The Effect of Introducing a Tag Node in Solving Scheduling Problems Using Edge Histogram Based Sampling Algorithms

Transcription

1 Proceedings of he 2003 Congress on Evoluionary Compuaion (CEC' 03) The Effec of Inroducing a Tag Node in Solving Scheduling Problems Using Edge Hisogram Based Sampling Algorihms Shigeyoshi Tsusui Deparmen of Managemen Informaion Hannan Universiy Amamihigashi, Masubara, Osaka , Japan susui@hannan-u.ac.jp Tomoyuki Hiroyasu Misunori Miki Deparmen of Knowledge Engineering and Compuer Sciences, Doshisha Universiy l-3 Taara, Miyakodani, Kyo-anabe, Kyoo, , Japan {omo@is, mmiki@mail}.doshisha.ac.jp Absrac- We have proposed probabilisic modelbuilding geneic algorihms (PMBGAs) for solving sequence problems using edge hisogram-based sampling algorihms (EHBSAs) in a previous paper. In his paper, we explore he effec of inroducing a ag node in sring represenaions for solving flow shop scheduling problems wih EHBSAs. The resuls showed EHBSAs using srings wih a ag worked beer han EHBSAs using srings wihou a ag node. 1 Inroducion There has been a growing ineres in developing evoluionary algorihms based on probabilisic models [Pelikan 99], [Larranaga 02]. In his scheme, he offspring populaion is generaed according o he esimaed probabilisic model of he paren populaion insead of using radiional recombinaion and muaion operaors. The model is expeced o reflec he problem srucure, and as a resul i is expeced ha his approach provides a more effecive mixing capabiliy han recombinaion operaors in radiional GAs. These algorihms are called probabilisic model-building geneic algorihms (PMBGAs) or esimaion of disribuion algorihms (EDAs). Many sudies on PMBGAs have been performed in discree (mainly binary) domains [Baluja 94], [Baluja 97], [Serve 97], [Harik 98], [Harik 99], [Pelikan 00] and here are several aemps o apply PMBGAs in coninuous domains [Mühlenbein 96], [Sebag 98], [Bosman 99], [Bosman 00], [Larranaga 99], [Larranaga 00], [Tsusui 01]. However, few sudies on PMBGAs in permuaion represenaion domains are found [Robles 02], [Larranaga 02]. In [Tsusui 02a], we have proposed an approach o PMBGAs in permuaion domains, focusing on solving he Traveling Salesman Problem (TSP) and compared is performance wih radiional recombinaion operaors. In his approach, we developed a symmerical edge hisogram marix from he curren populaion, where an edge is a link beween wo nodes in a sring. We hen sampled nodes of a new sring according o he edge hisogram marix. We called his mehod he edge hisogram-based sampling algorihm (EHBSA). We proposed wo ypes of EHBSAs, an edge hisogram-based sampling algorihm wihou emplae (EHBSA/WO) and an edge hisogram-based sampling algorihm wih emplae (EHBSA/WT). The resuls showed EHBSA/WT worked fairly well on he es suie. In [Tsusui 02b], we applied EHBSAs o flow shop scheduling problems. Since in a scheduling problem, each edge is direcional, we used an asymmerical edge hisogram marix from he curren populaion. The resuls showed EHBSA/WT worked well also in flow shop scheduling problems. In his paper, we explore he effec of inroducing a ag node (TN) in a chromosome represenaion and show he TN can improve he performance of EHBSA wih an asymmerical edge hisogram marix. Here, he TN is a viual node and is used o indicae which node is he he firs node (i.e., firs job) in a sring represenaion. Normally, in a sequence problem, he node in he firs posiion in a sring is assumed o be he firs node (job). In our approch, we assume each posiion of nodes in a sring conaining a TN o be relaive. Absolue posiion is obained as follows: in a sring wih a TN, he node immediaely following he TN is inerpreed as he firs node, i.e., job in a sring. The TN is virual because i does no correspond o any real nodes or jobs. However, he TN works as if i is a normal node in modeling and sampling of he EHBSA. In he remainder of his paper, a brief review of EHBSA is described in Secion 2, ag nodes and corresponding sampling EHBSAs are discussed in Secion 3, and he empirical analysis is given in Secion 4. Secion 5 concludes he paper. 2 EHBSAs This secion reviews how he edge hisogram based sampling algorihm (EHBSA) can be used o (1) model promising soluions and (2) generae new soluions by simulaing he learned model. 2.1 The Basic Descripion of he Algorihm An edge is a link or connecion beween wo nodes and has imporan informaion abou he permuaion sring. Some crossover operaors, such as Edge Recombinaion (ER) [Whiley 89] and enhanced ER (eer) [Sarkweaher, 91], which are used in radiional wo-paren recombinaion, use 22

2 Proceedings of he 2003 Congress on Evoluionary Compuaion (CEC' 03) he edge disribuion only in he wo parens srings. The basic idea of he edge hisogram based sampling algorihm (EHBSA) is o use he edge hisogram of he whole populaion in generaing new srings. The algorihm sars by generaing a random permuaion sring for each individual populaion of candidae soluions. Promising soluions are hen seleced using any popular selecion scheme. An edge hisogram marix (EHM) for he seleced soluions are consruced and new soluions are generaed by sampling, based on he edge hisogram marix. New soluions replace some of he old ones and he process is repeaed unil he erminaion crieria are me. 2.2 Developing Edge Hisogram Marix A Symmerical Edge Hisogram Marix A symmerical edge hisogram marix is inended o apply o problems such as a symmerical TSP where each edge has no direcion. In his case, we assume edges 1 >2 and 2 >1are idenical. If here is an edge 1 >2, hen we assume he edge 2 >1 also exiss. Here, 1 and 2 are nodes. Le sring of k-h individual in populaion P() a generaion represen s k = (π k(0), π k(1),..., π k(l-1)). (π k(0), π k(1),..., and π k(l-1)) are he permuaion of (0, 1,..., L 1), where L is he lengh of he permuaion. Then, a symmerical edge hisogram marix EHM (s) (e i,j) (i, j =0,1,.., L 1) of populaion P() consiss of L 2 elemens as follows: e i, j = N ( δ s k= ij ( k ) + δ 1 j, i 0 ( sk )) + ε if i j if i = j where N is he populaion size, δ ij (s k) is a dela funcion defined as δ 1 ) = 0 if h [ h {0,1, LL 1} π k ( h) = i π k (( h + 1) mod L) = ohersise i, j ( sk j and ε (ε>0) is a bias o conrol pressure in sampling nodes, jus like hose used for adjusing he selecion pressure in he proporional selecion in GAs. The average number of edges of elemen e i,j in EHM (s) is 2LN/(L 2 -L) = 2N/(L 1). Values e i,j will be laer be used in a varian of proporionae selecion, and he value of ε hus influences selecion pressure oward edges. To achieve comparable selecion pressure for all problems and parameer seings, ε should be proporional o he expeced value of e i,j. Therefore, 2N ε = Braio (3) L 1 where B raio (B raio > 0), he bias raio, is a consan relaed o selecion pressure An Asymmerical Edge Hisogram Marix An asymmerical edge hisogram marix is inended o apply o problems such as asymmerical TSP and scheduling problems where each edge has direcion. For example, in flow shop scheduling problems in his paper, each sring represens a sequence of jobs o be processed. A ] (1) (2) sring s = {1, 2 3, 0} s ha firs, job 1 mus be processed, hen jobs 2, 3, and 0 follow in his sequence. In his case, here are four edges, i.e., 1 >2, 2 >3, 3 >0, and 0 >1. Thus in a scheduling problem, each edge is direcional and he edge-hisogram marix becomes asymmerical. An asymmeric edge-hisogram marix EHM (A) (e ij) (i, j =0,1,.., L 1) of populaion P() consiss of L 2 elemens as follows: N + = = δ ( s ) ε if i j k 1 ij k e i, j (4) 0 if i = j where δ ij (s k) is a dela funcion defined by Eq. 2. The average number of edges of elemen e i,j in EHM (A) is LN/(L 2 -L) = N/(L 1). So he bias ε for EHM (A) is defined as in Eq. 3 as follows: N ε = 1 B (5) raio L 2.3 Sampling Mehods In his subsecion, we review how o sample a new sring from he edge hisogram marix. There are wo ypes of sampling mehods; one is an edge-hisogram based sampling algorihm wihou emplae (EHBSA/WO), and he oher an edge-hisogram based sampling algorihm wih emplae (EHBSA/WT) EHBSA/WO In a symmerical EHM such as used in he symmerical TSP, he absolue posiions (loci) of a sring have no ing. For example, sring s 1 = (0, 1, 2, 3, 4) and sring s 2 = (4, 0, 1, 2, 3) represen he same our. However, in a asymmerical EHM such as for scheduling problems, hese wo sring represen wo compleely differen soluions. Thus, we mus consider how o deermine he iniial posiion and wha node we assign o he posiion. In [Tsusui 02b], we have proposed wo ypes of EHBSA/WO, EHBSA/WO/T and EHBSA/WO/R. EHBSA/WO/T Le us represen a new individual permuaion by c[]. In EHBSA/WO/T, he iniial sampling posiion is always he firs posiion, i.e. c[0]. The value for c[0] is aken from a pseudo emplae individual PT[] which is aken from he curren populaion P() randomly, and a new individual permuaion c[] is generaed sraighforwardly as follows: 1. Se he posiion couner p 0 2. Choose a pseudo emplae PT[] from P() 3. Obain firs node as c[0] PT[0] 4. Consruc a roulee wheel vecor rw[] from EHM (A) as rw[j] = e (j=0, 1,.., L 1) c[p],j 5. Se o 0 previously sampled nodes in rw (rw[ c[i] ] = 0 for i =0, 1,.., p) 6. Sample he nex node c[p+1] wih probabiliy L 1 rw [ x] rw[ j] using roulee wheel rw[] j= 0 7. Updae he posiion couner p p+1 2

3 Proceedings of he 2003 Congress on Evoluionary Compuaion (CEC' 03) 8. If p<l 1, go o Sep 4 9. Obain a new individual sring c[] emplae T[] cu[0] cu[1] cu[2] segmen0 segmen1 segmen2 EHBSA/WO/R In EHBSA/WO/T, he iniial sampling posiion is fixed o 0. On he oher hand, in EHBSA/WO/R, he iniial sampling posiion is chosen from [0, L-1] randomly as follows: EHM (A) sampling 1. Obain random iniial sampling posiion p iniial from [0, L 1] 2. Choose a pseudo emplae PT[] from P() 3. Obain firs node as c[p iniial ] PT[p iniial ] 4. Se he posiion couner p p iniial 5. Consruc a roulee wheel vecor rw[] from EHM (A) as rw[j] = e c[p],j (j=0, 1,.., L 1) 6. Se o 0 previously sampled nodes in rw (rw[ c[i] ] = 0 for i = p iniial, (p iniial +1) mod L,..., p) 7. Sample he nex node c[(p+1) mod L] wih probabiliy L 1 rw [ x] rw[ j] using roulee wheel rw[] j= 0 8. Updae he posiion couner p (p+1) mod L 9. If ( p+1) mod L p iniial, go o Sep Obain a new individual sring c[] Here, noe boh EHBSA/WO/T and WO/R are he same for problems which have a symmerical EHM EHBSA/WT EHM is a marginal edge hisogram and has no graphical srucure. EHBSA/WT is inended o make up for his disadvanage by using a emplae in sampling a new sring. In generaing each new individual, a emplae individual is chosen from P() (normally, randomly). The n (n>1) cu poins are applied o he emplae randomly. When n cu poins are obained for he emplae, he emplae should be divided ino n segmens. Then, we choose one segmen randomly and sample nodes for he segmen. Nodes in oher n 1 segmens remain unchanged. We denoe his sampling mehod as EHBSA/WT/n. Since he average lengh of one segmen is L/n, EHBSA/WT/n generaes new srings which are differen L/n nodes on average from heir emplaes. Fig. 2 shows an example of EHBSA/WT/3. In his example, nodes of new sring from afer cu[2] and before cu[1] are he same as he nodes of he emplae. New nodes are sampled from cu[1] up o, bu no including, cu[2] based on he EHM. new sring c[] Fig. 1. An example of EHBSA/WT/3 The sampling mehod for EHBSA/WT/n is as follows: 1. Choose a emplae T[] from P(). 2. Obain a sored cu poin array cu[0], cu[1],.., cu[n 1] randomly. 3. Choose a cu poin cu[l] by generaing random number l [0, n 1]. 4. Copy nodes in T[] o c[] from afer cu[(l+1) mod n] and before cu[l] 5. Se he posiion couner p cu[l] 1 6. Consruc a roulee wheel vecor rw[] from EHM (A) as rw[j] = e (j=0, 1,.., L 1) c[p],j 7. Se o 0 copied and previously sampled nodes in rw[] (rw[ c[i] ] = 0 for i =cu[(l+1) mod n],.., p) 8. Sample he nex node c[(p+1) mod L] wih L probabiliy 1 rw[ x] rw[ j] using roulee wheel j= 0 rw[] 9. Updae he posiion couner p (p+1) mod L 10. If (p+1) mod L = cu[(l+1) mod n], go o Sep Obain a new individual sring c[] Le f(x) be he probabiliy densiy funcion of he lengh of a segmen o be sampled in EHBSA/WT/n. Then, f(x) is obained as [Tsusui 03] n 2 ( n 1) x f ( x) = 1. (6) L L Fig. 2 shows he probabiliy densiy funcion f(x). For n = 2, f(x) is uniformly disribued on [0, L]. Thus, he lengh of a segmen o be sampled in EHBSA/WT/2 is uniformly disribued on [0, L]. When he lengh of he segmen is small, he EHBSA/WT/2 samples a small number of nodes, performing a kind of local search improvemen over he emplae individual. On he oher hand, when he lengh is large, he EHBSA/WT/2 samples a large number of nodes, performing a kind of global search improvemen over he emplae individual or produces a new sring. Thus, we can expec he EHBSA/WT/2 o work by balancing global and local improvemens. For n > 2, shor segmens are more likely o occur. 3

4 Proceedings of he 2003 Congress on Evoluionary Compuaion (CEC' 03) f(x) L L x n=2 n=3 n=4 n=5 Fig. 2. Probabiliy densiy funcion f(x) 3 Inroducing he Tag Node (TN) o EHBSAs 3.1 Tag Node and Virual Sring In a scheduling problem where a soluion is represened by a permuaion sring, he performance of each sring is ighly linked no only o he relaive sequence of nodes (jobs) bu also o he absolue posiion of nodes in he sring. Since EHM, described in Secion 2, has no explici informaion on he absolue posiions of nodes in each sring, i may be useful o inroduce some addiional informaion on he absolue posiion of each node in a sring. The ag node (TN) proposed in his paper is an approch o inroduce informaion on he absolue posiion of each node in a sring. In addiion o normal nodes, we add a TN o each permuaion sring. We call a sring wih a TN a virual sring (VS). Sring lengh of a VS is L VS = L+1, where L is he lengh of real sring (RS; sring wihou TN). The TN in VS works as a ag in a permuaion sring o indicae he firs node (job), i.e., he nex node following he TN is assumed o be he firs node (job) in he soluion. The TN is a virual node because i does no correspond o any real nodes, or jobs. However, he TN in a VS is reaed as if i is a normal node in modeling and sampling of he EHBSA. Fig 3 shows how o obain RS from VS. In his case, L = 6. VS RS (real sring) TN Fig. 3 An example of a virual sring L VS = 7 L = Sampling Mehods Sampling mehods for VS are he same as sampling in srings wihou TN, referred o in Secion 2. Only a small difference exiss in EHBSA/WO/T. In EHBSA/WO/T, sampling sars wih fixing he node in posiion 0. In VS, his may correspond o sampling by fixing VS in posiion 0. Sampling for VS in EHBSA/WO/R and EHBSA/WT is compleely he same as sampling for srings wihou TN. Here, noe ha he TN is sampled in precisely he same manner as normal nodes and sampling is performed from a random posiion in a VS. 3.3 Represenaion of he TN in a VS We can use any symbol o represen he TN in a VS. However, for implemenaion convenience, we use an ineger number o represen i in a VS. Le us consider he sring in Fig. 3, for example. In his example, sring lengh L = 6 and he lengh of VS L VS = 7. Nodes 0, 1,, 5 represen real nodes. Then, we assign number 6 o he TN (see Fig. 4). Thus, a VS of lengh L+1 is represened as a permuaion {0, 1,, L 1, L} corresponding numbers 0, 1,, L 1 o real nodes and number L o he TN. Wih his represenaion, we do no need any special modificaion in basic EHBSAs VS RS (real sring) Fig. 4 Represenaion of he TN in a VS 4 Empirical Sudy L VS = 7 L = 6 In his secion, we explore he effeciveness of inroducing he TN node in solving flow shop scheduling problems using EHBSAs. 4.1 Experimenal Mehodology Evoluionary models The evoluionary model is he same as he model used for symmerical EHM (A) in [Tsusui 02b] as follows: Evoluionary model for EHBSA/WT Le he populaion size be N, and le i, a ime, be represened by P(). The populaion P(+1) is produced as follows (Fig. 5): 1. Edge disribuion marix EHM (A) described in Subsecion 2.2 b) is developed from P() 2. A emplae individual T[] is seleced from P() randomly 4

5 Proceedings of he 2003 Congress on Evoluionary Compuaion (CEC' 03) 3. EHBSA/WT described in Subsecion 2.3 b) is performed using EHM (A) and T[], and generaes a new individual c[] 4. The new c[] individual is rearranged, removing he VN (see Figs xx and xx) and hen evaluaed 5. If c[] is beer han T[], hen T[] is replaced wih c[], oherwise T[] remains, forming P(+1) = +1 Selec beer one P() N c T T EHM (A) Fig. 5. Evoluionary model for EHBSA/WT The evoluionary model for EHBSA/WO/T and EHBSA/WO/R The evoluionary model for EHBSA/WO is basically he same as he model for EHBSA/WT, excep EHBSA/WO uses a pseudo emplae PT[] Flow shop problems and performance measures General assumpions of flow shop scheduling problems can be described as follows: Jobs are o be processed on muliple machines sequenially. There is one machine a each sage. Machines are available coninuously. A job is processed on one machine a a ime wihou preempion, and a machine processes no more han one job a a ime. In his paper, we assume ha L jobs are processed in he same order on m machines. This s ha our flow shop scheduling is he L-job and m-machine sequence problem. The purpose of his problem is o deermine he sequence of L jobs which minimizes he makespan (i.e., he compleion ime of all jobs). This sequence is denoed by a permuaion sring of {0, 1,..., L 1}. As es problems, we generaed wo flow shop scheduling problems, [20-job 10-machine], [40-job 10- machine] problems. In designing each problem, we specified he processing ime of each job a each machine as an random ineger in he inerval [1, 99]. 50 runs were performed. Each run coninued unil he populaion was converged, or evaluaions reached E max. E max was se o 200,000. The performance is measured by he minimum makespan (bes), of he minimum makespan () in 50 runs and sandard deviaion of he minimum makespan (sd). Populaion sizes of 60, 120, 240 were used for EHBSAs. As o he bias raio B raio in Eq. 5, a B raio values of was used. As o he cu poins, n for EHBSA/WT, n = 2, 3, and 4 were esed Blind search In solving scheduling problems using GAs, muaion operaors play an imporan role. Several ypes of muaion operaors are proposed. Also, i is well known ha combining GAs wih local opimizaion mehods or heurisics grealy improve he performance of he algorihms. However, in his experimen, we use no muaion and no heurisic o see he pure effec of applying proposed algorihms. Thus, he algorihm is a blind search. 4.2 Empirical analysis of resuls Resuls on 20-jon and 10-machine flow shop problem Resuls on he 20-job and 10-machine problem are shown in Table 1. As was found in [Tsusui 02a] wih TSP, we can confirm clearly ha EHBSA/WT shows much beer performance, i.e. smaller makespan, han EHBSA/WO in all resuls in he able. Boh EHBSA/WO/T and EHBSA/WO/R showed obviously worse performance han EHBSA/WT/n (n = 2, 3, 4). In EHBSA/WO, we can see ha here is ingful difference beween EHBSA/WO/T and EHBSA/WO/R. EHBSA/WO/R, which samples nodes from a randomly chosen posiion of he sring, shows beer performance han EHBSA/WO/T, which fixes he iniial sampling posiion o he firs posiion of he sring. Nex, le s explore he effec of inroducing he ag node (TN) o EHBSA/WT. The bes value of he in mehods wihou TN was observed in EHBSA/WT/2 wih populaion size = 60 and he value = However, wih TN, he value decreases o Alhough he difference is no so remarkable, he imporan observaion is ha in all experimens wih EHBSA/WT, values of he wih TN always showed beer han hose wihou TN. On average, he value of he wih TN is smaller han hose wihou TN by 0.31%. Table 1 shows only resuls a evaluaions = Fig. 6 shows he convergence processes of EHBSA/WT wih and wihou TN for (a) EHBSA/WT/2, (b) EHBSA/WT/3, and (c) EHBSA/WT/4, wih populaion size = 60. From his figure, we can see ha values of wih TN always converge faser han hose wihou TN, aking significan smaller values. Table 1 Resuls on 20-job and 10-machineflowshop problem (makespan) 5

6 Proceedings of he 2003 Congress on Evoluionary Compuaion (CEC' 03) EHBSA EHBSA/WO/T EHBSA/WO/R EHBSA/WT/2 EHBSA/WT/3 EHBSA/WT/4 TN wihou wih wihou wih wihou wih wihou wih wihou wih evaluaions (b) EHBSA/WT/3 Performance Populaion Size Mesure bes sd bes sd bes sd bes sd bes sd bes sd bes sd bes sd bes sd bes sd wiou TN wih TN evaluaions (a) EHBSA/WT/2 wihou TN wih TN wihou TN wih TN evaluaions (c) EHBSA/WT/4 Fig. 6 Convergence process of EHBSA/WT on 20-jobs and 10-machine Resuls on 40-job and 10 machine flow shop problem Resuls on he 40-job and 10-machine problem are shown in Table 2. Again, we can confirm clearly ha EHBSA/WT shows much beer performance, i.e. smaller makespan, han EHBSA/WO in all resuls in he able. In EHBSA/WO, we can see EHBSA/WO/R shows beer performance han EHBSA/WO/T, as in he case of he 20-job and 10-machine problem. The effeciveness of inroducing he TN for EHBSA/WT is also clearly observed in his problem, oo. The bes value of he wihou TN was observed in EHBSA/WT/2 wih populaion size = 120, he value being However, wih TN, he value decreases o As was observed in he 20-job and 10-machine problem, he difference is again no so remarkable. However, he imporan observaion is ha in all experiences of EHBSA/WT, values of he wih TN always showed beer han hose wihou TN. On average, he wih TN is smaller han wihou TN by 0.34%. This effec can be observed for EHBSA/WO, oo. Fig. 7 shows he convergence processes of EHBSA/WT wih and wihou TN for (a) EHBSA/WT/2, (b) EHBSA/WT/3, and (c) EHBSA/WT/4, wih populaion size = 60. From his figure, we can see he values of wih TN always converge faser han hose wihou TN, resuling in significanly smaller values as was observed in Fig 6 on he 20-jobs and 10-machine problem Consideraions As shown in subsecion and 4.2.2, inroducing he TN wih EHBSA consisenly improved he performance of he EHBSA on flow shop scheduling problems. Now le us confirm he saisical difference of values beween he models wih he TN and wihou he TN using -es. Values of for EHBSA/WT in Tables 1 and 2 are in he range of [3.2, 9.5] excep for EHBSA/WT/2 where N = 120 on he 20-job and 10-machine problem (in his case, = 0.629). Since he value of df = 99, we can se he level of significance below Since EHM has no explici informaion on he absolue posiion of each node, he TN in EHBSA works as a ag ha indicaes he iniial posiion in a sring, hus as we inended, i improves performance of a problem where in addiion o he relaive posiion of each node, he absolue posiion of each node effecs he performance. Here, we mus noe ha wihou using he TN, EHBSA can somehow mainain informaion abou absolue posiion of each node in a sring of he populaion. Tha is because, for example in EHBSA/WT, nodes in a new sring ha are no sampled, inheri he same absolue posiions from is emplae sring. Inroducing he TN helps he managing of informaion on absolue posiion more, and hus increases he performance. Table 2 Resuls on 40-job and 10-machine flowshop problem (makespan) 6

7 Proceedings of he 2003 Congress on Evoluionary Compuaion (CEC' 03) EHBSA EHBSA/WO/T EHBSA/WO/R EHBSA/WT/2 EHBSA/WT/3 EHBSA/WT/4 TN wihou wih wihou wih wihou wih wihou wih wihou wih Performance Populaion Size Mesure bes sd bes sd bes sd bes sd bes sd bes sd bes sd bes sd bes sd bes sd From above discussion, we may expec ha he TN works well for radiional wo-paren recombinaion operaors. Fig. 8 shows he convergence processes wih and wihou TN in he original order crossover OX [Oliver 87] and he enhanced edge recombinaion operaor eer [Sarkweaher 91]. The evoluionary model used was similar o ha described in Fig 5 and populaion size = 480 was used. From his figure, we can see ha inroducing he TN is effecive for radiional recombinaion operaors, oo. Especially, eer showed a big difference in performance. Since he TN is reaed as jus a normal node in algorihms, here is no special processing necessary in inroducing he TN. The difference of compuaional complexiy wih and wihou TN is influenced only by he sring lengh; wihou TN he lengh is L, and wih TN he lengh is L+1. evaluaions (a) OX wihou TN wih TN evaluaions (b) eer wihou TN wih TN Fig. 8 Convergence processes of (a) OX and (b) eer on 20-jobs and 10-machine evaluaions wihou TN wih TN 2600 evaluaions wiou TN wih TN (b) EHBSA/WT/3 (a) EHBSA/WT/ wihou TN wih TN 2600 evaluaions (c) EHBSA/WT/4 Fig. 7 Convergence process of EHBSA/WT on 40-jobs and 10-machine 5 Conclusions EHBSAs (edge hisogram based sampling algorihms) have been proposed for permuaion represenaion problems based on probabilisic model-building geneic algorihm (PMBGAs) schemes. There are wo ypes of EHBSAs, EHBSA/WT and WHBSA/WO. EHBSA/WT works fairly well on TSPs, symmerical sequencing problems. In a previous sudy, we applied EHBSAs o flow shop scheduling problems. Since in a scheduling problem, each edge is direcional, we used an asymmerical edge hisogram marix from he curren populaion. The resuls showed EHBSA/WT worked fairly well in flow shop scheduling problems. In his paper, we explored he effec of inroducing a ag node (TN) wih EHBSA using 20-jobs and 10- machine, and 40-jobs and 10-machine flow shop problems. The resuls showed ha inroducing he TN could improve he performance of EHBSA/WT. We were also able o show ha inroducing he TN is effecive for radiional wo-paren recombinaion operaors. Since inroducing he TN does no increase compuaional complexiy, he approach can be applicable 7

8 Proceedings of he 2003 Congress on Evoluionary Compuaion (CEC' 03) o no only EHBSA bu also o oher permuaion-based recombinaion operaors in solving scheduling problems such as flow shop problems. Bu a deailed sudy of his remains for fuure work. Acknowledgmens This research is parially suppored by he Minisry of Educaion, Culure, Spors, Science and Technology of Japan under Gran-in-Aid for Scienific Research number , and a gran o RCAST a Doshisha Universiy from he Minisry of Educaion, Culure, Spors, Science and Technology of Japan. References [Baluja 94] Baluja, S.: Populaion-based incremenal learning: A mehod for ineracing geneic search based funcion opimizaion and coempive learning, Tech. Rep. No. CMU-CS , Carnegie Mellon Universiy (1994). [Baluja 97] Baluja, S. and Davies: Using opimum dependency-rees for combinaorial opimizaion: learning he srucure of he search space, Tech. Rep. No. CMU-CS , Carnegie Mellon Universiy (1997) [Bosman 99] Bosman, P. and Thierens, D.: An algorihmic framework for densiy esimaion based evoluionary algorihms, Tech. Rep. No. UU-CS , Urech Universiy (1999). [Bosman 00] Bosman, P. and Thierens, D.: Coninuous ieraed densiy esimaion evoluionary algorihms wihin he IDEA framework, Proc. of he Opimizaion by Building and Using Probabilisic Models OBUPM Workshop a he Geneic and Evoluionary Compuaion Conference GECCO-2000, pp (2000). [Harik 98] Harik, G., Lobo, F. G., and Goldberg, D. E.: The compac geneic algorihm, Proc. of he In. Conf. Evoluionary Compuaion 1998 (ICEC 98), pp (1998). [Harik 99] Harik, G: Linkage learning via probabilisic modeling in he ECGA, Tecnical Repor IlliGALRepor 99010, Universiy of Illinois a Urbana-Champaign, Urbana (1999). [Larranaga 99] Larranaga, P., Exeberria, R., Lozano, J.A., and Pena, J.M.: Opimizaion by learning and simulaion of Bayesian and gaussian neworks, Universiy of he Basque Counry Technical Repor EHU-KZAAIK -4/99 (1999). [Larranaga 00] Larranaga, P., Exeberria, R., Lozano, J. A., and Pena, J. M.: Opimizaion in coninuous domains by learning and simulaion of Gaussian neworks, Proc. of he 2000 Geneic and Evoluionary Compuaion Conference Workshop Program, pp (2000). [Larranaga 02] Larranaga, P. and Lozano, J. A. (eds): Esimaion of disribuion algorihms, Kluwer Academic Publishers (2002). [Mühlenbein 96] Mühlenbein, H and Paa, G.: From recombinaion of genes o he esimaion of disribuion I. Binary parameers, Proc. of he Parallel Problem Solving from Naure - PPSN IV, pp (1996). [Oliver 87] Oliver, I, Smih, D., and Holland, J.: A sudy of permuaion crossover operaors on he ravel salesman problem, Proc. 2nd In. Conf. on Geneic Algorihms, pp (1987). [Pelikan 99] Pelikan, M., Goldberg, D. E., and Lobo, F. G.: A survey of opimizaion by building and using probabilisic models, Technical Repor IlliGAL Repor 99018, Universiy of Illinois a Urbana-Champaign (1999). [Pelikan 00] Pelikan, M., Goldberg, D. E., and Canu-Paz, E.: Linkage problems, disribuion esimae, and Bayesian nework, Evoluionary Compuaion, Vol. 8, No. 3, pp (2000). [Robles 02] Robles, V., Miguel, P. D., and Larranaga, P.: Solving he raveling salesman problem wih EDAs, Esimaion of Disribuion Algorihms, Larranaga, P. and Lozano, J. A. (eds), Kluwer Academic Publishers, Chaper 10, pp (2002). [Sebag 98] Sebag, M. and Ducoulombier, A.: Exending populaion-based incremenal learning o coninuous search spaces, Proc. of he Parallel Problem Solving from Naure - PPSN V, pp (1998). [Serve 97] Serve, I. L., Trave-Massuyes, L., and Sern, D.: Telephone nework raffic overloading diagnosis and evoluionary compuaion echniques, Proc. of he Third European Conference on Arificial Evoluion (AE 97), pp (1997). [Sarkweaher, 91] Sarkweaher, T., McDaniel, S., Mahias, K, Whiley, D, and Whiley, C.: A comparison of geneic sequence operaors, Proc of he 4h In. Conf. on Geneic Algorihms, Morgan Kaufmann, pp (1991). [Tsusui 01] Tsusui, S., Pelikan, M., and Goldberg, D. E.: Evoluionary Algorihm using Marginal Hisogram Models in Coninuous Domain, Proc. of he 2001 Geneic and Evoluionary Compuaion Conference Workshop Program, pp (2001). [Tsusui 02a] Tsusui, S.: Probabilisic Model-Building Geneic Algorihms in Permuaion Represenaion Domain Using Edge Hisogram, Proc. of he 7h Parallel Problem Solving from Naure - PPSN VII, pp (2002). [Tsusui 02b] Tsusui, S. and Miki, M.: Solving Flow Shop Scheduling Problems wih Probabilisic Model-Building Geneic Algorihms using Edge Hisograms, Proc. of he 4h Asia-Pacific Conference on Simulaed Evoluion And Learning -SEAL02 (2002). [Tsusui 03] Tsusui, S., Pelikan, M., and Goldberg, D. E: Using Edge Hisogram Models o Solve Permuaion Problems wih Probabilisic Model-Building Geneic Algorihms, IlliGAL Repor No , Universiy of Illinois a Urbana-Champaign (2003). [Whiley 89] Whiley, D., Sarkweaher, T., and Fuquay, D.: Scheduling problems and raveling salesman problem: The geneic edge recombinaion operaor, Proc. of he 3rd In. Conf. on Geneic Algorihms, Morgan Kaufmann (1989). 8