Struct Multidisc OptimDOI 10.1007/s00158-006-0078-yRESEARCH PAPER2D decision-making for multicriteria design optimizationAlexander Engau Margaret M. WiecekReceived: 9 May 2006 / Revised manuscript received: 12 September 2006 Springer-Verlag 2006Abstract Thehighdimensionalityencounteredinen-gineeringdesignoptimizationduetolargenumbersof performance criteria and specications leads tocumbersome and sometimes unachievable trade-offanalyses. To facilitate those analyses and enhancedecision-makinganddesignselection, weproposetodecomposetheoriginal problembyconsideringonlypairs of criteriaat atime, therebymakingtrade-offevaluationthesimplestpossible. Forthenal designintegration, wedevelopanovel coordinationmecha-nismthat guaranteesthat theselecteddesignisalsopreferredfor theoriginal problem. Thesolutionofan overall large-scale problem is therefore reduced tosolvingafamilyofbicriteriasubproblemsandallowsdesigners to effectively use decision-making in merelytwo dimensions for multicriteria design optimization.KeywordsMulticriteria design optimization Interactive decision-making Decomposition Coordination Trade-off visualization SensitivityMargaret M. Wiecek is on leave from the Department ofMathematical Sciences, Clemson University, South Carolina29634, USA.A. Engau (B)Department of Mathematical Sciences, Clemson University,Clemson, SC 29634, USAe-mail: aengau@clemson.eduM. M. WiecekDepartment of Mechanical Engineering,Clemson University, Clemson, SC 29634, USANomenclatureMOP multiobjective problemMOPii-th multiobjective subproblemCOP coordination problemCOPii-th coordination problemSEP sensitivity problemSEPijj-th sensitivity problem for COPif vector objective function for MOPfivector objective function for MOPi, COPig, h vector constraint functionsx vector of design variablesxivector of design variables selected in COPiivector of tolerance values in COPiijvector of Langragean multipliers in SEPijConventionsubscripts denote vector components superscripts distinguish different vectors1 Introduction and literature reviewStructural designoptimizationdeals withthedevel-opment of complex systems andstructures suchascars, airplanes, spaceships, or satellites. On the basis oftremendous gain in experience and knowledge togetherwith the rapid progress in computing technologies, theunderlyingmathematical models anddesignsimula-tions become better and better and provide designerswith growing amounts of data that need to be analyzedforchoosinganaloptimaldesign. Inparticular, thesteadily increasing number of specications and criteriaA. Engau, M. M. Wiecekusedtoevaluate the performance of the simulateddesigns leads to cumbersome and sometimes unachiev-able trade-off analyses, thus resulting in complex anddifcult, if not unsolvable, decision-making problems.1.1 Design optimization under multiple performancemeasuresWhen evaluated by multiple criteria, it is long known(Zadeh 1963) that an overall optimal design, in general,doesnot exist, but aset of nondominatedorParetooptimal solutions (Pareto 1896). In principle, these so-lutionscanbefoundinthreedifferent ways. Inthetraditional butstill widelyappliedrstapproach, thedesign spaceis sampled and acertainnumber offea-sible designs are evaluated using simulation codes thatdescribetheunderlyingmodel (Vermaet al. 2005).Thereafter, these designs are ltered based on pairwisecomparisonsoftheirassociatedperformancessothatonlynondominateddesignsremainsubjecttofurtherconsideration. Other methods, such as the smart Paretolter (Mattson et al. 2004), exist to further reduce theremainingpointstothosemostinterestingtothede-signer. In enhancement of the rst, a second approachmakesadditionaluseofgeneticorevolutionary algo-rithms to further improve the initial designs comparedto a mere sampling (Gunawan et al. 2004; Narayananand Azarm 1999). Both approaches are similar in thatthey provide the designer with a set of solutions that arenondominated within the current sample, but generallydo not guarantee that any of these solutions also lies onthe Pareto optimal front of the original problem. Onlythe traditional third approach is based on classical opti-mization and nds one Pareto point at a time by solvingan auxiliary single objective problem. The typical for-mulation of this problem uses a linear combination (orweighted sum) that aggregates all participating criteriainto a single objective, so that, by varying the weightsassignedtoeachcriterion, different Paretosolutionscanbegenerated. Althoughcommonlyused, numer-ousdrawbacks ofthismethod, including itsfailuretogenerate points in nonconvex regions of the Pareto set,are well-recognized (Das and Dennis 1997). Recently,a family of aggregation functions more appropriate forengineering design is presented in Scott and Antonsson(2005). The concept of an aggregate objective functionisalsousedinthecontext of physical programming(Messac 1996; Messac and Ismail-Yahaya 2002; Messacet al. 1996) that can successfully be integrated into bothcollaborative and multidisciplinary design optimization(McAllister et al. 2005).1.2 Decomposition of the design optimization problemThecombinationof multipleperformancemeasuresintoonesingleindexisused, notonlytoreducethenumber of criteria, but moreimportantly, toreducethe complexity of the underlying optimization problem.Alternatively, areductioninthecomplexityof mostdesignproblemsistypicallyachievedbyvariousde-composition strategies (Blouin et al. 2004). These areparticularly well-suited for design optimization as mostcomplexengineeringsystemsusuallyconsistofmanysubsystemsandcomponentshavingsmallercomplex-ity(Chanronetal.2005).Decompositionthenmeanstodividethelargeandcomplexsystemintoseveralsmallerentities, whileresponsibilitiesforthevarioussubsystems or components areassignedtodifferentdesignersordesignteamswithautonomyintheirlo-cal optimization and decision-making. However, alongwith the benet of reduced complexity comes the dif-cultyof coordinatingthedifferent designdecisionsto eventually arrive at a single overall design solutionthat is feasible, thus meeting the design requirements,preferably optimal and acceptable for all participatingdesigners or decision-makers. A survey on existing co-ordination approaches is given in Coates et al. (2000)and Whiteld et al. (2000 and 2002, with a special focuson decentralized design). The issue of converging to acommonoverall designisaddressedinChanronandLewis (2005) and Chanron et al. (2005), who used gametheoretic concepts to model and analyze the competinginterest of the different decision-makers. Most recently,newstrategiesforthecoordinationbetweenmultiob-jective systems are also proposed for collaborative op-timization (Rabeau et al. 2006) and NASAs systemofsystems approach (Mehr and Tumer 2006).1.3 Relevance of this paperAddingtotheserecentresultsonthedecompositionof decision-makingproblems, thispaperpresentsaninteractive decision-making procedure for replacing theintuitive selection of the overall design (Agrawal et al.2004, 2005)byamoresystematicdesignintegration.This nal design integration is based on a novel coor-dinationmechanismthatguaranteesthattheselecteddesign is also preferred for the original problem. Previ-ous interactive methods are described in Tappeta andRenaud (1999), Tappeta et al. (2000), and Azarm andNarayanan(2000), butnoneofthesemethodsmakesuseofadecompositionassuggestedbytheapproachpresented in this paper. Motivated by the several recent2D decision-making for multicriteria design optimizationstudiesemphasizingtheimportanceofvisualizingtheoptimization process (Messac and Chen 2000), the de-signdata(EddyandLewis2002; Stumpet al. 2002,2003, 2004), and Pareto frontiers (Agrawal et al. 2004,2005; MattsonandMessac2005), thenewmethodol-ogyincludes thefeatureof visualizingthetrade-offcurves for every subproblem to support the choice ofan optimal design. In addition to the trade-off betweenthose objectives participating in the same subproblem,informationonthetrade-offsbetweendifferent sub-problems is obtainedfromasensitivityanalysis andusedfor the subsequent coordination. This methodthereby offers an alternative to the trade-off analysesfor the complete problem as suggested in Tappeta et al.(2000) and Kasprzak and Lewis (2000) or, in the contextofrobustmulticriteriaoptimization, inGunawanandAzarm(2005) andLi et al. (2005). Onlyonepaper(Verma et al. 2005) relies on a similar decompositionand visualization strategy in order to successively ltersolutions fromaset of simulateddesigns. However,therein all decisions are based merely on the intuitionof the designer and, as no optimization is involved, themethod cannot guarantee that the nal solution is alsooptimal for the overall problem. The procedure in thispaper, on the other hand, is capable of revealing trade-offs, generating new solutions based on the designerschoices, and always guaranteeing Pareto optimality ofthe nal and all intermediate solutions.1.4 Objectives of this paperTheobjectiveof this paper is thus threefold. First,assumingthatParetooptimal solutionscanbefoundby either a traditional approach or genetic algorithms,an interactive procedure is proposed for the selectionofanoptimal designforacomplexandmulticriteriadesign optimization problem. Second, this selection isfacilitatedusingadecompositionstrategy, sothatallintermediate decisions are made on smaller-sized sub-problems involving only two performance measures ata time. Anewcoordination mechanismthen guaranteesthat the nal selection leads to a common design thatisoptimal fortheoverall problem. Bychoosingthisdecompositioncoordination framework, the method isalso well-suited for decentralized design processes anddecision-making situations involving multiple decision-makers. Finally, thethirdobjectiveistoenhancetheprocedure by providing the designer or decision-makerwithtrade-off informationintheformofsensitivitiesanda2Drepresentationof thetrade-off curves forevery subproblem.2 Problem statement and preliminariesForthescopeof thispaper, weconsiderthemathe-matical model of a multiobjective optimization problem(MOP)MOP: minimize f (x) = [ f1(x),f2(x), . . . ,fp(x)]subject to g(x) = [g1(x), g2(x), . . . , gm(x)] 0h(x) = [h1(x), h2(x), . . . , hl(x)] = 0xLi xi xUi, i = 1, . . . , nwith vector objectivefand vector constraints g and h.The vectors xLand xU denote lower and upper boundsonthevectorofdesignvariablesx,andthesetofallfeasible designs is denoted byX :=_x Rn: g(x) 0, h(x) = 0, xL x xU_.2.1 Pareto optimal designsAdesignis Paretooptimal if it is not possible toimprove its performance with respect to one criterionwithout deterioration in at least one other criterion. Ifit is possible to improve a design with respect to somebutnotallcriteriaatthesametime, thenthisdesignis not Pareto optimal but only weakly Pareto optimal(Chankong and Haimes 1983).It thenfollowsthat aParetooptimal solutionforsomeset of criteriaremains at least weaklyParetooptimal upon enlarging this set with additional perfor-manceindices. Toseewhythisistrue, notethatfora Pareto optimal solution, no other design is availablethat is better in some but not worse in any other crite-rion. Then, in particular, no other solution can be betterinallcriteria, andclearlythismuststillbetrueafteradditional criteria are added. According to the abovedenition, this means that the solution is at last weaklyParetooptimalforthelargersetofcriteria;thus, thefollowing result holds.If adesignis Paretooptimal for asubset of allperformancemeasures, thenitis(atleastweakly)Pareto optimal for the overall problem involving allcriteria.Therefore, it is possible to identify Pareto optimal so-lutions for a large-scale problem by merely consideringsmaller-sized problems with a reduced number of cri-teria. Clearly, such approach requires to decide whichcriteriatochooseandwhichtodrop. InMatsumotoet al. (1993) and Dym et al. (2002), it is shown how todeveloparankingamongallcriteria, andtherelatedissueof decomposingamultiobjectivedesignbasedA. Engau, M. M. WiecekFig. 1 Traditionaldecomposition andintegration withoutcoordinationoncriteriainuenceisaddressedinYoshimuraetal.(2002, 2003). After these issues have been resolved, thereducedproblemcommonlytakestheformshowninFig. 1.2.2 Traditional decomposition and integrationFigure 1 illustrates the typical decompositionintegra-tion scheme for an overall MOP withp criteria that isdivided intoksubproblems ofsmaller size.Notethatonlytheperformancevector f isdecomposed, whilethe set of feasible designs X remains the same for everysubproblem. Therefore, eachproblemcanbesolvedseparatelyandthencommunicatesone(oraset of)optimal solutionstotheoverall MOP. Sinceall sub-problems are formulated over the same feasible designset, everysuchsolutionisalsofeasiblefortheothersubproblems and, in particular, for the overall problem.Therefore, the only task remaining is to select a nal so-lution among the designs proposed by the subproblems,andtheaboveresultthenguaranteesthatthisdesignis also (at least weakly) Pareto optimal for the overallproblem.However, by focusing on only one subproblem at atime, we risk to signicantly degrade the performancesof theothers. For illustration, assumethat wehavefound four designs x1, x2, x3, and x4that are evaluatedby the criteriaf1, f2, andf3, which we wish to minimize,as shown in Table 1.ItiseasytoverifythatallfourdesignsareParetooptimalforthecompleteproblemwithcriteria f1, f2andf3. For every combination of two criteria, however,onlyoneofthedesignsx1, x2, orx3remainsoptimalfortheassociatedsubproblem. Thedesignx4, ontheother hand, is not Pareto optimal for any subproblem,although it is the best compromise solution for the over-all problem. Moreover, this design constitutes the over-all minmaxsolution, that isthebest designforthestrategy to minimize all worst performances (the worstperformance of x1, x2, and x3is 9; of x4only 2). Hence,focusing on only one subproblem suffers from the ma-jor drawbackof regularlymissingthebest compro-mise solution for the overall problem and is, therefore,highly insufcient for the design of complex systems.2.3 Importance of tolerances and suboptimal designsTheaboveobservationindicatesthat solvingthein-dividualsubproblemstooptimalitybeforeselectinganal design generally risks to overemphasize one sub-problemwhile signicantly degrading the performancesinothers. Moreover, it showsthat thebest compro-misesolutionisnot necessarilyoptimal foranysub-problemandthus remainsunknowntothedesignerwho follows the traditional decompositionintegrationapproach.Therefore,wesuggestthatbettersolutionsfor the overall problemcanbe foundby enlargingasubproblems solutionset withslightlysuboptimaldesigns. Inotherwords, asolutionisallowedtode-viate by some tolerance from an optimal design for asubproblem, as long as the improvement with respectto another subproblem guarantees that this solution isstill optimal for the overall problem. It then is better ifa design meets all tolerances, although possibly beingTable 1 Drawback oftraditional decompositionand integration withoutcoordinationDesign f1f2f3Observationx11 1 9 Unique optimal design for subproblem with criteriaf1 andf2x21 9 1 Unique optimal design for subproblem with criteriaf1 andf3x39 1 1 Unique optimal design for subproblem with criteriaf2 andf3x42 2 2 Not optimal for any subproblem, but overall minmax solution2D decision-making for multicriteria design optimizationsuboptimal in all subproblems, rather than it is optimalfor one subproblembut violates the tolerances for someor all the others.Again consider the situation in Table 1. The optimalvalue for each of the three performance criteria equals1, and for every combination of two criteria there existsauniqueoptimal designthat achievestheseoptimalperformances. This also means that no design is optimalfor all criteria pairs, or similarly, that no design is pre-ferred in every subproblem. However, if we also acceptsuboptimal solutions and allow an additional toleranceof1thatisaddedtotheoptimalperformancevaluesof1,thenthedesignx4satisesallnewperformanceexpectationsofatleast2. Infact, x4thenistheonlysolutionthatisacceptedforall pairsofcriteriaand,thus, the preferred overall design. In accordance withtheliteratureandcommonpracticetodenotesuchtolerancevaluesbytheGreekletterorepsilon, weconform to this notation for the exposition of our newmethod in the subsequent section.3 Decomposition and integration with coordinationBecause of the broad familiarity of designers with thetraditional decompositionintegration scheme in Fig. 1,our procedure uses a similar setup and initially decom-poses the overall performance function into subsets ofcriteria, thus reducing the complexity in the individualsubproblems. In particular, as one of our objectives isto enable decision-making based on 2D visualization ofthe underlying trade-off curves, we suggest to divide theperformance indices into pairs, thereby making trade-off evaluationthesimplestpossible. Althoughasuit-able grouping is best to be chosen depending on expertknowledge of the specic problem, in principle, everygrouping is possible and may also use the same objec-tiveinmorethanonesubproblem. Thisisdiscussedinsomemoredetailincontextofthetwoillustratingexamples.3.1 Coordination between different subproblemsWhile the decomposition into pairs gives a convenientway to handle and reveal trade-offs within every sub-problem, the trade-offbetween different subproblemsthen is to be accomplished by some other mechanism.For thecoordinationbetweensubproblems, weusethe lexicographical ordering approach for multicriteriaoptimization (Rentmeesters et al. 1996) but introducethefollowingtwoessential modicationstotheorig-inal formulation. First, insteadof orderingall singleobjectives, weaccountforthespecicfeatureofourapproach and apply the lexicographic order to pairs ofcriteriaat atime(Ying1983). Secondandbasedonthediscussionintheprevioussection,weincludead-ditional epsilon tolerances to reect the implicit trade-offbetween two different subproblems, similar to theclassical epsilon-constraintmethodformultiobjectiveprogramming(ChankongandHaimes1983).Inaddi-tiontostressingtheimplicit preferenceinformationarticulatedbythevalues chosenfor epsilon, inthispaper, this numerical parameter is usedfor theco-ordinationoftheseveral decomposedbutstill multi-objective subproblems, rather than for the generationof Paretopoints usingsingleobjectiveoptimization.This substantially extends the original approach and, inparticular, guarantees that all Pareto optimal solutionsfor the complete problem can still be found through asuitable coordination. For the description of the overallprocedure, see Fig. 2.3.2 Description of procedureThe performance functionf =_ f1,f2, . . . ,fp_ for theoverall MOPis decomposed into k pairs of crite-riathatareusedasnewperformancefunctions f j=_ f j1,f j2_, j = 1, . . . , k. As described in the above para-graph, the canonical subproblem formulationsMOPj: minimize f j(x) =_ f j1(x),f j2(x)_subject to x Xaremodiedbyadditional epsilon-constraints toac-count for the additional tolerances imposed by the de-signer and to sequentially coordinate the performancetrade-offsbetweendifferentsubproblems. Therefore,we also call these new subproblems coordination prob-lems (COP)COPj: minimize f j(x) =_ f j1(x),f j2(x)_subject to fi(x) fi _xi_+ifor all i =1,. . . , j1x X,where i=_i1, i2_ R2, andi = 1, . . . , j 1aretheperformance tolerances specied by the designer.Since each coordination problem COPj is, in partic-ular,abicriteriaproblem,wecanvisualizethetrade-off betweenoptimal solutions intheformof a2DParetofrontier.Thechoiceofapreferredsolutionxjcan, thus, be basedona visual representationandonlyrequiresdecision-makingwithrespecttotwodi-mensions. The chosen designxjis communicated as aA. Engau, M. M. WiecekFig. 2 Decomposition andintegration with coordinationbaselinedesigntothesubsequentcoordinationprob-lemCOPj+1, together withtwovalues j=_j1, j2_that specify acceptable tolerances for the two optimalperformance values f j1_xj_ andf j2_xj_. This provides amechanism to also accept solutions with slightly worseperformances than the previously selected design and,eventually, to achieve better compromise solutions forthe overall problem. We later discuss possible choicesof and show how the designer, through the choice of, gains close control on the desired trade-offbetweenthedifferent coordinationproblems. If notrade-offis desirable, thedesigner mayalsochoosej= 0toguarantee that all subsequently found designs meet theperformances f j _xj_=_ f j1_xj_,f j2_xj__achievedbythe previously selected baseline design xj. Moreover, toprovide the designer with maximal exibility, all toler-ancesjcan still be updated throughout the remainingdecision-making cycle, as indicated on the left of Fig. 2.As a special feature of this procedure, the de-signer does not actually need to solve all coordinationproblems. Basedontheprevious results, it is guar-anteedthat all intermediate designs xjare alreadyat least weaklyParetooptimal fortheoverall prob-lem, althoughpossiblyinfavorofthosesubproblemsCOP1, . . . , COPj so far participating in the coordinationprocess, compared to COPj+1, . . . , COPk that are omit-ted due to early termination. Nevertheless and differentfrommostotherdecompositionanddecision-makingschemes, thedesigner maythus choosetostopthedecision-making process after every iteration and stillobtainan(atleastweakly)Paretooptimaldesignforthe overall problem.3.3 Setting and changing the tolerance values Tosupportthedesignerwiththetaskofsettingandchanging the tolerance values , or to reveal remainingtrade-off benetsanddecideuponearlytermination,we propose to performa trade-off and sensitivity analy-sisatthecurrentdesignxjwithrespecttoitsperfor-mance in different coordination problems.Toexplainthedetailsofthisanalysis, assumethatthedesigner has solvedCOP1throughCOPj1, thatis, the designer has selecteddesigns x1, x2, . . . , xj1and tolerances 1, 2, . . . , j1for all previous coordina-tion problems and arrives at COPj, as dened before.Onthebasisof2DvisualizationoftheParetocurvefor COPjandpossiblysupportedbytheunderlying2D decision-making for multicriteria design optimizationFig. 3 a Pareto curve forCOPi withfi=_ fi1,fi2_ andb its image for COPj withf j=_ f j1,f j2_numerical data, the designer should then select a newdesignxjandproceed tothenextcoordination prob-lem. By feasibility, it is always guaranteed that this newdesign satises the tolerances1, . . . , j1specied forall previousdesigns x1, . . . , xj1foundinCOP1, . . . ,COPj1. However, it might happen that the designer isnot satised with the achievable performance of xjwithrespect to the two objectives f j=_ f j1,f j2_ in COPj. Inotherwords, thedesignermight bewillingtoacceptafurtherrelaxationof oneormoreof theprevioustolerances1, . . . , j1toimproveoneorbothoftheperformancevalues f j1_xj_or f j2_xj_. Selecting i=_i1, i2_, for some 1 i j 1, this situation is depictedinFig. 3that illustrates theideaof thetrade-off atthe current designxjbetween the criterion pairs fi=_ fi1,fi2_ in COPi andf j=_ f j1,f j2_ in COPj.At rst, when i= 0, the epsilon-constraint pairfi(x) fi _xi_+iinCOPjguaranteesthatanyopti-mal solution xjfor COPj still meets the performance ofthe optimal design xiselected for coordination problemCOPi, fi _xj_fi _xi_. Then xjmust particularly lie onthe Pareto curve for COPi, as shown in Fig. 3a, with thepossible consequence that a satisfactory performance ofxjwith respect to the two criteriaf j=_ f j1,f j2_of COPjcannot anymore be achieved (see Fig. 3b). Therefore, ifwe specify some additional tolerancesi=_i1, i2_ andallowxj, but in a controlled way, to move away fromthe Pareto curve for COPi, say to xjnew in Fig. 3a, thenweexpecttogainimprovementofthepreviouslyun-satisfactory performance in COPj, indicated in Fig. 3b.Then the resulting achieved trade-offs can be computedasf j1_xj_fi1_xj_ =f j1_xj_f j1_xjnew_fi1_xjnew_fi1_xj_=f j1_xj_f j1_xjnew_1(1)and similarly forf j1(xj)fi2(xj),f j2(xj)fi1(xj), andf j2(xj)fi2(xj). In general,however, itisnotobvioushowfarweneedtomoveaway fromthe Paretocurve inCOPitoachieve asatisfactory improvement in COPj, that is, howlarge weneed to choose the value. To help the designer withthisdecision, weproposetoexaminethesensitivitiesat the current design with respect to the criteria fiinCOPiand f jinCOPj. Thehigherthesesensitivities,thehigherthepotential trade-off, sothatevensmalladditional tolerances for fi=_ fi1,fi2_ are expected toyieldasignicant improvementwithrespect to f j=_ f j1,f j2_. To enable meaningful trade-off comparisons,we then need to assume that all objectives are reason-ably scaled to the same order of magnitude.3.4 More guidance on choosing the tolerance values Toprovidethedesignerwithadditional informationon how to choose, we employ the sensitivity resultsfromnonlinear programming (Fiacco 1983; Luenberger1984), for which we formulate the auxiliary sensitivityproblem (SEP)SEPj1: minimize f j1(x)subject to fi(x) fi _xi_+ifor all i =1,. . . , j1f j2(x) f j2_xj_x X.Other than COPj, this problem minimizes only the rstof the two objectives f j=_ f j1,f j2_ while including thesecondintotheadditional constraint f j2(x) f j2_xj_.This guarantees that improvement inf j1 is not achievedthrough degradation of f j2 and, thus, that the sensitivityanalysis strictly distinguishes trade-offs that, ontheone hand, occur between the two criteria in the samesubproblem and, on the other hand, occur between twocriteria in two different coordination problems.As the formulation of this problem is only auxiliary,we do not need to actually solve it. Instead, it can easilyA. Engau, M. M. Wiecekbe shown that the design xj, that is Pareto optimal forCOPj, mustalsobeoptimalforSEPj1. Thereasontostill consider this problem is that it allows to computethesensitivitiesbetweenitsobjective f j1anditscon-straintvalues fiattheoptimal designxjintermsofthe associated Lagrangean multipliersji. Then, undersometechnical assumptions (ChankongandHaimes1983; Luenberger 1984), the sensitivity theorem statesthat f j1(x) fi1(x)x=xj= ji11 and f j1(x) fi2(x)x=xj= ji12for all i = 1, . . . , j 1 (2a)and similarly, if we change the roles of f j1andf j2andformulate the corresponding SEPj 2, f j2(x) fi1(x)x=xj= ji21 and f j2(x) fi2(x)x=xj= ji22for all i = 1, . . . , j 1. (2b)As almost all common optimization software providesLagrangeanmultiplierstogetherwithanalsolution,these sensitivities can be very efciently computed us-inganarbitraryoptimizationroutineandchoosingxjas the initial design. Then, since xjis known to be alsooptimal, we obtain the desired sensitivities in at mostone iteration and, in view of (2), the associated trade-off estimatesf j1_xj_fi1_xj_ ji11,f j1_xj_fi2_xj_ ji12,f j2_xj_fi1_xj_ ji21,f j2_xj_fi2_xj_ ji22. (3)As a rule of thumb, we can state that the larger the mag-nitude of a computed Lagrangean multiplier, the largerthetrade-off wecanexpect. Forexample, if ji11 1,then this tells us that even a small additional tolerancei1on fi1_xj_canbeexpectedtoyieldasignicantimprovement inf j1_xj_. On the other hand, if, for ex-ample, ji21< 1, then the improvement inf j2_xj_ is com-parably smaller than the allowed tolerancei1that wewould be willing to give up forfi1_xj_. In general, a sen-sitivity threshold less than 1 indicates the an additionaltolerance exceeds the expected improvement and, thus,that the corresponding trade-off is not favorable at thecurrent design. Then, if all sensitivities becomelessthan1, thissuggeststoterminatetheprocedurewiththe current design as the nal preferred design for theoverall problem. Ifsuchadesigncannotberevealedinadesirednumber of iterations, thedesigner maydecidetoallowmoreiterations,modifytheproposedsensitivity threshold, or start with a different baselinedesign.Theonlyremaininglimitationof theprocedureisthat, ingeneral, wearenot abletopreciselypredictthe actual trade-offs to be achieved. This is because allformersensitivitiesonlyprovideuswithlocal trade-off informationat xjand, thus, areaccurateonlyifthedesignerdecidestoallowaverysmall tolerance.Nevertheless,ifthechosentolerancei1isverysmall,by (1) and (3) it then can be expected that the resultingimprovement inf j1_xj_ or f j2_xj_ is approximatelyf j1_xj_ ji11fi1_xj_= ji11i1,orf j2_xj_ ji21fi1_xj_= ji21i1, (4a)respectively. Similarly, if the designer decides to choosea very small tolerance i2, thenf j1_xj_ ji12fi2_xj_= ji12i2,orf j2_xj_ ji22fi2_xj_= ji22i2. (4b)4 ExamplesFor a demonstrationof the proposed2Ddecision-making procedure and the underlying coordinationmechanism, we adopt the role of a hypotheticaldecision-makerandapplythismethodtotwoexam-plesfrommultiobjectiveprogrammingandstructuraldesign optimization. By the nature of this approach, itisunavoidablethatallthedecisionswemakeremainsubjectiveand,inpractice,wouldalsodependontheexpertise, preferences, andperformanceexpectationsof the actual designer.4.1 A mathematical programming problemTherstproblemconsistsoffourquadraticobjectivefunctionsf1, f2, f3, andf4, which need to be minimized,subject to three inequality constraints g1, g2, and g3 intwo variables x1 and x2.Minimize _ f1 (x1, x2) =(x12)2+(x21)2,f2 (x1, x2) = x21+(x23)2,f3 (x1, x2) =(x11)2+(x2+1)2,f4 (x1, x2) =(x1+1)2+(x21)2_subject to g1 (x1, x2) = x21 x2 0g2 (x1, x2) = x1+ x2 2 0g3 (x1, x2) = x1 0.The feasible set for this problemis X = {x R2:gi(x) 0, i = 1, 2, 3}, andinspiteof amissingphysi-calmeaningwecall Xthesetofallfeasibledesigns.2D decision-making for multicriteria design optimizationWithout practical interpretation, however, the four ob-jectives alsolackrelativeimportances and, thus, donotallowforpriorityrankingaccordingtotheirrele-vanceontheoverallperformance. Instead, wegroupthe objectives into the canonical pairs f1=_ f11,f12_=( f1,f2)and f2=_ f21,f22_= ( f3,f4)and, accordingly,reduce further notational burden by replacing all dou-ble indices11,12,21, and22 by1,2,3, and4, respectively.Tondapreferreddesigntotheaboveproblemusing the proposed procedure, we would then start bysolvingCOP1: minimize f1= [ f1(x),f2(x)]subject to x X.Recall that solving COP1 means to nd a set of Paretosolutions by either sampling, genetic algorithms, orclassical optimization, andthentoselect onedesignx1that will beusedas thebaselinedesignfor thesecondcoordinationproblem. Forexample, byusingthe classical epsilon-constraint method (Chankong andHaimes 1983), wecanndthetenParetosolutionsthataredepictedontheleftinFig.4andthenselectthehighlightedpointasbaselinedesignx1=(0.4471,1.5529). Inapractical context, this choicewouldbejustied by the fact that this design yields comparableperformancesof f1_x1_=2.7173and f2_x1_=2.2939and, thus, isoneofthebestcompromisedesignsforCOP1. Also, for later reference, we calculate f3_x1_ =6.8232 andf4_x1_ = 2.3997.Notehow, sofar, ourdecisionismerelybasedonthevisualizationofthetwoobjectives f1and f2and,hence, found by 2D decision-making. The visualizationof the Pareto designs for COP1 with respect to the twoothercriteria f3and f4ontheright inFig. 4isnotneeded, butisprovidedhereforconvenience. More-over, although usually not readily available, both plotsdepictasampleofthecompletefeasiblesettoshowhowall Paretosolutionsfortherstsubproblemareamong the worst solutions for the second. Recall thattraditional decompositionmethodsthatdonotallowfor abetter coordinationwouldalreadystopat thispoint and, although clearly undesirable, propose thesesolutions as nal designs for the overall problem.Before we formulate the coordination problemCOP2, weinvestigatethesensitivities of thecurrentdesign x1with respect to the two criteria f3andf4bycomputing the Lagrangean multipliers for the sensitiv-ity problemsSEP3: minimize f3(x)subject to f1(x) f1_x1_+1f2(x) f2_x1_+2f4(x) f4_x1_x XSEP4: minimize f4(x)subject to f1(x) f1_x1_+1f2(x) f2_x1_+2f3(x) f3_x1_x Xwhere, initially, 1 = 2 = 0. Thenchoosingtheopti-mal designx1as the initial design in our optimizationroutine, this immediately (after one iteration) conrmsoptimality of x1for both SEP3 and SEP4 and, accordingtothesensitivitytheorem(2), provides us withthesensitivity information f3(x) f1(x)x=x1= 31 = 0.1706, f3(x) f2(x)x=x1= 32 = 1.8294, f4(x) f1(x)x=x1= 41 = 1.1706, f4(x) f2(x)x=x1= 42 = 0.8294.Hence, we see that only two trade-off values are greaterthan1, therebysuggestingthat improvement in f3,or f4, is best achievedbyallowingsomeadditionalFig. 4 Pareto solutions forCOP1 withf1= ( f1,f2) andtheir images for f2= ( f3,f4)A. Engau, M. M. WiecekFig. 5 On the right, Paretosolutions for COP2 withf2= ( f3,f4) and, on the left,their images for f1= ( f1,f2),with tolerances = (1, 2) =(1, 2)tolerance2on f2, or 1on f1, respectively, beforesolving the second coordination problemCOP2: minimize f2= [ f3(x),f4(x)]subject to f1(x) f1_x1_+1 = 2.7173 +1f2(x) f2_x1_+2 = 2.2939 +2x X.If we decide to focus on the more promising trade-offbetweenf3 andf2, we might set the new tolerance2 = 1 and, after solving COP2 with = (1, 2) = (0, 1),select the new design xnew = (0.4402, 1.2393) with per-formancesf(xnew) = (2.4903, 3.2939, 5.3278, 2.1314). Inparticular,theactuallyachievedtrade-off between f3andf2 can now be computed asf3_x1_f2_x1_ =f3_x1_f3 (xnew)f2 (xnew) f2_x1_=6.8232 5.32783.2939 2.2939 = 1.4954.Note that although the chosen tolerance 2 = 1 is ratherlargecomparedtotheperformancevalue f2 _x1_=2.2939, the local trade-off 32 = 1.8294 at x1providesa quite reasonable estimate for the actual trade-offof1.4954.Howeverandasemphasizedbefore, thelocaltrade-offvalues should not be mistaken as predictionsfor the trade-offs that are achieved globally, especiallywhen we decide to change the tolerance values1 and2 simultaneously.Forillustration, assumethatwenowdecidetoset = (1, 2) =(1, 2). Based on the initial trade-offvalues computed at x1and the trade-offestimates (4),wemightexpecttogainimprovementsof f3_x1_322 = 1.8294 2 = 3.6588 and f4_x1_ 411 =1.1706 1 = 1.1706. ThensolvingCOP2for, say, venewdesigns,weobtaintheParetosolutionsofCOP2depicted in Fig. 5, which, together with their sensitivi-ties, are listed in Table 2. For convenience, we also cir-cle all those sampled outcomes from Fig. 4 that satisfythenewperformanceboundsof f1_x1_+1=3.7173andf2_x1_+2 = 4.2939 and, thus, form the underlyingset of feasible points for COP2.Note howthe maximal improvements with respect tof3andf4,

max f3 _x1_=f3 _x1_f3(0.5026, 0.9897)= 6.8232 4.2064= 2.6168 (expected: 3.6588)

max f4 _x1_=f4 _x1_f4(0.0720, 1.0000)= 2.3997 1.1491= 1.2506 (expected: 1.1706)areachievedbytwoof thevedifferent Paretoso-lutionsand, thus, arenot achievablesimultaneously.Therefore, a preferred solution must also compromisebetweenthesetwoimprovements, whichthencanbeaccomplished by 2D decision-making for COP2. Againselecting the best compromise design, we nd the thirddesigninTable2, x2=(0.2400, 0.9418), highlightedinFig. 5, withperformances f _x2_=(3.1009, 4.2939,4.3481, 1.5410) as the preferred overall design. In par-ticular, notethat at this designall sensitivities and,hence, all remainingtrade-offsarereducedtovaluesless than 1, suggesting the termination of the algorithm.Table 2 Pareto solutions forCOP2 as depicted in Fig. 5and their updated sensitivitiesx1x2f1f2f3f4313241420.5026 0.9897 2.2424 4.2939 4.2064 2.2578 0 0.9898 1.0034 00.3838 0.9637 2.6133 4.2939 4.2358 1.9163 0 0.9898 0.8558 00.2400 0.9418 3.1009 4.2939 4.3481 1.5410 0 0.9898 0.7036 00.1220 0.9314 3.5317 4.2939 4.5013 1.2635 0 0.9898 0.5964 00.0720 1.0000 3.7173 4.0052 4.8613 1.1491 0 1.0602 0.5560 02D decision-making for multicriteria design optimizationFig. 6 Three different loading conditions on a four-bar plane truss structureToconcludethisexample, alsoobservethatwhileour nal and all highlighted designs in Fig. 5 are Paretooptimal for the COP2 and, thus, (at least weakly) Paretooptimal for the original MOP, none of these solutionsactually lies on the Pareto curve for either the rst orsecond subproblem. Hence, as expected, our method-ology is capable to nd a best compromise design thatcannot befoundusingthetraditional decompositionapproach.4.2 A structural optimization problemWhilethediscussionof therst examplefocusedindetail on the evaluation of trade-offs, we omit some ofthose details for our more practical second test problemto design a four-bar plane truss structure (Koski 1984,1988; Stadler and Dauer 1992).Theoriginal mathematical model isabi-objectiveprogram with the two conicting objectives of minimiz-ing both the volume V of the truss and the displacementd1of the node joining bars 1 and 2, given the loadingcondition depicted for the leftmost truss in Fig. 6.In addition, we use the two additional loading con-ditions(Koski1984)thataredepictedforthemiddleand right truss in Fig. 6, for which one can compute thetworespectivedisplacementsd2andd3(Blouin2004,private communication). The length L = 200 cm of thestructure, the acting force F = 10 kN, Youngs modulusof elasticity E = 2 105kN/cm2, and the only nonzerostresscomponent =10kN/cm2areassumedtobeconstant. The cross-sectional areas x1, x2, x3, and x4 ofthe four bars are subject to physical restrictions yieldingthe feasible design setX =_x = (x1, x2, x3, x4) : (F/) x1, x4 3(F/),2(F/) x2, x3 3(F/)_.Theoverallproblemthenbecomestheselectionofafeasible preferred design that minimizes the structuralvolume V(x) and joint displacements d1, d2, and d3 forthe three different loading conditionsMinimize_V(x) =L_2x1+2x2+2x3+ x4_,d1(x) =FLE_2x1+22x222x3+2x4_,d2(x) =FLE_2x1+22x2+42x3+6x4_,d3(x) =FLE_62x3+3x4__ subject to xX.This problem can be viewed as a multi-scenario multi-objective problem(Fadel et al. 2005), with each loadingconditionactingas oneof threepossiblescenarios.Based on the assumption that the rst loading scenariooccurs most oftenandthethirdscenarioonlyveryrarely, we decompose the overall problemintothethree associated criteria pairs f1=_ f11,f12_= (d1, V),f2=_ f21,f22_= (d2, V), and f3=_ f31,f32_= (d3, V).Notehowthevolumecriterionparticipatesineveryscenarioand, thus, is repeatedineachsubproblem.Inparticular, byincludingadditional scenarios, thisexample readily extends to problems with ve or anyother number of objectives, and the restriction to threeismerelyfortheeaseofdemonstration. Then, usingthe proposed procedure to nd a common design thatis preferred for all given scenarios, we start by solvingCOP1: minimizef1(x) =_d1(x), V(x)_subject to x X.Again, recall that solving COP1 means to nd a numberof Paretodesigns, fromwhichwe needtoselect arst baselinedesignforthesubsequent coordinationproblems. As before, we choose the epsilon-constraintmethod and nd the ten solutions that are highlightedin Fig. 7.A. Engau, M. M. WiecekFig. 7 From left to right,Pareto solutions for COP1(rst scenario) and theirimages for the second andthird scenarioBased on the 2Ddecision-making as proposed by ourprocedure, we only focus on the rst subproblem andselectthehighlightedpointasthepreferredcompro-mise design for COP1. The corresponding designx1=(1.7459, 2.4732, 1.4142, 2.4730)isillustratedinFig. 9andhasperformancevaluesof V _x1_=2,292.5cm3,d1_x1_ = 0.0110 cm, d2_x1_ = 0.0721 cm, and d3_x1_ =0.0872 cm.Before we solve the next coordination problems, weformulate the two sensitivity problemsSEP21: minimized2(x)subject tod1(x) d1_x1_+d1V(x) V _x1_+Vx XSEP31: minimized3(x)subject tod1(x) d1_x1_+d1V(x) V _x1_+Vx Xbut, forconceptual simplicity, restrictouranalysistothe trade-offbetween the different deection criteria.Solving the two sensitivity problems SEP21and SEP31withx1as initial design, we obtain the associated La-grangean multipliersd2(x)d1(x)x=x1= 21 = 413.77 andd3(x)d1(x)x=x1= 31 = 538.03.Theverylargemagnitudesofthesevaluesclearlyin-dicatethatthecurrentlyselecteddesignx1shouldbefurther improved. However, as emphasized before andquite obvious at this point, these values do not give usan actual prediction on the improvement that we shouldexpect.Consideringthat thecurrent rst nodedeectionof d1_x1_= 0.0110 cm is signicantly smaller than thesecondandthird, d2_x1_=0.0721cmandd3_x1_=0.0872 cm, we assume that we are still willing to acceptadesignthat yields arst nodedeectionof upto0.03 cm, provided that there is a reasonable trade-offwithrespect tod2or d3. Thus, whilesettingV = 0tomaintainthecurrent volume, weonlychangetheFig. 8 In the center, Paretosolutions for COP2 (secondscenario) and, on the left andright, their images for the rstand third scenario,respectively2D decision-making for multicriteria design optimizationFig. 9 Selected Paretodesigns x1for COP1 (rstscenario, left) and x2forCOP2 (second scenario, right)tolerance valued1= 0.02 cm and then solve the nextcoordination problemCOP2: minimizef2=_d2(x), V(x)_subject to d1(x) d1(x1) +d1 = 0.0310V(x) V _x1_+V = 2, 292.5x X.By solving COP2 for the second scenario, we nd itsset of Pareto optimal designs that is depicted in the mid-dle plot of Fig. 8, with the corresponding performancesfor the rst and third scenario depicted on the left andright, respectively.Note fromthe left plot that all these newsolutions, infact, meet the specied upper performance bound on d1for the rst subproblem, while now visualizing a trade-off betweenthevolumeandsecondnodedeectionin COP2. Assuming that the main incentive is still theimprovement withrespect tonodedeectiond2, wedecidenot tofurtherimprovethestructural volumeand, thus, selectthehighlightedbottommostpointasthe improved second design x2= (1.1754, 1.6582, 2.7837,2.8298), as depicted in Fig. 9.TheperformancesofthisnewdesignaregivenbyV _x2_= 2, 292.5 cm3, d1_x2_= 0.0310 cm, d2_x2_=0.0411 cm, and d3_x2_ = 0.0756 cm. Hence, the actualtrade-offs achieved ared2(x)d1(x)=d2_x1_d2_x2_d1_x2_d1_x1_=0.0721 0.04110.0310 0.0110 = 1.5500,d3(x)d1(x)=d3_x1_d3_x2_d1_x2_d1_x1_=0.0872 0.07560.0310 0.0110 = 0.5800.As expected, the trade-off between d1and d2achievedavaluegreaterthan1and, thus, wasafa-vorableone. Thesmallertrade-off valuebetweend1and d3 is not surprising either as we only solved COP2which, in fact, did not minimize with respect to d3. Nev-ertheless, from the right plot in Fig. 8, we see that thecurrentdesignx2alreadygivesareasonablecompro-mise solution with respect to the third loading scenariothat also involves d3. In particular, upon computing theupdatedsensitivity21fromSEP21atthenewdesignx2and, in addition, including the additional constraintd2(x) d2_x2_+d2 with_d1, d2, V_= (0.02, 0, 0) intoa new SEP31minimize d3(x)subject to d1(x) d1_x1_+d1 = d1_x2_= 0.0310d2(x) d2_x2_+d2 = 0.0411V(x) V _x2_+V = 2, 292.5x X,we obtain a set of new Lagrangean multipliers, yieldingthe updated sensitivitiesd2(x)d1(x)x=x2= 21 = 0.7877,d3(x)d1(x)x=x2= 31 = 0,d3(x)d2(x)x=x2= 32 = 0.2481.In particular, since all Lagrangean multipliers are nowlessthan1, thesesensitivitiesreveal that solvingthethirdcoordinationproblemisunlikelytoyieldsignif-icant further improvement and, thus, suggests the ter-mination of this problem with x2from Fig. 9 as the naldesign.5 ConclusionIn this paper, we propose an interactive decision-making procedure to select an overall preferred designfor acomplexandmulticriteriadesignoptimizationproblem. Taking into account that these problems areA. Engau, M. M. Wiecekusuallysolvedbymultipledesignersormultidiscipli-nary design teams, the selection of a nal design is fa-cilitated using a decompositionintegration framework,together witha novel coordinationmechanismthatguaranteesthat thechosendesignis alsocommonlypreferred for the overall problem.The method requires that only one subproblemneeds to be solved at a time, upon which the designercommunicates a new baseline design and a set of newtolerances totheother designteams tosequentiallycoordinatethenal designintegration. Moreover, inorder to not overload the individual designers with an-alytical analyses and to make trade-offevaluation anddecision-making the simplest possible, all decisions arereduced to merely two dimensions. As one of the mainfeaturesofthisprocedure, thisalsoenablesthecom-pletevisualizationofall designdecisionsintheformof 2D trade-off curves.The specication of tolerances for a selected designis supportedbyanadditional trade-off analysis thatprovides a very efcient way to compute design sensi-tivitieswithrespecttodifferentperformancecriteria.In particular, this extends the consideration of perfor-mance trade-offs within one subproblem to trade-offsthatoccurbetweendifferentsubproblems, infurtherenhancementof theproposedcoordinationandnaldesign integration.Based on several theoretical results and the illustra-tion of the procedure on two examples, this method hasbeen shown to1. Offerthecapabilityofreplacingthecumbersometrade-off analysis for an overall multicriteria prob-lemwith trade-off analyses for a collection ofsmaller-sized bicriteria problems2. Makedesigners judgment andknowledgeaboutsmaller-sizedproblemssufcientforthenal de-sign integration3. AllowdesignerstovisualizetheParetocurvesineach subproblem and for each decision4. Provide a general framework for the independentparticipation of multiple designersIn our future work, we intend to further investigatethe information that can be obtained fromthe proposedtrade-off and sensitivity analysis. In view of the currentapproach, weareawareof theremainingweaknessthatthisinformationonlyallowsalocaltrade-off as-sessment and, thus, cannot be used for more accurateestimates ina larger regionof the outcome space.Wewouldalsoliketoaddressremainingissuessuchas computational benchmarking or further analysis ofeffects from grouping and ordering of objectives usingexamples fromtheindustry.Webelieve thatsuchfu-ture efforts will further improve the recognized featuresof the current method and eventually provide an effec-tive and exible decision-making tool for multicriteriadesign optimization.Acknowledgements ThisresearchhasbeensupportedbytheNational Science Foundation, grant number DMS-0425768, andby the Automotive Research Center, a US Army TACOMCenterofExcellenceforModelingandSimulationofGroundVehicles at the University of Michigan. The authors thank V.Y.BlouinandG.M. 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