1 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and AstronauticsMultidisciplinary System Design Optimization (MSDO)Multiobjective Optimization (I)Lecture 14byDr. Anas Alfaris2 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and AstronauticsWhere in Framework ?Discipline A Discipline BDiscipline CTradespaceExploration(DOE)Optimization AlgorithmsNumerical Techniques(direct and penalty methods)Heuristic Techniques(SA,GA, Tabu Search)12nxxxCoupling12zJJJApproximationMethodsCouplingSensitivity AnalysisMultiobjectiveOptimizationIsoperformanceObjective Vector3 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and AstronauticsLecture Content• Why multiobjective optimization?• Example – twin peaks optimization• History of multiobjective optimization• Weighted Sum Approach • Pareto-Optimality• Dominance and Pareto Filtering4 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and AstronauticsMultiobjective Optimization Problem Formal Definition,,1,..., )min ,s.t. , 0 , =0(i LB i i UBinx x xJ x pg(x p)h(x p)Design problem may be formulatedas a problem of Nonlinear Programming (NLP). WhenMultiple objectives (criteria) are present we have a MONLP121111where ( ) ( )( ) ( )TzTinTmTmJJx x xgghhJ x xxg x xh x x5 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and AstronauticsMultiple Objectives123cost [$]- range [km]weight [kg]- data rate [bps]- ROI [%]izJJJJJJThe objective can be a vector J of z system responsesor characteristics we are trying to maximize or minimizeOften the objective is ascalar function, but forreal systems often we attempt multi-objectiveoptimization: x J(x)Objectives are usuallyconflicting.6 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and AstronauticsWhy multiobjective optimization ?While multidisciplinary design can be associated with the traditional disciplines such as aerodynamics, propulsion, structures, and controls there are also the lifecycle areas of manufacturability, supportability, and cost which require consideration.After all, it is the balanced design with equal or weighted treatment of performance, cost, manufacturability and supportability which has to be the ultimate goal of multidisciplinary system design optimization.Design attempts to satisfy multiple, possiblyconflicting objectives at once.7 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and AstronauticsExample: F/A-18 AircraftDesignDecisionsObjectivesAspect RatioDihedral AngleVertical Tail AreaEngine Thrust Skin Thickness# of EnginesFuselage SplicesSuspension PointsLocation of MissionComputerAccess Door LocationsSpeedRangePayload CapabilityRadar Cross SectionStall SpeedStowed VolumeAcquisition costCost/Flight hourMTBFEngine swap timeAssembly hoursAvionics growthPotential8 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and AstronauticsMultiobjective ExamplesProduction Planningmax {total net revenue}max {min net revenue in any time period}min {backorders}min {overtime}min {finished goods inventory}Aircraft Designmax {range}max {passenger volume}max {payload mass}min {specific fuel consumption}max {cruise speed}min {lifecycle cost}12zJJJJDesignOptimizationOperationsResearch9 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and AstronauticsMultiobjective vs. Multidisciplinary• Multiobjective Optimization– Optimizing conflicting objectives– e.g., Cost, Mass, Deformation – Issues: Form Objective Function that represents designer preference! Methods used to date are largely primitive.• Multidisciplinary Design Optimization– Optimization involves several disciplines– e.g. Structures, Control, Aero, Manufacturing– Issues: Human and computational infrastructure, cultural, administrative, communication, software, computing time, methods• All optimization is (or should be) multiobjective– Minimizing mass alone, as is often done, is problematic10 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and AstronauticsMultidisciplinary vs. Multiobjectivesingle discipline multiple disciplinessingle objectivemultiple obj.single disciplinemultiple disciplinesMinimize displacements.t. mass and loading constraintFlmcantilever beamsupport bracketMinimize stamping costs (mfg) subjectto loading and geometryconstraintFD$airfoil(x,y)Maximize CL/CDand maximizewing fuel volume for specified voVfuelvoMinimize SFC and maximize cruisespeed s.t. fixed range and payload commercial aircraft11 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and AstronauticsExample: Double Peaks OptimizationObjective: max J= [ J1J2]T(demo)2212221222122( 1)1135112( 2)12311053 0.5 2xxxxxxJ x exx x ee x x22212 2 2 22 1 2 12( 1)22(2 )352213110 35xxx x x xJ x exx x e e12 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and AstronauticsDouble peaks optimizationOptimum for J1alone:Optimum for J2alone:x1* =0.05321.5973J1* = 8.9280J2(x1*)= -4.8202x2* =-1.58080.0095J1(x2*)= -6.4858 J2* = 8.1118Each point x1* and x2* optimizes objectives J1and J2individually.Unfortunately, at these points the other objective exhibits a low objective function value. There is no single point that simultaneouslyoptimizes both objectives J1and J2!13 © Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxEngineering Systems Division and Dept. of Aeronautics and AstronauticsTradeoff between J1and J2• Want to do well with respect to both J1and J2• Define new objective function: Jtot=J1+ J2• Optimize JtotResult:Xtot*=0.87310.5664J(xtot*) =3.0173 J13.1267 J2Jtot* = 6.1439=max(J1)max(J2)tradeoffsolutionmax(J1+J2)14 © Massachusetts Institute of Technology - Prof. de