Response surface methodology

Transcription

1 CHAPTER 3 Respose surface methodology 3. Itroducto Respose surface methodology (RSM) s a collecto of mathematcal ad statstcal techques for emprcal model buldg. By careful desg of epermets, the objectve s to optmze a respose (output varable) whch s flueced by several depedet varables (put varables). A epermet s a seres of tests, called rus, whch chages are made the put varables order to detfy the reasos for chages the output respose. Orgally, RSM was developed to model epermetal resposes (Bo ad Draper, 987), ad the mgrated to the modellg of umercal epermets. The dfferece s the type of error geerated by the respose. I physcal epermets, accuracy ca be due, for eample, to measuremet errors whle, computer epermets, umercal ose s a result of complete covergece of teratve processes, roud-off errors or the dscrete represetato of cotuous physcal pheomea (Guta et al., 996; va Campe et al., 990, Toropov et al., 996). I RSM, the errors are assumed to be radom.

2 Respose surface methodology 6 The applcato of RSM to desg optmzato s amed at reducg the cost of epesve aalyss methods (e.g. fte elemet method or CFD aalyss) ad ther assocated umercal ose. The problem ca be appromated as descrbed Chapter wth smooth fuctos that mprove the covergece of the optmzato process because they reduce the effects of ose ad they allow for the use of dervatve-based algorthms. Veter et al. (996) have dscussed the advatages of usg RSM for desg optmzato applcatos. For eample, the case of the optmzato of the calcato of Roma cemet descrbed Secto 6.3, the egeer wats to fd the levels of temperature ( ) ad tme ( ) that mamze the early age stregth (y) of the cemet. The early age stregth s a fucto of the levels of temperature ad tme, as follows: y = f (, ) + ε (3.) where ε represets the ose or error observed the respose y. The surface represeted by f(, ) s called a respose surface. The respose ca be represeted graphcally, ether the three-dmesoal space or as cotour plots that help vsualze the shape of the respose surface. Cotours are curves of costat respose draw the, j plae keepg all other varables fed. Each cotour correspods to a partcular heght of the respose surface, as show Fgure 3..

3 Respose surface methodology Fgure 3. Three-dmesoal respose surface ad the correspodg cotour plot for the early age stregth of Roma cemet where s the calcato temperature ( C) ad s the resdece tme (ms). Ths chapter revews the two basc cocepts RSM, frst the choce of the appromate model ad, secod, the pla of epermets where the respose has to be evaluated. 3. Appromate model fucto Geerally, the structure of the relatoshp betwee the respose ad the depedet varables s ukow. The frst step RSM s to fd a sutable appromato to the true relatoshp. The most commo forms are low-order polyomals (frst or secod-order). I ths thess a ew approach usg geetc programmg s suggested. The advatage s that the structure of the appromato s ot assumed advace, but s gve as part of the soluto, thus leadg to a fucto structure of the best possble qualty. I addto, the complety of the fucto s ot lmted to a polyomal but ca be geeralsed wth the cluso of ay mathematcal operator (e.g.

4 Respose surface methodology 8 trgoometrc fuctos), depedg o the egeerg uderstadg of the problem. The regresso coeffcets cluded the appromato model are called the tug parameters ad are estmated by mmzg the sum of squares of the errors (Bo ad Draper, 987): P ~ ( ) m G( a ) = wp Fp Fp ( a) (3.) = p where w p s a weght coeffcet that characterzes the relatve cotrbuto of the formato of the orgal fucto at the pot p, p=,...,p. The costructo of respose surface models s a teratve process. Oce a appromate model s obtaed, the goodess-of-ft determes f the soluto s satsfactory. If ths s ot the case, the appromato process s restarted ad further epermets are made or the GP model s evolved wth dfferet parameters, as eplaed Chapter 4. To reduce the umber of aalyses computer smulatos, sestvty data may be used the model fttg, although ths formato s ot always avalable at low cost. If addto to the values of the orgal fucto F p = F( p ) ther frst order dervatves at pot p Fp, = F p (=,,N, p=,,p) are kow, the problem (3.) s replaced by the followg oe (Toropov et al., 993):

5 Respose surface methodology 9 ~ ( F F ( a) ) N P,, p p ~ ( ) ( ( ) ) G a = w + = p Fp Fp a γ m (3.3) N p=, Fp = where γ >0 s the parameter characterzg a degree of equalty of the cotrbuto of the respose ad the sestvty data. I ths thess, γ s take as 0.5, followg recommedatos by Toropov et al. (993). Va Keule et al. (000) have preseted a methodology for the costructo of resposes usg both fucto values ad dervatves o a weghted least-squares formulato. The authors coclude that the use of dervatves provdes better accuracy ad requres a reduced umber of data. 3.3 Desg of epermets A mportat aspect of RSM s the desg of epermets (Bo ad Draper, 987), usually abbrevated as DoE. These strateges were orgally developed for the model fttg of physcal epermets, but ca also be appled to umercal epermets. The objectve of DoE s the selecto of the pots where the respose should be evaluated. Most of the crtera for optmal desg of epermets are assocated wth the mathematcal model of the process. Geerally, these mathematcal models are polyomals wth a ukow structure, so the correspodg epermets are desged oly for every partcular problem. The choce of the desg of epermets

6 Respose surface methodology 0 ca have a large fluece o the accuracy of the appromato ad the cost of costructg the respose surface. I a tradtoal DoE, screeg epermets are performed the early stages of the process, whe t s lkely that may of the desg varables tally cosdered have lttle or o effect o the respose. The purpose s to detfy the desg varables that have large effects for further vestgato. Geetc Programmg has show good screeg propertes (Glbert et al., 998), as wll be demostrated Secto 6., whch suggests that both the selecto of the relevat desg varables ad the detfcato of the model ca be carred out at the same tme. A detaled descrpto of the desg of epermets theory ca be foud Bo ad Draper (987), Myers ad Motgomery (995) ad Motgomery (997), amog may others. Schoofs (987) has revewed the applcato of epermetal desg to structural optmzato, Ual et al. (996) dscussed the use of several desgs for respose surface methodology ad multdscplary desg optmzato ad Smpso et al. (997) preseted a complete revew of the use of statstcs desg. As troduced Secto 3., a partcular combato of rus defes a epermetal desg. The possble settgs of each depedet varable the N- dmesoal space are called levels. A comparso of dfferet methodologes s gve the et secto Full factoral desg To costruct a appromato model that ca capture teractos betwee N desg varables, a full factoral approach (Motgomery, 997) may be ecessary to

7 Respose surface methodology vestgate all possble combatos. A factoral epermet s a epermetal strategy whch desg varables are vared together, stead of oe at a tme. The lower ad upper bouds of each of N desg varables the optmzato problem eeds to be defed. The allowable rage s the dscretzed at dfferet levels. If each of the varables s defed at oly the lower ad upper bouds (two levels), the epermetal desg s called N full factoral. Smlarly, f the mdpots are cluded, the desg s called 3 N full factoral ad show Fgure Fgure 3. A 3 3 full factoral desg (7 pots) Factoral desgs ca be used for fttg secod-order models. A secod-order model ca sgfcatly mprove the optmzato process whe a frst-order model suffers lack of ft due to teracto betwee varables ad surface curvature. A geeral secod-order model s defed as y = = = a0 + a + a + a j (3.4) j = = < j where ad j are the desg varables ad a are the tug parameters. The costructo of a quadratc respose surface model N varables requres the study at three levels so that the tug parameters ca be estmated. Therefore, at

8 Respose surface methodology least (N+) (N+) / fucto evaluatos are ecessary. Geerally, for a large umber of varables, the umber of epermets grows epoetally (3 N for a full factoral) ad becomes mpractcal. A full factoral desg typcally s used for fve or fewer varables. If the umber of desg varables becomes large, a fracto of a full factoral desg ca be used at the cost of estmatg oly a few combatos betwee varables. Ths s called fractoal factoral desg ad s usually used for screeg mportat desg varables. For a 3 N factoral desg, a 3 p fracto ca be costructed, resultg 3 N-p pots. For eample, for p= a 3 3 desg, the result s a oe-thrd fracto, ofte called 3 3- desg, as show Fgure 3.3 (Motgomery, 997) Fgure 3.3 Three oe-thrd fractos of the 3 3 desg 3.3. Cetral composte desg A secod-order model ca be costructed effcetly wth cetral composte desgs (CCD) (Motgomery, 997). CCD are frst-order ( N ) desgs augmeted by addtoal cetre ad aal pots to allow estmato of the tug parameters of a secod-order model. Fgure 3.4 shows a CCD for 3 desg varables.

9 Respose surface methodology 3 3 Factoral pots Aal pots Cetral pot Fgure 3.4 Cetral composte desg for 3 desg varables at levels I Fgure 3.4, the desg volves N factoral pots, N aal pots ad cetral pot. CCD presets a alteratve to 3 N desgs the costructo of secod-order models because the umber of epermets s reduced as compared to a full factoral desg (5 the case of CCD compared to 7 for a full-factoral desg). CCD have bee used by Escheauer ad Mstree (997) for the multobjectve desg of a flywheel. I the case of problems wth a large umber of desgs varables, the epermets may be tme-cosumg eve wth the use of CCD D-optmal desgs The D-optmalty crtero eables a more effcet costructo of a quadratc model (Myers ad Motgomery, 995). The objectve s to select P desg pots from a larger set of caddate pots. Equato (3.4) ca be epressed matr otato as:

10 Respose surface methodology 4 Y = X * B + e (3.5) where Y s a vector of observatos, e s a vector of errors, X s the matr of the values of the desg varables at pla pots ad B s the vector of tug parameters. B ca be estmated usg the least-squares method as: B = ( X T * X ) - X T Y (3.6) The D-optmalty crtero states that the best set of pots the epermet mamzes the determat X T X. "D" stads for the determat of the X T X matr assocated wth the model. From a statstcal pot of vew, a D-optmal desg leads to respose surface models for whch the mamum varace of the predcted resposes s mmzed. Ths meas that the pots of the epermet wll mmze the error the estmated coeffcets of the respose model. The advatages of ths method are the possblty to use rregular shapes ad the possblty to clude etra desg pots. Geerally, D-optmalty s oe of the most used crtera computer-geerated desg of epermets. Several applcatos are descrbed Guta et al. (996) for the wg desg of a hgh-speed cvl trasport ad Ual et. al. (996) for a multdscplary desg optmzato study of a lauch vehcle. Haftka ad Scott (996) have revewed the use of D-optmalty crtera for the optmzato of epermetal desgs Taguch's cotrbuto to epermetal desg Taguch's methods (Motgomery, 997) study the parameter space based o the fractoal factoral arrays from DoE, called orthogoal arrays. Taguch argues that t

11 Respose surface methodology 5 s ot ecessary to cosder the teracto betwee two desg varables eplctly, so he developed a system of tabulated desgs whch reduce the umber of epermets as compared to a full factoral desg. A advatage s the ablty to hadle dscrete varables. A dsadvatage s that Taguch gores parameter teractos Lat hypercube desg Lat hypercube desg (McKay et al., 979) ca be vewed as a N-dmesoal eteso of the tradtoal Lat square desg (Motgomery, 997). O each level of every desg varable oly oe pot s placed. There are the same umber of levels as rus ad the levels are assged radomly to rus. Ths method esures that every varable s represeted, o matter f the respose s domated by oly a few oes. Aother advatage s that the umber of pots to be aalyzed ca be drectly defed. A eample of the use of such plas ca be foud Schoofs et al. (997) Audze-Eglas' approach Audze ad Eglas (977) suggested a o-tradtoal crtero for elaborato of plas of epermets whch, smlar to the Lat hypercube desg, s ot depedet o the mathematcal model of the problem uder cosderato. The put data for the elaborato of the pla oly clude the umber of factors N (umber of desg varables) ad the umber of epermets K. The ma prcples ths approach are as follows:

12 Respose surface methodology 6. The umber of levels of factors (same for each factor) s equal to the umber of epermets ad for each level there s oly oe epermet. Ths s smlar to the Lat hypercube desg.. The pots of epermets are dstrbuted as uformly as possble the doma of varables. There s a physcal aalogy wth the mmum of potetal eergy of repulsve forces for a set of pots of ut mass, f the magtude of these repulsve forces s versely proportoal to the dstace squared betwee the pots: P P p= q = p+ L pq m (3.7) where L pq s the dstace betwee the pots havg umbers p ad q (p q). The elaborato of the plas s tme cosumg, so each pla of epermet s elaborated oly oce ad stored a matr characterzed by the levels of factors for each of P epermets. For eample, for a umber of factors (desg varables) N = ad P = 0, the matr s (3.8) The pla (3.8) s represeted Fgure 3.5 ad compared wth a CCD for two desg varables wth 9 rus.

13 Respose surface methodology 7.5 CCD (9 pots).5 Lat hypercube samplg (0 pots located at radom) (a) (b).5 Audze-Eglas (0 pots) (c) Fgure 3.5 Comparso betwee CCD (a), Lat hypercube desg (b) ad Audze- Eglas desg (c) The advatages of ths method are the space-fllg property as show Fgure 3.5 ad the presetato of the data as tabulated desgs. A dsadvatage s that oce a desg has bee defed, o etra pots ca be added to the tal set. Ths approach has bee used by Rkards (993) to desg composte materals wth predcted propertes (weght, prce, etc.).

14 Respose surface methodology Va Keule's approach I the course of a teratve optmzato process modelled by appromatos, ew pots must be geerated specfed domas of the desg varable space. A ew scheme for the desg of epermets (Va Keule ad Toropov, 999) has bee formulated wth the followg characterstcs:. The scheme works effcetly eve f oly a sgle addtoal desg pot s geerated to the estg pla. For a umber of ew desg pots, the algorthm s used several tmes.. The scheme remas effectve f dfferet types of fuctos are used wth the same optmzato task to appromate the objectve fucto ad the costrats. The approach dstrbutes pots as homogeeously as possble the subdomas of terest. Ths s doe by the troducto of the followg cost fucto: Q = + P p= = p d + + p= = ( d ) ( d d ) P = j = + + ( [ ] d ) ( d ) p j = + (3.9) whch s mmzed wth respect to the locato of the ew pot d. Symbols deoted refer to coordates whch are ormalzed the sub-doma of terest. The frst term the epresso attempts to mamze the dstace betwee pots, ad the secod term promotes a homogeeous dstrbuto alog the coordate aes. The thrd ad fourth terms esure that pots do ot belog to the boudary of the sub-

15 Respose surface methodology 9 doma. The last term prevets pots from algg alog the dagoal of the search sub-rego whe oly a few pots are avalable. 3.4 Cocluso The respose surface methodology aalyss has bee revewed. RSM ca be used for the appromato of both epermetal ad umercal resposes. Two steps are ecessary, the defto of a appromato fucto ad the desg of the pla of epermets. As cocluded Chapter, geetc programmg s the method of choce to fd a sutable appromato fucto ad wll be descrbed Chapter 4. A revew of dfferet desgs for fttg respose surfaces has bee gve. A desrable desg of epermets should provde a dstrbuto of pots throughout the rego of terest, whch meas to provde as much formato as possble o the problem. Ths "space-fllg" property s a characterstc of three plas: Lat hypercube samplg, Audze-Eglas ad va Keule. All three plas are depedet of the mathematcal model of the appromato. However, Lat hypercube samplg dstrbutes the pots radomly the space, whle Audze-Eglas uses a dstrbuto based o mamum separato betwee pots. The Audze-Eglas pla has bee chose ths thess. It should be oted that f the model buldg s to be repeated wth a teratve scheme (e.g. wth md-rage appromatos), va Keule s pla would become a attractve alteratve as t adds pots to a estg pla. Ths thess s prmarly focused o buldg global appromatos.