser was solved by he adapaion of he Open SSI (hp//www.openssi.org) sofware. Regarding he demonsraed example applicaions: 3
5 he programs of he benchmark es asks has been wrien by myself; he simulaion of he deailed example applicaions has been solved by he generic simulaor based on he direc compuer mapping of processes, developed in he research school of process informaics, using he version running under Windows wih EXCEL inerface. In he soluion of he various pracical problems and experimenal case sudies from he fields of idenificaion, opimal conrol, opimal design and opimal scheduling in he pas decades, he coninuously developing geneic algorihm collaboraed also wih he various adapaions of he generic process simulaor, based on Direc Compuer Mapping. 3 RESULTS AND DISCUSSION The mehods, applied for he economic opimizaion of complex sysems in pracical problem solving, have o saisfy many crieria. One of he wo mos imporan demands is supporing of he mulicrieria evaluaion in decision making. The oher is, he capabiliy for he represenaion of he complex possibiliy spaces, characerizing he economic and/or echnological processes. In he developmen of he geneic algorihm, prepared for he mulicrieria economic opimizaion, was moivaed by he above crieria. 3.1 Developmen of a complex geneic coding The represenaion of complex sysems by srucure laices is based on he works of Blickle. The essenial feaure of he corresponding heory is ha an acual se of complex objecs (sysems) can be described by properies, classified ino equivalency classes. The individual propery classes correspond o he fundamenal properies, characerizing he objecs. The laice is deermined by he following principle: he objecs can be deermined by propery combinaions, consising of one, and only one propery from each equivalency class. Of course, a well defined par of he pair wise combinaion is forbidden. Having defined his pair wise incompaibiliy relaion, we can generae all of he compaible 4
6 Propery classes: Properies of classes: T 6 T 7 T 8 6,1 6,2 7,3 7,4 7,2 6,0 7,0 7,1 8,4 8,5 8,2 8,3 8,0 8,1 T 3 T 4 T 5 3,2 3,3 3,0 3,1 4,6 4,7 4,4 4,5 4,2 4,3 5,3 4,0 4,1 5,1 5,2 5,0 T 0 T 1 T 2 0,4 0,5 0,2 0,3 0,0 0,1 1,1 1,2 2,3 2,4 2,2 1,0 2,0 2,1 Two objecs: 6,1 7,3 8,4 6,1 7,1 8,1 3,2 4,6 5,2 3,1 4,1 5,1 0,4 1,1 2,3 0,1 1,1 2,1 Figure 1. Represenaion of complex sysems by srucure laices.the form of incompaibiliy relaion is I = { ( 0,1, 3,2 ),( 2,0, 5,3 ),( 8,0, 7,4 ) }, where i,j is he jh propery of he ih class. propery combinaions auomaically. The essenial feaures of he srucure laice are illusraed in Fig. 1. From he viewpoin of geneic coding, he possible codes are deermined by he ensemble propery classes. The individual genes of he chromosome correspond o he respecive propery classes. The locus of he gene refers o he index of he corresponding propery class. The possible alleles of he genes can be described by he properies of he given propery class. A he same ime, we can declare all of he incompaible pairs of he alleles. The respecive geneic coding can be seen in Fig. 2. When inroducing he geneic algorihm, I emphasized ha hey can be applied for he coding of discree and coninuous characerisics, as well as of full permuaions. In describing he complex sysem, he genes (i.e. he propery classes) can be discree (e.g. ype of process unis), coninuous (e.g. economic or echnical parameers) or permued discree variables (e.g. producion sequences). Accordingly, he descripion of hese complex sysems needs he simulaneous 5
7 The geneic code: G 0 G 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 Alleles: g g0,3 g0,4 0,5 g0,1 g0,2 g g2,3 g 2,4 g0,0 g1,0 g1,1 1,2 g2,1 g2,2 g3,2 g 3,3 g2,0 g3,0 g3,1 g g4,5 g4,6 4,7 g4,3 g4,4 g4,1 g4,2 g5,2 g 5,3 g4,0 g5,0 g5,1 g g7,3 g 7,4 g8,4 g 8,5 g6,0 g6,1 6,2 g7,1 g7,2 g8,2 g8,3 g7,0 g8,0 g8,1 Two chromosomes: g 0,1 g 1,1 g 2,1 g 3,1 g 4,1 g 5,1 g 6,1 g 7,1 g 8,1 g 0,4 g 1,1 g 2,3 g 3,2 g 4,6 g 5,2 g 6,1 g 7,3 g 8,4 Figure 2. Geneic code of he complex sysem using propery classes. The incompaible alleles of he g i,j allele characerized by he I i,j = { g l,m, g k,n }. use of hese various gene ypes. Considering he represenaion of real numbers in he compuer, all of he coninuous properies are ransformed o discree values. In addiion, he coninuous parameers, characerizing he pracical problems, have also a given accuracy (e.g. granulariy). Accordingly, all of he gene ypes can be described by discreized values in geneic coding. The essenial difference beween he coninuous and discree geneic operaors is wheher hey can follow he Euclidean or he Hammingian merics. Finally, he coninuous and discree cases can be disinguished by he appropriae merics. Having deermined he gene ypes, we can declare he possible alleles of hem. In he case of he discree genes, i can be solved by an enumeraion, while for he coninuous characerisics we can give a leas he lower and upper bounds, as well as he refinemen of he discree seps. In his work, hese properies were supplied by he lower and upper limis, awaied from he expers. In his geneic coding, he elemens of he ih gene ype, are characerized by he D i = { (g i,1, I i,1,k i,1 ),(g i,2, I i,2,k i,1 ),...,(g i,n, I i,n,k i,n ) } 6
8 riple, where g i,j N ha in he majoriy of he cases corresponds o he ordinal number of he elemen in he caalogue of he algorihm, couning of he evaluaing poin of views. The I i,j = { g l,m,..., g k,v } corresponds o he se of he incompaible alleles, while K i,j deermines he laer defined gene ype of he chromosome, corresponding o he furher elemens of he caalogue (e.g. he parameers or changeable pars of he process uni). The coninuous gene ypes are deermined by he C i = ( l i,h i,e l i,eh i,dmax,d min ) i i sexuples, where l i,h i and e l i,eh deermine he lower and upper bounds of he domain, A and B i describe he bounds of he awaied inerval, while d max and d min deermine he i i maximal and minimal discreizaion of he inerval. The wo above gene ypes are supplemened by hree combined gene ype. This corresponds o he discree gene sequences, o permuable gene sequences, as well as o he chromosome. The gene sequences are defined as he ordered ses of he discree or discree permuable genes, while he elemens of he chromosome may be all of he above defined simple or complex genes. The inroducion of complex genes is moivaed of he fac, ha he iniializaion, recombinaion and he muaion of he full permuaions can be reaed only wih he knowledge of he whole ordered permuable gene sequences. On he oher hand, he complex genes, having recursively embedded, can form also hierarchical daa srucures. The soluion is conform wih he laer described efficiency increasing mehods. The above defined geneic elemens are visualized in Fig. 3. The exension of geneic coding makes possible he respecive exension of he algorihm, oo. The elemens of he algorihm, elaboraing he code, are he iniializaion, as well as he recombinaion and muaion operaors. These aciviies are defined for all of he gene ypes, according o heir properies. Accordingly, he coninuous, discree or permuable ypes can use he corresponding recombinaion and muaion operaors, described in he lieraure, while he iniializaion is carried ou mosly by random generaion. In he soluion of a given ask, he geneic coding corresponds one of he above defined gene ypes. Having generaed he alleles, we have o describe he incompaibiliy relaions for he discree genes. All of he gene ypes have 7
9 Chromosome: D 0 C 1 GP 2 GD 3 K 4 D 5 Alleles: Figure 3. The complex and he simple gene ypes. defaul iniializaion, recombinaion and muaion operaors, however, we can replace hem for anoher embedded one. 3.2 Elaboraion of a grid mehod The necessiy for he use of small populaion size causes many problems o be solved. One of hem is ha he soluion is very sensiive for he qualiy of he iniial populaion. The oher is ha he diversiy of he populaion can decrease rapidly. The random number generaors produce really uniform random numbers only for larger populaion size, while in he case of small populaions he disribuion of he variance is no equal in he differen segmen of he search space. In he case of he coninuous gene ypes, he generaion of random numbers can be conrolled by he awaied limis e l i,eh, esimaed by he exper. i In addiion, wih he knowledge of he (d max ) maximal discreizaion of he i genes (given in he iniializaion), he values, suggesed by he random number generaor, can be rounded for he respecive preciosiy. This grid, generaed in he search space, suppors he more uniform disribuion of he iniial populaion. The mehod is illusraed in Fig. 4. Anoher useful applicaion of he grid is he filering of he idenical varians. 8
10 x2 x2 x 1 a) Random populaion wihou grid mehod. x 1 b) Random populaion wih grid mehod. Figure 4. The modified iniializaion. Wihou he grid he equaliy of wo varians canno be idenified simply because of he floaing poin number represenaion. However, he grid suppors he conrol of he minimal disance beween a given varian, and he neighboring ones. The developed complex geneic coding effecively suppors he descripion of he search space for he geneic algorihm. However, comparing wih he simple sringlike and homogeneous chromosome coding, i is disadvanageous, ha he evaluaion of he similariy measures is more difficul. By means of he Euclidean and Hammingian disances, i is no possible o define a consisen similariy measures for he opionally hierarchical combinaion of coninuous and discree gene sequences. This siuaion limis he applicaion of he mehods, used for he conservaion of he diversiy in he populaion. The inroduced grid mehod makes possible anoher new soluion o converse he diversiy. Accordingly, he new varians are o be generaed only ino he grid poins, while having modified he grid size, we can conrol he equilibrium beween he exploraion and exploiaion ha conrols also he diversiy. For his soluion, I have modified he recombinaion and muaion operaors appropriaely. The modificaion of he operaors for he coninuous genes is similar o he mehod, applied for he iniializaion, i.e. he values produced by he convenional operaors, are rounded o he neares grid poins. In he case of 9
11 discree genes, a simplified grid mehod has been realized. Accordingly he gene sequences are associaed wih an ineger, ha less hen he half of he sequence lengh ha deermines he awaied minimal Hammingian disance, beween he paren and child gene sequences in he execuion of he crossover and muaion. This soluion is similar o he consideraion of precision for coninuous genes. The algorihm of he operaor should no be modified, only he Hammingian disance beween he gene sequences is compared wih he awaied value, and he operaors are repeaedly applied unil eiher he appropriae child will have appeared, or he number of repeiions reaches he prescribed upper bound. This upper bound is he funcion of he lengh of he given sequence, and is defaul value is he half of he sequence. I is advanageous, ha he inroduced developmen does no change he convenional algorihm of he operaors, only he resuls are modified. Accordingly, in a marginal case he precision and/or he awaied minimal disance can be se o zero, and in such a way he operaors behave according o he convenional siuaion. Saring wih a rough grid and refining he grid along he subsequen generaions, we can avoid he oo early convergence. The criical elemen of he mehod is he scheduling of he changing decomposiion of he grid. In he case of a single objecive funcion, he adapive scheduling has been proposed ha follows he change of he acually bes soluion. In he case of he mulicrieria evaluaion, only he a priori, deerminisic scheduling can be applied. According o he experiences, wih he use of an exponenial decrease, o ge he awaied preciosiy a he 75% of he planned generaions, seems o be an effecive and safe scheduling sraegy. 3.3 Managemen of he mulicrieria evaluaion Supporing of he mulicrieria decisions has a keynoe role in he elaboraion of he new algorihm. In he soluion I made possible fiing he preferences of he decision maker by each of he a priori, ineracive and a poseriori mehods. In he declaraion of he evaluaing crieria, we can define he opional properies. These properies are he prioriy, as well as he apparen lower bounds of he given objecive. By means of hese properies he decision maker can guide he 10
12 1. Consrains (by prioriy) (number or sum of violaion) P 1 For example # of consrain violaion P 2 dominaing? Normalized sum of violaions dominaing? yes yes nondominaed dominaed dominaed yes dominaing? no 2. Fulfill of evoluion goals nondominaed P 1 yes dominaing? no 3. Objecive values (by prioriy) Objecives wih P 1 prioriy dominaing? P i Objecives wih P i prioriy (i {2,...,m}) yes dominaed dominaing? yes yes dominaing? no nondominaed Figure 5. The calculaion of modified Pareodominance. opimizaion process o he preferred regions of he Pareofron. The flexible possibiliy of he unified consideraion of consrains, prioriies and objecives, makes possible he applicaion of a ranked finess calculaion, according o a modified Pareodominance in he mulicrieria geneic algorihm. The ranking can be used boh for a single objecive and in he mulicrieria evaluaion wih he possibiliy of he reamen of condiions. There are various sraegies for he finess calculaion, and he majoriy of he mehods, published in he lieraure can be applied. For example we can use he number of he dominaed varians or he number of he varians, dominaing a given varian, as well as he deph of he Pareofron. In he calculaion of he dominance, firs he condiions are evaluaed one afer he oher. Opionally, a prioriy ranking can be defined for he condiions. This makes possible he combined use of 11
13 he condiion violaions (i.e. how many condiions are violaed by he given varian), followed by he consideraion of he summarized or maximal measure of consrain violaions (i.e. how long is he disance from he awaied range). The evaluaing crieria have a prioriy ranking, oo. In a given prioriy group, firs he fulfillmen of he objecives are compared (i.e. he varians ha fulfill he objecives are beer). Nex, he varians ha fulfill he objecives are compared according o he value of he crieria. The procedure of he calculaion is illusraed in Fig. 5. The necessary condiions for he good esimaion of he Pareofron are ha he proposed soluions have a uniform disribuion along he fron. Therefore, in addiion o he finess values, he varians are characerized by a crowding parameer. There are many opions for hese crowding parameers. In he presen work, he muliplicaion of he disance from he k neares neighbors was used as a crowding parameer, where k was he number of he evaluaing crieria. In he applied selecion algorihm, his parameer was used for he selecion and replacemen of he comparison of he parens for he varian, having he idenical finess value. This helps he uniform disribued idenificaion of he Pareofron. Because of he small populaion size, he number of he nondominaed soluions in he las populaion is also small. Consequenly, i is advanageous o save all of he known Pareoopimal soluions in ouer archive sorage. The size of he archives can be configured freely. Having reached he maximal size of he archives, he saved Pareoopimal varians are deleed, according o heir crowding parameers. 3.4 Implemenaion properies Because of he high compuaional demand of he evaluaions, usually only a couple of housand or en housand varians can be calculaed in problem solving. Nowadays, he usually applied compuers make possible he fas enough sorage of geneic code wih a rapid enough reaing of hem, wihou slowing down he execuion of he algorihm. 12
14 Consequenly, I made possible he sorage for all of he evaluaed individuals. The role of his sorage is similar o he well known abu lis. The sorage helps o avoid he repeaed evaluaion of he varians. Accordingly, he modules of he algorihm, responsible for he generaion of he new varians (iniializaion, recombinaion and muaion), are prepared for he invesigaion, wheher he new geneic code paricipae in he sorage of he evaluaed ones. In he case of a possible repeiion, he operaors ry o generae a new geneic code. The number of he repeaed generaions is limied, while he defaul value is he half of he number of simple genes in he geneic coding. Having arranged his limi value, he usual recombinaion and muaion are replaced by an exra muaion. This exra muaion is execued for each gene wih he probabiliy of 0.5, according o he respecive gene ype. Consrains are reaed in wo levels in he mehod. In he firs level he incompaibiliy relaions declared for he discree gene ypes, are considered in he iniializaion, recombinaion and muaion. The forbidden combinaions are managed by he mehod, describe for he abu lis. In he second level, consrains are aken ino consideraion in he calculaion of he exended Pareodominance. The developed geneic coding is conform wih he objec oriened programming paradigm. The objec oriened design and implemenaion of he program made possible he opional runime change of he program componens. This characerisic helps in he adapaion of he algorihm for he soluions of various pracical problems, as well as in he esing of he operaors, finess calculaions, and crowding parameers. The highes class of he objec hierarchy is responsible for he calculaion of he evaluaing crieria. The children of his class are he mos frequenly used classes, communicaing wih he collaboraing simulaor, as well as he classes execuing he various es and benchmark problems. In he former version, he EXCEL, as a DDE server was used for he communicaion wih he simulaor, running under Windows. The new version uses a plaform independen communicaion proocol. This proocol suppors he parallel work wih muliple simulaors in sense of server/clien model. The main funcionaliies of his proocol are he configuraion of he source space, he configuraion of 13
15 1. Algorihm: The pseudocode of he new algorihm. Inpu: cmp F : he modified Pareodominance operaor Inpu: ogrid: he round o grid funcion Inpu: s, a: he populaion and archive size Inpu: r,m: he recombinaion and muaion facor Daa: : he generaion couner Daa: Pop: he populaion Daa: Arc: he archive wih he acual bes individuals Daa: Par: conainer wih he acual parens Daa: Off: conainer wih he new children Daa: v: finess access funcion Oupu: X : he opimum se 1 begin 2 X creaedecisionspace(x) //creae he decision space 3 Y creaeobjecivespace(y) //creae he objecive space 4 C creaeconsrainspace(c) //creae he consrain space Arc Pop ogrid ( creaepop(s, X ) ) 7 //iniialize he populaion Pop evaluaeindividuals ( Pop, X,Y,C ) 8 //evaluae v assignfiness ( ) 9 Pop,Arc,cmp F //he Pareo ranking funcion 10 while erminaioncrierion() do Arc updaeopimalsen ( Arc,Pop ) 11 //updae he archive 12 Arc pruneopimalse(arc, a) //pruning he archive 13 X updaegrid(x, ) //updae he grid Par selec ( Pop,Arc, v, s ) for i 0 up o Par 2 do if rand u () r hen Off[i,i+1] ogrid(recombine(par[i],par[i+1])) if rand u () m hen Off[i,i+1] ogrid ( muae ( )) Off[i,i+1] Off evaluaeindividuals ( Off, X,Y,C ) 18 //evaluae v assignfiness ( ) 19 Pop,Arc,cmp F //he Pareo ranking funcion Pop reproducepop ( Par,Off,Pop, v ) 20 // reproduce he populaion reurn exracopimalse ( Pop Arc ) end 14
16 he evaluaing crieria, he sending of he code of varians o be evaluaed, as well as he receiving of he evaluaions. The simple pseudocode of he above describe geneic algorihm is describe in algorihm 1. 4 CONCLUSIONS Considering he above described research resuls, as well as he experiences, obained from he applicaion of he coninuously developing geneic algorihm, he mos imporan conclusions are he followings: The elaboraed complex geneic coding can effecively be applied for he common represenaion of discree and coninuous genes, while he a priory deerminaion of he incompaibiliy relaions helps o avid he elaboraion specific individual geneic coding for he various complex asks. The flexibiliy and efficiency of he complex geneic coding and of he respecive exended se of operaors are proven by illusraing examples and es problems. In addiion, he elaboraed mehods were successfully applied for he idenificaion and model based opimal design, conrol and scheduling of various complex pracical problems wih hybrid models. The synergic applicaion of he elaboraed complex coding and he grid mehod can effecively be applied for descripion of he genes for he coninuous propery classes. The developed mehodology gives possibiliy o uilize he heurisic knowledge of he exper in he descripion of he search space. Simulaneously, anoher advanages feaure is, ha having recognize he soluions in he viciniy of he prescribe bounds, he algorihm auomaically exen he search space. The elaboraed grid mehod suppors he adequae iniializaion and effecive run of he geneic algorihm, also in he case of small populaion size and of small generaion number. This makes possible o increase he compuaional demand of he evaluaion, consequenly he applicaion of he possibly mos deailed models in he soluion of many pracical problems. This is very imporan, because he applicabiliy of he economic and echnical opimizaions is deermined 15
17 by he exhausiveness. Accordingly, he less number of calculaions wih a more deailed model is preferred o he more number of evaluaions wih a simplified one. The new mehod elaboraed for he reamen of Pareodominance, conribues o he correc and powerful soluion of he mulicrieria problems. The new developmens help he adequae choice and ranking of he consrains and evaluaions, as well as he evoluion of he mulicrieria good enough soluion. In he soluion of he pracical problems, he prioriy ranking of he consrains and evaluaions combined wih he new grid mehod, help o focus on he very par of he Pareofron, where he good soluions are awaied. In pracical applicaions o deermine he evaluaions, he exper ofen ends inuiively or consciously o define objecives of approximaely idenical imporance. Consequenly, he soluions wih almos commeasurable values used o be preferred. One of he lessons, coming from he pracical resolved opimizaion and idenificaion problems was ha i is no possible or i does no worh o aggregae he evaluaion ino a single objecive funcion. In opimizaion, almos everybody wans o make an economic evaluaion (i.e. minimizing he cos or maximizing he profi), however he daa for he calculaion of he economic goal funcion are no known. I is a ypical case, when we have o opimize one, byiself also complex par of a echnological process, consising of many seps. The economic parameers of he inpu and oupu maerials are ofen no known. Consequenly, he sudy ough o be exended o a greaer sysem consising of his par. On he oher hand, he field expers can declare very good naural objecive funcions. Neverheless, on ongoing mehodological developmen ends o bridge he exising gap beween he echnological and economical processes, and his makes possible he more and more correc economic evaluaion. In accordance wih he resuls, obained from he logisical example of he presen work, he combined applicaion of he economic and naural evaluaions seems o be a feasible mehod, emporarily. The experiences, obained wih he coninuously developing and presenly furher developed, inegraed geneic algorihm, proved ha he applied coding and operaors, as well as he archived sorage of he invesigaed varians suppor 16
18 he opimizaion process wih small populaion and generaion number. The opimizaion of he pracical asks wih grea compuaional demand for he evaluaion can be solved by he macrogranularly parallel simulaion and evaluaion of he varians. The mehod, implemened in a PC cluser, can accelerae he geneically conrolled evoluionary process almos proporionally wih he number of CPUs of he cluser. 5 NEW SCIENTIFIC RESULTS The new scienific resuls of my work are he followings: 1. I have been developed a new complex geneic coding, based on he srucure laice for he combinaion of opionally hierarchic, discree/ coninuous and permuable gene sequences. The srucure laice makes possible he a priory definiion of he opional incompaibiliy relaions beween he discree properies, classified ino equivalency classes. The developed coding suppors he uniform reamen of he discree and coninuous propery classes. The alleles of he coninuous propery classes are described by auomaically generaed discree elemens, deermined by he prescribed composiion of he given domain. Wihin a domain, he user can define he lower and upper bounds of he awaied subinerval. The new coding suppors he hierarchic coding, deermined by he ree srucure of he combined genes. Also he mehod makes possible he applicaion, auomaic recogniion and reamen of he full permuaions. I have inroduced new, exended geneic operaors for he iniializaion, recombinaion and muaion, which auomaically consider he ype of he given gene sequence wihin he scope of he previously described coding. I have exended he geneic algorihm wih a new, global operaor, which wih he knowledge of he evaluaions auomaically decreases he grea decomposiion of he properies for boh he coninuous and discree gene sequences. 2. I have elaboraed a userfriendly, complex mehod for he calculaion of he Pareodominance ha conains he published mehods for he 17
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