Modification on Search Algorithm for Computer Simulated Experiment

Transcription

1 การประช มว ชาการด านการว จ ยด าเน นงานแห งชาต ประจ าป 255 ว นท กรกฎาคม พ.ศ. 255 Modificaion on Search Algorihm for Comuer Simulaed Exerimen Thammara Thamma, Jarasri Rungraanaubol 2* and Anamai Na-udom 3 Absrac Currenly comuer simulaed exerimens (CSE) have been exensively used in sciences and engineering alicaions. Selecing a roer design o run CSE is very criical for reliabiliy of ouu resonse. Normally he bes design is generaed by using a search algorihm along wih a re-secified oimaliy crierion. This aer erforms he modificaion on SA (Simulaed Annealing) for CSE o imrove he efficiency. From he emirical sudies, he resuls indicae ha he ime required for modified SA o reach he same level of oimal design is significanly lower han ha of he original SA. Keywords: comuer simulaed exerimen, search algorihm, oimaliy crieria 2* 3 Maser suden, Dearmen of Comuer Science and Informaion Technology, Naresuan Universiy Lecurer, Dearmen of Comuer Science and Informaion Technology, Naresuan Universiy (Corresonding Auhor) Lecurer, Dearmen of Mahemaics, Naresuan Universiy 7

2 การประช มว ชาการด านการว จ ยด าเน นงานแห งชาต ประจ าป 255 ว นท กรกฎาคม พ.ศ Inroducion Recenly comuer simulaed exerimens (CSEs) have been exensively used o invesigae a sohisicaed comlex henomena, esecially when classic (hysical) exerimens are no feasible. For insance, he use of reservoir simulaor o redic ulimae recovery of oil [3], he use of finie elemen codes o redic behavior of a meal srucure under sress [8], and so on. Tyically he rocess of CSE is considered as a black box and no known a riori. Simson e al. [9] classified he basic ses of CSE ino 3 ars: he choice of exerimenal design, simulaion rouine (black box) and develoing an aroximaion model, resecively. These hree ses are visualized in Figure. Figure Comuer simulaed exerimen [9] Usually comuer simulaed exerimens are comuaionally exensive and ime consuming. Hence many effors have gone o develoing an accurae aroximaion model based on handful of runs. A selecion of design (design of exerimen) o run CSE (Figure (a)) is criical for he accuracy of he develoed aroximaion model (Figure (b)). As comuer simulaed exerimens are deerminisic in naure, he characerisics of classical exerimenaions such as blocking, relicaion and randomizaion are irrelevan [8]. Therefore, he key sraegy of designs for CSE is focused on sreading he design oins over he region of ineres. Such designs are called sace filling designs. The examle of a classical design and a sace filling design is shown in Figure 2. In a classical design, he design oins are concenraed on he boundary and cener whereas in a sace filling design, design oins are uniformly sread. Normally he sace filling designs can be consruced by searching for a good design hrough he search algorihms under a re-secified oimaliy crierion. In he conex of CSE, Lain hyercube designs (LHD), roosed by Mackay e al. (979) are widely used since is consrucion is very simle in racice. Figure 2 Classical and Sace Filling Designs The CSE are usually comlex and consis of many inu variables o invesigae so a large number of runs are required o esimae all relaed arameer in he model. For insance, if he roblem of ineres consiss of d inu variables and n number of runs, he oal number of all ossible LHD is ( n!) d. Obviously his number exlodes as values of n and d increase. Therefore he full sace of LHD canno be exlored. In such cases we need search algorihm o lead us o a good design wih resec o an oimaliy crieria. The key idea of all exising search algorihms is o use some kinds of exchange rocedures o move owards beer a design. In racice, for a given oimaliy crierion, he search algorihm will look for a beer LHD, unil re-secified olerance level is achieved. A range of search algorihms and oimaliy crieria have been roosed in he conex of CSE. For examle, Johnson e al. [3] roosed he oimaliy crieria called minimax and maximin disance crieria o measure he goodness of he oimal LHD. Morris and Michell [6] adaed a version of Simulaed annealing algorihm (SA) o search for he oimal LHD by considering oimaliy crieria. Ye e al [0] roosed he consrucion of oimal symmeric Lain hyercube design (SLHD) and modified he columnwise-airwise exchange algorihm (CP) o consruc he oimal SLHD by considering maximin disance crierion and oimaliy crieria. Jin e al. [] develoed a new algorihm called Enhanced Sochasic Evoluionary (ESE) and modified he calculaion of various oimaliy crieria (including crieria). Rungraanaubol and Na-udom [7] resened a comarison on he erformance of Geneic algorihm (GA) [4] and Simulaed annealing algorihm (SA) wih resec o oimaliy crieria. The resuls indicae ha SA erforms much beer han GA in erms of he simliciy of arameers seing and rae of 8

3 การประช มว ชาการด านการว จ ยด าเน นงานแห งชาต ประจ าป 255 ว นท กรกฎาคม พ.ศ. 255 convergence. Though SA has shown high erformance in searching for an oimal LHD, SA sill requires such a long ime o search for he oimal LHD, esecially when he dimension of he roblems is large. Therefore his aer aims o imrove he erformance of SA by using he idea of seing he olerance level o erminae SA and adjusing he calculaion of he oimaliy crieria. In he nex secion, we will resen he research mehod including he design used in his sudy, deails of SA and he modificaions of SA, followed by he resuls and conclusion. 2. Research Mehod This aer aims o imrove he erformance of SA o search for he bes LHD under oimaliy crieria. We firs develoed boh of original SA (SA) and modified SA (MSA) in MATLAB codes and simulaed hem, and hen he oimal value of crieria for secified dimensions of roblem are recorded and comared. The deails of design used, LHD, he ses of an original SA, and he modificaions of SA based on avoiding of recalculaing of crieria are resened as follows. 2. Lain hyercube design (LHD) LHD has been exensively used in he conex of CSE. I was originally roosed by McKay e al. [5]. LHD is a marix ( ), of n rows and d columns where n is he number of runs and d is he number of inu variables. LHD can be consruced based on he idea of sraified samling o ensure ha all subregions in he divided inu variable sace will be samled wih equal robabiliy. A Lain hyercube samling has π ij Uij ij = () n, where π ij are he elemens of an n d marix comrising of columns π i ( i =,2,, d). Each column π i ( i =,2,, d) is indeenden random ermuaion of number, 2,, n and U ij are n d values of i.i.d. uniform U[0,] random variables indeenden of he π ij. An examle of LHD is resened in Table. Table The 5 4 random LHD As can be seen from Table, each column in he design marix is referred o each inu variable, while each row is referred o he design run. If an exchange akes lace in any wo airs of runs wihin randomly seleced column, a new design is occurred. LHDs are widely used in racice since is roery sill remains afer any exchange rocedures. 2.2 Simulaed annealing algorihm (SA) Morris and Michell [6] adaed simulaed annealing search algorihm o find he oimal design wih reseced o crieria. The ses of SA are based on he idea of exchanging wo random oins (Perurbaion) of a dimension in he design. However, in he SA algorihm, an exchange wih no imrovemen or deerioraion in he crieria may be also reained during search rocess. The ses are resened below and is flowchar is dislayed in Figure 3. Se : Se iniial value of I max (maximum number of erurbaion o seek imrovemen), 0 (iniial cooling emeraure) and C (facor by which 0 is reduced when no imrovemen in is aained). The facor 0 conrols he robabiliy of reaining a erurbaion ha resuls in no imrovemen of. In aricular, higher value of 0 leads o higher robabiliy of reaining a nonimroving erurbaion. As search rocess coninues 0 is changed by a facor C resuling in lower value of 0 and hence lower robabiliy of reaining non imroving erurbaion of same order. Secify he dimension of he roblem n and d. Se 2: Generae a random LHD of given order n d. Le bes =, = 0 Se 3: Se I =, Label = 0 Se 4: Le ry = Randomly selec a column say j, of marix ry and exchange wo randomly seleced elemens of column j, say aj bj 9

4 การประช มว ชาการด านการว จ ยด าเน นงานแห งชาต ประจ าป 255 ว นท กรกฎาคม พ.ศ. 255 Se 5: Se =, Label =, ry / ry If ry l or wih robabiliy e Se 6: If ry < bes, se I = and bes = ry, else I= I +. Se 7: If I< I max go o Se 4. Se 8: If Lable =, se = C, go o Se 3. Se 9: So and reor. bes d d2 L d j d2 d22 L d 2 j D = M M O M di di2 d L ij Le a Euclidean disance lis ( d, d2, K, d m), is he elemens lis from he smalles o he larges. Then define an index lis ( J, J2, K, J m), J j is he number of airs of sies in he design searaed by disance d j. Thus is a maximin design if among available designs, i maximizes d while J is minimized. The scalar crierion can be exressed as m = Jd j j j = (3), where is a osiive ineger, J j and d j are secified from. The design ha minimizes is he oimal LHD in he class. In his sudy, he adaive form of [] which is simler han equaion (3) is considered. / n n = i= j= i+ d ij (4) Figure 3 A flowchar of SA 2.3 The crierion Morris and Michell [6] roosed a modificaion class of maximin disance crierion in order o search for he oimal design. For a given design, he Euclidean inersie disance beween he design oins can be calculaed from (2) d / 2 dx ( i, xj) = ( xik x jk) k= By using equaion (2) o calculae all inersie disance for every airs of design oins in he design marix will resul in a symmeric marix of inersie disance as follows. The behavior of SA rocess wih resec o he value of for each ieraion is dislayed as Figure 4. Figure 4 The aern of SA rocess wih resec o 20

5 การประช มว ชาการด านการว จ ยด าเน นงานแห งชาต ประจ าป 255 ว นท กรกฎาคม พ.ศ Modified simulaed annealing (MSA) This aer will modify he calculaion of crieria by using he mehod ha avoids re-calculaing value. As menioned before, SA uses he exchange rocedure beween wo airs of oins wihin he randomly seleced column. Hence, afer an exchange beween rows i and i 2 wihin column k ( x ik x ) i2k, only elemens in rows i and i 2, and columns i and i 2 are changed in he disance marix D []. For any j n and j i, i 2 le: si (, i2, k, j) = xi 2k x jk xik x jk (5) hen and d = = + ij dji d ( ij s i, i2, k, j ) (6) d = = + i2j dji d ( 2 i2j s i, i2, k, j ) (7) Thus new is comued by + ( dij) ( dij) + j n, j i, i2 = ( d i2j) ( di2j) j n, j i, i2 Figure 5 A flowchar of MSA / (8) The flowchar dislays he rocess of MSA is resened in Figure 5. As shown in equaion (5) o (8), only some rows and columns are udaed o calculae crierion in MSA. Hence he comlexiies or BigO of MSA is much smaller han SA as resened in Table 2. Table 2 The comlexiies o calculae crierion in SA and MSA 2 2 BigO(SA) Odn ( ) + On ( log 2( )) BigO(MSA) On () + On (log()) Daa Qualiy The erformance of SA and MSA is invesigaed via oimaliy crieria. The comarison is made on he dimension of he roblems as secified in Table 3. For each d, number of runs (n) is fixed a a maximum number of unknown arameers ha are required in he second order olynomial model d ( n= 2d ). 2 Table 3 Dimension of he roblem d n For each dimension of he roblem in Table 3, SA and MSA are simulaed 0 imes for a secified olerance and he algorihmic arameers wih resec o oimaliy crierion [6]. The wo algorihms are imlemened via codes wrien in MATLAB o search for an oimal LHD wih resec o crieria. SA requires inu arameer seings for 0, I max, C and l. We use he heurisics resened in [6] and our own exerience of using SA o se he iniial values resened below. is se o average Euclidean inersie disance of he 0 iniial LHD. n n d 2 d 2 0 = ( il jl ) / i= j= i+ l= 2 I is he maximum number of exchanges before he emeraure is reduced. The number should be large enough o allow SA o move away from local minimum. Imax is se o 000 for d 6, and I max is se o 500 for d 7. max 2

6 การประช มว ชาการด านการว จ ยด าเน นงานแห งชาต ประจ าป 255 ว นท กรกฎาคม พ.ศ. 255 C is he rae a which he emeraure is cooled down. C is se a 0.95, o lower he emeraure slowly as he search rogresses. The olerance level l is se a as our emirical sudies indicaed ha seing smaller olerance levels would no resul imrovemen in oimal value bu require longer ime o erminae. 3. Resul The values of crieria a he erminaion se of SA and MSA from each dimension of roblems saed in Table 3 are resened in Table 4. The descriive saisics on he values obained from each search echnique (SA and MSA) are dislayed in columns 4-7. The resuls in columns 4-7 indicae ha MSA erforms similarly o SA in erms of minimizaion of crieria. Furher, he SD values aeared in column 7 dislays a slighly larger amoun of variaion over 0 relicaions in SA han MSA. This indicaes he consisency in he search rocess for MSA when differen saring oins are considered. The resuls in columns 8-9 dislay ime elased for each search algorihm and he ercenage of ime reducion in MSA over SA. I can be concluded ha from hese ercenages MSA converges much more quickly han SA. The maximum ercenage of imrovemen over SA is aroximaely 97%. I should be noed ha larger amoun of imrovemen for MSA over SA occurs when he dimensions of roblems are increased. Table 4 Resuls on and ime elased for SA and MSA 4. Conclusion According o he resuls resened in Table 4, i can be concluded ha he modified SA (MSA) erforms much beer han an original SA (SA) as he ime required for MSA o reach he same level of oimal design is significanly smaller han ha of SA. Hence MSA is recommended for he consrucion of he oimal LHD for CSE. Furhermore, oher classes of design can be develoed and collaboraed wih SA o search for he bes design in he class. Oher yes of search algorihm like Paricle swarm oimizaion (PSO) or An colony can be furher develoed in consrucing an oimal LHD for CSE. References R. Jin, W. Chen, and A. Sudjiano. An efficien algorihm for consrucing oimal design of comuer exerimens., Journal of Saisical Planning and Inference, Vol. 34, 2005, M. Johnson, L. Moore, and D. Ylvisaker. Minimax and maximin disance designs., Journal of Saisical Planning and Inference, Vol. 26, 995, J. Koehler, and A.B. Owen. Comuer exerimen., Handbook of Saisics, Vol. 3, 996, M. Liefvendahl, and R. Socki. Sudy on algorihms for oimizaion of Lain hyercubes., Journal of Saisical Planning and Inference, Vol. 36, 2006, M.D. McKay, R.J. Beckman, and W.J. Conover. A comarison of hree mehods for selecing values of inu variables in he analysis of ouu from a comuer code., Technomerics, Vol. 2, 979, M.D. Morris, and T.J. Michell. Exloraory design for comuer exerimens., Journal of Saisical Planning and Inference, Vol. 43, 995, J. Rungraanaubol and A. Na-udom. Comarison of Evoluionary Search Algorihms in Comuer Simulaed Exerimens. Naional Comuer Science and Engineer Conference, Vol., 2007,

7 การประช มว ชาการด านการว จ ยด าเน นงานแห งชาต ประจ าป 255 ว นท กรกฎาคม พ.ศ J. Sacks, S.B. Schiller, and W.J. Designs for Comuer Exerimens. Technomerics, Vol. 3, No., Feb 989, T.W. Simson, D.K.J. Lin, and W. Chen. Samling sraegies for comuer exerimens: Design and analysis. Inernaional Journal of Reliabiliy and Alicaions, Vol. 2, No.3, 200, K.Q. Ye, W Li,. and A Sudjiano. Algorihmic consrucion of oimal symmeric Lain hyercube designs. Journal of Saisical Planning and Inference, Vol 90, 2000,