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

Ma h w a e&So C om pu i ng 3( 19 94)23 9-25 1 TheU s eo F uzz yC onnec i es oD esign Real-CodedGene icA lgo i hm s 3 F.He e a,M.Lozanoa nd J.L.V e dega y Dep . o Com pu e Sci enceandA ici alIn e ll igence Uni e si y o G anada,18071-G anada,S pa in e-mail:he e a,loz ano, e de gay@ obinson.ug .es Ab s ac Gen e icalgo i hmsa eadap i e me hods ha usep incip lesi nspi edb y na u alpopu la iongene ics oe ol esol u ions osea c handop imiza i on p oble ms.Gene icalgo i hmsp o cessapopula iono sea c hspac esolu ions wi h h eeope a ions:sele c ion,c osso e andm u a io n. Ag ea p o blemin heuseo gene icalgo i hmsisp ema u econ e gence; h esea c hbecomes appedinal ocalop im umbe o e hegloba lop im um is ound.F uzzylogic ec hniquesma ybeused o sol in g hisp oblem.This pape p esen soneo hem: hedesign o c osso e ope a o s o eal-coded gene icalgo i hmsusing uz zyconnec i esandi sex ens ionbasedon he useo pa ame e ized uzzyconnec i esa s ools o ac kling hep ema u e con e gencep oblem. Keyw o ds: Gene icAlgo i hms,RealC oding,F uzzyConne c i es. 1In oduc i on G ene i cal go i hm s( GAs)a esea c halgo i hms ha useope a io ns oundi n na u algen e ics og ui de he ek h oughase a ch space.G Asa e heo e ical ly a ndem pi ical ly p o en op o i de obus sea c hi nco m plexs paces,g i i nga al id app oa ch o p o blem s equi i ng ecien and e ec i esea c h([6] ). A GA s a s wi hapopula i on o ando mly ge ne a e d sol u i ons, c h o mosom es, and ad ance o wa ds be e so lu io ns b yapply ing g e ne ic op e a o s m odel ed o n he g e ne i c p o ce sse s o cc u ing i n na u e. In hes e al go i hm swemain ai n a p opula ion o sol u ions o a gi e n p oblem; his p opul a io n unde go es e olu i on in a o m o na u al s elec i on. In eac h ge ne a ion, ela i el y go o d sol u i ons ep o duc e o gi e o sp ing ha eplace he ela i el ybadso lu ions whic hdi e. An e al ua io no  nes s unc i on pla ys he ol eo heen i o nmen odi s i nguish b e wee n good and badsolu ions. 3 This esea chhasbeen suppo ed by DGICY T PB92- 0933. 239 240 F.He e a,M .L oz ano& J. L. V e dega y Al h o ugh he ea em a n ypossi ble a i an so hebasi cGA, he undam en al un de lyi ngm echa nismope a es on apo pul a i ono c h o mosome s( ep esen i ngpo s- si blesol u ions o hep oblem) andco nsi s so he o llo wing op e a io nswhic ha e appli eddu ingea chg ene a i on : 1.e alua ion o in di idua l ness, 2. o m a iono ag enepoo lb yc hoo si ngi ndi i dualsi np opo ion o hei ela i e  ness, 3. ec om bi na io nb ym eanso heg ene i cope a o c o sso e andm u a ion. Thep ocessisi e a e du n il hesys emceases oim p o eo ag i engene a ion T is ea ched. Fi xe d-l eng handbi na yencodeds ings o ep esen a ion so lu io nha edom - ina edG A esea c h,since he e exis heo e ical esul s ha sho w hem obe he m os a pp op i a e,a nd hey a eam ena bl e osim ple im plem en a io n.Bu heGA's goodp ope iesdono s em om heuseo bi s i ngs( [ 1] ).F o his eason, hepa hha sbeenlai n ow a ds heuseo al phabe swi h ahi ghe ca dinal , ol- lo w edb y hede el opme n o newgene icope a o s(c osso e andm u a ion)on hesealpha be s.Nonbina yencodingsi ncl ude ealnum be ep esen a i ons,whi ch w ouldseempa i cul a lyna u alwhenw ea e a ckl ingop i m iza i onp oble m so pa am e e swi h a ia blesi ncon i n uousdo mai ns.Thenac h o m osom ei sa ec o o oa ingpoi n n um be swhosesi zei sk ep hes am eas heleng ho he ec o whic hi s hesol u ion o hep obl em .G Aswi h his ypeo encodi ngwil lbecall ed eal -codedGAs(R CGAs). Theuseo ealpa am e e sm a k esi possi bl e ou se la gedom a ins(e enunknown dom ai ns ) o he a ia bles,whi c hi sd icul oac hie ei nbi na yim plem en a ions, whe e inc easing hedo m ai nw oul dm ea ns ac icingp eci si on,assumi ng axed leng h o hec h om osom es.Ano he a d an ag ewhenusi ng ealpa a me e sis hei capa ci y oe xplo i heg aduali yo he unc ionswi hcon i n uous a iabl es(whe e heco nc ep o g adual i y e e s o h e ac ha sl igh c hangesi n he a i ables co e spond osl igh c hangesi n he unc i on).Las l y , h ey al lo w he oo ls ha handlen on- i i al es ic ions ob ed esi gnedm o eeasi ly ,a nd he ep esen a ion o heso lu io nsi s e yc lo se o hena u al o m ul a io no m an yp obl em s. The m u a io no pe a o a bi a i ly al e s one o m o e c o mp onen s (c a lled ge nes) o asel ec ed s uc u e s o as o i nc e a se he s uc u al a i abil i y o he p opula i on, e. g., i explo es he se a c hspac e. Unde bi na y c od i ng, gi enag enewi h al ue 0 ", i is e pl ace d b y 1", and i ce e sa. Unde e a l co ding di e e n e sions o hi s op e a o w e e p ese n ed ([ 9] ). Eac h gene o e ac hc h om osom ein he p opul a ion un de go es a andom chang eacco di ng o a p obabil i y dene d by he m u a ion a e, he m u a ion p oba bi li y , p m . The c osso e o p e a o e xpl oi s he a ai la ble i n o m a ion om he p opul a ion ab ou he se a chspace. I combines he ea u e s o wopa en s uc u es o o m wosimila o sp ing. The classical c osso e op e a o unde bina y co ding builds TheUseo F uzzyC onne c i es o Desi gn. .. 2 41 an osp i ngb yl inki ng oge he w ogenesegm en s,eac ho ne be longi ng oadi e en pa e n .Thi sope a o i sappl iedw i hap obabi li yo pe o ma nce, hec osso e p obabi li y , p c , ha de e m i nes hen um be o c h om o som esin hepopula ion o be c ossed.[ 7,9 ] epo c osso e o pe a o m odelsunde b ina yand ealcodi ng . Thec osso e op e a o pla ysacen a l ole in h eGA' spe o m ance.I could beconside ed o be o neo healg o i hm 's deningc ha ac e is ics,andi i so ne o hecom po ne n s obea inm ind o im p o i ng heG A'sbeha io u ([ 12] ). The m u a ion op e a o isacom plem en o hec o sso e o pe a o .I isneed ed oa oid helo sso u se ulin o m a io np oduce db y hec osso e . Ani m po an p o blemin heu seo G Asis p ema u ec on e genc e ; hesea ch becom es appedi naloc al op i m umbe o e hegl obalo p im um i s ound.Thisi s p oducedb y hel ac ko di e si y in hepopul a ionandadi sp opo io na e exploi a- ion / explo a ion ela i onship;anadequa eba lancebe w ee nab oa dsea c handa sucien e nem en isno es a bl ished . F uzzy logi c ec hniquesm a ybeused os ol e hesep o blem s.Ana em p con- sis s o heu seo uzzyl ogi cbased sys em s o hedy nam i ccon ol pa am e e s o R CGA( p m , p c ,popula i onsize,e c)insuc haw a y h a heco ec exploi a- i on/explo a i on el a ionsh i pandsui abl edi e si yl e elsbecam ees abli shed([8 , 1 1]).Ano he o neuses uzzy co nnec i es odesig nc osso e o pe a o s([8] ). In hispape w ep esen c osso e ope a o s o R CGAsba sedo n heus eo uzzyco nnec i es,and hei e x ensi onbasedo n heuseo pa am e e i zed uzzy connec i es o designi ngdynam i cc osso e ope a o swi h hem ai nobj ec i eo in oducingpopula i ondi e si yi n heGAsea c h. 2 D esi gno C osso e O pe a o s o R CGAus- i ngF uzz yConne c i es Ash as a l ea dybeenpoi n edo u ,a ch om o som eisa ec o o e al nu m be s,and i s p ec i si onwi llbem a k edby h a o hecom pu e unde whi ch h ea l go i hmi s ca iedou .Thesi ze o hec h om osom ei sk ep hes am ea s hel eng ho he ec o ha is heso lu io n o he p oblem ;in hisw a y ,e ac hg ene ep esen sa a i abl e o hep oblem .The a lueso hec h om o som egenesa e o ced o em ai nin he i n e al e s abli s hed b y he a i able ha he c h om oso me e p e sen s , so he gene i c op e a o s m us p ese e his e qui em en . In[3]i w as p oi n ed ou ha he c osso e op e a o i sak ey poi n o sol ing he p em a u e con e genc e p oblem .Th us , s ol u ions o hi sp ob lem m aybe ound b y desi gni ng new al e na i es o hi sope a o . He e, he de el opm en o suc h c oss o e op e a o s is a em p e d. W e p e sen c osso e op e a o s o R CG A base d o n he use o uzz y connec i es: - no m s , -c o no m s, a e age unc ions and gene al ize d com p ens a ion op e a o s ([14 ,1 5]) whic h induce di  e e n di e si y le els in he p opula ion , and he e o e he p ema u e c on e genc e p oblem maybe con olled. Le usassume ha he ch omos 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.He e a,M .L oz ano& J. L. V e dega y a es elec ed oapply hec osso e ope a o o hem ,a nd w og en es c 1 i and c 2 i o be c ossedo e , 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 ). I se em s easo nable oim agine he po ssibi li yo ob ai ningg oodd escenden sou si de h is in e al.Insho , hei n e alo ac io no heg en e i ,[ m i ;M i ], ma ybedi ided in o h ee egio ns[ m i ;x i ],[ x i ;y i ] ,a nd[ y i ;M i ],whe egooddescende n sm ay be ob ained;e enconside inga egi on[ x 0 i ;y 0 i ]wi h x 0 i  x i and y 0 i  y i w ouldseem easonabl e.G a phi ca ll y Fi gu e1 :Ac io nin e al o ag ene W esh al lno wg oon opu o w a dase o c osso e ope a o s ha al lo w desce nden s obeob ai nedin hep e i ousin e al s.Ino de o do ha ,w euse ou unc ions F , S , M a nd L dened om[ a;b ] 2 [ a;b ]i n[ a; b ] , a;b 2< ,whi ch ul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 a e mono onenon-dec easing. Le usassum e ha Q 2 F;S;M;L g ,and C 1 =( c 1 1 :: :c 1 n )a nd C 2 =( c 2 1 :::c 2 n ) a e wo ch om osom es ha ha e b ee n s elec ed o apply he c osso e o p e a o o hem .W em ay gene a e he c h om 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 i h he -no m o p e a o s , -cono m s, a e a ging unc io ns and gene al ize d co mp e nsa io nope a o s used as uz zy c onnec i es , w e shall a s so cia e F wi h a - no m , S wi h a -cono m , M wi h ana e a ging o p e a o and L wi h a ge ne al- ize d comp ens a ion op e a o . Fi s , weneedase o linea ans o ma ions o be able o apply he se op e a o s unde he gen e deni ionin e als. TheUseo F uzzyC onne c i es o Desi gn. .. 2 43 Le O beano pe a o belonging o hese o m edb y he -no m s, -co no m s, a e a ging unc io nsandgene ali zedco m pensa io no pe a o s. F o eac hpo si ion i 2 1 ;: :: n g , he oll o wi ng op e a i onswil lbeca i edou : 1.T ans o m c 1 i and c 2 i in o he alues s 1 i ;s 2 i 2 [0 ; 1]s uc h ha s k i = c k i 0 m i M i 0 m i ;k =1 ; 2 : Usin g hi ss ep,w e ans o m he alueso hegenesso ha heo pe a o m a ybea ppl ied o hem . 2.Appl y heope a o O ( s 1 i ;s 2 i )andca lcula e he alu e h i : h i = m i +( M i 0 m i ) 1 O ( s 1 i ;s 2 i ) ; so ha hegene h i i si n el a ion oi s o ig inalli m i s, h i 2 [ m i ;M i ] . 3. h i wi llbe he alue o heg enea posi i on i o hec h o m osom e esul ing om hec o sso e o hec h o m osom es C 1 and C 2 . Com plyi ngwi hase o uzzyconnec i es,( T j ;G j ;P j ; ^ C j ), j =1 ;:: :;k ,ase o unc io ns F j , S j , M j and L j j =1 ;:: :;k isbuil a sw edesc i 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 ) Thesec oss o e op e a o sha edi e e n ea u es: he F -and S -c os so e s show explo a i on, he M -c osso e ope a o ssho wexpl oi a i onand he L -c osso e sho w elaxed exploi a io n. 3Exa m pl e W eha e ca i e d ou die en ex pe im en s h a help o com pa e hebeha iou o ab ina y c o ded GA, and som eR CG As wi h c os so e op e a o s h a ha ebeen p op ose d i n o he publi ca ions, wi h a s e o alg o i hm s bas ed on he c oss o e op e a o s p op ose d, which us e he uzzy connec i es ( T j ;G j ;P j ; ^ C j ), j =1 ;:::; 5 showed inTable1. 244 F.He e a,M .L oz ano& J. L. V e dega y -no m -cono mA e agingO pe a o LogicalP o duc 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 P o duc Hamac he S um ( 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 Algeb aicP oduc Algeb aicSum ( 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 Eins einP oduc Eins einSu m ( 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 BoundedP oduc BoundedSum ( 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: Se o Op e a o s Eac h -cono mi sdual o he -no msho wn oi sl e . Thea e a ging unc ion iscal cula ed om he o m ulao hequasi-a i hme i ca e ages,usingas he unc i on hea ddi i e gene a o unc io no he -no mp la cedin hesa m eline, excep o he s one wh i c h,no bei ngA c him edean,d oesno ha eag ene a o unc i on,andw eshall us e ( x )= x .Thi s unc io ni ss ho wni n heuppe pa o hecel lswhe e hesea e agi ngope a o sa especi ed.F o ea ch am i lyo o pe a o s inT abl e1 ,agene al izedcom pensa ionope a o ^ C j wil lbeconside ed,denedas ol lo ws: ^ C j = P j ( T j ;G j ) Fo he am i lyo Lo gicalope a o s,w eshal lconside ^ C 1 = T 1 0 p 1 :S p 1 . TheGA am il iesa edi e en i a edacco di ng oho w heyca y ou he ol lo wing w os eps: 1.Gene a i on o osp ingusi ng hedie en c osso e ope a o s. 2.Sel ec iono osp i ng esu l ing om hec osso e whic hwil l o mpa o he po pula i on. Ap opo sali s he o llo wing :F o ea ch pai o c h om oso m es om a o al o 1 2 1 p c 1 N ( p c c osso e p obabi li y , N po pul a ionsize) , ou osp inga eg ene a ed, he es ul o applyi ng sp e cic u nc io ns F , S , M ,a nd L o hem .The wom os p o misi n g osp i ng o he ou epla ce hei pa en s in he p opula i on. Thi ss el e c ion s a egy in oduce sahi gh explo i a i on l e el wi h a n unde l ying expl o a ion c ause d b y he use o he die en Q -c oss o e o p e a o s . Nex , he esul s o nR ose n b o c k's Gene al ize d un c io n([ 4]) a e sho wn. The anal y i ca l and g aphi ca l o m ul a io n oge he wi h he elem en h a ep es en s he glo bal o p i mum(m inim um ) a e: ( ~x )= n 0 1 X i =1 (100 1 ( x i +1 0 x 2 i ) 2 +( x i 0 1) 2 ) TheUseo F uzzyC onne c i es o Desi gn. .. 2 45 0 5 : 12  x i  5 : 12 min( )= (1 ;:::; 1)=0 Fig u e2: whe e n =5. Ase o nineR CGAbasedo nc osso e ope a o sp es en edi n heli e a u e w e econside ed o he expe i m en s(R GA1-R GA9 ). Belo w he ei sa a bleindica - i ng he y peo c osso e a ndm u a io nusedb yeac ho hem , og e he wi h hei na m es. Algo i hmsMu a ionC os so e R GA 1 RandomSimpl e[13,17] R GA 2 Non- Uni o mSimple[13,17] R GA 3 RandomUni o mA i hme i cal[13] a =0 : 35 R GA 4 Non- Uni o mUni o mA i hme ical [13] a =0 : 35 R GA5-  Non- Uni o mBLX-  [5] (  =0 ;: 15 ;: 3 ;: 5) R GA 6 Non- Uni o mDisc e e [16] R GA 7 Non- Uni o mLinea [17] R GA 8 Non- Uni o m Ex endedI n e medi a e[16] R GA 9 Non- Uni o mEx endedLine[16] Tabl e2 : Real Co ded Gen e i cA lgo i hms By BGA w edeno eabi na y co ded GA wi h 30 genes pe a i able,m ul ipl e c oss o e wi h wopoi n s and p o po i onal se l ec i on p obabi li y. By NRG A 1 ; ::: ; N RGA 5w edeno e a a mil yo G A based on he uzzy conne c- i e s c oss o e and he gene a i on a nd sel ec i on o os p ing p op osals, using he amilie s o uz zy conne c i es: Log ical, Ham am che , Al ge b aic , Eins e i n and Bound, es pec i el y.Inal l case s w euse he s o cha s i c uni e sa lsam pl ing ([ 2]) se l ec ion p o ce du e and he e li is model. Weca ied ou ou exp e imen s u sing h e ol- lowing pa ame e s: h e p opula ion size is 61 indi iduals, he c osso e p obabili y 246 F.He e a,M .L oz ano& J. L. V e dega y p c =0 : 6,and hep obabi li yo c h om osom eupda e p u = p m 1 5=0 : 6,and hepa- a me e b usedb y henon-un i o mm u a i onis5.W eex ecu ed al l heal go i hm s3 i mes,ea chonewi h10 0,500a nd 5000gene a ions,a ndp es en hea e age al ue o hemi nT a bl e3. Al go i hms10 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: Resul s Thebes beha i ou co espo nds o heLog icalc osso e o pe a o .Thi sop- e a o oge he wi h heosp i ngsel ec i onm ec hanismoe asui a bleexplo i a- i on/explo a i onbal ance, al houghw em us poi n ou ha hi sm ec hani smi sm o e im eexpensi ebecauseneed mo e e a lua io ns.O he osp i ng sel ec ionm ec ha- nism sa ep oposedin[ 10] . 4 D esi gno Dynam i cC oss o e Op e a o sUsi ng Pa a me e i ze dF uzz yC on ne c i es A n idea o a oiding he p e m a u e c on e genc e consis s in al lo wi ng he e xplo a ion in he b e gi nning o he sea ch p o c es s and he e xpl oi a i on a he end o i . Wi h he explo a i on he di e si ybeca mes g ea e , i nc easing he p o babil i yo nding z ones whic ha e clo sed o op i mal s o lu io ns . Then, supp os i ng ha he p o pula- ion ha ei n o m a i on ab ou hese zones, he co n e genc e o wa ds he op i mumi s p o duce d h ough e xpl oi a i on. Am u a i on op e a o o R CG Acal le d non-uni o m m u a io n ([1 3]) is base d on he a o em en ioned p i nc i ple.T h e p o po i on in whic ha ea lgene is m u a e d de c ease s as he GA's e xec u ion ad ance. Thu s, he changes p o duc ed on he ge nes a e smalle in he las ge ne a ions p o duc ing a local uning. TheUseo F uzzyC onne c i es o Desi gn. .. 2 47 We ma yex end heuseo hec o ss o e o pe a o sp esen e di no de o oll o w hea o em en i onedi deas. Die en uzzyconnec i esc ould be useddu ing he GA's un.Fi s l y ,w eshall us e uz zyconnec i es ha p od uceh ighdi e si yle els, andl a e o he onesp oducingasu  ci en di e si yle el oa llo w heco n e g ence obe ea ched. W ep esen ase o d ynam ic c osso e ope a o sb ased on heuseo pa am e e - i zed -no m s, -co no m sa nda e ag ing unc i ons( [14, 15 ]).In he s s ag es, we shal luse -no m sand - co no m sdis an om hemi nimu m -no ma ndm a xim um -cono m espec i ely ,sohi ghdi e si yisi nduced.La e , - no m scl ose o he m ini m umand -cono m sclo se o hem a xim uma e conside ed.Theco n e g ence i scausedand hego od beha io u o helog ical uzz yconnec i eswil lbek ep . As w assho wninT able3, hes eope a o sa eingene al he m os p o  able. T odo his,w ep op osease o c o sso e ope a o sbasedon he unc io ns am ilie s: F p g p =1 ;:::;G , S p g p =1 ;:::; G ,and M p g p =1 ;: ::;G , G 2 N de ned om[ a;b ] 2 [ a;b ]in [ a;b ] , a;b 2< ,whic h ul ll hec o espondingP1 -P5p op e ies 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 o 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 Le us conside aGAwi ham axim um num be o gene a io ns  a nd C 1 = ( c 1 1 ;:: :c 1 n )and C 2 =( c 2 1 ;:::c 2 n ) w oc h om o som es ha w e es el ec e din hegene - a i on oa ppl y hec osso e o pe a o o hem .I Q p 2 F p ;S p ;M p g p =1 ;:::;  w em a yg ene a e hech om osome H =( h 1 ;:: :;h n )a s H = Q ( C 1 ;C 2 ) ; h i = Q ( c 1 i ;c 2 i ) ; i =1 ;: ::;n: W esha ll buil d unc i ons a mil ies wi h he (P6) and (P7) p op e i e s using he pa a me e ize d -no m sa nd - co no ms desc ib ed in T able 4. Ta ble 5 sho ws he p ope i e s o he pa am e e i zed -no m si n hi s a ble. The p op e ies o he pa- ame e iz ed -cono ms a e analogous. Wemus p oin ou ha T 6 is he d as ic -no m ( he smalles -no m).