Crossover in Probability Spaces

Transcription

1 Crossover n Probablty Spaces Sddhartha Bhattacharyya Inormaton and Decson Scences Dept.. College o Busness Admnstraton Unversty o Illnos at Chcago 60 S. Morgan Street (MC 9) Chcago, IL sdb@uc.edu Marvn D. Troutt Department o Admnstratve Scences Graduate School o Management Kent State Unversty P.O. Bo 590 Kent, Oho -000 mtroutt@bsa3.kent.edu Abstract Ths paper proposes a new crossover operator or searchng over dscrete probablty spaces. The desgn o the operator s consdered n the lght o recent theoretcal nsghts nto genetc search provded by orma analyss. A non-trval test problem n enorcng coherency o probablty estmates n cross-mpact analyss hghlghts the utlty o the desgned operators. The presented operators wll be useul or a varety o probablty-ttng and optmzaton applcatons. INTRODUCTION Probablty estmates are essental nput data or many decson support technques, ncludng epert systems, long range orecastng, data mnng, decson analyss and cross-mpact analyss, among others. Ths paper suggests a genetc algorthm (GA) based approach or searchng over dscrete probablty spaces, that wll be useul or a varety o probablty ttng and optmzaton applcatons. The ocus, n partcular, n on the desgn o a crossover operator or eectve search over a space o probablty dstrbutons. The desgn o operators s consdered n the lght o recent theoretcal nsghts nto genetc search provded by orma analyses (Radcle, 99). Forma theory provdes an etenson o the orgnal schema analyss or bt-strng representatons -- o genetc algorthms to arbtrary representatons, and suggests prncples or the desgn o eectve genetc search operators. We consder a drect real-number encodng o dscrete probablty dstrbutons. Two crossover operators are developed, one n eplct consderaton o orma processng prncples, and the other a more ntutve operator along the lnes o tradtonal sngle-pont crossover. Both operators are analyzed n terms o the ormae that they process, and ther perormance s evaluated on a non-trval test problem. The desgned genetc search approach s llustrated on a problem o enorcng coherency o probablty estmates n cross-mpact analyss (see Dalkey, 97; Duperrn and Godet, 975; Gordon and Hayward, 96; Helmer, 977; Martno, 93). The problem also allows soluton through mathematcal programmng technques, thus provdng useul comparson or the genetc algorthm approach. The net secton brely ntroduces orma analyss and the related desgn prncples or genetc search operators. The desgn o the crossover operator and a mutaton operator or searchng over probablty spaces are then consdered n the ollowng secton. The thrd secton then eplans and develops the test problem, and s ollowed by a perormance evaluaton o the genetc search operators mplemented FORMA AND DESIGN PRINCIPLES Schema analyses and the schema theorem (Holland, 975) orm the bass or most tradtonal theoretcal analyses o genetc algorthms. In a recent generalzaton, Radcle (99) etends the noton o schema to general equvalence classes, called orma. The schema theorem and ntrnsc parallelsm, key to the power o genetc search, s shown to apply to such general-purpose equvalence classes o solutons also. Developed n a seres o papers (Radcle, 99a, 99b, 99, 99), orma theory ocuses on the desgn o representatons and search operators such that the schema (orma) theorem makes accurate estmates o the orma tness, and thereby gudes the search approprately. Doman knowledge s brought to bear on the problem through speccaton o ormae that group together solutons o related perormance. Forma theory then establshes certan propertes or the genetc search operators so as to oster eectve search over the gven space, consderng the

2 dened orma as buldng blocks n the search process. These propertes can be used to analyze heurstc operators too. Forma theory ders rom tradtonal schema theory where the approach taken s to dene a lnear strng representaton o the search space so that standard crossover and mutaton operators can be appled. Forma theory nstead emphaszes an eplct characterzaton o approprate orma n consderaton o problem doman normaton, and consders search operators workng drectly on the search space, rather than on a ed strng encodng o t. Radcle (99) eplans how the two approaches are not equvalent; the ormer s shown to be more general. In orma theory, doman knowledge s epressed through equvalence relatons dened on the search space. Any relaton between pars o solutons that satsy relevty, symmetry and transtvty can be used. These equvalence relatons then nduce a parttonng o the search space nto equvalence classes, or ormae. Solutons wthn a orma are thus equvalent under the equvalence relaton. Equvalence relatons dene the genes, wth correspondng orma denng the alleles n the strng representaton. The ollowng eample, adapted rom Radcle, 99) llustrates. Eample: For a space o humans, eye color could be an equvalence relaton, useul ound to be an mportant determnant o some perormance. Formae then specy sets o people wth the same eye-color. Assumng only three colors, { ξ brown, ξ bgreen, ξ bblue } denes the set o ormae nduced by ths equvalence relaton. Here ξ denotes the set o people satsyng. Smlarly, consderng an equvalence relaton or har-type gves another set o orma { ξ bcurly, ξ strnght }. We can now have some composte equvalence classes lke ξ dened by the equvalence relaton o people brown,curly wth brown eyes and curly har. Such composte equvalence relatons do not orm genes; but note that composte equvalence relatons can be ormed as the ntersecton o the basc equvalence relatons (genes). I these equvalence relatons unquely denty each ndvdual n the search space, they orm a representaton o the space. Obvously, eye-color and har-type are nadequate or denng a representaton snce, n general, two people can have same eye-color and har-type. Radcle (99) provdes a rgorous denton o orma theory. Here we consder only those aspects necessary or our analyss n ths paper. Consder S the search space, letξ denote some orma on S and let Ξ be the set o ormae consdered. Then Ξ s a subset o the power set o S. A set o ormae Ξ s sad to cover a search space any soluton s unquely dented by the orma. Formally, Ξ covers S and only S, ξ Ξ wth I ξ =. Coverage s an mportant property snce t ensures that orma analyss can dstngush between solutons. Two ormaeξ andξ are compatble a soluton can belong to both. Two ormae wll thus be compatble ther ntersecton s non-empty,.e. ξ I ξ. It s desrable that compatble ormae be closed under ntersecton. Ths allows ormae to specy solutons wth derent degrees o accuracy and thereby gradually rene the search. Any genetc operator R that recombnes two parents n S to produce an osprng n S can be dened through a uncton R: S X S P R S where P R s the control set that determnes eactly whch o the varous possble osprng results rom applcaton o the operator. The non-determnstc nature o typcally used genetc operators can be speced through some approprate choce o control parameters n P R. The control set or an operator need not be eplctly dened; any recombnaton operator that manpulates two parents to produce osprng n a reasonable manner can be used. Note that ths denton o operators s representaton ndependent, and thus allows the ormulaton o operators drectly on the search space. Such operators, however, need careul desgn so as to process the dened ormae n a meanngul manner operators should be able to eplot the doman knowledge nherent n the orma speccaton. In addton to the coverage and closure propertes dened above, orma analyss provdes three desgn prncples or the genetc recombnaton operators: respect, assortment, and ergodcty. These are dened below. Respect: A recombnaton operator R s sad to respect a set o ormae Ξ recombnng two nstance o a partcular orma produces an nstance o the same orma. Formally R respects Ξ and only ξ Ξ,, g ξ, p PR : R(, g, p) ξ. Respect seeks to preserve genes common to both parents n ther osprng as well. It s eectve at reducng orma dsrupton, specally towards the later stages o the search when populaton dversty s low. Assortment: A recombnaton operator R s sad to assort a set o ormae Ξ, gven two compatble ormae, t s possble to recombne any two nstances o these to produce an osprng that s n nstance o both ormae. Formally, R assorts Ξ and only ξ,ξ Ξ such that ξ Iξ, ξ g ξ, p P : R(, g, p), R ξ Iξ. The assortment property relates to the buldng block hypothess, and seeks to ensure that good buldng blocks rom two parents can be eectvely combned.

3 Ergodcty: Gven any populaton, t should be possble to access any pont n the search space through a nte sequence o applcatons o the genetc search operators. The mutaton operator s generally used to provde or ergodcty. A set o ormae Ξs sad to be separable a recombnaton (crossover) operator capable o respectng and assortng t ests..e. the propertes o respect and assortment are not contradctory. Whle desrable, the twn propertes o respect and assortment, as ndcated earler, are ncompatble n certan practcal problems. Notng that most standard crossover operators respect schemata, Radcle (99) wrtes: Thus a respectul operator s used, the non-separablty mmedately means that assortment must al to be acheved. The sgncance o ths s that even when parents are chosen that contan all the genetc materal (apparently) necessary to buld some gven chld, t may be mpossble or a respectul operator to construct that chld. As well as beng n conlct wth reason, ths would seem to make navgaton around the search space unnecessarly dcult. Thus, where ormae are non-separable, assortment would seem to be the more desrable property sought; respect ncreases n relevance towards the later stages o search, as the populaton begns to converge.the orma propertes and GA desgn prncples gven above, however, should not be consdered as provdng necessary or sucent condtons or eectve genetc search (Radcle, 99). Rather, they provde a set o condtons whch, satsed, s epected to yeld mproved search capabltes. 3 CROSSOVER AND FORMA IN PROBABILITY DISTRIBUTIONS The operators are dened drectly on the search space. A populaton member thus represents a soluton drectly as a strng o real-numbers, each number correspondng to the probablty o a specc event. Probablty dstrbutons mpose the constrant that each value be n [0,] and that all values sum to one. Thus, consderng a problem wth N events, a soluton s represented as the strng < s, s,, s N > wth s [0, ], =,..,N, and s =. N = We consder a crossover operator, obtaned ntutvely along the lnes o tradtonal GA crossover operators that swap alleles n two parents to obtan two osprng. Here, the osprng are ormed by echangng the ront and tal ends o the parent dstrbutons. A cumulatve dstrbuton uncton (CDF) representaton acltates the desgn o ths operator. Each populaton member represents a CDF correspondng to the probablty dstrbuton = <,,, n >. Then rom any populaton member S(), the probabltes can be determned as = S S, =,...,n n n wth S(0)=0 and S()= beng ed. Thus each populaton member s represented as a n-component vector (, n ) whose coordnates are monotone ncreasng to the rght. 3. DISTRIBUTION SWAP CROSSOVER Gven two parents F() and G(), the dstrbuton-swap crossover operator obtans two osprng P() and Q() as ollows:. Select a crossover ste c = U[0,] (unormly n [0,]).. P() = F( 3. Q( ) = G( c F() c G() G(c ) ) + ( F(c ))[ ] > c G( ) c G() c F() F(c ) ) + ( G(c)[ ] > c. F( ) (Ths CDF decomposton was employed earler, n non- GA related work (Troutt and Pane, 990). Note that ths operator consders dstrbutons as contnuous, and crossover stes can all n between two dened postons on a strng. Ths s easly avoded, where desrable, by choosng c = (r/n) where r = U[,n]; ths ensures that the crossover ste corresponds to some strng poston. The operator also ehbts a hgh postonal bas as assocated wth tradtonal sngle-pont crossover. Ths can be overcome n a unorm verson o the operator:. Consderng probabltes (,, n ), select crossover postons unormly ([ 0,] ). Let C be the set o selected crossover postons.. Rearrange the orderng o the probabltes such that { : C} appears to the let o { : C}. Transorm ths rearranged probablty dstrbuton to CDF and perorm crossover wth crossover ste ma{ : C} c =. I c s not requred to n correspond to any dened strng poston, t may be selected unormly n [ c, c+ ]. The Dstrbuton Swap crossover preserves the probablty ratos amongst events n the ront and back ends o a dstrbuton,.e., on ether sde o the crossover ste c. For eample, gven two parent probablty dstrbutons =<,, 3, > and g=<g, g, g 3, g >, and wth the crossover ste chosen to be, say, at the second strng poston, osprng p=<p, p, p 3, p > wll mantan n c c

4 p /p = / and p 3 /p =g 3 /g, and smlarly or the second osprng. Ths operator may then be consdered as processng Probablty-Rato orma. 3. PROBABILITY-RATIO FORMA Probablty-rato ormae relate dstrbuton strngs havng the same probablty-ratos or the denng events. A probablty dstrbuton =<,.., n > can be represented wth ts probablty-rato equvalent =<...,,, n,... >. The set o probablty-rato orma can be dened as Ξ= F j where k F j = {, k, r P} U{*} ; here P denotes the set o events r wth speced probabltes n the dstrbuton, and * s a don t care symbol correspondng to events wth unspeced probabltes. Then, two strngs and g (dstrbutons, or ther probablty-rato equvalents) are equvalent or nstances o a ormaξ R accordng to: R p g p, g ξ P {,.., n}, = p, q P. g For eample, consderng probablty dstrbutons = <.,.3,.,.> and g = <.,.6,.,.>, ther probablty-rato equvalents are =<,,,,, > and g =<,,,,, >. 3 6 Thus, both and g are nstances o the orma < *, *,*,*,*>. Dstrbuton-swap crossover does not respect probabltyrato orma, as seen n the ollowng eample: consder parents :<0.,0.3, 0., 0.> and g:<0., 0., 0.3, 0.>. Note that / =g /g = 0.5, and a probablty-rato respectng operator wll preserve ths rato n osprng as well. However, dstrbuton swap crossover, wth crossover ste at, say, the second poston yelds osprng <0., 0.3, 0.6, 0.3> and <0., 0., 0.7, 0.3>, none o whch mantan the rato between the rst and ourth event probabltes. Respect here, however, rses wth ncreasng smlarty n gene values as the search progresses towards convergence. Snce respect s an mportant property n the later stages o search, dstrbuton-swap crossover can thus stll be epected to perorm well. The unorm verson o the dstrbuton swap crossover operator also assorts probablty-rato orma. Consder parents :<0.,0.3, 0., 0.> and g:<0., 0., 0.3, 0.>, or ther probablty rato equvalents q q j : <,,,,, > and g < :,,,,, >. Here s an nstance o orma ξ : <,*,,*,*, > and g s an 3 nstance o the orma ξ : < *,*,,*,,* >. Now, nstances and g o the above ormae can be recombned by dstrbuton-swap crossover to gve osprng that are nstances o both ormae. Note that ξ I ξ =< *,*,,*,*,*>,.e., havng / =0.5. Instances o ths ntersecton orma are always obtanable rom parents and g by consderng events and (rst and ourth postons o the strng) n the crossover set C (see descrpton o the unorm verson o the operator above). 3.3 MUTATION As noted earler, mutaton plays a key role n mplementng the ergodcty property. Wth both the crossover operators above, mutaton s desgned to randomly change the probablty o any event. The probabltes o the other events wll also then need to be adjusted so as to satsy the probablty dstrbuton restrcton (summaton to unty). These adjustments are made proportonal to ther estng values. In addton, notng that the crossover operator carres an nherent bas away rom the end-ponts (0 and probablty values), the mutaton operator s mplemented to nsert a 0-value to randomly selected events. Ths 0- value mutaton shares equally wth the regular mutaton. Snce a mutaton to a value o would reduce the probabltes o all the other events to 0, and s thus, n general, not epected to yeld good solutons, we do not mplement a -value mutaton. TEST PROBLEM: ENFORCING COHERENCY OF PROBABILITY ESTIMATES We eamne the perormance o the genetc search operator on a problem o obtanng coherent probabltes estmates or a set o related events. Obtanng hgh qualty probablty estmates rom decson makers s a problem that occurs wth many decson support models and methods. Such estmates, partcularly those o related probabltes, can al to be consstent wth the laws o probablty theory, a stuaton known as ncoherency. The desrablty or coherence has been argued by many authors (Hogarth, 975; Martno, 93; Spelzler and von Holsten, 975; Tversky and Kahneman, 93). Here, we rst develop a search space or representng related probabltes n such a way that coherent scenaro and event probablty estmates can be obtaned. Genetc

5 search then provdes a general approach or ndng estmates whch are eactly coherent whle beng close to the decson maker s estmates as well. The constructed problem s also amenable to mathematcal programmng soluton approaches, and provdes an means or comparng wth the desgned genetc search operators.. TEST PROBLEM DESCRIPTION The problem addressed n ths paper may be descrbed as ollows. Gven data o the type shown n Table, nd coherent probablty estmates or the varous possble scenaros and events whch are as nearly consstent as possble wth the assgnments o Table. Our startng pont s an nterpretaton o the constrants n Table as relatonshps between condtonal probabltes and margnal probabltes as ollows: the rst constrant s nterpreted as requrng P(B A) = P(B) (-P(B)). () Smlarly the net two gve rse to P(C A) = P(C) - 0.0, () and P(C D ) = P(C) + 0.3(-P(C)) (3) respectvely. The remanng three equatons are respectvely P(A D) = 0.5 P(A), () P(D C) = P(D) + 0. (-P(D)), (5) and P(D B) = 0.3 P(D) (6) Here we use the notaton P( ) to denote the probablty o event ( ) and use the overbar to denote complementary events. To develop the representaton, consder a specal type o sample space, S, whose sample ponts are potentally elements o one or more o the event sets A, B, C, and D. A partcular pont n S can belong to one and only one o the = 6 mutually eclusve subsets or scenaros characterzed accordng to whether the pont s or s not an element o each o the our sets A, B, C, and D. In act, we may lst and number these scenaros as n Table. Further, n Table, we assocate a varable to represent the relatve requency o sample ponts n scenaro. We call S a leble sample space (FSS). That s, nstead o each sample space pont havng equal probablty, these ponts are allowed to have leble probabltes denoted by the. The FSS approach appears to be a relatvely straghtorward modelng tool whch may have been used elsewhere. It appears to be essentally the same as what some have called the mnmal relevant sample space. 6 = Snce must be unty, the reader may check that the relevant margnal and jont events nvolved n system ()- (6) may be epressed as ollows: P(A) = (7) = 6 P(C) = () = =5 0 =9 =3 P(B) = + (9) = =9 P(D) = (0) = P(BI A) = () 6 P(CI A) = + () = =5 P(CI D ) = (3) P(AI D) = () P(DI C) = (5) P(DI B) = (6) The requrements o Table may now be epressed as ollows ater some smple algebra: = ( + )( ) (7) = = = = 9 = = ( + + )( ) () = = 5 = = = 5 = 3 = =.3 ( ) = = 5 = 9 = ( )( ) (9) = 0.5( )( ) (0) = = ( )) () ( = 0.3( = = )( + ) ( ) () = = 5 = 9 = 3 = 0.5 (3) = + = 0. () = = = 0. (5)

6 The last three requrements, (3)-(5), are due to the ntal margnal probablty estmates n the lower hal o Table. Thus the FSS decomposton descrbes a hypothetcal sample space n terms o the possble elementary scenaros related to the partcular set o events o nterest. All possble jont, margnal and condtonal probabltes o these events can thereore be modeled n a coherent way. It s mportant to emphasze here that coherency s strctly enorced by ths scheme. That s, all normaton about the events A, B, C, D, ther ntersectons, complements, and other set operatons, s contaned n the, =, 6. The reader may check that coherency requrements, such as P(AI B) = P(A B) P(B), reduce to denttes n the. Moreover the above ponts reman true even the system (7)-(5) s solved only appromately. Thus the FSS model used here precsely enorces coherency but permts some possble lack o t wth the decson maker s data such as gven by Table. The equatons (7)-(5) consttute a set o nne constrants, some o whch are nonlnear, n steen varables. In addton, t s necessary that =. 0 and 6 = 0, =,...,6. Hence, t s not known a pror whether an eact soluton ests. We mght thereore seek scenaro probabltes whch come as close to these condtons as possble n the least squared error sense. Dene e as the derence o let and rght hand sdes o (7), e smlarly or () and so on to e 9 or (5). Ths lne o reasonng would lead to the constraned programmng problem (6)- (7) as ollows: mn 9 e = 6 (6) s.t. =, 0, =,..6. (7) = Ths problem was coded and solved usng GAMS. Table 3 shows the results or both the scenaro probabltes and the probabltes o events A, B, C and D. Note that the objectve uncton n (6) can not easly be checked or convety. The GAMS soluton thus mght not guarantee a global optmum. In order to compute a global mnmum o (6), one would need a startng pont grd search procedure. However, the grd mesh s based on dvdng the nterval [0, ] nto k ponts, then gven the 6 varable problem at hand, the number o such search ponts s o the order o k5. Thus ths strategy s thereore unattractve even or the small eample at hand. A genetc algorthm approach proves much more desrable.. SOLUTION USING GENETIC ALGORITHMS Snce the GA seeks solutons wth hgh tness, the ollowng tness-uncton, consstent wth the objectve o (6) s used: 9 j = ( e ) = where j s the tness o the j th populaton member, and e, =,..,9 are the errors as determned by (7)-(5). The second power n the above epresson provdes or greater dscrmnaton between solutons wth close values o the total sum o squared errors (SSE). Table presents the SSE, the scenaro probabltes and the probabltes o A, B, C, and D as obtaned usng genetc search wth the dstrbuton-swap operators. These results correspond to the ollowng settngs or the GA parameters: a populaton sze o 00 was mantaned or each smulated generaton, and the best tness number retaned ntact n the net generaton (eltst selecton). The mutaton rate, was set at 0.05 per strng poston and the crossover rate used was 0.7. The search was termnated ater 500 generatons. Table presents three derent solutons obtaned wth derent random number streams. All the GA solutons are ound to outperorm the GAMS soluton. 5 CONCLUSION Ths paper consders the desgn o a crossover operator or search over dscrete probablty spaces. A test problem llustrates an applcaton to the estmaton o coherent probablty assgnments. More generally, these procedures apply to optmzaton over such spaces, or smplees descrbed by barycentrc coordnates. The crossover operator s an ntutve heurstc that orms osprng through an echange o the ront and tal ends o dstrbutons. Ths dstrbuton swap crossover operator s shown to process what may be called probablty-rato orma. The operator assorts, but does not ully respect probablty-rato orma; however, respect holds to a degree whch ncreases as convergence n the populaton advances. Genetc search usng the desgned crossover operator s ound to perorm sgncantly better than a conventonal mathematcal programmng approach or the test problem. It should be noted, however, that the orma desgn prncples o assortment and respect are nether necessary nor sucent or eectve genetc search; rather, they should be looked upon as propertes epected to acltate successul search. The success o the dstrbuton-swap operator adds credence to the argument that n stuatons where the twn propertes o respect and assortment are n conlct, orma assortment s more desrable, wth respect ganng n relevance as the populaton converges. The dstrbuton-swap crossover, analyzed as processng probablty-rato orma, possesses precsely these characterstcs.

7 Reerences Dalkey, N.C., An Elementary Cross-Impact Model, Technol. Forecast. Soc. Change, 3, 97, Davs, L. ed., Handbook o Genetc Algorthms, 99, Van Nostrand Renhold. Duperrn, J.C. and M. Godet, SMIC 7 - A Method or Constructng and rankng Scenaros, Futures, 7 (), 975. Eshelman, L.J., R.A. Caruna and J.D. Schaer, Bases n the Crossover Landscape, Proc. o the Thrd Internatonal Conerence on Genetc Algorthms, J. D.Schaer (ed.), 99, 0-9. Gordon, T.J. and H. Hayward, Intal Eperment wth the Cross-Impact Matr Method o Forecastng, Futures,, No., Dec. 96. Hogarth, R.M., Cogntve processes and the assessment o subjectve probablty dstrbutons, Journal o the Amercan Statstcal Assocaton, 975. Helmer, O., "Problems n Futures Research", Futures, 9 (), 977, -3. Holland, J. H., Adaptaton n Natural and Artcal Systems, Unversty o Mchgan Press, Ann Arbor, 975. Mchalewcz, Z., Genetc Algorthms + Data Structures = Evolutonary Programs, nd Edton, 99, Sprnger- Verlag. Martno, J.P., Technologcal orecastng or decson makng, nd Edton, North-Holland, New York, NY, 93. Moskowtz, H., and Sarn, R.K., Improvng the consstency o condtonal probablty assessments or orecastng and decson makng, Management Scence, 9, 93, Radcle, N.J., The Algebra o Genetc Algorthms, Annals o Maths and Artcal Intellgence, vol. 0, 99, pp Radcle, N.J., Genetc Set Recombnaton, Foundatons o Genetc Algorthms,, L.D. Whtley (ed.), 99, Morgan Kaumann. Radcle, N.J. Forma Analyss and Random Respectul Recombnaton, Proc. Fourth Internatonal Conerence on Genetc Algorthms, 99a, pp.-9, Morgan Kaumann. Radcle, N.J., Equvalence Class Analyss o Genetc Algorthms, Comple Systems, 5, 99b, pp Radcle, N.J. and P.D. Surry, Ftness Varance o Forma and Perormance Predcton, Foundatons o Genetc Algorthms, III, M.D. Vose and L.D. Whtley (eds.), pp. 5-7, 99, Morgan Kaumann. Spetzler, C.S., and Staël von Holsten C.A.S., Probablty Encodng n Decson Analyss, Management Scence, 975,, Surry, P.D. and N.J. Radcle, Fomal Algorthms + Formal Representatons = Search Strateges, Parallel Problem Solvng rom Nature, IV, 996, Sprnger Verlag. Syswerda, G., Unorm Crossover n Genetc Algorthms, Proc. o the Thrd Internatonal Conerence on Genetc Algorthms, 99, Morgan Kaumann. Troutt, M.D., and Pane, T.B., A monotone varatonal method or optmal probablty dstrbutons, OR Spektrum, :0-06, 990. Tversky, A. and Kahneman, D., Etensonal versus ntutve reasonng: The conjuncton allacy n probablty judgment, Psychologcal Revew, 90, 93, Table : Data or our hypothetcal nterrelated events. I event A occurs then the probablty o event B s ncreased by 5% o ts allowable ncrease.. I event A occurs then the probablty o event C s decreased 0% (or as near to 0% as possble). 3. I event D does not occur, then the probablty o event C s ncreased by 30% o ts allowable ncrease.. I event D occurs, then the probablty o event A s decreased 50%. 5. I event C occurs, then the probablty o event D s ncreased 0% o ts allowable ncrease. 6. I event B occurs, then the probablty o event D s decreased 30% o ts allowable decrease. Intal Probablty Estmates: P(A) =.50, P(B) =.0, P(C) = Not Speced, P(D) =.0

8 Table : Scenaros and ther relatve requency varables Scenaro Relatve Scenaro* Scenaro Relatve Scenaro* Scenaro Relatve Scenaro* Number Frequency Number Frequency Number Frequency A B C D 7 7 A B C D 3 3 A B C D A B C D A B C D A B C D 3 3 A B C D 9 9 A B C D 5 5 A B C D A B C D 0 0 A B C D 6 6 A B C D 5 5 A B C D A B C D 6 6 A B C D A B C D * For eample, A B C D may be descrbed as the scenaro n whch events A and C both occur, whle B and D al to occur. Table 3: GAMS soluton SSE P(A) P(B) P(C) P(D), =, 6 (by row order) Table : Genetc Algorthm solutons (Dstrbuton-Swap Crossover) SSE P(A) P(B) P(C) P(D), =, 6 (by row order)