The use of fuzzy connectives to design real-coded genetic algorithms

Ma th w are&Sof tC om puti ng 3( 19 94)23 9-25 1 TheU s eofF uzz yC onnec ti vest oD esign Real-CodedGeneticA lgorithm s 3 F.He rrera,M.Lozanoa nd J.L.V erdega y Dept. ofCom puterSci enceandA rtici alIn te ll igence Uni v ersi ty of Gr anada,18071-Granada,S pa in e-mail:her r er a,loz ano,verde gay@r obinson.ugr .es Ab stract Gen eticalgorithmsareadapti ve methodsthatuseprincip lesi nspiredb y naturalpopu la tiongeneticstoev olv esol ut ionstosearc handoptimizati on proble ms.Geneticalgorithmspro cessapopulationofsearc hspac esolutions withthreeoperations:sele ction,crosso verandm utat io n. Agreatpro blemintheuseofgeneticalgorithmsisprematurecon vergence; th esearc hbecomestrappedinal ocaloptim umbef orethegloba loptim um isfound.F uzzylogictec hniquesma ybeusedf orsolvin gthisproblem.This paperpresentsoneofthem:thedesign ofcrosso v eroperat orsforreal-coded geneticalgorithmsusingf uz zyconnectiv esanditsextens ionbasedonthe useofparameterizedf uzzyconnectiv esa st oolsfortac klingtheprema ture con vergenceproblem. Keyw ords: GeneticAlgorit hms,RealC oding,F uzzyConne ctives. 1In troduc ti on G eneti cal gorithm s( GAs)aresea rc halgo ri thms thatuseoperatio nsfoundi n na turalgen eticst og ui dethetrekthroughase arch space.G Asaretheor etical ly a ndem pirical ly pro v entoprovi derobustsearc hi nco m plexs paces,g ivi nga v al id app roa ch t o pr o blem srequi ri ng ecien t and e ect i vesea rc h([6] ). A GA starts wi thapopulati on of r ando mly ge nerate d sol ut i ons, c hr o mosom es, and adv ance to wards better so lutio ns b yapply ing g e netic op e rato rs m odel ed o nthe g e net i c pr o ce sse s o cc urring i n natur e. In thes e al gorithm swemain tai n a p opula tion o f sol utions f or a gi ve n problem; this p opul atio n unde rgo es ev olut i on in a f orm of na t ural s electi on. In eac h ge neration, relati vel y go o d sol ut i ons r epro duc e t o giv e o spring tha tr eplace the relati vel ybadso lutions whic hdi e. An ev al uatio nof tnes s func ti on pla ys the rol eoftheen vi ro nmenttodi sti nguish b et wee n good and badsolutions. 3 This researchhasbeen supported by DGICY T PB920933. 239 240 F.Herrera,M .L oz ano& J. L. V erdega y Al th o ughtherearem a n ypossi blev a ri an tsofthebasi cGA,t he fundam en tal un derlyi ngm echa nismoper at es on apo pul ati onofc hro mosome s( represen ti ngpo ssi blesol utionstotheproblem) andco nsi stsofthefo llo wing op eratio nswhic hare appli edduringea chg enera ti on t : 1.ev alua tion of in di vidua ltness, 2.form ationofag enepoo lb yc hoo si ngi ndivi dualsi np ropo rtiontotheirrelati v e tness, 3.r ec om bi natio nb ym eansoftheg enet i coperatorcro sso v erandm utation. Theprocessisitera te du n tilthesystemceasestoim pr o v eo rag iv engeneration T isrea ched. Fi xe d-l eng thandbi naryencodeds tringsf orrepresen tation so lutio nha v edom - inatedG Aresearc h,sincethe re existtheoreticalresults that sho wthemtobethe m osta ppropri ate,a nd they aream ena bl etosim ple im plem en tatio n.ButtheGA's goodpropertiesdonot stemfromtheuseo fbitstri ngs( [ 1] ).F orthisreason, thepathha sbeenlai nt ow ardstheuseofal phabetswi th ahi ghercardinal ,f ollo w edb ythedev el opme nt ofnewgeneticoperato rs(c rosso v erandm uta tion)on thesealpha bets.Nonbinaryencodingsi ncl uderealnum berrepresen tati ons,whi ch w ouldseempart i cul arlynaturalwhenw ea ret a ckl ingopti m izati onproble m sof param eterswi th v aria blesi ncon ti n uousdo mai ns.Thenac hro m osom ei sav ecto r ofoatingpoi n tn um berswhosesi zei sk eptthes am easthelengtho fthev ector whic hi sthesol utiontotheprobl em .G Aswiththist ypeofencodi ngwil lbecall ed real -codedGAs(R CGAs). Theuseo frealpar am etersm a k esitpossi bl etou se la rgedom a ins(ev enunknown dom ai ns )f orthev aria bles,whi c hi sd iculttoac hiev ei nbi naryim plem en tations, where incr easingthedo m ai nw oul dm ea ns acricingpreci si on,assumi ng axed lengthf or thec hrom osom es.Anothera dv an tag ewhenusi ngrealpa ra mete rsisthei r capa cit yt oe xplo itthegraduali t yof thef unctionswithcon ti n uousv ariabl es(where theco nc epto fg radual it yref erstoth ef actt ha tsl igh tc hangesi nt he vari ables corre spondtosl igh tc hangesi nt he functi on).Lastl y ,th ey al lo wthetoo lstha t handlen on-tri vi alrestrictionst ob ed esi gnedm oreeasi ly ,a nd ther epresen tation oftheso lutio nsi sv eryc lo setothenaturalf orm ul atio nofm an yprobl em s. The m utatio no pe rator a rbi tr a ri ly al te rs one or m ore c o mp onen ts (c a lled ge nes) ofasel ected structur e s o as to i nc re a se the structu ral v ari abil it y of t he p opulati on, e. g., i t explores the se arc hspac e. Unde r bi nary c od i ng, givenag enewithv al ue \0 ", it is re pl ace d b y\ 1", and vi ce ver sa. Unde r re a l co ding di e re n tve rsions of t hi s op e rator w er e pr ese n ted ([ 9] ). Eac h gene of e ac hc hrom osom ein the p opul ation un dergo es a random chang eacco rdi ng to a probabil it y dene d by the m utation ra te, the m ut a tion proba bi li ty , p m . The c rosso ver o p e rator e xpl oi ts the a vai la ble i nform ation f rom the p opul ation ab out the se archspace. It combines the feature s of twoparent st ruct ures to form twosimilar o spring. The classical c rossover op er ator unde r binary co ding builds TheUseofF uzzyC onne ctiv es to Desi gn. .. 2 41 an ospri ngb yl inki ngtoge ther tw ogenesegm en ts,eac ho ne be longi ngtoadi eren t pa re n t.Thi sopera tori sappl iedw i thaprobabi li t yo fpe rforma nce,thecrosso v er probabi li t y , p c ,t hatdeterm i nesthen um bero fc hrom o som esinthepopulation to be cr ossed.[ 7,9 ]repo rtcrosso ve ro peratorm odelsunderb inaryandrealcodi ng . Thecrossov er op eratorpla ysacen tra lrole in th eGA' sperform ance.It could beconsidered to be o neofthealg orithm 's deningc haracteristics,andi ti so ne o fthecom po ne ntst obearinm indforim pro vi ng theG A'sbeha vio ur([ 12] ). The m utation op er a to risacom plem en tofthec ro sso v ero perator.Itisneed ed toa v oid thelo sso fu se fulinfo rm atio nproduce db yt hecrosso v er. Ani m portan tpro blemin theu seofG Asis pr emat ur ec onvergenc e ;thesearch becom estrappedi naloc al opti m umbefo rethegl obalo ptim um i sf ound.Thisi s producedb ythel ac kof diversi ty inthepopul ationandadi sproportio nate exploitation / explor ation rela ti onship;anadequateba lancebet w ee nabroa dsearc handa sucient re nem en tisnotesta bl ished . F uzzy logi ct ec hniquesm a ybeusedt os olve thesepro blem s.Anattem pt consists of theu seo ffuzzyl ogi cbased system sfo rthedy nam i ccon trol param eters o fR CGA( p m , p c ,popula ti onsize,etc)insuc haw a yth at theco rrectexploi tati on/explorati onrel ationsh i pandsuitabl ediv ersi t yl ev elsbecam eestabli shed([8 , 1 1]).Anothero neusesf uzzy co nnectiv estodesig ncrosso vero perators([8] ). Inthispaperw epresen tcrosso v eroper a to rsforR CGAsba sedo ntheus eof f uzzyco nnectiv es,andthei re xtensi onbasedo ntheuseofparam eteri zedfuzzy connectiv esfordesigni ngdynam i ccrosso v eroperatorswiththem ai nobj ectiv eof in troducingpopulati ondi v ersit yi ntheGAsearc h. 2 D esi gnofC rosso v erO perato rs forR CGAusi ngF uzz yConne ctiv es Ash as a lrea dybeenpoi n tedo ut ,a chrom o som eisav ectorofre al nu m bers,and i ts pr ec i si onwi llbem ark edby th at of thecom puterunderwhi ch th ea l gori thmi s carriedout.Thesi ze of thec hrom osom ei sk eptthes am ea st hel engtho fthev ector thatistheso lutio nt ot he problem ;inthisw a y ,e ac hg ener ep resen tsav ari abl e o ft heproblem .Thev a luesofthec hrom o som egenesarefo rcedto rem ai ninthe i nte rv al e stabli s hed b y the v ari able that t he c hrom oso mere pre sen ts , so the gene ti c op erators m ust pr ese rv ethisre qui rem en t. In[3]i tw as p oi nt ed out that the c rosso ver op er a to ri sak ey poi ntfor solv ing the prem ature con ve rgenc e problem .Th us , s ol utions to thi sprob lem m aybe f ound b y desi gni ng new alter na ti ves to thi sopera tor. Her e, the de v el opm ento fsuc h cross o ver op erators is attem pte d. W e pre sen t c rosso v er op e rators fo rR CG A base d o nthe use o f fuzz y connec tiv es: tno rm s , t-c o norm s, a ve rage functions and gene ral ize d com p ens a tion op er a tors ([14 ,1 5]) whic h induce di  ere n t div ersi ty lev els in the p opulation , and ther efore the premature c onve rgenc e problem maybe controlled. Letusassumethat the chromos omes C 1 =( c 1 1 ;c 1 2 ;:::; c 1 n )and C 2 =( c 2 1 ;c 2 2 ;:::; c 2 n ) 242 F.Herrera,M .L oz ano& J. L. V erdega y ares electedtoapplythecr osso veroperatortothem ,a nd tw og en es c 1 i and c 2 i to be cr ossedo v er, c 1 i ;c 2 i 2 [ m i ;M i ], bei ng x i =m in( c 1 i ;c 2 i )a nd y i =m ax( c 1 i ;c 2 i ). It se em sreaso nabletoim aginet he po ssibi lit yofobtai ningg oodd escenden tsoutsi de th is in terv al.Insho rt,thei nterv alofactio noftheg en e i ,[ m i ;M i ], ma ybediv ided in tothreeregio ns[ m i ;x i ],[ x i ;y i ] ,a nd[ y i ;M i ],wheregooddescende ntsm ay be obtained;ev enconsider ingaregi on[ x 0 i ;y 0 i ]wi th x 0 i  x i and y 0 i  y i w ouldseem reasonabl e.G ra phi ca ll y Fi gure1 :Ac tio nin terv alforag ene W esh al lno wg oontoputforw ardase tofcrosso v eropera torsthat al lo w desce nden tstobeobtai nedinthepr evi ousin terv al s.Inorderto dothat,w euse fourfunctions F , S , M a nd L denedf rom[ a;b ] 2 [ a;b ]i n[ a; b ] , a;b 2< ,whi ch full l: (P1 ) 8 x;y 2 [ a; b ] F ( x;y )  m in( x;y ), (P2 ) 8 x;y 2 [ a ;b ] S ( x ;y )  m ax( x ;y ), (P3 ) 8 x; y 2 [ a ;b ]m in( x;y )  M ( x;y )  ma x( x;y ), (P4 ) 8 x;y 2 [ a ;b ] F ( x;y )  L ( x;y )  S ( x ;y ), (P5 ) F , S , M ,and L are monotonenon-decreasing. Letusassum ethat Q 2f F;S;M;L g ,and C 1 =( c 1 1 :: :c 1 n )a nd C 2 =( c 2 1 :::c 2 n ) are t wo chrom osom es that ha v e b ee n s elec ted to apply the cr osso ver o p erator to them .W em ay gene rate the c hrom osom e H =( h 1 :::h n )as H = Q ( C 1 ;C 2 ) ;h i = Q ( c 1 i ;c 2 i ) ; i =1 ;: :: ;n: W ith the t-norm o p erators , t-conorm s, a ver a ging f unc tio ns and gene ral ize d co mp e nsatio nopera to rs used as fuz zy c onnec ti ves , w e shall a s so ciate F wi th a tnorm , S with a t-conorm , M wi th ana ver a ging o p erator and L wi th a ge neralize d comp ens ation op e rator. Firs t, weneedasetoflinear t ransformations to be able to apply the se op e rators unde r the gen e denitioninter vals. TheUseofF uzzyC onne ctiv es to Desi gn. .. 2 43 Let O beano peratorbelonging to thesetfo rm edb ythet-no rm s, t-co norm s, av er a gingf unctio nsandgene rali zedco m pensatio no perators. F oreac hpo si tion i 2f 1 ;: :: n g ,thef oll o wi ng op er at i onswil lbeca rri edou t: 1.T ransform c 1 i and c 2 i intot he v alues s 1 i ;s 2 i 2 [0 ; 1]s uc hthat s k i = c k i 0 m i M i 0 m i ;k =1 ; 2 : Usin gt hi ss tep,w etransf ormthev alueso fthegenessothattheo perator m a ybea ppl iedtothem . 2.Appl ytheoperator O ( s 1 i ;s 2 i )andca lculatethe valu e h i : h i = m i +( M i 0 m i ) 1 O ( s 1 i ;s 2 i ) ; sotha tthegene h i i si nrel ationtoi ts orig inalli m i ts, h i 2 [ m i ;M i ] . 3. h i wi llbethe v aluefo rtheg eneatposi ti on i o fthec hro m osom eresul ting fromthecro sso v erofthec hro m osom es C 1 and C 2 . Com plyi ngwi thaset offuzzyconnectiv es,( T j ;G j ;P j ; ^ C j ), j =1 ;:: :;k ,aseto f f unctio ns F j , S j , M j and L j j =1 ;:: :;k isbuil ta sw edescri bebelo w: F j ( c 1 i ;c 2 i )= m i +( M i 0 m i ) 1 T j ( s 1 i ;s 2 i ) S j ( c 1 i ;c 2 i )= m i +( M i 0 m i ) 1 G j ( s 1 i ;s 2 i ) M j ( c 1 i ;c 2 i )= m i +( M i 0 m i ) 1 P j ( s 1 i ;s 2 i ) L j ( c 1 i ;c 2 i )= m i +( M i 0 m i ) 1 ^ C j ( s 1 i ;s 2 i ) Thesecross ov er op er atorsha v edi ere nt featur es:the F -and S -cros so ve rs show explo ra ti on,the M -crosso v eroperatorssho wexpl oitati onandthe L -crosso v ersho w relaxed exploi tatio n. 3Exa m pl e W eha ve carr i e d out die ren tex perim en t s th a t help to com parethebeha viour of ab inary c o ded GA, and som eR CG As wi t h cros so ve r op e rators th a tha vebeen prop ose d i n other publi ca tions, with a s et o f alg orithm s bas ed on the cross o ver op erators prop ose d, which us e t he fuzzy connectives ( T j ;G j ;P j ; ^ C j ), j =1 ;:::; 5 showed inTable1. 244 F.Herrera,M .L oz ano& J. L. V erdega y t-normt-conormAv eragingO perator LogicalP ro duct LogicalS um T 1 ( x;y )=min( x;y ) G 1 ( x;y )=max( x;y ) P 1 ( x ;y )=(1 0 p ) x + py Hamac he rP ro ductHamac he rS um f ( x )= 1 0 x x T 2 ( x;y )= xy x + y 0 xy G 2 ( x;y )= x + y 0 2 xy 1 0 xy P 2 ( x;y )= 1 y 0 yp 0 xy + xp xy +1 AlgebraicProductAlgebraicSum f ( x )= 0 log x T 3 ( x ;y )= xyG 3 ( x ;y )= x + y 0 xyP 3 ( x;y )= x 1 0 p y p EinsteinProductEinsteinSu m f ( x )=log 2 0 x x T 4 ( x;y )= xy 1+(1 0 x )(1 0 y ) G 4 ( x;y )= x + y 1+ xy P 4 ( x;y )= 2 1+( 2 0 x x ) p ( 2 0 y y ) 1 0 p BoundedProductBoundedSum f ( x )=1 0 x T 5 ( x;y )= 0 _ ( x + y 0 1) G 5 ( x;y )=1 ^ ( x + y ) P 5 ( x ;y )= (1 0 p ) x + py T a ble1: SetofOp er ators Eac ht-cono rmi sdualt ot het-no rmsho wntoitsl ef t. Thea v era gingfunction iscal cula tedfromt he form ulaofthequasi-a ri thmeti ca v erages,usingasthe f functi onthea ddi tive gene ratorf unctio nofthet -normp la cedinthesa m eline, exceptf ortherstone wh i c h,notbei ngArc him edean,d oesno tha v eag enera to r functi on,andw eshall us e f ( x )= x .Thi sf unctio ni ss ho wni ntheupperpartof thecel lswhere thesea v eragi ngopera to rsarespeci ed.F orea chf am i lyof o perators inT abl e1 ,ageneral izedcom pensationoperator ^ C j wil lbeconside red,denedas fol lo ws: ^ C j = P j ( T j ;G j ) Forthef am i lyofLo gicaloper ators,w eshal lconsider ^ C 1 = T 1 0 p 1 :S p 1 . TheGAfam il iesaredi eren ti atedacco rdi ngtoho wtheycar ry outthefol lo wing tw osteps: 1.Generati on of osp ringusi ngthedieren tc rosso v eroperators. 2.Sel ectionofospri ng resu l tingfromthecrosso v er whic hwil lformpartofthe po pulati on. Ap ropo sali sthefo llo wing :F or ea ch pair ofc hrom oso m esfrom at ot al of 1 2 1 p c 1 N ( p c crosso v erprobabi lit y , N po pul ationsize) ,fourospringareg enera ted, t he res ul t of applyi ng sp e cic f u nctio ns F , S , M ,a nd L to them .Thet wom ost pr o misi n g osp ri ng of the f our r epla ce thei rparen ts in the p opulati on. Thi ss el e ction st rategy in troduce sahi gh explo itati on l ev el wi th a n unde rl ying expl oration c ause d b y the use of the dier en t Q -cross o ver o p erators . Next, the resul ts o nR ose n bro c k's Gene ral ize d f un ctio n([ 4]) are sho wn. The anal yt i ca l and graphi ca l form ul atio n togethe r wi th the elem en tth at repr es en ts the glo bal o pti mum(m inim um ) are: f ( ~x )= n 0 1 X i =1 (100 1 ( x i +1 0 x 2 i ) 2 +( x i 0 1) 2 ) TheUseofF uzzyC onne ctiv es to Desi gn. .. 2 45 0 5 : 12  x i  5 : 12 min( f )= f (1 ;:::; 1)=0 Fig ure2: where n =5. AsetofnineR CGAbasedo ncrosso v eroperatorspres en tedi ntheli terature w ereconsideredfort he experi m en ts(R GA1-R GA9 ). Belo wtherei sata bleindicati ngthet y peofcrosso vera ndm utatio nusedb yeac hoft hem ,tog etherwi ththeir na m es. AlgorithmsMutationC ros so v er R GA 1 RandomSimpl e[13,17] R GA 2 NonUniformSimple[13,17] R GA 3 RandomUnif ormArithmeti cal[13] a =0 : 35 R GA 4 NonUniformUnif ormArithmetical [13] a =0 : 35 R GA5-  NonUniformBLX-  [5] (  =0 ;: 15 ;: 3 ;: 5) R GA 6 NonUnif ormDiscrete [16] R GA 7 NonUniformLinear[17] R GA 8 NonUniform ExtendedI ntermedi at e[16] R GA 9 NonUniformExtendedLine[16] Tabl e2 : Real Co ded Gen eti cA lgor ithms By BGA w edeno teabi nary co ded GA with 30 genes per v ari able,m ultipl e cross o ver wi th t wopoi nts and pr o porti onal se l ecti on probabi lity. By NRG A 1 ; ::: ; N RGA 5w edeno te a fa mil yo fG A based on t he f uzzy conne ctive s c ross o ver and the gene rati on a nd sel ecti on of os pring prop osals, using the familie s of fuz zy conne ctiv es: Log ical, Ham am cher , Al ge braic , Einste i n and Bound, res pectiv el y.Inal l case s w eusethe sto cha sti c univ ersa lsam pl ing ([ 2]) se l ection pro ce dure and the e litist model. Wecarried out our exp eriments u sing th e following parame te rs: th e p opulation size is 61 individuals, the c rossover pr obability 246 F.Herrera,M .L oz ano& J. L. V erdega y p c =0 : 6,andtheprobabi li t yo fc hr om osom eupdate p u = p m 1 5=0 : 6,andthepara meter b usedb ythenon-un iformm utati onis5.W eex ecut ed al lt heal gorithm s3 ti mes,ea chonewith10 0,500a nd 5000generations,a ndpres en tt hea v eragev al ue of themi nT a bl e3. Al gori thms10 050050 00 BGA 1.1262e+013.6069e+ 001.9045e+00 R GA1 3.0446e+012.0070e +0 16.0669e+00 R GA2 6.9230e+002.1448e+ 004.7343e-01 R GA3 4.1941e+00 8.6031e+ 00 6.3745e+00 RGA4 3.7915e+002.9624e+ 008.9244e-01 R GA5-0.0 4.5504e+002.3454e+ 009.1602e-01 R GA5-0.15 3.215 7e+002.9556e+ 007.092 9e-01 R GA5-0.3 3.7477e+001.5844e+ 004.8854e-01 R GA5-0.5 8.2653e+002.1099e+ 001.7329e+00 R GA6 5.5393 e+002.8379 e+ 003.5106 e-01 R GA7 4.7115e+001.8487e+ 005.1499e-01 R GA8 4.6861e+003.5337e+ 005.3325e-01 R GA9 4.2196e+002.9374e+ 003.8014e-02 NR GA1 4.1431e+001.9687e+ 004.9364e-03 NR GA2 1.2378e+017. 38 68e-014.0848e-02 NR GA3 1.003 1e+013.5037e+ 001.0099e+00 NR GA4 1.2425e+014.3985e + 001.8347e+00 NR GA5 6.0121e+002.6930e+ 001.2150e+00 T a ble3: Results Thebestbeha vi ourcorrespo ndstotheLog icalcrosso v ero perator.Thi soperatortogetherwiththeospr i ngsel ecti onm ec hanismoer asuita bleexplo itati on/explo ra ti onbal ance, al thoughw em ustpoi n touttha tt hi sm ec hani smi sm ore tim eexpensiv ebecauseneed more ev a luatio ns.Otherospri ng sel ectionm ec hanism sareproposedin[ 10] . 4 D esi gnofDynam i cCross ov er Op e ratorsUsi ng Para met eri ze dF uzz yC on ne ctiv es A n idea for a voiding the pre m atur e c on ve rgenc e consists in al lo wi ng the e xploration in the b e gi nning of the search pro c es s and t he e xpl oitati on at the end of it. With t he explorati on the di versi tybeca mes greate r, i nc reasing the pr o babil it yo f nding z ones whic ha r e clo sed to opti mal s o lutio ns . Then, supp os i ng t ha t the p o pulation ha vei nf orm ati on ab out these zones, the co nve rgenc e to war ds the opti mumi s pr o duce d through e xpl oitati on. Am utati on op er a tor fo rR CG Acal le d non-unif orm m utatio n ([1 3]) is base d on t he aforem en tioned pr i nc i ple.T h e pr o po rti on in whic harea lgene is m utate d de cr ease s as the GA's e xec ution advance. Thu s, the changes pro duc ed on the ge nes are smaller in the las t ge ner ations pro duc ing a local tuning. TheUseofF uzzyC onne ctiv es to Desi gn. .. 2 47 We ma yextendtheuseofthec ro ss ov ero peratorspresen te di no rdertof oll o w thea forem en ti onedi deas. Die ren tf uzzyconnectiv esc ould be usedduringthe GA's run.Firstl y ,w eshall us ef uz zyconnecti v esthatprod uceh ighdiv ersit ylev els, andl a terotheronesproducingasu  ci en tdi v ersit ylevelt oa llo wtheco n v erg ence toberea ched. W epresen tasetofd ynam ic crosso v eroperatorsb ased on theuseofparam eteri zedt-norm s,t-co no rm sa nda v erag ingfuncti ons( [14, 15 ]).Intherststag es, we shal luset-no rm sand tco no rm sdistan tfromthemi nimu mt-no rma ndm a xim um t-conormrespecti vely ,sohi ghdi v ersit yisi nduced.Later,tno rm scl ose to the m ini m umandt-conorm sclo setothem a xim uma re considered.Theco n v erg ence i scausedandthego od beha vio uro fthelog icalfuzz yconnectiv eswil lbek ept. As w assho wninT able3,thes eoperatorsareingeneralthe m ostpro table. T odothis,w eprop oseasetofcro sso v eroperatorsbasedonthef unctio ns f am ilie s: f F p g p =1 ;:::;G , f S p g p =1 ;:::; G ,and f M p g p =1 ;: ::;G , G 2 N de nedfrom[ a;b ] 2 [ a;b ]in [ a;b ] , a;b 2< ,whic hful ll thec orrespondingP1 -P5prop er ties and: (P6) 8 x;y 2 [ a;b ] ; l im p ! G F p ( x; y )  = m in( x;y ) (P7) 8 x;y 2 [ a;b ] ; l im p ! G S p ( x;y )  = m ax( x ;y ) (P8) 8 x;y 2 [ a;b ] ; m i n( x; y )  M p ( x ;y )  x + y 2 or x + y 2  M p ( x;y )  m ax( x ;y ) and 8 x;y 2 [ a; b ] ; l im p ! G M p ( x ;y )  = x + y 2 Let us consideraGAwi tham axim um num berofgeneratio ns  a nd C t 1 = ( c 1 t 1 ;:: :c 1 t n )and C t 2 =( c 2 t 1 ;:::c 2 t n )tw oc hrom o som esthatw eres el ec te dinthegenera ti on t toa ppl ythecrosso vero pe ratort othem .If Q p 2f F p ;S p ;M p g p =1 ;:::;  w em a yg enera te thechrom osome H t =( h t 1 ;:: :;h t n )a s H t = Q t ( C t 1 ;C t 2 ) ; h t i = Q t ( c 1 t i ;c 2 t i ) ; i =1 ;: ::;n: W esha ll buil d funct i ons fa mil ies with the (P6) and (P7) p rop ert i e s using the pa ra me terize d t-norm sa nd tco no rms desc rib ed in T able 4. Ta ble 5 sho ws the properti e s of t he param eteri zed t-norm si nthi sta ble. The prop e rties of the parameteriz ed t-conorms are analogous. Wemust p oint out that T 6 is the dr astic t-nor m (the smallest t-norm).