Case Studes n Pareto Set Identfcaton Usng Pseudo Response Surfaces Anoop A. Mullur Achlle Messac Srsha Rangavajhala Correspondng Author Achlle Messac, Ph.D. Dstngushed Professor and Department Char Mechancal and Aerospace Engneerng Syracuse Unversty, 63 Lnk Hall Syracuse, New York 1344, USA Emal: messac@syr.edu Tel: (315) 443-341 Fax: (315) 443-3099 https://messac.expressons.syr.edu/ Bblographcal Informatonn Mullur, A. A., and Messac, A., Case Study n Pareto Set Identfcaton Usng Pseudo Response Surface,, 4th AIAA Multdscplnary Desgn Optmzaton Specalst Conference, AIAA-008-147, Schaumburg, Illnos, Aprl 7-10, 008.
49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamcs, and Materals Conference <br>16t 7-10 Aprl 008, Schaumburg, IL AIAA 008-147 4 th AIAA Multdscplnary Desgn Optmzaton Specalst Conference, Aprl 008, Schaumburg, IL Case Studes n Pareto Set Identfcaton Usng Pseudo Response Surfaces Anoop A. Mullur 1 and Achlle Messac Rensselaer Polytechnc Insttute, Troy, NY 1180 Srsha Rangavajhala 3 Rensselaer Polytechnc Insttute, Troy, NY 1180 Multobjectve optmzaton of functons nvolvng complex and expensve smulatons can be a computatonally prohbtve task wthout the use of response surface models. Ths paper presents multobjectve optmzaton case studes usng a newly developed method called Pseudo Response Surface (PRS) approach by the authors. Ths approach offers several computatonal benefts over tradtonal response surface approaches, such as sequental approxmate optmzaton and constructon of approxmatons over the global nput space. Numercal examples are presented, whch show a sgnfcant reducton over tradtonal approaches n the number of functon evaluatons requred to obtan the Pareto fronter of the multobjectve problem. We also explore the ablty of the PRS approach to handle the specal case of a dsjont Pareto fronter, and comment on the certan numercal ssues durng the use of ths approach. Nomenclature, j = desgn varable or objectve functon subscrpt x = vector of desgn varable values g(x) = vector of constrant functon values μ = vector of objectve functon values μ = vector of normalzed objectve functon values I. Introducton umercal smulatons are used by desgners and developers n a wde range of ndustry and research settngs, Nwhere there s nvarably a need for computatonal cost reducton. The subject paper deals wth the ntegraton of advanced optmzaton-based desgn tools wthn a smulaton-based decson makng framework, whle mantanng the computatonal effort to a mnmum. In ths context, buldng computatonally bengn approxmate models of expensve computer smulatons, or metamodelng, s becomng wdely accepted as an effectve tool for comprehensve desgn space exploraton 1-4. Pursts often vew the lack of physcs wthn metamodels as a hndrance to physcally ntutve decson makng. However, metamodels can dramatcally reduce computatonal cost, allowng desgners to seamlessly and practcally ntegrate advanced computatonal tools wthn expensve smulatons. In ths paper, we nvestgate the performance of surrogate models from the perspectve of multobjectve optmzaton a crtcal area of practcal desgn optmzaton, whch presents new challenges for effcently constructng metamodels 5,6. As part of ths nvestgaton, we present case studes n effcent Pareto set dentfcaton usng varous types of surrogate models, whle focussng on the recently developed Pseudo Response Surface (PRS) approach by the authors 7-9. Ths new approach offers sgnfcant computatonal benefts over tradtonal response surface approaches for solvng multobjectve problems. Under ths approach, the desgn space s effcently explored such that the expensve functon evaluatons are concentrated n the regon of Pareto optmalty. 1 Currently Sr. Research Engneer, ExxonMobl Upstream Research Company, Houston, TX, AIAA Member Professor, Mechancal and Aerospace Engneerng, AIAA Fellow 3 Ph.D., Mechancal Engneerng, AIAA Member 1 Copyrght 008 by Achlle Messac. Publshed by the, Inc., wth permsson.
Addtonally, the resultng metamodels have the unque capablty of yeldng hghly accurate Pareto optmal ponts upon optmzaton. Thus, the PRS approach can potentally lead to accurate optmal solutons at a fracton of the computatonal cost of current methods 7,9. PRS models are zero-th order surrogates (constructed usng only functon values), whch combne the appealng characterstcs of local and global approxmatons through a judcous choce of samplng locatons. The PRS approach follows three basc steps: () anchor pont dentfcaton: ths entals obtanng an estmate of the end ponts of the Pareto fronter, () explorng the Pareto fronter usng the normal constrant (NC) exploraton process to obtan the sample ponts for response surface buldng, and fnally () constructng the pseudo response surfaces usng the obtaned ponts. These PRS models can now be used n the place of the orgnal objectve functons and constrants to solve the multobjectve optmzaton problem. For more detals of PRS modelng, the reader s referred to recent publcatons on ths topc 7-9. In ths paper, we present several numercal examples and case studes (multobjectve optmzaton problems from the lterature 9 ), whch demonstrate the effcency of the PRS approach. Specfcally, through these examples, we strengthen our orgnal proposton of developng a computatonally effcent metamodelng approach for multobjectve optmzaton. The examples presented n ths paper do not nvolve any expensve analyss; however, they are complex enough n terms of the number of desgn varables, objectves, and constrants that we can draw strong conclusons about the effcacy of the PRS approach usng the results obtaned. The frst two problems nvolve the well-known Golnsk Speed Reducer problem whch has also been used as an MDO benchmark problem. We use the two and three objectve varatons of ths problem. Problem 3 s another well-known benchmark problem from the lterature the ten bar truss, and the last problem s an 18-varable portal frame desgn problem. The latter two problems are borrowed from the GENESIS (a commercal structural analyss and optmzaton software) user s manual 10, and ther analyses are performed wthn Geness to mmc a smulatonbased optmzaton problem. Dong so also allows us to use some of the effcent bult-n sequental approxmate optmzers n Geness for comparson purposes. For these last two problems, we also compare the results usng the PRS approach and those obtaned usng Geness. We also dscuss potental numercal ssues that we experenced n applyng the PRS approach, and conclude the paper wth possble areas of further development. II. Numercal Examples In ths secton, we frst compare the performance of the PRS approach wth the tradtonal surrogate modelng approach for solvng multobjectve optmzaton problems. The tradtonal approach nvolves buldng global approxmatons for each objectve functon and then obtanng the Pareto set by solvng the resultng approxmate multobjectve problem. We then compare (for problems 3 and 4) the performance of the PRS approach wth that of the sequental approxmate optmzaton (SAO) approach usng GENESIS. Secondly, we present our experence of usng PRS models for some specalzed cases of multobjectve problems, such as those nvolvng dsjont Pareto fronters. Experence wth PRS parameter settngs and ther senstvty to the fnal outcome s also dscussed n ths secton. A. Pareto set dentfcaton usng PRS Example 1: Golnsk speed reducer Ths s a well-known test problem (MDO Test Sute Problem.4) concernng the desgn of a smple gear box that can be used n a lght arplane between the engne and the propeller to allow optmum rotatng speed for each. The orgnal test sute problem s a sngle objectve problem to mnmze the weght of the gear box, subject to constrants mposed by typcal desgn practces. We use a multobjectve formulaton as solved by Kurpat et al. 11, where mnmzng volume s one objectve, whereas the other s to mnmze the stress n one of the gear shafts. The complete problem descrpton can be found n references 9,11. NC Exploraton To begn the NC exploraton, we assume the knowledge of the two anchor ponts (one for each objectve). We construct an ntal sample of 10 ponts n the regon of the anchor pont correspondng to the objectve, μ. Ths s followed by the normal constrant exploraton process that yelds an approxmately Pareto set, whch s used as a sample to construct the objectve functon and constrant functon metamodels. The number of sample ponts requested s 40, whch s equal to the number of ponts on the utopa lne (the lne jonng the anchor ponts). Note, however, that even though the spacng between the ponts may be equal n the begnnng, the NC exploraton
algorthm s capable of adaptng tself, and can result n a hgher number of ponts by the tme the algorthm termnates. The entre exploraton algorthm requres 50 functon evaluatons not ncludng the expense for anchor pont calculaton. However, the anchor ponts can be obtaned by computatonally effcent approaches, such as sequental approxmate optmzaton, as wll be seen n problems 3 and 4 n ths secton. PRS Model Buldng The nearly Pareto sample s passed through the E-RBF (Extended Radal Bass Functon 1 ) nterpolator after the dentfcaton of pseudo data ponts and the assgnment of pseudo response values. The E-RBF nterpolator produces pseudo response surfaces of the objectve and constrant functons, and helps us defne the approxmate multobjectve optmzaton problem. It can be solved usng any preferred multobjectve technque, but here we show the results of usng the normal constrant approach to generate the b-objectve Pareto fronter. Fgure 1 shows the Pareto optmum ponts obtaned upon usng the normal constrant approach on the PRS models. The ponts shown n the fgure are the ponts valdated from the actual models (that s, the actual objectve functons evaluated at the obtaned desgn ponts). Clearly, one can see that a hghly accurate Pareto optmum set s obtaned. Fgure 1. Actual and obtaned Pareto ponts Golnsk Speed Reducer (two objectves) The percentage metamodel error s calculated at four dstnct ponts along the fronter. Also, the percent constrant volaton s also obtaned for each obtaned Pareto pont. The errors are all n the range of 3% or less, ndcatng the hgh accuracy of the PRS approach. Table 1 compares the computatonal costs n terms of functon evaluatons per Pareto pont wth the global approxmaton approach. Thus, we observe that by smply constructng metamodels n the regon of Pareto optmalty, one can drastcally reduce the number of functon evaluatons of the exact functons wthout loss n accuracy of the fnal Pareto ponts. For the next example, we convert the above two objectve problem nto a three-objectve one by treatng one of the constrants as an addtonal desgn objectve. By so dong, we wll demonstrate the effcacy of the PRS approach for multobjectve problems. Example : Golnsk speed reducer, 3 objectves The three-objectve problem descrpton can be found n Ref. 9. Here we concentrate on the results of applyng the PRS approach to solve the multobjectve problem. Fgure. Actual and obtaned ponts- Golnsk speed reducer (three objectves) NC Exploraton The ncluson of a thrd objectve requres us to follow the more nvolved exploraton procedure detaled n Ref. 7. Ths nvolves creatng a grd of ponts on the utopa plane. Each vertex of the trangle represents an anchor pont (deal pont wth respect to each desgn objectve). The number of ponts on ths grd for the current problem s 64. Each of these ponts wll yeld a nearly Pareto sample pont durng our exploraton process. Note that the process requres as many functon evaluatons as the 3
number of grd ponts (64), and n addton 10 sample ponts created near the vcnty of the anchor pont correspondng to objectve 1. PRS Model Buldng 38 pseudo ponts 7 (n a 7-dmensonal space) were dentfed usng the pseudo data pont dentfcaton process. These ponts along wth the sample ponts above are used to construct the metamodels for each objectve functon and constrant functon. As before, we use the E-RBF approach for solvng the fnal optmzaton problem. Metamodel-based Optmzaton The MATLAB gradent-based optmzer fmncon s used to solve the multobjectve optmzaton nvolvng the metamodels. The multobjectve technque used to generate the Pareto fronter s the normal constrant approach. 100 Pareto ponts are obtaned on the fronter and plotted n Fg.. Note that the plotted ponts are the exact objectve functon values at the Pareto ponts yelded by the metamodelbased optmzaton. The metamodel error and constrant volaton are less than %. The computatonal expense s compared wth that of constructng global approxmatons n Table 1. The PRS approach results n a sgnfcant reducton n computatonal effort, as can be seen. Example 3: Ten-bar truss The ten-bar truss s a well-known problem 13 often used n the desgn optmzaton lterature for benchmarkng the performance of varous optmzaton algorthms. The structure s shown n Fg. 3 (top). These members are connected to each other at sx nodes, represented by 1 to 6 n Fg. 3. We assume that the truss s n one plane, and every node has only two degrees of freedom (dsplacements along the horzontal and vertcal axes). The left hand sde of the truss s fxed to the wall, and therefore nodes 1 and 4 have zero dsplacements. F 1 to F 8 represent loads appled to these nodes. Cross sectonal areas of the truss members are gven by x 1 to x 10. For the current problem, the loadng condton s: F 6 = F 8 = 100, 000 lb; all other F s = 0. Fgure 3. Ten Bar Truss (top); Portal Frame Desgn (bottom) A multobjectve problem s defned to mnmze the mass (m) of the structure, and smultaneously mnmze the stress n beam (σ ). The beam number s chosen arbtrarly, as the am s to smply demonstrate the effcacy of the PRS approach. Notce that the two objectves are conflctng: ncreasng the areas wll ncrease the mass, but decrease the stress, and decreasng the areas wll lead to lower structural mass and hgher stress. The problem formulaton s gven as mn x st.. [ m σ ] max σ σ, = 1,..,10 0.1 x 0 n, = 1,..,10 m 100 lb σ 500 lb/n (1) 4
where σ max s the maxmum allowable stress = 5,000 ps (tenson and compresson). The Young s modulus, E = 1 107 ps and the materal densty, ρ = 0.101 lb/n3. The am s to construct metamodels for the volume and all of the 10 stress responses wth a mnmal number of exact analyses (to calculate the stress and the mass responses). The analyss s performed n GENESIS, smply because we can use ts approxmate optmzer to yeld us the anchor ponts usually wth only a few (10-0) functon evaluatons. Sequental Approxmate Optmzaton For comparson purposes, we use the sequental optmzer of Geness to obtan the Pareto fronter for ths 10 varable problem. The computatonal expense for obtanng 0 Pareto ponts usng ths approach s 145 functon evaluatons n addton to any gradent calculatons that Geness performs. Ths soluton s shown n Fg. 4(a) as sold black dots. (a) (b) Fgure 4. (a) Pareto ponts Ten bar truss (b) Pareto ponts Portal frame NC Exploraton Anchor ponts for each objectve were frst obtaned usng the Geness approxmate optmzer. The anchor pont calculaton requres 1 analyses n total usng the Geness optmzer. A grd of 40 ponts s generated on the utopa lne, and 10 ntal sample ponts are generated n the vcnty of the stress anchor pont. Fnally, we use sequentally updatng local approxmatons to obtan a nearly Pareto sample set. PRS Model Buldng Usng the 40 explored sample ponts from the above set, we use the pseudo data pont dentfcaton procedure to dentfy 103 pseudo data ponts n the 10-dmensonal desgn space. The pseudo data ponts are assgned pseudoresponse values (for constrants and objectve functons) usng the scheme descrbed n Ref. 9. Fnally, we use the E- RBF approach to construct PRS metamodels for each objectve functon and constrant functon usng the sample ponts. These metamodels are then used to obtan the Pareto ponts usng the NC procedure. The obtaned ponts are shown n Fg. 4 (a). The error between the metamodel values and the actual analyss values at some of the obtaned Pareto ponts was calculated. The error for the stress objectve at pont 1 s hgh, but acceptable. The reason for the hgh error can be attrbuted to the large range of values (00-500) that the stress objectve takes possbly causng hgh (relatvely speakng) error n the regons of low stress. Thus, the PRS approach yelds accurate Pareto ponts n 6 functon evaluatons compared to 145 requred usng Geness. Experence wth krgng models To further compare the performance of the PRS approach, we constructed globally accurate krgng models by samplng the desgn space usng a 300 pont latn hypercube sample. We observed that even wth a large number of sample ponts, the resultng Pareto set was not satsfactory, compared wth the SAO and the PRS methods. As such, we confrm our ntal proposton that constructng globally accurate metamodels n a multobjectve problem may not be computatonally attractve, and may not guarantee an accurate Pareto set upon optmzaton. 5
Example 4: Portal frame desgn Ths problem s an example problem from the user s manual of the structural optmzaton software GENESIS 10. The motvaton behnd choosng ths problem s ts moderately hgh number of desgn varables 18, and constrants 1. The problem nvolves desgnng the cross-sectonal dmensons of each member of the structure shown n Fg. 3 (bottom). Each component has an unsymmetrcal I-beam cross secton (secton A-A), and the structure s loaded wth a pont load (P = 50, 000) as shown. The analyss s performed wthn Geness, whch s also used to calculate the locatons of the anchor ponts. The cross-sectonal dmensons of each component (sx each) are the desgn varables. The bendng stress response at four ponts (two at the top and two at the bottom of the cross-secton) for each component s constraned to be less than the maxmum permssble stress. We formulate a multobjectve desgn problem to mnmze the volume of the frame and the stress n beam 1, subject to the stress constrants, as follows. mn x st.. [ V σ ] max 1 σ σ, = 1,..,1 V 50000 () NC Exploraton As mentoned above, the sequental approxmate optmzer wthn Geness was used to obtan the anchor ponts wth respect to each desgn objectve. A total of 14 exact analyses were requred for ths purpose. An ntal sample of 10 ponts was then constructed n the vcnty of the anchor pont correspondng to the stress objectve. A grd of 30 ponts s then constructed on the utopa lne followed by the NC exploraton procedure. Fgure 4(b) shows the Pareto optmal ponts obtaned usng the SAO approach (usng the bult-n sequental optmzer n Geness). The computatonal expense (functon evaluatons) for each approach s provded n Table 1. One can see that the PRS approach requres almost an order-of-magntude fewer functon evaluatons compared to SAO. We can further construct metamodels for the objectve functons and constrants n order to provde better control over the decson makng process. However, snce the PRS results appear to be hghly accurate after the exploraton phase tself, we do not need to construct the PRS metamodels for ths case. Table 1. Functon evaluaton comparson: Tradtonal and PRS Example # vars. Globally accurate/sao PRS functon evaluatons 1 7 150 70 7 150 94 3 10 145 6 4 18 15 54 Example 5: Mathematcal example fve objectves We demonstrate the PRS approach on a fve objectve mathematcal problem. Fgure 5 shows the PRS exploraton process appled to the fve-objectve multobjectve problem defned n Eq. 3. Each radal plot n the fgure corresponds to a sngle pont n a fve-dmensonal objectve ( μ ) space. The top row shows fve representatve Pareto ponts, whle the bottom row shows the correspondng ponts obtaned usng the PRS normal constrant exploraton process. As the fgure shows, the explored ponts closely match the actual Pareto ponts. 6
mn x st.. [ μ μ μ μ μ ] 1 3 4 5 ( ) μ = x, = 1..5 ( μ ) ( μ ) ( μ ) ( μ ) ( μ ) 4 4 4 4 4 1 3 4 5 1 + 1 + 1 + 1 + 1 1 (3) Fgure 5. Fve-objectve example: Actual Pareto ponts (top); obtaned Pareto ponts (bottom) B. Dsjont Pareto fronters Dsjontedness n the objectve and/or the desgn space can occur due to multmodalty of the objectve functons or the constrants or both. Multmodalty, especally n the Pareto regon, translates nto dsjont Pareto sets n the objectve space or the desgn varable space. Interestngly, the PRS approach can be used, wth due care, for problems wth dsjont Pareto fronters, as well. A smple way to understand ths s through a b-objectve example where the entre Pareto fronter s the constrant boundary: mn x st.. [ x x ] 1 x ( ) 1 0.5 x1 3 gx ( ) = 5e + e x 0 0 x 5 (4) However, because of the nature of the constrant functon, part of the constrant boundary s non-pareto (see Fg. 6) gvng rse to a dsjont Pareto fronter. Such a case needs careful handlng usng the NC exploraton approach. We need to ensure that the regon of non-pareto optmalty s also captured durng the exploraton (samplng) phase. No sgnfcant changes need to be made to the exploraton process, however, a fner resoluton of the utopa lne ponts s recommended, so that the entre constrant boundary s captured. Fgure 6 (left) shows the explored ponts for the above problem, where the actual constrant boundary s plotted for reference. 7
Fgure 6. PRS approach for obtanng dsjont Pareto ponts; explored ponts (left); obtaned ponts (rght) The PRS model for the constrant functon does not requre any specal handlng, and t s constructed usng the metamodel constructon procedure for constrants dscussed before. Fnally, the results of usng a multobjectve optmzer to obtan the fnal Pareto ponts are shown n Fg. 6 (rght). As we can see, the PRS model s able to accurately capture the dsjont nature of the Pareto fronter. From the results of the smple example presented above, we can make a few general observatons. The PRS approach can be effectvely used for multobjectve optmzaton problems wth dsjont Pareto fronters. However, one requrement appears to be that the feasble space of the problem (g(x) 0) should be contnuous. In other words, the regon of dscontnuty of the Pareto fronter should correspond to the feasble regon (as was the case n the example above). Ths would allow the NC exploraton approach to traverse the dscontnuty n the Pareto fronter although ths may result n some non-pareto sample ponts. These may or may not be ncluded n the sample ponts used to construct the PRS models. C. Numercal Experence Wth Parameter Settngs Sample sze An mportant ssue n the mplementaton of any metamodelng approach s the specfcaton of an approprate sample sze. In most practcal cases, the experence of the engneer needs to be trusted n order to specfy an approprate sample sze. The metamodelng framework presented n ths paper also requres the specfcaton of a sample sze. Recall that the sample sze n the NC exploraton approach s equal to the number of grd ponts generated on the utopa plane. Typcally, t s well-understood that more sample ponts are needed n the regons where the responses are more senstve to the change n the desgn varables. For example, n a regon where the response s predomnantly lnear, we may need only a few data ponts to capture the lnear trend, whereas f the response s hghly nonlnear n a partcular regon, t may need more sample ponts to accurately model the behavor. However, we note that the NC exploraton under the PRS approach s unlke most other samplng-based metamodelng approaches. In ths case, we generate a sample pont by solvng a sequentally updatng local optmzaton problem n the vcnty of the Pareto fronter. As we shall see, although an ntal estmate of the number of sample ponts on the utopa plane s requred, t can be changed as the exploraton progresses along the fronter. For example, say, we have ntally specfed a dstance of δ between two consecutve utopa lne ponts. As we progress along the fronter, we mght encounter a stuaton where the soluton obtaned by the local optmzaton problem may not be accurate, that s, the dfference between the exact and approxmate objectve functon and/or constrant functon values may be more than some pre-specfed tolerance. If so, we smply reject ths sample pont, and solve another local optmzaton problem wth a reduced value of δ. The normal constrant n ths case wll lkely yeld a sample pont that s more accurate. The exploraton process can then progress from ths newly obtaned nearly Pareto pont. Such an approach would automatcally result n more sample ponts n the regons that are more senstve than those where the objectve functons and constrants do not change sgnfcantly. Ths topc can be pursued n more detal n the future extensons of the PRS approach. In general, however, the sample sze s hghly dependent on the senstvty of the ndvdual responses to the desgn varables along the Pareto fronter. 8
Optmzaton parameters For all of the example problems n ths paper, we have used the gradent-based optmzer n Matlab to solve the metamodel-based optmzaton problem. The type of optmzer s generally not expected to have a sgnfcant mpact on the qualty of sample ponts obtaned durng the exploraton process. However, the Matlab optmzer fmncon s known to be occasonally unstable. Therefore, we dscuss our computatonal experence wth some of the mportant optmzaton parameters. The examples solved n ths paper are vared enough n terms of the number of desgn varables and the type of analyss (lnear elastc, bucklng) for us to make the generc observatons that follow. The NC exploraton approach reles on the exstence of an optmum such that the approxmate normal constrant s actve at the optmum. Thus, n most cases, ths constrant wll be a crtcal constrant that wll gude the exploraton process. As such, specfyng an approprate constrant satsfacton tolerance could be crtcal from the perspectve of convergence. Note that for all the examples, we lnearly transform the multobjectve space to a unt hypercube. Thus, the normal constrant s normalzed by default, and t s numercally advsable to normalze all other constrants as well. Under such normalzed crcumstances, a constrant tolerance of 0.001-0.01 s generally adequate to ensure acceptable results. A smaller tolerance could potentally yeld more accurate results. However, accuracy of the Pareto ponts s not crtcal durng the exploraton stage, as we merely desre what we term nearly Pareto ponts. Another beneft to usng a loose tolerance on the constrants s that the resultng sample wll contan both feasble and nfeasble ponts all n the vcnty of the Pareto fronter. We can explot ths property of the sample durng metamodel constructon, such that the fnal Pareto ponts wll be able to satsfy the constrants more closely. A loose constrant tolerance wll generally promote a more evenly-spread sample, and wll lead to more accurate PRS models. Normalzaton We comment on the advantages of normalzaton (whch s mplemented for all of the examples n ths paper) for effectve NC exploraton and model constructon. A major source of problems n multobjectve optmzaton can be traced to the uneven scalng between the objectve functons and/or desgn varables. Thus, for the algorthms presented n ths paper to perform as expected, normalzaton (or scalng) of the relevant quanttes s crtcal, and s always recommended. A smple and commonly used scheme for objectve functon normalzaton s to mnmze each objectve functon to obtan the anchor ponts. Usng the coordnates of these anchor ponts, the objectve functon space (that ncludes the Pareto fronter) s lnearly transformed to le wthn a unt hypercube. Mathematcally, the transformed doman ( μ ) s gven by μ j* μ ( x) mnμ j ( x) = j* j* max μ j mn μ j j (5) where the LHS denotes the coordnates of the anchor pont for the j-th objectve functon. Ths step should be performed before the NC exploraton algorthm s mplemented. Thus, the nearly Pareto sample set generaton would be performed n the normalzed space, and the solutons would be mapped back to the orgnal doman usng the above transformaton n reverse. We note that obtanng anchor ponts (although approxmate) s one of the frst steps under the NC exploraton process, makng such normalzaton possble. Desgn varable scalng s also crtcal for the success of the NC exploraton procedure. We follow the usual gudelnes recommended for optmzaton algorthms, that s, all varable values should be between 1-10. Ths scalng s also benefcal from the perspectve of metamodel constructon especally usng RBFs (as used n ths paper). Pseudo data pont dentfcaton also shows more stablty f the desgn varables are scaled approprately. III. Concludng Remarks and Future Work Ths paper provded some addtonal multobjectve desgn optmzaton examples that are solved usng the PRS approach developed by the authors. Two of the structural optmzaton problems were analyzed usng Geness, whch allowed us to compare the results of the PRS approach wth those usng the sequental optmzer n Geness. Overall, the results of ths paper echo the dscusson presented n the ntroducton, n that the PRS approach s a computatonally effcent and accurate approach for solvng expensve multobjectve problems. For the structural 9
problems of ths paper, we have demonstrated a computatonal savng of almost 70% n terms of the number of functon evaluatons over tradtonal approaches. Future work nvolves applyng the PRS approach to large-scale multobjectve problems, and on problems wth numercal nose n the objectve functon and constrants. Acknowledgments Sponsorshp of ths work by the Natonal Scence Foundaton through awards numbers ENG-CMMI-0533330 and ENG-CMMI-0333568 s gratefully acknowledged. References 1 Gunta A.A., Balabanov V., Ham D., Grossman B., Mason W.H., Watson L.T., and Haftka R.T., Multdscplnary Optmsaton of a Supersonc Transport Usng Desgn of Experments Theory and Response Surface Modelng, The Aeronautcal Journal, Vol. 101, No. 1008, 1997, pp. 347 356. Mullur A.A. and Messac A, Extended Radal Bass Functons: More Flexble and Effectve Metamodelng, AIAA Journal, Vol. 43, No. 6, 005, pp. 1306 1315. 3 Sacks J., Welch W., Mtchell T.J., and Wynn H.P, Desgn and Analyss of Computer Experments, Statstcal Scence, Vol. 4, No. 4, 1989, pp. 409 43. 4 Smpson T.W., Peplnsk J.D., Koch P.N., and Allen J.K, Metamodels for Computer-based Engneerng Desgn: Survey and Recommendatons, Engneerng wth Computers, Vol. 17, No., 001, pp. 19 150. 5 Wlson B., Cappeller D., Smpson T.W., and Frecker M., Effcent Pareto Fronter Exploraton Usng Surrogate Approxmatons, Optmzaton and Engneerng, Vol., 001, pp. 31 50. 6 Yang B.S., Yeun Y.N., and Ruy W.S., Managng Approxmaton Models n Multobjectve Optmzaton, Structural and Multdscplnary Optmzaton, Vol. 4, No., 00, 141 156. 7 Messac A. and Mullur A.A., A Computatonally Effcent Metamodelng Approach for Expensve Multobjectve Optmzaton, Optmzaton and Engneerng, Vol. 9, No. 1, 008, pp. 1-93. 8 Messac A. and Mullur A.A., Pseudo Response Surface (PRS) Methodology: Collapsng Computatonal Requrements, 10th Multdscplnary Analyss and Optmzaton Conference, Albany, NY, Paper No. AIAA-004-4376, Aug. 30 Sept. 1, 004. 9 Mullur A.A., A New Response Surface Paradgm For Computatonally Effcent Multobjectve Optmzaton, Ph.D. Dssertaton, Mechancal Engneerng, Rensselaer Polytechnc Insttute, Troy, NY, 005. 10 Vanderplaats R&D, Geness: Structural Analyss and Optmzaton, Verson 7.0, User Manual, Volume II, August 001. 11 Kurpat A., Azarm S., and Wu J., Constrant Handlng Improvements for Multobjectve Genetc Algorthms, Structural and Multdscplnary Optmzaton, Vol. 3, No. 3, 00, pp. 04 13. 1 Mullur A.A. and Messac A., Extended Radal Bass Functons: More Flexble and Effectve Metamodelng, AIAA Journal, 005, Vol. 43, No. 6, pp. 1306 1315. 13 Zhang W.H. and Yang H.C., Effcent Gradent Calculaton of the Pareto Optmal Curve n Multcrtera Optmzaton, Structural and Multdscplnary Optmzaton, 00, Vol. 3, No. 4, pp. 311 319. 10