A MOVING MESH APPROACH FOR SIMULATION BUDGET ALLOCATION ON CONTINUOUS DOMAINS

Transcription

1 Proceedngs of the Wnter Smulaton Conference M E Kuhl, N M Steger, F B Armstrong, and J A Jones, eds A MOVING MESH APPROACH FOR SIMULATION BUDGET ALLOCATION ON CONTINUOUS DOMAINS Mark W Brantley Chun-Hung Chen Dept of Systems Engneerng and Operatons Research 44 Unversty Drve, MS 4A6 George Mason Unversty Farfax, VA 3, USA ABSTRACT We ntroduce a movng mesh algorthm for smulaton optmzaton across a contnuous doman Durng each teraton, the mesh movement s determned by allocatng smulaton runs to parttons of the doman that are crtcal n the process of dentfyng good solutons The partton boundares are then adjusted to equally dstrbute the allocaton runs between each partton To test the movng mesh algorthm, we mplemented t usng the OCBA method to allocate smulaton runs to each partton But, the smplcty of the procedure should provde flexblty for t to be used wth other smulaton optmzaton technques Results are presented for several numercal experments and suggest that the technque has potental gven further development INTRODUCTION Smulaton optmzaton s a method to fnd a desgn consstng of combnaton of nput decson varable values of a smulaton system that optmzes a partcular output performance measure of the system When presented wth a stochastc, contnuous doman wth an nfnte number of values for each nput decson varable and a fnte smulaton budget, we must effcently allocate our smulaton runs n order to nvestgate the combnatons of nput decson varable values (Law and Kelton ) Most smulaton optmzaton methods use ponts n the doman to represent desgns (Swsher et al ) These methods typcally requre ndfference zones to not only represent solutons wthn a certan dstance of the best soluton but to also ensure that smulaton runs are not wasted by comparng two desgns that are essentally the same Sh and Olafsson present a randomzed method for global optmzaton called Nested Partton (NP) that seeks to effcently concentrate the computatonal effort n parts of the doman that may be most lkely to contan the global optmum (Sh and Olafsson, ) NP aggregates the nformaton from desgns to allocate addtonal runs and then parttons the doman to search the most promsng regon Ths paper nvestgates a dfferent approach of usng parttons of the doman that s motvated by mesh movng technques for fnte dfference and fnte element schemes Smlar to nested partton, these numercal technques can use what s typcally called local mesh refnement to dvde the mesh n certan regons of the doman to reduce the error or to adapt to nonunformty (Arney and Flaherty 986) An alternate approach for adaptng the mesh s to keep a fxed number of parttons on the doman but to move the mesh to have a fne grd where needed and a course grd elsewhere (Adjerd and Flaherty 986) The method we ntroduce mrrors ths alternate approach Instead of parttonng the doman by refnng the mesh lke nested parttonng, we wll move the mesh to concentrate the search n the most promsng regon As we move the mesh, we reduce the sze of the parttons n the most promsng regons and ncrease the sze of the parttons elsewhere By changng our problem from fndng the most promsng pont n our doman to searchng for the most promsng regon, we have transformed our contnuous stochastc optmzaton problem to a dscrete stochastc optmzaton problem We now seek to dentfy the best partton b out of k competng parttons Whle dscrete stochastc optmzaton s stll an actve feld of research, recent advances provde technques that greatly reduce the number of smulaton runs requred to obtan a good or the best soluton Gven the smplcty of the movng mesh algorthm, we expect that t can be coupled wth many of the dscrete stochastc optmzaton technques However, we focused our efforts on usng a hghly effcent technque developed by Chen et al () called the Optmal Computng Budget Allocaton (OCBA) method Ther numercal comparsons have shown that OCBA can acheve a speedup factor of approxmately 4 for a small number of competng desgns and can be as much as tmes faster than tradtonal approaches for a much larger number of desgns 699

2 THE OCBA ALGORITHM Snce the mplementaton of our movng mesh technque n ths paper utlzes OCBA to award runs to parttons durng each teraton, we present a summary of the technque OCBA allocates smulaton runs by consderng the followng optmzaton problem: max N, K, N k P{ CS} s t N + N + L+ N k = T () Under a Bayesan model, OCBA approxmates the probablty of correctly selectng the best desgn, P{CS}, usng the Bonferron nequalty and offers an asymptotc soluton to ths approxmaton In partcular, OCBA allocates smulaton runs accordng to: N σ /, δ b =, N / j σ j δ b, j, j {,, L, k}, and j b, () N Chen et al () denote by b k =, b N = σ b (3) σ N : the number of smulaton runs for desgn, X j : the j-th ndependent and dentcally dstrbuted sample of the performance measure from desgn, J : the sample average of the smulaton output for N runs n desgn, J = Xj, N j= σ : the varance of the smulaton output for runs n desgn, approxmated by the sample varance of the smulaton output, N e, σ S = ( Xj X), N j= b: the desgn havng the smallest sample mean performance measure, e, Jb mn J, δ J J b, b For our dscusson, we wll adopt ths same conventon 3 THE MOVING MESH ALGORITHM 3 Movng Mesh Algorthm Steps Lmtng our dscusson to a one-dmensonal doman wthout loss of generalty, we have k contnuous parttons (ntervals) on a doman of length L and we denote Φ : the -th partton where by conventon we order the parttons such that X, j X( + ), m j, m, X, j : the j-th ndependent and dentcally dstrbuted sample of the performance measure from the regon of the doman currently assgned to partton, Ω, j : the j-th boundary for partton For the onedmensonal case each Φ wll have two boundares Ω, and Ω, wth coordnates x, and x, constructed such that x x k, = ( + ), < Although the mplementaton of the movng mesh algorthm s dependent upon the allocaton method that we use, the basc algorthm s very smple: Determne k, the number of parttons, and then construct the mesh unformly across the doman such that x, x, = x j, x j,, j, and the ntervals span the entre doman such that x k, x, = L Randomly generate n ntal runs on each partton For ths paper, we used a unform dstrbuton to select the locaton of the run on the partton The number of ntal runs for each Φ wll be dependent upon the allocaton method that we use 3 Allocate more runs to each Φ In order to do ths, we aggregate the nformaton for all of the runs n each Φ to calculate the sample statstcs requred by the allocaton method that we are usng When coupled wth OCBA, we estmate σ and calculate δ b, for each partton as presented n Secton We then allocate addtonal runs, Δ N, to each Φ accordng to Equatons () and (3) 4 Keepng k, the number of parttons, fxed, move Ω, j so that each Φ has the same number of runs or N = N j, j By conventon, we establsh the boundary by merely equally dvdng the dstance between the last pont n one nterval and the frst pont n the next nterval 7

3 Repeat steps 3 and 4 untl we exhaust the computng budget 6 After exhaustng the computng budget, determne a pont to represent the partton havng the smallest sample mean performance measure For ths paper, we used the mdpont of the partton but we could use other conventons such as selectng the pont n the best partton that has the best performance measure A one-dmensonal example of the frst fve steps usng OCBA can be seen n Fgures - 3 below The underlyng functon used s f ( x) = ( x ) + 4U (,) where x (,) and the optmal soluton s located at x = f(x) = (-x)^ + 4*U(,) As shown n Fgure, the doman s dvded nto equal ntervals We set n = for each Φ and randomly dstrbute (unform dstrbuton) the ntal runs across x x 3 each separate nterval ( ) Iteraton Desgn <x< Iteraton Desgn <x<4 Iteraton Desgn 3 4<x<6,,, Iteraton Desgn 4 6<x<8 Iteraton Desgn 8<x< Fgure : Movng Mesh Example, Steps and 3 We calculated the mean and standard devaton of the runs n each nterval and used OCBA to allocate a total of more runs across the entre doman In ths case, ΔN = 3, ΔN =, ΔN 3 = 7, ΔN 4 =, and ΔN = 6 These runs are shown n Fgure wth the new runs portrayed by trangles f(x) = (-x)^ + 4*U(,) f(x) = (-x)^ + 4*U(,) 3 Iteraton Desgn <x< Iteraton Desgn <x<4 Iteraton Desgn 3 4<x< Iteraton Desgn 4 6<x<8 Fgure : Movng Mesh Example, Step 3 Iteraton Desgn 8<x< 4 Keepng the number of ntervals fxed, we then adjusted Ω Snce we have allocated a total of 3, j runs among the fve desgns, we move Ω, j so that N = 3 Usng ths construct, we now have the new ntervals shown n Fgure 3 Iteraton Desgn <x<4 Iteraton Desgn 4<x<46 Iteraton Desgn 3 46<x<64 Iteraton Desgn 4 64<x< Fgure 3: Movng Mesh Example, Step 4 Iteraton Desgn 78<x< From here, we would repeat Step 3 whch calculates the new mean and standard devaton for each nterval and then uses OCBA to allocate new runs to each nterval untl we exhaust our computng budget 3 Algorthm Convergence OCBA, and other methods, concentrate the smulaton effort on desgns that are promsng and do not allocate to desgns that are not promsng Mathematcally, we do not change the OCBA method descrbed by Chen et al () Instead of desgns consstng of ponts, we merely compete desgns consstng of a group of runs dstrbuted across a partton aganst each other n order to maxmze the probablty of dentfyng the best partton We obtan our con- 7

4 vergence by dynamcally redefnng the desgns The mesh wll get smaller for the parttons that receve more runs allocated n an teraton Intutvely, we expect these boundares to converge on the optmal desgn partton (or pont) In fact, as the mesh sze for a partton gets smaller, the mean and standard devaton of the runs of the partton begn to resemble those from a pont snce these measures are less affected by the dstrbuton of the runs across the partton and more heavly nfluenced by the varance of the underlyng functon However, as Chen et al () menton, when usng OCBA, the number of runs allocated to a partcular desgn ncreases as the mean the desgn decreases or the standard devaton of the desgn ncreases It s ths property that enables our movng mesh method to mantan a global perspectve By teratvely wdenng the nterval boundares of a less desrable partton, we expect that the mean of ths desgn wll decrease and the standard devaton of the desgn wll ncrease untl t becomes compettve for addtonal smulaton runs Fgure 4 shows the convergence map for an experment usng the movng mesh method wth parttons coupled wth OCBA for the example functon n Secton 3 of ths paper Boundary Locaton (x-value) Total Smulaton Runs Allocated Fgure 4: Convergence Example ( Parttons) 4 NUMERICAL TESTING FRAMEWORK In ths secton, we descrbe how we tested our new movng mesh approach and compared t wth a seres of numercal experments t aganst two allocaton procedure: Equal Allocaton-Unform Mesh (EA-UM) and OCBA-Unform Mesh (OCBA-UM) The next secton wll provde the results of these experments 4 Equal Allocaton Unform Mesh (EA-UM) Ths s a brute force method for allocatng the number of runs, N, to each desgn Gven a smulaton budget T and k desgns, we space the k desgns unformly across the doman and allocate the runs equally such that N = T / k for each The effcency of ths method s dependent upon two nversely proportonal parameters: the number of runs allocated to each desgn and the sze of the mesh (number of desgns) A small mesh enables the method to potentally have a desgn close to the optmal soluton but a lmted computng budget for each desgn may prevent the method from dfferentatng the best possble desgn from others A large mesh provdes enough runs to dfferentate the desgns under consderaton but the best possble soluton may be relatvely removed from the optmal soluton Through expermentaton, we found that ths method performed best usng about desgns durng our tests 4 OCBA Allocaton Unform Mesh (OCBA-UM) Gven a smulaton budget T and k desgns, we space the k desgns unformly across the doman for ths method However, nstead of equally allocatng the runs between the desgns we use OCBA Lke EA-UM method, the effcency of ths method s dependent upon two nversely proportonal parameters: the number of runs ntally allocated to each desgn, n, and the sze of the mesh (number of desgns) Based upon the dscusson by Chen et al () and a lttle expermentaton, we used n = for all of our testng A small mesh enables the method to potentally have a desgn close to the optmal soluton but leaves a lmted computng budget for OCBA after provdng each desgn wth ts ntal allocaton However, lke EA-UM, a large mesh provdes enough runs to dfferentate the desgns under consderaton but the best possble soluton may be relatvely removed from the optmal soluton Through expermentaton, we found that ths method enabled us to use a fner mesh than EA-UM and performed best usng about 4 - desgns durng our tests Ths method also requres us to specfy an addtonal parameter, Δ, for the total number of runs allocated durng each teraton We used Δ = for all of our testng 43 Movng Mesh OCBA (MMO) As prevously dscussed, the convergence of ths method s dependent upon the number of parttons we used In addton, t s also dependent upon parameters for the allocaton method t uses Snce we used OCBA, we had to decde values for n and Δ Through expermentaton, we found that the method performed best durng our tests usng 8 parttons, n =, and Δ = k, where k s the number of parttons Allocatng too few runs per teraton does not provde enough new ponts to cause sgnfcant movement n the mesh boundares Allocatng too many obvously wastes runs that could be concentrated n the most promsng parttons 7

5 44 Test Procedures Gven the dfferng parameters for each of these methods, we constructed our experments to provde a far comparson The EA-UM and OCBA-UM have fxed mesh szes whle the MMO method obvously has a dynamc mesh In order to farly compare these methods, each of our experments ncorporates a randomly selected optmal soluton and our comparson metrc s the dstance from our best soluton to the randomly generated optmal soluton In addton, the methods have varyng fxed costs assocated wth them To mtgate ths dfference, we calculate the error for each method durng each teraton of the method untl the total smulaton budget s exhausted For each experment, we lmted the smulaton budgets to, runs snce t was a suffcent number to dfferentate between the dfferent methods We repeat ths whole procedure, tmes and then calculate the average error obtaned for each method durng these, ndependent applcatons of each method Ths average error obtaned from each dfferent procedure serves as our measurement of ts effectveness NUMERICAL TESTING RESULTS To ntally test our method, we conducted the followng experments on a one-dmensonal doman Each of the numercal experments was constructed to see f the movng mesh method had convergence problems relatve to the two other methods we tested The frst experment s a baselne experment where the underlyng functon s a quadratc The next experment has an underlyng functon wth two optmal solutons and that s relatvely flat when compared to ts varance The thrd experment has two quadratc functons constructed on each half of the doman wth one of the crtcal ponts havng a lower functonal value than the other Experment : Convex Functon Ths experment s our baselne and uses the followng underlyng functon: f ( x) = ( x ξ ) + 4U (,) where ξ ~ U (,) and where x (,) The optmal soluton s ξ so the error for each teraton s measured as x b ξ Note that for the portons of doman n the nterval ( ξ, ξ + ), the varance of the 4U (,) term clearly domnates a change n the underlyng functon f ( x) = ( x ξ ) Fgure contans the smulaton results for the three methods We can see that MMO obtans rapd convergence n the frst few teratons and then slowly mproves after that Compared to the other two methods, MMO performs the best for when the smulaton budget s less than 4 runs, performs about the same as OCBA-UM when the budget s between 4 and, runs, and performs worse than OCBA-UM after, runs Norm for, Experments Total Smulaton Budget (Runs) EA-UM OCBA-UM MMO Fgure : Results for Experment (Convex Functon) Experment : Two Optmal Solutons Ths experment s constructed to see f the two optmal solutons cause the MMO method to dverge and uses the followng underlyng functon: 3ξ f ( x) = cos( x ) + 4U (,) where ξ ~ U (,) and where x (,) There are two optmal solutons on the nterval (,) at x a = π 3ξ / and xb = 3π 3ξ / so the error for each teraton s measured as the mnmum of x b x and x b x For the underlyng functon f ( x) = cos( x 3ξ /), we obtan values n the nterval [-,] so the varance of the 4U (,) term agan domnates a change n the underlyng functon The results of ths experment are very smlar to those from the frst experment and are shown n Fgure 6 MMO makes most of ts convergence n the frst 7 teratons of the method (7 total runs allocated) At ths pont t begns to converge slowly at best and OCBA-UM begns to provde better results 73

6 Norm for, Experments Total Smulaton Budget (Runs) EA-UM OCBA-UM MMO Fgure 6: Results for Experment (Two Optmal Solutons) 3 Experment 3: Competng Near Optmal Alternate Soluton Ths experment s an extenson of Experment Instead of determnng f MMO can fnd one of two optmal solutons, we test to see f t can dfferentate between an optmal soluton and another near optmal soluton We defne the underlyng dfference between the optmal soluton and the near optmal soluton as the constant λ For ξ ~ U (,4 ) and x (,), ths experment uses the followng underlyng functon: and f ( x) = ( x ξ ) + 4U (,) when x f ( x) = ( x + ξ ) + 4U (,) + λ when x The optmal soluton s ξ so the error for each teraton s measured as x b ξ Just as n Experment, for the portons of doman n the nterval ( ξ, ξ + ), the varance of the 4U (,) term clearly domnates a change n the underlyng functon f ( x) = ( x ξ ) The results for ths experment wth λ = are shown n Fgure 7 and are not encouragng Whle MMO agan converges rapdly, OCBA-UM performs better than ths method after only 4 runs and EA-UM performs better after about 3 runs Norm for, Experments Total Smulaton Budget EA-UM OCBA-UM MMO Fgure 7: Results for Experment 3 (Competng Near Optmal Alternate Soluton) However, these results are very senstve to the value we use for the constant λ If we use λ = nstead of λ = as shown above, MMO s better than OCBA-UM untl we exceed 8 runs and remans better than EA-UM throughout the, runs If we use λ =, MMO s clearly superor to OCBA-UM untl we allocate about 8 runs When compared to EA-UM for when λ =, t only takes MMO 3 runs to obtan better results than those obtaned by EA-UM n, runs In order to see f modfcatons to MMO mght mprove ts performance, we repeated Experment 3 wth λ = However, for the MMO method we trmmed smulaton runs from the upper porton of the doman for each teraton after we had awarded runs The results for trmmng 3 ponts each teraton (MMO-T3) and 7 ponts each teraton (MMO-T7) are compared aganst our orgnal results for ths experment n Fgure 8 Ths naïve trmmng approach clearly mproves the performance However, t ntroduces a cyclng pattern that s clearly evdent n the MMO-T7 results Ths pattern s ntroduced because we regon we are trmmng from reaches the partton that covers the near optmal alternate soluton When we trm from ths partton, we have fewer runs allocated to the partton and are more prone to select t as the best partton n error However, the OCBA method coupled wth the movng mesh method recognzes that we need to allocate more runs n ths area Ths mproves our soluton untl we trmmed from ths partton agan 74

7 Norm for, Experments Total Smulaton Budget EA-UM OCBA-UM MMO MMO- T3 MMO- T7 Fgure 8: Experment 3 wth MMO Trmmed Methods 6 EXTENDING THE METHOD TO TWO DIMENSIONAL PROBLEMS 6 Movng Mesh Algorthm Modfcatons The movng mesh method does not change dramatcally when movng from a one dmensonal problem to a two dmensonal problem The man dfference s that, wth two dmensonal problems, there are numerous methods to construct the mesh However, the purpose of ths paper s to ntroduce the movng mesh method Therefore, we used a smple rectangular mesh and a basc accountng scheme to move the mesh between each teraton We stll have k contnuous parttons on a doman of length L and wdth W and we denote Φ : the -th partton Ω, j : the,j-th boundary For the two dmensonal case, each Φ wll have four boundares wth coordnates ( x (,), y(,) ), ( x (,), y(,) ), x, y ), and x, y ) ( (,) (,) ( (,) (,) The new algorthm now becomes: Determne k, the number of parttons, and then construct the mesh unformly across the doman such that the parttons span the entre doman such that x k, x, = L and y k, y, = W Randomly generate n ntal runs on each partton Φ As n the one dmensonal case, we used a unform dstrbuton to select the locaton of the run on the partton 3 Allocate more runs to each Φ In order to do ths, we aggregate the nformaton for all of the runs n each Φ to calculate the sample statstcs requred by the allocaton method that we are usng When coupled wth OCBA, we estmate σ and calculate δ b, for each partton as presented n Secton We then allocate addtonal runs, Δ N, to each Φ accordng to Equatons () and (3) 4 Keepng k, the number of parttons fxed, move Ω, j so that each Φ has the same number of runs or N = N j, j For ths paper, we kept our mesh constructon method smple We frst equally dvded the runs n the x drecton and establshed boundares by equally dvdng the dstance between the last pont n one sub-secton and the frst pont n the next sub-secton We then took each x drecton sub-secton and equally dvded the runs n the y drecton and agan establshng the boundares equdstant from the last and frst ponts of the resultng parttons Repeat steps 3 and 4 untl we exhaust the computng budget 6 After exhaustng the computng budget, determne a pont to represent the partton havng the smallest sample mean performance measure An example of the frst fve steps usng OCBA can be seen n Fgures 9 and below The underlyng functon used s f ( x, y) = ( x 66) + ( y 43) + 4U (,) where x, y (,) and the optmal soluton s located at ( x, y) = (66,43) y value As shown n Fgure 9, the doman s dvded nto 6 equal parttons We set n = for each Φ and randomly dstrbute (unform dstrbuton) the ntal runs across each partton Fgure 9: D Example, Steps and 3 We calculated the mean and standard devaton of the runs n each nterval and used OCBA to allocate a total of 6 more runs across the entre do- 6 7

8 y value man In ths case, the 6 parttons receved ΔN = (,,,,4,8,3,,6,84,8,3,4,3, 7,3) new runs respectvely These runs are shown n Fgure wth the new runs portrayed by trangles 4 Keepng the numbers of parttons fxed, we then adjusted Ω, j Snce we have allocated a total of 3 runs among the 6 desgns, we adjust Ω, j so that N = Usng ths construct, we now have the new parttons shown n Fgure 7 4 Optmal 6 Soluton Fgure : D Example, Steps 3 and 4 We would then repeat Step 3 whch calculates the new mean and standard devaton for each nterval and then uses OCBA to allocate new runs to each nterval 6 Experment 4: Two Dmenson Convex Functon Ths experment s a two dmensonal verson of Experment and uses the followng underlyng functon: f ( x, y) = ( x ξ ) + ( y ξ) + 4U (,) where ξ, ξ ~ U (,) and where x, y (,) ( ξ The optmal soluton s ξ, ) so the error for each teraton s measured as ( x b ξ ) + ( yb ξ) We agan conduct, experments and use EA-UM and OCBA-UM for comparson purposes For EA-UM, we constructed a x, x, and x grds provdng,, and 4 total desgns respectvely By extendng our smulaton budget for each experment to, runs, we also used x, x, and x grds for the OBCA-UM method For MMO, we used a 4x4 construct for 6 total parttons Fgure contans the smulaton results for EA-UM x, OCBA-UM x, OCBA-UM x, OCBA x, and MMO 4x4 The EA-UM results were very smlar for the three dfferent grds we used and converged very slowly After applyng the ntal runs, the OCBA-UM methods converge rapdly to the best possble soluton but are ultmately lmted n performance by the wdth of the unform mesh used However, unformly refnng the mesh to obtan a better soluton comes at a large cost n terms of ntal runs for each desgn We can see that MMO obtans rapd convergence n the frst few teratons and then contnues to slowly mprove after that Compared to OCBA- UM, MMO obtans the same results n vastly fewer runs Norm for, Experments Total Smulaton Budget (Runs) OCBA-UM x OCBA-UM x OCBA-UM x EA-UM x MMO 4x4 Fgure : Results for Experment 4 (D Functon) 7 CONCLUDING REMARKS In ths paper we ntroduced a movng mesh algorthm for smulaton optmzaton across a contnuous doman Durng each teraton, the mesh movement s determned by allocatng smulaton runs to parttons of the doman that are crtcal n the process of dentfyng good solutons The partton boundares are then adjusted to equally dstrbute the allocaton runs between each partton whch reduces the sze of promsng parttons and ncreases the sze of less desrable parttons Comparsons wth smulaton optmzaton methods usng ponts n the doman on a unform mesh as desgns show that our approach s promsng However, as we refne our approach, we may have to develop trmmng heurstcs to ensure the method contnues to converge and mprove the effcency of our teratve mesh constructon as we expand to hgher dmensons ACKNOWLEDGMENTS Ths work has been supported n part by NSF under Grant IIS-374, by NASA Ames Research Center under Grants NAG--6 and NAG--643, by NASA Langley Research Center and NIA under task order NNL4AA7T, by FAA under Grant -G-6, and by George Mason Unversty Research Foundaton 76

9 REFERENCES Adjerd, S and JE Flaherty 986 A movng-mesh fnte element method wth local refnement for parabolc partal dfferental equatons, Computer Methods Applcatons for Mechancs and Engneerng, : 3-6 Arney, DC and JE Flaherty 986 A two-dmensonal mesh movng technque for tme-dependent partal dfferental equatons, Journal of Computatonal Physcs, 67: 4-44 Chen, CH, JLn, EYücesan, and SEChck Smulaton budget allocaton for further enhancng the effcency of ordnal optmzaton, Journal of Dscrete Event Dynamc Systems: Theory and Applcatons, : -7 Law, AM and WD Kelton Smulaton Modelng and Analyss, McGraw-Hll, Inc Sh, L and S Olafsson Nested parttons method for global optmzaton, Operatons Research, 48 (3): Swsher, J, S Jacobson, P Hyden, and L Schruben A survey of smulaton optmzaton technques and procedures, Proceedngs of the Wnter Smulaton Conference, ed JA Jones, RR Barton, K Kang, and PA Fshwck, 9-6 Pscataway, NJ: Insttute of Electrcal and Electroncs Engneers AUTHOR BIOGRAPHIES MARK W BRANTLEY s n the Department of Systems Engneerng and Operatons Research at George Mason Unversty He receved hs BS degree n Mathematcal Scences at the Unted States Mltary Academy and receved MS degrees n Appled Mathematcs and Operatons Research from Rensselaer Polytechnc Insttute Hs research nterests nclude smulaton and optmzaton Hs e-mal address s mbrantl@gmuedu CHUN-HUNG CHEN s an Assocate Professor of Systems Engneerng at George Mason Unversty He served as the Co-Edtor of the Proceedngs of the Wnter Smulaton Conference and the Methodology Analyss track coordnator for the 3 and 4 Wnter Smulaton Conference He receved hs PhD from Harvard Unversty n 994 Hs research nterests are manly n development of very effcent methodology for smulaton and optmzaton, and ts applcaton to engneerng desgn and ar traffc management He s a member of INFORMS and a senor member of IEEE Hs emal address s cchen9@gmuedu 77