M 16888 S077 Multidisciplinary System Design Optimization(MSDO) Multiobjective Optimization ( ecture 16 31 March 2004 Prof. olivier de Weck o Massachusetts Institute of Technology -Prof. de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
1 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Multidisciplinary System Multidisciplinary System Design Optimization (MSDO) Design Optimization (MSDO) Multiobjective Optimization (I) Lecture 16 31 March 2004 by Prof. Olivier de Weck
M Where in Framework 16888 E077 Objective Vector Discipline A Discipline B Discipline c Coupling Multiobjective Optimization Approx Optimization Algorithms Methods Numerical Techniques Sensitivit Tradespace (direct and penalty methods Analysis Exploration Heuristic Techniques Coupling (DOE) SA, GA, Tabu Search) performance o Massachusetts Institute of Technology -Prof. de Weck and Prof. Willco Engineering Systems Division and Dept of Aeronautics and Astronautics
2 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Where in Framework ? Where in Framework ? Discipline A Discipline B Discipline C nI put Ou pt ut Tradespace Exploration (DOE) Optimization Algorithms Numerical Techniques (direct and penalty methods) Heuristic Techniques (SA,GA, Tabu Search) 1 2 n x x x ª º « » « » « » « » « » ¬ ¼ # Coupling 1 2 z J J J ª º « » « » « » « » « » ¬ ¼ # Approximation Methods Coupling Sensitivity Analysis Multiobjective Optimization Isoperformance Objective Vector
M Lecture content 16888 E5077 Why multiobjective optimization? EXample-twin peaks optimization History of multiobjective optimization Weighted Sum Approach(Convex Combination Dominance and Pareto-Optimality Pareto Front Computation -NBI o Massachusetts Institute of Technology -Prof. de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
3 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Lecture Content Lecture Content • Why multiobjective optimization? • Example – twin peaks optimization • History of multiobjective optimization • Weighted Sum Approach (Convex Combination) • Dominance and Pareto-Optimality • Pareto Front Computation - NBI
M Multiobjective Optimization Problem.888 S077 Formal Definition Design problem may be formulated as a problem of Nonlinear programming(NLP). When Multiple objectives(criteria) are present we have a MONLP minJ(x, P) wheJ=[(x)…J:( s.g(x,p)≤0 X h(x, p) g =g1(x)…gn(x) m1 i LB UB X o Massachusetts Institute of Technology -Prof. de Weck and Prof. Willco Engineering Systems Division and Dept of Aeronautics and Astronautics
4 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Multiobjective Optimization Problem Multiobjective Optimization Problem Formal Definition Formal Definition ( ) , , 1, ..., ) min , s.t. , 0 , =0 ( i LB i i UB x xx i n = ≤ ≤ ≤ - [S J [ S K [ S Design problem may be formulated as a problem of Nonlinear Programming (NLP). When Multiple objectives (criteria) are present we have a MONLP () () [ ] 1 2 1 1 1 1 where () () () () = ª º ¬ ¼ = = ª º ¬ ¼ = ª º ¬ ¼ " " " " " T z T i n T m T m J J x x x g g h h -[ [ [ J[ [ K[ [
M Multiple objectives 16888 S077 The objective can be a vector j of z system responses or characteristics we are trying to maximize or minimize cost Often the objective is a scalar function but for range [km]I real systems often we attempt multi-objective J 3 weight[kg] optimization JiI-data rate [bps XHJX) Objectives are usually ROi[% conflicting o Massachusetts Institute of Technology -Prof. de Weck and Prof. Willco Engineering Systems Division and Dept of Aeronautics and Astronautics
5 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Multiple Objectives Multiple Objectives 1 2 3 cost [$] - ran ge [km] wei ght [k g ] - data rate [bps] - ROI [%] i z J J J J J ª º ª º « » « » « » « » « » « » = = « » « » « » « » « » « » « » « » « » ¬ ¼ ¬ ¼ - # # The objective can be a vector J of z system responses or characteristics we are trying to maximize or minimize Often the objective is a scalar function, but for real systems often we attempt multi-objective optimization: [ - [ 6 Objectives are usually conflicting