OPTIMIZATION BY SIMULATION METAMODELLING METHODS. Russell C.H. Cheng Christine S.M. Currie

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1 Proceedngs of the 4 nter Sulaton Conference R G Ingalls, M D Rossett, J S Sth, and B A Peters, eds OPIMIZAION BY SIMULAION MEAMODELLING MEHODS Russell CH Cheng Chrstne SM Curre School of Matheatcs Unversty of Southapton Southapton, SO7 BJ, UK ABSRAC e consder how sulaton etaodels can be used to optze the perforance of a syste that depends on a nuber of factors e focus on the stuaton where the nuber of sulaton runs that can be ade s lted, and where a large nuber of factors ust be ncluded n the etaodel Bayesan ethods are partcularly useful n ths stuaton and can handle probles for whch classcal stochastc optzaton can fal e descrbe the basc Bayesan ethodology, and then an extenson to ths that fts a quadratc response surface whch, for functon nzaton, s guaranteed to be postve defnte An exaple s presented to llustrate the ethods proposed n ths paper INRODUCION In sulaton output analyss, t s often ore convenent to use a sple statstcal (usually regresson) odel of the sulaton output rather than to work drectly wth the sulaton odel hese splfed odels are tered etaodels (see Barton 998) Let the output of one run of our sulaton odel be y e assue that ths depends n a contnuous way on p factors x = (x, x,, x p ), and can therefore be regarded as a functon of these factors e treat these factors as decson varables that are set by the controller of the syste under nvestgaton e thus assue the followng sulaton etaodel y = q(x, θ) + ε, () where q(x, θ) s a regresson functon descrbng the underlyng dependence of the response y on the factor varables x, and ε s a rando nose varable, to allow for the nherent rando varablty n the observed output value he vector θ = (θ, θ,, θ ) s a set of coeffcents on whch the regresson functon depends Soe, possbly all, of these coeffcents wll be unknown and have to be estated by fttng the regresson functon to observed sulaton output Geoetrcally the regresson functon can be thought of as defnng a surface n p-densonal space that descrbes our estate of the dependence of the sulaton response on the p factors (x, x,, x p ) hs nterpretaton gves rse to the nae response surface ethodology Good basc references are Barton (998) and Hood and elch (993) Response surface ethodology s useful f we wsh to fnd the factor settng x* for whch the syste response s optal For exaple, f the etaodel descrbes the perforance of the syste, so that q(x, θ) s sply a perforance ndex, then x* wll be the factor settng that optzes q(x, θ) For a dscusson of sulaton optzaton and the use of response surface ethodology for ths, see Azadvar (999), Kelton (999), Glover, Kelly and Laguna (999), Fu et al (), Gonda et al (), Law and McCoas () and Swsher et al () One serous and coon dffculty reans If the nuber of factors s large but the sulaton odel s coplex so that only few runs can be ade, then t ay not be possble to estate all the coeffcents of the etaodel usng for exaple classcal ethods of axu lelhood Here a Bayesan approach has a great advantage In the Bayesan approach we have a pror probablty dstrbuton π(θ) that expresses our pror belef about the lely value of the paraeters, whch gves us a vable startng pont for the sulaton runs As we ake our sulaton runs we can progressvely odfy and ake ore precse our belef about the dstrbuton of θ e can stop the procedure at any pont, wth the knowledge about the behavor of the syste output provng wth the nuber of sulaton runs ade One further potental proble that can occur wth ths ethodology s that the response surface ftted s not postve (negatve) defnte when searchng for a nu (axu) For stuatons where we are farly certan that there s an optu (whch wll be assued to be a nu n what follows) we present here a sple ethodology for a quadratc q(x, θ), based on a restrcton of the paraeter values, such that the response surface s always postve defnte

2 hus n suary we exane the Bayesan approach for fttng q(x, θ) and then obtanng the factor settng x* for whch q(x, θ) s optzed he basc ethodology was gven by Cheng (4) For copleteness we outlne the ethodology agan here for the case where we ft a quadratc q(x, θ) e then go on to dscuss the ssue of ensurng how to ensure that the ftted regresson s postve defnte, when t s known at the outset that the proble s well-posed so that q(x, θ) can be expected to satsfy ths condton BAYES MEHOD thout loss of generalty, we shall assue that the objectve s to estate the value of x whch nzes q(x, θ) e further assue that we can only perfor a lted nuber of sulated trals but that there s soe pror expert understandng of how the perforance ndex q ght vary as a functon of x e use the specfc exaple frst gven by Cheng (4) to show how the approach works hs allows for a clearer explanaton of the ethods For further dscusson of Bayesan ethodology n sulaton see Cheng (999) and Chck () Our exaple s based on a real engneerng applcaton n whch the a of the odelng was to optze the setup of a pece of coplex equpent so that the te that t took to perfor a gven task was nzed he te taken to perfor the task was therefore taken to be the perforance easure here were 6 factor varable settngs to be set: A, B, C, D, E and F, and we denote ther values by x = (x, x,, x 6 ) he sulaton odel was an accurate odel of the real syste and took soe te to run For our analyss only 7 sulaton runs were ade, each at a dfferent value of the factor settngs he output fro the sulaton runs can be descrbed by y = q(x, θ) + ε, =,,, 7 () where ε s ntally taken to have ean zero wth a constant varance As we wsh to carry out an unrestrcted optzaton, we only requre q to be a good approxaton to the unknown true syste perforance n the regon of ts nu e therefore use a quadratc expresson for q and put q(x, θ) = θ + θ x + θ 3 x + + θ 7 x 6 + θ 8 x + θ 9 x x + + θ 8 x 6 he Bayesan approach treats θ = (θ, θ,, θ 8 ) as beng a rando varable, wth a pror dstrbuton e set the E[θ j ] equal to the best estates of the factor values pror to the sulaton experents, and the standard devaton StdDev[θ j ] s set to a value ndcatve of the confdence that we have n ths estate (3) Bayes' heore allows updatng of the dstrbuton of θ, as we get new data ponts y, usng the forula p(θ y) = K - p(y θ) π(θ) (4) where, assung a contnuous dstrbuton for splcty, ( y θ) π ( θ) dθ K = p (5) s a noralzng constant to ensure that p(θ y) s a proper probablty densty Here p ( θ y) s the posteror dstrbuton of the paraeters θ gven the observed sulaton values y, and p ( y θ) s the lelhood of observng y gven that the paraeters have value θ he an advantage of the Bayesan approach over the classcal ethods of axu lelhood s that t allows the ncorporaton of pror knowledge A further attracton s that t allows ncreental updatng of θ, and hence q(x, θ), as addtonal values of x are obtaned he choce of desgn ponts x, x,, x n at whch to perfor sulatons s a classcal statstcal experental desgn proble he a s to nze the nuber of desgn ponts needed to estate q In our case, we need to estate frst and second order ters of q A classcal non-central coposte desgn (see Cochran & Cox (957) for detals) would use (6-) + *6 + = 45 desgn ponts, where there are 3 ponts to estate the lnear and xed ters of q, and 3 ponts to estate the squared ters hus the upper bound on the nuber of tral runs requred s 45 he nuber of desgn ponts can be reduced f soe of the xed ters are assued to be zero e do not consder desgn aspects further here Bayesan updatng of θ, and hence q(x, θ), can be perfored wth any nuber of data ponts, akng t partcularly attractve n ths exaple, where only 7 data ponts are avalable 3 NUMERICAL EXAMPLE e ntend to nvestgate the propertes of the ethod ore fully ncludng tests on probles where the soluton s known However we lt ourselves n ths paper to the exaple ntroduced n the prevous secton A pror odel for q(x) was specfed as follows e take, as our ntal guess of the nu pont, the values x = (7586, 794, 58, 33, 6, 73) hese were obtaned fro dscusson wth engneers falar wth the syste he xed ters x x j of q(x, θ) were all taken to be zero and the reanng ters of q(x, θ) were selected so

3 that q(x, θ) ncreases by at a dstance + or fro x along any axs hs gave q(x, θ) = 893-5x -56x -6x 3-46x 4-3x 5-6x 6 + x + x +x 3 + x 4 + x 5 + x 6 hs selecton was ade to gve a startng shape for the regresson surface q(x, θ) wth a defnte, but not sharply defned, nu at the pont x ndcated by the engneers he uncertanty about the coeffcents s not so dependent on expert opnon e therefore assue that each coeffcent has a noral dstrbuton, wth standard devaton hs bulds n a far aount of uncertanty about paraeter values he 7 seven sulaton output results (y) are dsplayed n able, wth the correspondng factor settngs (x) he selecton of the desgn ponts was ade n a rather splstc way In the frst run the settngs were selected to be x, the engneers best estates of the optal settngs Subsequent factor settngs were chosen by varyng one paraeter at a te and n a drecton that was thought ght gve an proved perforance able : Results of 7 Sulaton Runs Run A B C D E F Y Only factors C and D could not be vared ndependently, and these were nstead vared sultaneously n runs and 3, n a way that was physcally possble Updatng the odel, usng ths sulaton data, was carred out by calculatng the posteror dstrbuton as gven by Bayes' forula (5) here are any ways of dong the calculaton and any packages exst for dong ths e used the nbugs package avalable on the Internet he updated values of the paraeters are gven n able he updated functon q has a nu of 743 at the pont x* = (7394, 35, 59, 39, 84, 36) here are two ponts to note about ths procedure Frstly, the updatng oved the nu fro x to x* hs shows the nfluence of the observatons n changng our vew of the nu pont Note also that the value of the updated nu s q(x*) = 743 hs s slght proveent copared wth the value at the ntal guess of q(x ) = 755 Secondly, the standard devatons of all the θ coeffcents are reduced hs s reassurng, and suggests that our (6) able : Pror and Posteror Paraeter Dstrbutons Pror Posteror Paraeter Mean Std Dev Mean Std Dev θ[] θ[] θ[3] θ[4] θ[5] θ[6] θ[7] θ[8] θ[9] 84E θ[] θ[] θ[] θ[3] θ[4] θ[5] θ[6] θ[7] θ[8] θ[9] θ[] θ[] θ[] θ[3] θ[4] θ[5] θ[6] θ[7] θ[8] knowledge about the coeffcents has been proved by the nforaton obtaned fro the sulaton data ponts he posteror estate x* depends on x hs dependence wll decrease as we get ore data However, the better our ntal guess, the faster we converge to the true optu 4 POSIIVE DEFINIENESS A weakness of the above ethodology s that there s no guarantee that the posteror dstrbuton of the regresson etaodel wll possess a nu, even f we are confdent that a nu exsts n the true odel e therefore consder a odfcaton of the above procedure where we use a odel for whch the quadratc regresson functon s guaranteed to have a nu e wrte θ as 3 separate paraeters: µ, a and he regresson functon q(x, θ) can then be wrtten as q ( x, θ) = µ + ( x a) ( x a) (7)

4 where s syetrc hs s a slghtly dfferent defnton fro that used n (3) whch used an essentally upper trangular for e now ensure that s postve defnte by defnng the eleents of the atrx to be = δ j p k δ jk, (8) = where δ = k = for p If we assue noral errors, the lelhood of the data, gven the paraeters µ, a and s gven by L ( y θ) σ n n exp [ y µ ( x a) ( x a)] σ = (9) e now have to consder the pror for a and If we consder the as beng the ean and varance of a p- ultvarate noral dstrbuton, then n ths case, convenent prors for a and - would be a noral and a shart dstrbuton: a ~ N ξ p, n ; () ~ shart p ( χ,), () he pror dstrbuton for a s then π ( a ξ, n ) / {det[( n )]} n exp ( a ξ) ( a ξ) whlst the pror for s π ( χ, ) ( χ + p + ) / {det } exp trace{ } () (3) e also need a pror for the other two paraeters µ and σ For µ we have used a noral pror so that π µ µ, τ exp [ µ µ ] (4) τ whlst for σ we have used the Jeffreys reference pror π ( σ ) σ (5) he posteror dstrbuton s proportonal to the product of (9), (), (3), (4) and (5), e π ( a, µ, σ y, x) L( y θ) e also have that and where π ( a ξ, n) π ( χ, ) π ( µ µ, τ ) π ( σ ) E( ) = (6) χ p ( ω ) Var = ( χ p 3)( χ p ) ω = ( ) (7) he hyperparaeters, n, ξ, χ and, µ and τ are set by the user, gven the avalable pror nforaton (see Robert () for ore detal) In our case they are chosen as follows n order to correspond reasonably closely wth the values used n the orgnal forulaton of the proble gven n Secton e let E ( ) = Moreover we set Var ( ) = and because we use a syetrc atrx for rather than an upper trangular for we set Var = / when j hen use of (6) and (7) where p = 6 n our exaple yelds ω = ω = / 48 when j and E = /, agan when j, and χ = Also we set µ = 75 so that the ean corresponded approxately to the pror estate of the optu as gven by the engneers Slarly the value of ξ was taken to be the pror estates of the optu factor values as gven by the engneers o allow for our uncertanty about the pror values we set n = - and τ = 5 e dd not carry out a full Bayesan analyss for ths odel but nstead carred out a nuercal optzaton to dentfy the ode of the posteror dstrbuton π ( a, µ, σ y, x)

5 Mnzaton of logπ ( a, µ, σ y, x) was carred out usng the Nelder-Mead nuercal ethod e dd not attept to alter drectly but nstead, together wth a, µ, and σ, treated the δ n (8) as beng varables n the Nelder-Mead procedure, subject to δ = k = for p he choce of was based on the followng consderaton e need to have suffcent flexblty to allow a free choce of the eleents of Now, as t s syetrc, there are 5p(p+) dstnct eleents n Now we have p(-) dstnct choces for the p δ, as they are subject the restrctons δ = k =, p o retan the sae degree of freedo of choce we therefore need to have p(-) 5p(p+), e (3+p)/ th p = 6 we therefore ust have 5 In the exaple we took = 7 hs gave a nu at x* = â = (83, 86, 53, 53, 76, 9); wth 8 ˆ = wth nu value 754 he value of σ was σ = 545E-8 ndcatng a good ft to the data ponts whch s to be expected he nzed value n ths case was actually stll close to the ntal estate provded by the engneers An nterestng aspect of the odel s that f we had been able to obtan a set of noral devates Z, wth suffcent statstcs z and S, drectly then the posteror dstrbuton for a and s gven drectly by where n ξ + nz a ξ, z, S ~ N p, n n n n + + z, S ~ p ( χ + n, ( z, S) ) nn ( )( ) ( z, S) = + S + z ξ z ξ n + n whch s wrtten n ters of the suffcent statstcs for Z (8) z = z = S ( z = z)( z z) (9) = 5 SUMMARY AND CONCLUSIONS In suary, we have shown how sulaton etaodels can be used to fnd the optal factor settngs for a syste he exaple consdered llustrates how Bayesan ethods can be used n practce when te perts only a few runs of a sulaton odel to be ade, but the nuber of factors to be consdered s large hen the response surface s constraned to be postve defnte, the optu obtaned s slar to that obtaned wthout ths constrant Havng a postve defnte response surface also ensures that a nu does exst Although we have not constructed a full posteror dstrbuton for the data, we have carred out an estate of optal coeffcent values (usng odal estators) whch ncorporates pror knowledge to better nfor the optzaton routne used to fnd the odal estators In the future, we hope to refne ths ethod to fnd the true posteror dstrbuton of the paraeters REFERENCES Azadvar, F 999 Sulaton Optzaton Methodologes In Proceedngs of the 999 nter Sulaton Conference, P A Farrngton, H B Nebhard, D Sturrock, and G Evans, eds, Pscataway, NJ: Insttute of Electroncs and Electrcal Engneers 93- Barton, R R 998 Sulaton Metaodels In Proceedngs of the 998 nter Sulaton Conference, DJ Mederos, EF atson, JS Carson and MS Manvannan, eds, Pscataway, NJ: Insttute of Electroncs and Electrcal Engneers Cheng, R C H 999 Regresson Metaodellng usng Bayesan Methods In Proceedngs of the 999 nter Sulaton Conference, P A Farrngton, H B Nebhard, D Sturrock, and G Evans, eds, Pscataway, NJ: Insttute of Electroncs and Electrcal Engneers Cheng, R C H 4 Optzaton of Systes by Sulaton Metaodellng Methods In Proceedngs of OR Socety Sulaton orkshop 4, S Robnson and S aylor, eds, Brngha: OR Socety Chck, S E Bayesan Methods for Sulaton, In Proceedngs of the nter Sulaton Conference, J A Jones, R R Barton, K Kang, and P A Fshwck, eds, IEEE, Pscataway, NJ 9-8 Cochran, G and Cox, G M 957 Experental Desgns, nd Ed, New York: John ley Fu, M C, S Andradottr, JS Carson, F Glover, CR Harrel, Y-C Ho, JP Kelly and SM Robnson Integratng Optzaton and Sulaton: Research and Practce In Proceedngs of the nter Sulaton Conference, J A Jones, R R Barton, K Kang, and P A Fshwck, eds, Pscataway, NJ: Insttute of Electroncs and Electrcal Engneers 6-66

6 Glover, F, JP Kelly and M Laguna 999 New advances for weddng optzaton and sulaton In Proceedngs of the 999 nter Sulaton Conference, P A Farrngton, H B Nebhard, D Sturrock, and G Evans, eds, Pscataway, NJ: Insttute of Electroncs and Electrcal Engneers 55-6 Gonda, H, HG Neddereer, J Gerrt, GJ van Oortarssen, N Persa and RA Dekker Fraework for Response Surface Methodology for Sulaton Optzaton In Proceedngs of the nter Sulaton Conference, J A Jones, R R Barton, K Kang, and P A Fshwck, eds, Pscataway, NJ: Insttute of Electroncs and Electrcal Engneers 9-36 Hood, S J and PD elch 993 Response surface ethodology and ts applcaton n sulaton In Proceedngs of the 993 nter Sulaton Conference, G Evans, M Mollaghase, EC Russell, and E Bles, eds, Pscataway, NJ: Insttute of Electroncs and Electrcal Engneers 5 Kelton, D 999 Desgnng Sulaton Experents In Proceedngs of the 999 nter Sulaton Conference, P A Farrngton, H B Nebhard, D Sturrock, and G Evans, eds, Pscataway, NJ: Insttute of Electroncs and Electrcal Engneers Law, A M and MG McCoas Sulaton-Based Optzaton In Proceedngs of the nter Sulaton Conference, J A Jones, R R Barton, K Kang, and P A Fshwck, eds, Pscataway, NJ: Insttute of Electroncs and Electrcal Engneers Robert, C he Bayesan Choce: fro Decson- heoretc Motvatons to Coputatonal Ipleentaton New York: Sprnger-Verlag Swsher, J R, SH Jacobson, PD Hyden and L Schruben A Survey of sulaton optzaton technques and procedures In Proceedngs of the nter Sulaton Conference, J A Jones, R R Barton, K Kang, and P A Fshwck, eds, Pscataway, NJ: Insttute of Electroncs and Electrcal Engneers 9-8 CHRISINE CURRIE s a PhD student at the Unversty of Southapton She has an MPhys fro Oxford Unversty and an MSc n Operatonal Research fro the Unversty of Southapton Her research nterests nclude atheatcal odelng of epdecs, Bayesan statstcs and varance reducton ethods Her eal address s: <CSMCurre@athssotonacuk> AUHOR BIOGRAPHIES RUSSELL CH CHENG s Professor, Head of Operatonal Research, and Deputy Dean of the Faculty of Matheatcal Studes at the Unversty of Southapton He has an MA and the Dploa n Matheatcal Statstcs fro Cabrdge Unversty, England He obtaned hs PhD fro Bath Unversty He s a forer Charan of the UK Sulaton Socety, a Fellow of the Royal Statstcal Socety, Meber of the Operatonal Research Socety Hs research nterests nclude: varance reducton ethods and paraetrc estaton ethods He was a Jont Edtor of the IMA Journal of Manageent Matheatcs Hs eal and web addresses are RCHCheng@athssotonacuk and <www athssotonacuk/staff/cheng>