DECISION BASED COLLABORATIVE OPTIMIZATION

Transcription

1 8 th ASCE Specialty Cnference n Prbabilistic Mechanics and Structural Reliability PMC DECISION BASED COLLABORATIVE OPTIMIZATION X. Gu and J.E. Renaud University f Ntre Dame, Indiana, IN Renaud.2@nd.edu Abstract In this research a Cllabrative Optimizatin (CO) apprach fr multidisciplinary systems design is used t develp a decisin based design framewrk fr nn-deterministic ptimizatin. T date CO strategies have been develped fr use in applicatin t deterministic systems design prblems. In this research the decisin based design framewrk prpsed by Hazelrigg (1996, 1998) is mdified fr use in a cllabrative ptimizatin framewrk. The Hazelrigg framewrk as riginally prpsed prvides a single level ptimizatin strategy that cmbines engineering decisins with business decisins in a single level ptimizatin. By transfrming the Hazelrigg framewrk fr use in cllabrative ptimizatin ne can decmpse the business and engineering decisin making prcesses. In the new multilevel framewrk f Decisin Based Cllabrative Optimizatin (DBCO) the business decisins are made at the system level. These business decisins result in a set f engineering perfrmance targets that disciplinary engineering design teams seek t satisfy as part f subspace ptimizatins. The Decisin Based Cllabrative Optimizatin framewrk mre accurately mdels the existing relatinship between business and engineering in multidisciplinary systems design. Intrductin Increasing attentin has been paid t the ntin that engineering design is a decisin making prcess. This ntin is cnsistent with the definitin f decisin as a chice frm amng a set f ptins and as an irrevcable allcatin f resurces. The apprach f decisin based design (DBD) is built upn this ntin. Rted frm mre than tw hundred years f research in the field f decisin science, ecnmics, peratins research and ther disciplines, decisin-based design (DBD) prvides a rigrus fundatin fr design, which enables engineers t identify the best trade-ff and fcus n where the payffs are greatest. Decisin Based Design (DBD) Framewrk Applicatin f decisin based design within an ptimizatin dmain, requires practitiners t frmulate valid bjective functins fr prper decisin making. An ptimizatin slutin btained using any search methd is n better than the bjective functin chsen fr the ptimizatin. If a mathematically defective bjective functin is used (Hazelrigg 1996) then there are n guarantees f any srt f slutin. Therefre a primary cncern in DBD is the develpment f a mathematically sund bjective functin. Recgnizing that design is a decisin-making prcess, the decisin based design framewrk f Hazelrigg (1996, 1998) implements the cncept f ratinal decisins. Ratinal decisins fllw the rule that the preferred decisin is the ptin whse expectatin has the highest value. Due t the nature f engineering design, expectatins n design alternatives can never be determined with certainty. It is imperative that the bjective functin (r utility functin in the cntext f ecnmy) be valid under cnditins f uncertainty and risk. The vn Neumann- Mrgenstern (vn-m) utility (vn Neumann, 1953) is such a value measure. The DBD framewrk f Hazelrigg acknwledges the limitatins impsed by Arrw s Impssibility Therem (Arrw, 1963) and views the bjective f systems design as ne f maximizing prfit. Therefre a valid bjective functin fr ptimizatin r decisin making under Gu and Renaud 1

2 uncertainty and risk is established: the ptimizer shuld seek t maximize the expected vn-m utility f the prfit. Prfit is als referred t as net revenue (NR). The relatinship between prfit (net revenue NR), demand q, ttal cst C T (cst f manufacture and all ther life cycle csts), and the price P can be summarized in Equatin (1). NR = ( P C T ) q (1) Multidisciplinary Enterprise Mdel Design is inherently a multidisciplinary prcess. The DBD framewrk f Hazelrigg, 1996, 1998 cmbines bth engineering and business perfrmance simulatins in a single level all-at-nce ptimizatin apprach. In this research the Hazelrigg framewrk has been decmpsed int the multidisciplinary enterprise mdel shwn in Figure 1. The decmpsed system cnsists f tw majr rganizatins: the engineering disciplines and the business discipline. The wrk in the engineering disciplines fcuses n predicting the perfrmance f the prduct fr different design cnfiguratins, as well as satisfying perfrmance targets set in the business discipline (i.e., management). The rle f the business discipline, centers n prviding targets fr perfrmance imprvements in rder t yield higher prfit. These tw rganizatins are cupled thrugh attributes a, ttal cst C T and demand q. In this research it is assumed that the demand fr the prduct, the number f the prduct manufactured and the amunt f the prduct sld are equal. Attributes a refer t the features f a prduct that custmers tend t be interested in. Examples f attributes include speed, acceleratin, quality, reliability r safety, etc. Nte that the price f the prduct nt nly directly affects the amunt f prfit r net revenue (see Eqn. 1), it is als an imprtant factr driving the demand q. Since price is free t be chsen by the decisin maker, demand q can be mdeled as a functin f attributes a and the price, P Variabilty x Engineering Design Variables x CA1 Tl A Predictin Errr y a CA2 Tl B Preditctin Errr Engineering Discipline Perfrmance y c y b CA3 Tl C a Manufacturing cst and ther life cydle csts CT CT a attributes q demand C T ttal cst u utility y states q Price Demand q Business Discipline q Utility f Prfit u Figure 1. Multidisciplinary Enterprise Mdel Gu and Renaud 2

3 (see Equatin (2)). q = q( a, P) (2) Cllabrative Optimizatin (CO) The Cllabrative Optimizatin (CO) strategy was first prpsed by Kr and Sbieski (1994) and has been successfully applied t a number f different design prblems. Tappeta and Renaud (1997) extended this apprach and develped three different frmulatins t prvide fr multibjective ptimizatin f multidisciplinary systems. Cllabrative Optimizatin (CO) is a tw level ptimizatin methd specifically created fr large-scale distributed-analysis applicatins. The system level ptimizer attempts t minimize a system level bjective functin F while satisfying all the cmpatibility cnstraints. System level design variables cnsist f nt nly the shared variables but als auxiliary variables. These variables are specified by the system level ptimizatin and are sent dwn t subspaces as targets t be matched. Each subspace, as a lcal ptimizer, perates n its wn set f design variables with the gal f matching target values psed by the system level as well as satisfying lcal cnstraints. The matching can be attained by minimizing the discrepancy between sme f the lcal design variables and/r lcal states and their crrespnding target values, in ther wrds, the bjective functins at subspace level are identical t the system level (cmpatibility) cnstraints. This frmulatin allws the use f pst-ptimal sensitivities at the subspace ptimum as the gradients f the system level cnstraints. This imprtant feature imprves the verall efficiency f CO by eliminating the need t execute subspace analyses fr the sle purpse f calculating system cnstraint gradients by finite differencing. Decisin Based Cllabrative Optimizatin (DBCO) Framewrk In Decisin Based Cllabrative Optimizatin (DBCO), the methd f cllabrative ptimizatin (CO) is used t determine the ptimal design f the multidisciplinary enterprise mdel (Fig. 1). The resulting decisin based cllabrative ptimizatin (DBCO) framewrk (shwn in Fig. 2) rigrusly simulates the existing relatinship between business and engineering in multidisciplinary systems design. In this framewrk, the business decisins are made at the system level. The system level ptimizer attempts t increase expected utility f net revenue E(u(NR)) while satisfying cmpatibility cnstraints d. Accrding t the analyses in the business discipline and the subspace ptimizatin results, the system level ptimizer determines price P and establishes a set f perfrmance targets including demand q and ttal cst C T. These targets are then sent dwn t apprpriate subspaces. The subspace ptimizer, based n his/her expertise in the discipline analysis, tries t match these targets as clse as pssible and reprts the discrepancy back t the system level. The subspace ptimizers are subject t lcal design cnstraints. In the field f engineering design, the design cnstraints nrmally guard against failure r ther unacceptable behavir. The use f cnstraints t prevent undesirable behavir requires designers t quantify what is undesirable. In DBD the market place is used t determine undesirability thrugh demand mdels and therefre cnstraints related t undesirability are eliminated. Therefre the lcal cnstraints in the decisin based cllabrative ptimizatin framewrk (DBCO) tend t be thse cnstraints that guard against system failure. Other Gu and Renaud 3

4 traditinal engineering cnstraints related t cnsumer preference are eliminated and instead incrprated in the demand mdel and/r cst mdel. System Level Optimizer Maximize: f=u Subject t: d i= D.V. : x = [ x, x, C, price ] shared aux T x u,q aux T, C, price Demand q Business Discipline q Utility f Prfit u 0 0 shared, xaux x d 1 SubSpace 2 Optimer SubSpace 3 Optimer d c shared aux T x, x, C, q 0 CA2 CA3 SubSpace 1 Optimizer Min: d1 D.V.: x = [ (x ), (x ), x ] ss1 sh 1 aux j1 1 SubSpace Cst Optimizer Min: d c D.V.: x = [ (x ), (x ), x ] ssc sh c aux jc c x ss1 y 1j x ss1 y 1j CA1 Manufacturing cst and ther life cydle csts C T Figure 2. Decisin-Based Cllabrative Optimizatin The system level ptimizatin prblem and the subspace ptimizatin prblem fr discipline 1 in Fig 2, in its standard frm, are given in Equatin (3). System Level Optimizatin Prblem Minimize F = E( u( NR) ) * Subject t: d i = 0 i = 12,,, n ss ( x sys ) min x sys ( x sys ) max x sys = ( x sh, x aux ) P > 0 Suspace 1 Optimizatin Prblem Minimize d 1 = (( x sh ) 1 ( x sh ) 1 ) 2 n ss + (( x aux ) j1 ( x aux ) j1 ) 2 j = 2 n ss + ( y 1 j ( x aux ) 1k ) 2 k = 2 Subject t g 1 0 ( x ss1 ) min x ss1 ( x ss1 ) max x ss1 = (( x sh ) 1, ( x aux ) j1, x 1 ) (3) Test Prblem: Aircraft Cncept Sizing (ACS) Prblem A preliminary applicatin f the decisin based cllabrative ptimizatin framewrk has been tested n the Aircraft Cncept Sizing (ACS) prblem. This prblem was riginally develped by the MDO research grup at the University f Ntre Dame (Wujek and Renaud, 1996, Tappeta, 1996). It invlves the preliminary sizing f a general aviatin aircraft subject t certain perfrmance cnstraints. The design variables in this prblem are cmprised f variables relating t the gemetry f the aircraft, prpulsin and aerdynamic characteristics, and flight regime. Apprpriate bunds are placed n all design variables. The prblem als includes a number f parameters which are fixed during the Gu and Renaud 4

5 design prcess t represent cnstraints n missin requirements, available technlgies, and aircraft class regulatins. The bjective in the ACS prblem is t determine the least grss take-ff weight within the bunded design space subject t tw perfrmance cnstraints. The first cnstraint is that the aircraft range must be n less than a prescribed requirement, and the secnd cnstraint is that the stall speed must n greater than a specified maximum stall speed. The riginal Aircraft Cncept Sizing (ACS) prblem had three disciplines: aerdynamics, weight and perfrmance. Tw disciplines have been added t fit in the DBD apprach: cst and business. A demand mdel and a cst mdel have been develped fr use in the business discipline and cst discipline, respectively. Optimizatin Results & Discussin This applicatin study is in its preliminary stage, and fcuses n the cllabrative ptimizatin features f the DBCO framewrk. The issues f prpagated uncertainty are neglected. The utility f prfit is assumed t be the prfit itself. Hence the bjective f the resulting deterministic ptimizatin prblem is t maximize prfit (r net revenue). During the ptimizatin, the demand q is treated as a cntinuus variable, ther than an integer. At the end f the system ptimizatin, q is runded t the nearest integer. A Sequential Quadratic Prgramming (SQP) methd was used fr ptimizatin at bth the system level and the subspace level. The SQP slver, fmincn, was btained frm the Matlab Optimizatin Tlbx. The system level ptimizer tries t maximize the negative prfit and minimize the cnstraint vilatin simultaneusly. At the beginning f the ptimizatin, aiming at achieving large prfit, the system level ptimizer set targets n high price, high level f perfrmance (t ensure high demand) and lw cst accrding t the business analyses. Hwever these targets cnflict with ne anther and lead t a large discrepancy at the subspace level. Thus the system level ptimizer, while trying t keep prfit as high as pssible, was frced t lwer price, dwngrade perfrmance and tlerate higher cst s that the subspace discrepancy culd be reduced. Gradually the system level ptimizer fund the best trade-ff amng the targets and reached a cnsistent ptimal design. The ptimizatin histry bserved in the ACS prblem resembles the existing relatinship between business and engineering in multidisciplinary systems design. The ptimal slutin is listed in Table 1. The demand mdel and cst mdel play an imprtant rle in the decisin based design apprach. In rder t illustrate the influence f demand and cst mdels, a cnventinal all-at-nce ptimizatin was perfrmed using the prblem frmulatin in Wujek and Renaud (1996). The cnventinal ptimum btained is als listed in Table 1. It can be bserved that the cnventinal ptimum utperfrms DBCO ptimal design n lwer weight (y 3, y 4 ). Hwever it pssess pr characteristics in many aspects such as smaller aircraft range (y 5 ), higher stall speed (y 6 ) and smaller fuselage vlume (y 7 ). Such an utcme is n surprise since the main cncern f the cnventinal ACS prblem is t minimize take-ff weight, while the DBD apprach takes int accunt ther perfrmance attributes, because f the DBD bjective f maximizing prfit. Cnclusins In this research a Decisin Based Cllabrative Optimizatin (DBCO) framewrk which Gu and Renaud 5

6 incrprates the cncepts f nrmative decisin based design (DBD) and the strategy f Cllabrative Optimizatin is develped. This bi-level nn-deterministic ptimizatin framewrk mre accurately captures the existing relatinship between business and engineering in multidisciplinary systems design. The business decisins are made at the system level, which result in a set f engineering perfrmance targets that disciplinary engineering design teams seek t satisfy as part f subspace ptimizatins. A preliminary applicatin f this apprach (deterministic case) has been cnducted n the Aircraft Cncept Sizing (ACS) test prblem. The crrespnding ptimizatin results are discussed. x 1 Name (Unit) aspect rati f the wing Table 1: Optimal Slutins fr ACS Prblem DV Bunds DBCO Cnven. Name (Unit) 5~ y 1 ttal aircraft wetted area (ft 2 ) DBCO Cnven x 2 wing area (ft 2 ) 100~ y 2 max lift t drag rati x 3 fuselage length (ft) 20~ y 3 empty weight (lbs) x 4 x 5 fuselage diameter (ft) density f air at cruise speed (slug/ft) 4~ y 4 grss take-ff weight (lbs).0017~ y 5 aircraft range (miles) x 6 cruise speed (ft/sec) 200~ y 6 stall speed (ft/sec) x 7 fuel weight (lbs) 100~ y 7 fuselage vlume (ft 3 ) Acknwledgements This multidisciplinary research effrt was supprted in part by the fllwing grants: NSF grant DMI and NASA grant NAG References Arrw, K.J., 1963, Scial Chice and Individual Value, 2nd Editin, Jhn Wiley and Sns, NY. Hazelrigg, G.A., 1996, Systems Engineering: An Apprach t Infrmatin-Based Design, Prentice Hall, Upper Saddle River, NJ. Hazelrigg, G.A., 1998, A Framewrk fr Decisin-Based Engineering Design, Jurnal f Mechanical Design. Kr, I., Altus, S., Braun, R., Gage, P., Sbieski, I., 1994, Multidisciplinary Optimizatin Methds fr Aircraft Preliminary Design, AIAA CP, Prceedings f the 5th AIAA/NASA/USAF/ISSMO Sympsium n Multidisciplinary Analysis and Optimizatin, Panama City, Flrida, September Tappeta, R.V., 1996, An Investigatin f Alternative Prblem Frmulatins fr Multidisciplinary Optimizatin, M.S. thesis, University f Ntre Dame, December Tappeta, R.V. and Renaud, J.E., 1997, Multibjective Cllabrative Optimizatin, ASME Jurnal f Mechanical Design, Vl 119, N. 3, September 1997, pp vn Neumann, J. and Mrgenstern, O., 1953, The Thery f Games and Ecnmic Behavir, 3rd Editin, Princetn University, Princetn, NJ. Wujek, B.A. and Renaud, J.E., 1996, Design Flw Management and Multidisciplinary Design Optimizatin in Applicatin t Aircraft Cncept Sizing, AIAA , 34th Aerspace Sciences Meeting & Exhibit, Ren, NY, January Gu and Renaud 6