DANTE The Combination between an Ant Colony Optimization Algorithm and a Depth Search Method

DANTE – The combina ion be ween an An Colony Op imiza ion algo i hm and a Dep h Sea ch me hod Ped o Ca doso – M´a io Jesus EST – Uni e sidade do Alga e 8005-139 Fa o, Po ugal [email p o ec ed] Albe o Ma quez Dep . Ma em´a ica Aplicada I ETSII – Uni e sidad de Se illa A da. Reina Me cedes s/n 41012 Se illa, Spain [email p o ec ed] Abs ac The -DANTE me hod is an hyb id me a-heu is ic. In combines he e olu iona y An Colony Op imiza ion (ACO) algo i hms wi h a limi ed Dep h Sea ch. This Dep h Sea ch is based in he phe omone ails used by he ACO, which allows i o be o ien ed o he mo e p omising a eas o he sea ch space. Some esul s a e p esen ed o he mul iple objec i e k-Deg ee Spanning T ees p oblem, p o ing he e ec i eness o he me hod when compa ed wi h o he al- eady es ed e olu iona y me hods. 1. In oduc ion I is well known ha , e y o en, he use o pu e E o- lu iona y Algo i hms has a lack o pe o mance, namely in he op imiza ion o la ge combina o ial p oblems ins ances. Mo eo e , he majo i y o he e olu iona y me a-heu is ics ha e some common cha ac e is ics. Fo example, hey usu- ally each good app oxima ions o he op imum solu ions (despi e he di icul ies o e ine hose app oxima ions, i.e., ge ing o he eal op imum). O en only he bes solu ion(s) a e kep . The o he solu ions a e disca ded wi hou u he explo a ion, which o ge s he expensi e compu a ional e - o necessa y o build hem. Examples o op pe o m- ing me a-heu is ics ha employ his me hodology a e he Simula ed Annealing, he Tabu Sea ch, o he An Colony Op imiza ion algo i hms [11]. The e o e, his s a egy does no allow a p ope local explo a ion o he e en ually mo e p omising egions o he sea ch space. In o he wo ds, he neighbo hoods o he ob ained solu ions do no go h ough an exhaus i e explo a ion. Algo i hmic hyb idiza ions appea ed as an e o o sol e some o he abo e men ioned p oblems. The imp o emen o he me hods by hose combina ions can gi e us he bes o se e al algo i hmic s a egies. In he implemen ed cases, di e en possible hyb idiza ions can be hough , like he use o ini ial solu ions ia ano he me hod han he main one (used many imes o ins ance in he Gene ic Algo i hms o ob ain he ini ial popula ion[9, 11]) o a local imp o emen o he ob ained solu ions. Usually, hyb id me hods a e con- side ed as he ones ha use wo o mo e me hods in he ollowing sense: he p ima y me hods a e used o gene a e mo e o less ough app oxima ions o he p oblems solu- ions, ollowed by o he (s) me hod(s) ha e ine he ea lie solu ions. Some examples o he applica ion o he second phase o he solu ion ob ained by some me a-heu is ic, a e he use o he 2-OPT o 3-OPT o he T a elling Salesman Pe son [5], he SOP-3-exchange o he Sequen ial O de - ing P oblem [6], o he I e a ed Local Sea ch o he Bin Packing P oblem [10]. In [1] is p esen ed he Beam–ACO, which is a combina ion o a Beam Sea ch heu is ic wi h an ACO algo i hm. In his case, he solu ion cons uc ion mechanism o s anda d ACO algo i hms is eplaced by he solu ion cons uc ion mechanism in which each a i icial an pe o ms a p obabilis ic Beam Sea ch. This is done by e- placing he de e minis ic choice o a solu ion componen a each cons uc ion s ep by a p obabilis ic choice based on ansi ion p obabili ies. In his pape , we p opose a hyb id me aheu is ic called –Dep h ANT Explo e (-DANTE), ha uses an e icien local sea ch (adap ed o a phe omone-o ien ed p ocedu e). Mo e p ecisely, in each one o he -DANTE cycles, se s o solu ions a e compu ed using a manda o y i s phase ha , in some cases, is ollowed by an elec i e second phase ha depends on he quali y o he o me solu ion. Mo e speci - ically, •In he i s phase, simila o mos ACO algo i hm, a solu ion is gene a ed based in a cons uc i e p ocedu e ha successi ely adds selec ed componen s acco ding o a pseudo- andom o mula. Eigh h In e na ional Con e ence on Hyb id In elligen Sys ems 978-0-7695-3326-1/08 $25.00 © 2008 IEEE DOI 10.1109/HIS.2008.115 36 •In he second phase, a limi ed dep h sea ch p ocedu e based on he bes i ing solu ions is made. In o he wo ds, i he i s phase ou come is wi hin  ange o he app oxima ion se ( he se o he bes known pe - o ming solu ions) o imp o es his se , hen a lim- i ed dep h sea ch p ocedu e is applied. The same phe omone ails, ha we e used in he i s phase, a e also used he e. This enhances he limi a ions o he dep h sea ch me hods by leading he p ocedu e o include he mo e p omising componen s in he con- s uc ed solu ions. The e o e, his pape is s uc u ed as ollows. The -DANTE is desc ibed in he nex sec ion. In he hi d sec- ion a e p esen ed some esul s and, in he las one, a e d awn some conclusions. 2. –DANTE – Dep h ANT Explo e The second phase local sea ches men ioned abo e (e.g., 2-OPT,3-OPT,...) a eused o e ine he solu ions, bu do no use he in o ma ion acqui ed by he e olu iona y me a- heu is ic, excep o he base solu ion i sel . Mo eo e , mos o he imes he local sea ch ope a o a e applied o all solu- ions e en in he cases whe e hey a e no much p omising. To y o explo e some o hese weaknesses we de el- oped he –Dep h ANT Explo e (–DANTE) which is a hyb id me hod. In ac , –DANTE is li le mo e since, i is a usion be ween an An Colony Op imiza ion me hod [4] and an Dep h Sea ch me hod [8]. F om his combina ion esul s mo e han a me a-heu is ic ollowed by he local sea ch, as we will see nex . The ke nel o he me hod, simila ly o he An Colony Op imiza ion me hod, has a se o cycles. In each cycle a se o solu ions is buil using he phe omone ails. Those so- lu ions a e hemsel es used o upda e he phe omone ails and o upda e he se o (bes ) app oxima ions o he p ob- lem solu ion: a single se o he single objec i e p oblems o he Pa e o se o he mul iple objec i e case [3] (see Sec- ion 3). Al e na i ely, he upda ing o he phe omone ails can be done in a mo e g eedy me hod using only he bes solu ions ha whe e ob ained du ing he en i e p ocess [2]. One o he main ideas o he -DANTE, is o make a local explo a ion o he bes solu ions. In o he wo ds, a solu ion is a ac i e whene e ce ain quali y pa e n is achie ed. This can be quan i ied has he solu ion being wi hin an  ange om he bes known solu ions. In his case, he Dep h Sea ch based on he ea lie buil solu ion is applied. This s a egy ies o a oid he, p obably useless, compu a ional e o associa ed wi h he explo a ion o neighbo hoods o he wo s solu ions. The Dep h Sea ch is limi ed bo h in he dep h i sel and in he numbe o possible b anches o he sea ch ee. Fu he mo e, he dep h sea ch is o ien ed 1. Ini ialize he phe omone ail. 2. While s opping c i e ion is no me do (a) Fo all an s do •Cons uc a new solu ion, S, using he cu en phe omone ail. •I he dis ance o S o he app oxima- ion se is in e io o o S imp o es he app oxima ion se hen apply a lim- i ed dep h sea ch p ocedu e, based on he phe omone ails, om ha solu ion. (b) Upda e he phe omone ail Figu e 1. -DANTE Algo i hm by he phe omone ails and he local heu is ics used in he An Colony me hod (Sec ion 2.1 desc ibes in mo e de ail he Dep h Sea ch phase). The main -DANTE algo i hm is ske ched in Figu e 1. 2.1 Dep h Sea ch phase As e e ed, he i ness o each gene a ed solu ion, S,is compa ed wi h he i ness o he elemen s in he app ox- ima ion se , P(Pis he app oxima ion se : in he single objec i e case i will be a singula se and in he mul iple objec i e case i will ha e he non-domina ed solu ions [3] – see sec ion 3.1). Then he limi ed dep h sea ch goes o Le el D–I Simp o es P, ha is,Sis no wo s han any elemen o P;o Le el d–I Sis wo s han some elemen o P, bu i s el- a i e dis ance o he elemen s o he app oxima ion se is smalle han . He e dand Da e algo i hm pa ame e s and should e i y d>D. Fu he mo e, he numbe o b anches used in each le el o he Dep h Sea ch, M, is also an algo i hm pa ame- e . Figu e 2 p esen s a high le el desc ip ion o he p ocess whe e Func ion -DANTE Solu ion equi es some pa ame e s besides he solu ion in cons uc ion T, namely •–Thepa ame e will in luence he numbe o imes ha he p ocess en e s in he Dep h Sea ch mode. Small alues will gua an ee ha only he solu ions wi h objec i e alue nea he bes known solu ions will go in o he o ien ed Dep h Sea ch. La ge alues will do he opposi e. •The algo i hm con ains a abu lis o each le el, abul, ha is ini ialized as emp y se . This abu lis a oids 37 Func ion -DANTE Solu ion(T) l←|T|/*de ines he le el by he numbe o added componen s*/ i Tis a solu ion hen Se Δas he ela i e dis ance om T o P i Δ<0 hen /*Timp o es P*/ Upda e he app oxima ion se wi h T Upda e Δk(k=1,2,...,m) wi h T† e u n D else i Δ< hen Upda e Δk(k=1,2,...,m) wi h T† e u n d else e u n 0 else sle el ←0 NB anches ←M o k=1 o NB anches i ∃e∈E− abul:T∪{e}is admissible hen Choose an edge, e, om E− abul‡ Tabu l←Tabu l∪{e}† T←T∪{e} /*Recu si e call*/ L←-DANTE SOLUTIONS(T) i L>0 hen NB anches ←NB anches +M sle el ←max{sle el, L} T←T−{e} else b eak /* o */ Tabu l←∅†/*clean he abu lis */ e u n max{sle el −1,0} Figu e 2. -DANTE’s solu ions explo a ion al- go i hm (whe e he s eps ma ked wi h †a e op ional and he ones wi h ‡a e p oblem spe- ci ic). ha he same solu ions a e ebuil in he Dep h Sea ch, by es ic ing he addi ion o he same compone s in he same le el. The use o he abu lis is implemen a- ion dependen and should be hough acco ding o he cleaning abuls ep, since i also es ic s he cons uc- ion o new solu ions con aining hose componen s. •The compu a ion o he dis ance om T o he app ox- ima ion se P e u ns a non-posi i e alue i Tim- p o es P; •The upda e p ocedu e consis s in inse ing Tin Pand emo ing he elemen s o P ha a e wo s han T. •The upda e o Δkis op ional since, i all solu ions con- ibu e o he a ia ion o he phe omones, he esul is a noisy ail wi h consequen los o pe o mance. Al e na i e s a egies include a g eedy upda e whe e only he bes solu ions con ibu e o he a ia ion o he phe omone ails, o e en a mo e es ic ed selec- ion o e he app oxima ion se (see sec ion 3.4). A mo e de ailed desc ip ion o he me hod can be ound in [2]. 3 Some compu a ional es s 3.1 Tes p oblems - Mul iple Objec i e k- Deg ee Minimum Spanning p oblem To es he -DANTE me hod i was conside ed he Mul- iple Objec i e k-Deg ee Minimum Spanning p oblem. A k-Deg ee Minimum Spanning T ee is a minimal weigh spanning ee such ha he maximum deg ee o any node is k. The ee is buil o e a ne wo k (V,E,W),whe e Vis he nodes se , Eis he edges se , W:E→ IR mis he weigh ec o - unc ion de ined as W(e)= (w1(e),w 2(e),...,w m(e)),andmis he numbe o objec- i es. He e, since we ha e a mul iple objec i e p oblem, min- imal weigh is conside ed in he Pa e o op imali y con ex . Tha is, gi en wo solu ions Xand Yo he easible se S,i issaid ha Xdomina es Y,X≺Y,i o all i∈{1,2,...,m}we ha e wi(X)≤wi(Y), and exis s j∈{1,2,...,m}such ha wj(X)<w j(Y). A solu ion Xis Pa e o (op imal) i i is no domina ed by any o he so- lu ions o S, ha is, o allY∈S−{X}i is e i ied ha Y≺ X. The se o all Pa e o solu ions is called Pa e o se o e icien se . 3.2 Me ics To e i y he pe o mance ou me hod we used wo me - ics: R1and R3[7]. The R1me ic measu es he p obabil- i y ha an app oxima ion se P1is be e han ano he P2 o e a amily o u ili y unc ions U.I R1(P1,P2,U)>1 2 hen, acco ding o his measu e, P1is be e han P2and i will be no wo se i R1(P1,P2,U,p)≥1 2.TheR3me ic measu es he expec ed p opo ion o supe io i y o one se o e ano he . The la ge R3(P1,P2,U)is, he wo s is P1 when compa ed wi h P2. The opposi e is alid i R3is neg- a i e. I R3(P1,P2,U)≈0 hen P1and P2ha e simila quali y o e U. 38 3.3 k-Deg ee Spanning T ees p oblem cons uc ion In he p oposed p ocess, o build he k-deg ee spanning ee, easible edges a e successi ely added un il a solu ion is comple e. Be o e he addi ion o any edge, i mus be e i ied ha such addi ion will no o m a cycle nei he he nodes maximum deg ee condi ion is iola ed. Mo e p ecisely, he p ocess s a s by andomly se- lec ing a node om V,s, and se ing TN={s} (TNis he se o he nodes al eady included i he ee). Then n−1admissible edges om A= {eu ∈E:u∈TN∧δ(u)<k∧ ∈V−TN},a e se- quen ially added, whe e δis he deg ee o node uin he sub- ee ha is being cons uc ed, and k he maximum deg ee al- lowed. The selec ion o he edges is pseudo- andomly made using o mula es =a g max ∈A m j=1 τj( )αjwj( )βji q≤q0 ei q>q 0, (1) whe e •τj(e)is he phe omone alue associa ed o he jweigh in edge e; •wj(e)is he j-weigh o edge e; •αjis an algo i hm pa ame e associa ed o he ele- ance o weigh j; •βjis an algo i hm pa ame e associa ed o he local heu is ic ha a ou s edges wi h lowe j-weigh ; •e∈Ais an edge pseudo- andomly chosen wi h p ob- abili y p(e)= m j=1 τj(e)αjwj(e)−βj  ∈A m j=1 τj( )αjwj( )−βj.(2) •qis a uni o m andom alue in [0,1];and •q0∈[0,1] is a pa ame e ha in luences which b anch o (1) is used mo e o en: a smalle alue o q0p o- duces a mo e explo a o y sea ch, since i implies he use o he pseudo- andom o mula (2) wi h highe p obabili y. When q0is nea 1, he easible edge wi h la ge p obabili y o en e ing he ee is used wi h g ea e equency, which sugges an exploi ing sea ch. I he solu ion sa is ies he dis ance cons ain hen he p ocess en e s Dep h Sea ch me hod desc ibed in Algo i hm 2. The selec ion o he edges o en e he sea ch ee in his phase also ollows he abo e o mulas, ha is, (1) and (2). 3.4 Phe omone upda e To upda e he phe omones ma ices i was used he Angle-Phe omone Upda e s a egy [2]. This s a egy can be conside ed g eedy in he sense ha i only uses elemen s o he app oxima ion se . The objec i e is o explo e small egions o he sea ch space by using, in he phe omone up- da ing o mula, only a subse o he solu ions con ained in ha se . This idea is mo i a ed by he ac ha , in mos o he cases, he numbe o elemen s in he app oxima ion se becomes e y la ge. Empi ical es s p o ed ha i all he so- lu ions in he app oxima ion se we e used, he phe omone based selec ion becomes e y noisy, which delays he con- e gence owa d he Pa e o se . Fo he single objec i e case, a se o he kbes solu ions can be kep , using a subse o ha se o do phe omones upda e. The e o e, he phe omone ec o upda e is made acco d- ing o o mula τ(e)=ρτ(e)+Δ(e),e∈E, whe e •τ(e)is a phe omone ec o associa ed o edge e; •ρ∈[0,1] is called he pe sis ence ac o (1−ρis he e apo a ion ac o ). The smalle he alues o ρa e, he smalle quan i y o in o ma ion, used in one cycle, is ansmi ed o ollowing cycle; •Δ(e)=(Δ 1(e),Δ2(e),...,Δm(e)) is he ein o ce- men phe omone ec o associa ed o edge eand is compu ed using he elemen s o he app oxima ion se , P, and o mula Δk(e)=  T∈Pe Q wk(T), whe e –Qis a alue wi h he same magni ude o he solu- ions. Fo example, i he weigh s a e balanced i can be used he a e age o he minimum weigh s, 1 m m  k=1 min T∈P wk(T); –Pea e he elemen s o he app oxima ion se ha con ain edge eand lie in a subangle o he angle de ined by he o igin and he ex eme solu ions (o weigh s kand k+1), ha is, Pe={T∈P:e∈T∧γ1≤φk(T)≤γ2} whe e φk(T) = a c an wk+1(T) wk(T), 39 Pa ame e Values D1 d3|V| 4 M2 0 αi,βi{0,0.03,0.06,...,3.0} ρ0.1 q0{0.5,0.6,0.7,0.8,0.9} k3 Numbe o an s pe cycle |V| Numbe o cycles 2 Maximum un ime min{60|V| log |V|,36000} Seconds Table 1. Used pa ame e s. γ1=φmin k+Ikφh k γ2=φmin k+(Ik+1)φh k , φmin k=min T∈P φk(T) φmax k=max T∈P φk(T), o k=1,2,...,m−1,φh kis he s ep in e al, and Ikis a alue ela ed o he egion o be ex- plo ed which is con olled by he main p ocess. 3.5 Resul s The algo i hm -DANTE was implemen ed in C++ and es swe e unonaPCwi ha3Ghz In el Pen ium IVTMp ocesso , 512Mbo RAM and Windows XP OS. Fo each p oblem he me hod was un 15 imes wi h he pa am- e e s epo ed in Table 1. The p oblems ins ances ha we e used o es he imple- men a ion we e de ined by [9] and he -DANTE solu ions a e compa ed wi h he solu ions ob ained wi h he Gene ic Algo i hms p esen ed by he same au ho s. Mo e speci i- cally, i was used as e e ence se he one composed by he non-domina ed elemen s o he union o he 30 app oxima- ion se s p esen ed in [9]. Table 2 p esen s a esume o he esul s o some o he es ed ins ances o e he 15 uns. This able con ains in o - ma ion abou he a e age alues o he me ics R1and R3, he a e age ca dinali y o he app oxima ion se buil wi h -DANTE (in pa en heses he ca dinal o he e e ence se ), and he a e age ime o he las upda e o he app oxima- ion se . F om he same able i is possible o obse e ha in mos o he cases, -DANTE imp o es he e e ence se since R1<0.5and R3≈0. The excep ions we e he 10 nodes ne wo ks whe e in one case i has achie ed he exac Pa e o se (ob ained wi h he B u e Fo ce me hod) and in P oblems μR1μR3μ|P| (μ|P e |)μTime 10 ac 0.57 0.00 183 (191†)7.6 10 -m-c 0.5 0.00 129 (129†)628 10 conc 0.54 0.00 128 (134†) 8 25 ac 0.47 0.00 517 (439) 3754 25 c 0.39 0.00 2496.6 (1480) 4351. 25 -m-c 0.13 0.01 2168.7 (820) 4743 50 ac 0.40 -0.05 6328.1 (1436) 11960 50 conc1 0.48 0.00 1748.6 (894) 11803 Table 2. Resume o he esul s o he k- Deg ee Minimum Spanning T ee p oblem. he o he wo cases i has ob ained 128 and 183 o he 134 and 191 solu ions, espec i ely. The ime e olu ion o he R1and R3me ics a e de- pic ed in Figu es 3 and 4, espec i ely, o some o he es ed ins ances (25 ac, 25 c and 50 ac). I is possible o obse e ha he con e gence o he solu ions owa d he e e ence se is made qui e as since R3quickly akes alues nea o ze o. Ne e heless, he local e inemen akes some ex a ime, as e lec ed in he las upda e ime o he app oxima- ion se (μTime), which as al eady e e ed is a cha ac e is- ic common o mos o he An Colony based algo i hms. 4 Conclusions This pape was de o ed o he s udy o he -DANTE me hod. This me hod is based in he An Colony Op imiza- ion pa adigm and appea s as an e o o p o ide a mo e e - ec i e way o u he explo ing he bes i ness pe o ming solu ions. Mo e p ecisely, whene e a solu ion is inse ed in o he app oxima ion se , o sa is ies an dis ance o he app oxima ion se c i e ion, i is pe o med a limi ed Dep h Sea ch using he phe omone alues o guide he sea ch. Resul s o a e sion o -DANTE applied o he Mul i- ple Objec i e k-Deg ee Minimum Spanning T ees a e p e- sen ed. I was e i ied ha he me hod apidly con e ges owa d he on s achie ed by o he au ho s and ends up by imp o ing, in gene al, hei esul s. As in many An Colony algo i hms, he -DANTE uses a g eedy s a egy, whe e only he bes pe o ming solu ions a e used o upda e he phe omone alue. Re e ences [1] C. Blum. Beam–ACO – hyb idizing an colony op imiza- ion wi h beam sea ch: an applica ion o open shop schedul- ing. Compu e s and Ope a ions Resea ch, 32(6):1565– 1591, 2005. 40 0 1000 2000 3000 4000 seconds 0.5 0.6 0.7 0.8 0.9 1 R1 25 ac 0 1000 2000 3000 4000 5000 seconds 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R1 25 c 0 2000 4000 6000 8000 1000012000 seconds 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R1 50 ac Figu e 3. Time e olu ion o R1me ic o e he15 uns. [2] P. Ca doso. An Colony Algo i hms o Mul iple Objec i e Combina o ial Op imiza ion: Applica ions o he Minimum Spanning T ees P oblems. PhD hesis, Uni e si y o Se ille, Spain, Ma 2007. [3] K. Deb. Mul i-objec i e Op imiza ion using E olu iona y Algo i hms. John Wiley & sons, 2001. [4] M. Do igo, E. Bonabeau, and G. The aulaz. Swa m In elli- gence: F om Na u al o a i icial Sys ems. Ox o d Uni e - si y P ess, 1999. [5] M. Do igo and T. S u zle. An Colony Op imiza ion.MIT P ess, 2004. [6] L. Gamba della and M. Do igo. An an colony sys em hy- b idized wi h a new local sea ch o he sequen ial o de ing p oblem. INFORMS, Jou nal on Compu ing, 12(3), 2000. [7] A. Jaszkiewicz. Mul iple Objec i e Me aheu is ic Algo- i hms o Combina o ial Op imiza ion. PhD hesis, Poznan Uni e si y o Technology, 2001. [8] D. Jungnickel. G aphs, Ne wo ks and Algo i hms, olume 5 o Algo i hms and Compu a ion in Ma hema ics. Sp inge - Ve lag, Be lin, 1999. 0 1000 2000 3000 4000 seconds R3 25 ac -0.74 -0.59 -0.44 -0.29 -0.15 0 0 1000 2000 3000 4000 5000 seconds R3 25 c -2.35 -1.88 -1.41 -0.94 -0.47 0 0 2000 4000 6000 8000 1000012000 seconds R3 50 ac -2.79 -2.24 -1.68 -1.12 -0.56 0 Figu e 4. Time e olu ion o R3me ic o e he15 uns. [9] J. Knowles and D. Co ne. Benchma k p oblem gene a o s and esul s o he mul iobjec i e deg ee-cons ained min- imum spanning ee p oblem. In P oceedings o he Ge- ne ic and E olu iona y Compu a ion Con e ence (GECCO- 2001), pages 424–431. Mo gan Kau mann Publishe s, San F ancisco, Cali o nia, 2001. [10] J. Le ine and F. Duca elle. An colony op imisa ion and lo- cal sea ch o bin packing and cu ing s ock p oblems. Jou - nal o he Ope a ional Resea ch Socie y, Special Issue on Local Sea ch, 55(7), 2004. [11] I. Pa mee. E olu iona y and Adap i e Compu ing in Engi- nee ing Design. Sp inge -Ve lag, London, 2001. ISNB:1- 85233-029-5. [12] J. Puchinge and G. R. Raidl. Combining me aheu is ics and exac algo i hms in combina o ial op imiza ion: A su ey and classi ica ion. In P oceedings o he Fi s In e na ional Wo k-Con e ence on he In e play Be ween Na u al and A - i icial Compu a ion, olume 3562 o LNCS. Sp inge , 2005. [13] G. R. Raidl. A uni ied iew on hyb id me aheu is ics. In P oceedings o he Hyb id Me aheu is ics Wo kshop, num- be 4030 in LNCS. Sp inge , 2006. 41