M 16888 S077 Multidisciplinary System Design Optimization(MSDO) Multiobjective Optimization () ecture 17 Apri5,2004 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 (II) Lecture 17 April 5, 2004 by Prof. Olivier de Weck
M Moo 2 Lecture Outline 16888 E5077 Lecture 2(today) Alternatives to Weighted Sum(WS) Approach Multiobjective Heuristic Programming Utility function Optimization Physical Programming(Prof Messac Application to Space System Optimization Lab preview Friday 4-9-2003- Section 1) o Massachusetts Institute of Technology -Prof de Weck and Prof. Willcox 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 MOO 2 Lecture Outline MOO 2 Lecture Outline Lecture 2 (today) • Alternatives to Weighted Sum (WS) Approach • Multiobjective Heuristic Programming • Utility Function Optimization • Physical Programming (Prof. Messac) • Application to Space System Optimization • Lab Preview (Friday 4-9-2003 – Section 1)
Mlesd Weighted Sum(WS)Approach 50.1 Min J2) MO ∑ miss this i =1 S concave region convert back to sop LP in J-space easy to implement > Pareto scaling important front weighting determines 0 Which point along pf is J-hyperplane utopia found misses concave PF Max(J1) 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 Weighted Sum (WS) Approach Weighted Sum (WS) Approach 1 z i MO i i i w J J = sf = ¦ utopia Max( J 1 ) Min( J 2 ) miss this concave region Pareto front • convert back to SOP • LP in J-space • easy to implement • scaling important ! • weighting determines which point along PF is found • misses concave PF w 2>w1 w1>w 2 J-hyperplane J*i J*i+1
MIlesd Weighted Square Sum Approach齡别 J=w1J12+w2 Obj Fun. Line Ref: messac J2 o Massachusetts Institute of Technology -Prof. de Weck and Prof Willcox 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 Weighted Square Sum Approach Weighted Square Sum Approach 2 2 11 2 2 J = + wJ wJ Obj. Fun. Line J1 J2 Ref: Messac
MIlesd Compromise Programming(CP)E5., J=W,,twJ Obj. Fun. Line This allows “ access" to the non-convex part of the pAreto front 3 5 6 55 Obiective 1 o Massachusetts istitute of Technology -Prof de Weck and Prof. Willcox 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 Compromise Programming (CP) Compromise Programming (CP) Obj. Fun. Line 11 2 2 n n J wJ wJ = + 55 This allows “access” to the non-convex part of the Pareto front
M d Multiobjective Heuristics 16888 E5077 Pareto Fitness- Ranking Recall: Multiobjective GA This number comes from the d-matrix Pareto ranking scheme Allows ranking of population 0 without assigning preferences …… or weights to individual 十 objectives :1:…+ Successive ranking and removal scheme i÷-.· Deciding on fitness of dominated solutions is more difficult Pareto ranking for a minimization problem e of Technology -Prof. de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
6 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Multiobjective Heuristics Multiobjective Heuristics • Pareto ranking scheme • Allows ranking of population without assigning preferences or weights to individual objectives • Successive ranking and removal scheme • Deciding on fitness of dominated solutions is more difficult. Pareto ranking for a minimization problem. Pareto Fitness - Ranking Recall: Multiobjective GA This number comes from the D-matrix
Mlesd Example Multiobjective GA 16888 E5077 Minimization Objective 1 f(x,x)=1-p∑x n x p ex x+ Objective 2 No mating Generaton 100 restrictions With mating restrictions nates individuals Nondominated individuals O Dominated ind ividuals -Best Radle eff found (cunt Best trade off found (cumulative) Artial Pareto set o Massachusetts Institute of Technology -Prof. de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
7 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Example Multiobjective GA Example Multiobjective GA ( ) 2 1 1 1 1 ,..., 1 exp n n i i fx x x = n ª º § · =− − − « » ¨ ¸ « » © ¹ ¬ ¼ ¦ Minimization Objective 1 Objective 2 ( ) 2 1 1 1 1 ,..., 1 exp n n i i fx x x = n ª º § · =− − + « » ¨ ¸ « » © ¹ ¬ ¼ ¦ No mating restrictions With mating restrictions
M esd Double Peaks Example: MO-GA ES 77 Multiobjective Genetic algorithm Generation 1 Generation 10 o Massachusetts Institute of Technology -Prof de Weck and Prof. Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
8 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Double Peaks Example: MO Double Peaks Example: MO -GA Multiobjective Genetic Algorithm Generation 1 Generation 10
Mlesd Utility Function Approach 16888 S077 Decision maker has utility function U: R2>R This function might or might not be known mathematically U maps objective vector to the real line MOP:max{(小)J=Cx,x∈S} MNP:ma(y(0)J=(x,x∈S Example 0,0 maxJ=x,+x21 (0,4) maX V2=x s. x3=(3, U=18 4x,+3x,<12 S 、U=24 x,x1≥0 Where U=2J, 12 o Massachusetts Institute of Technology -Prof de Weck and Prof. Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
9 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Utility Function Approach Utility Function Approach Decision maker has utility function This function might or might not be known mathematically U maps objective vector to the real line : z U \ \ → MOLP: MONLP: max , {U S ( ) J J Cx x = ∈ } max , {UfS () () JJ x x = ∈ } (0,0) (0,4) (3,0) = = = 1 2 3 x x x { } { } 112 2 1 1 2 1 2 1 2 max max s.t. 4x 3 12 , 0 where 2 J x x J x x x x U JJ = + = + ≤ ≥ = x 1 x 2 c1 x 2 c 2 x 3 x1 S U=24 U=18 Example:
Mlesd Utility Function Shapes 16888 E5077 Monotonic Strictly Concave Increasing Concave Convex Non-monotonic decreasing Convex Cook Smaller-is-better(SIB) Nominal-is Range Larger-is-better(LIB) -better(NIB) -is-better(RIB Messac. Class 1S Class 3s Class 4S Class 2S o Massachusetts Institute of Technology -Prof de Weck and Prof. Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
10 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Utility Function Shapes Utility Function Shapes Ji Ui Ji Ui Ui Ji Ui Monotonic Ji increasing decreasing Strictly Concave Convex Concave Convex Non-monotonic Cook: Smaller-is-better (SIB) Larger-is-better (LIB) Nominal-is -better (NIB) Range -is-better (RIB) - Messac: Class 1S Class 2S Class 3S Class 4S