M esd Example: Double Peaks Optimization ES. 71 Objective: max J=[, d2]' (demo) objective J,( x2) bjective 2(, x2) 23 x1 x2-(x2+1)2 x e J2=3(+x)e ,_30-(2-2) 5 X e -(x1+2)-x of techno +0.5(2x, +x, ision and Dept ot Aeronautics and astronautics
11 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Example: Double Peaks Optimization Example: Double Peaks Optimization Objective: max J= [ J 1 J 2 ] T (demo) ( ) ( ) 2 2 1 2 2 2 1 2 2 2 1 2 2 ( 1) 1 1 1 3 5 1 2 ( 2) 1 2 3 1 10 5 3 0.5 2 x x x x x x J xe x x xe e xx −− + − − −+ − = − § · − −− ¨ ¸ © ¹ − ++ ( ) 2 2 2 1 22 22 21 2 1 2 ( 1) 2 2 2 3 5 (2 ) 2 1 3 1 10 3 5 x x xx x x J xe x x xe e − + − −− − = + § · − −+− − ¨ ¸ © ¹
M d Double peaks optimization 16888 E5077 Optimum for J, alone Optimum for 2 alone X X 0.0532 5808 1.5973 100095 J*=89280 (×2)=-6:4858 J2(x1)=-48202 丿2=8.1118 Each point x1* and x2" optimizes objectives J, and J2 individually Unfortunately at these points the other objective exhibits a low objective function value. There is no single point that simultaneously optimizes both objectives J, and J2! o Massachusetts Institute of Technology-Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
12 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Double peaks optimization Double peaks optimization Optimum for J 1 alone: Optimum for J 2 alone: x 1 * = 0.0532 1.5973 J 1 * = 8.9280 J 2 ( x 1 *)= -4.8202 x 2 * = -1.5808 0.0095 J 1 ( x 2 *)= -6.4858 J 2 * = 8.1118 Each point x1* and x2* optimizes objectives J1 and J 2 individually. Unfortunately, at these points the other objective exhibits a low objective function value. There is no single point that simultaneously optimizes both objectives J1 and J 2 !
Mlesd tradeoff between J, and J 16888 E5077 Want to do well with respect to both J, and J 2 Define new objective function J,of =J,+j, max(J1) Optimize Jtot Objective J Result xto 09731 0.5664 tradeoff solution Jn*=6.1439 max(J,+J2 maX X 301731J1 0 13.1267=2 o Massachusetts Institute of Technology-Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
13 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Tradeoff between Tradeoff between J 1 and J2 • Want to do well with respect to both J 1 and J 2 • Define new objective function: Jtot = J 1 + J 2 • Optimize Jtot Result: Xtot* = 0.8731 0.5664 J ( xtot *) = 3.0173 J1 3.1267 J2 Jtot * = 6.1439 = max(J 1) max(J 2) tradeoff solution max(J 1+J 2)
M esd History (1)-Multicriteria Decision Making ESD.77 Life is about making decisions. Most people attempt to make the best decision within a specified set of possible decisions Historically, best was defined differently in different fields Life sciences the best referred to the decision that minimized or maximized a single criterion Economics: The best referred to the decision that simultaneously optimized several criteria In 1881, King,'s College (London) and later Oxford Economics Professor F.Y. Edgeworth is the first to define an optimum for multicriteria economic decision making. He does so for the multiutility problem within the context of two consumers, P and T It is required to find a point(x, y, such that in whatever direction we take an infinitely small step, P and r do not increase together but that, while one increases the other decreases Reference: Edgeworth, F.Y., Mathematical Psychics, P Keagan, London, England, 1881 o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
14 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics History (1) History (1) – Multicriteria Decision Making Multicriteria Decision Making • Life is about making decisions. Most people attempt to make the “best” decision within a specified set of possible decisions. • Historically, “best” was defined differently in different fields: – Life Sciences: The best referred to the decision that minimized or maximized a single criterion. – Economics: The best referred to the decision that simultaneously optimized several criteria. • In 1881, King’s College (London) and later Oxford Economics Professor F.Y. Edgeworth is the first to define an optimum for multicriteria economic decision making. He does so for the multiutility problem within the context of two consumers, P and π: – “It is required to find a point (x,y,) such that in whatever direction we take an infinitely small step, P and π do not increase together but that, while one increases, the other decreases.” – Reference: Edgeworth, F.Y., Mathematical Psychics, P. Keagan, London, England, 1881
M d History(2 ) -vilfredo Pareto 16888 S077 Born in paris in 1848 to a french mother and genovese Father Graduates from the University of Turin in 1870 with a degree in Civil Engineering Thesis Title: The Fundamental Principles of Equilibrium in Solid bodies While working in Florence as a civil Engineer from 1870 1893, Pareto takes up the study of philosophy and politics and is one of the first to analyze economic problems with mathematical tools In 1893, Pareto becomes the Chair of Political Economy at the University of Lausanne in Switzerland where he creates his two most famous theories Circulation of the elites The Pareto optimum The optimum allocation of the resources of a society is not attained so long as it is possible to make at least one others as well off as before in their own estimation, individual better off in his own estimation while keepi Reference pareto, V. Manuale di economia politica societa Editrice Libraria, Milano, Italy, 1906. Translated into English by A.S. Schwier as Manual of Political Economy, Macmillan, New o Massachusetts Institute of Technology -Prof de Weck and Prof Willco Engineering Systems Division and Dept of Aeronautics and Astronautics
15 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics History (2) History (2) – Vilfredo Pareto Vilfredo Pareto • Born in Paris in 1848 to a French Mother and Genovese Father • Graduates from the University of Turin in 1870 with a degree in Civil Engineering – Thesis Title: “The Fundamental Principles of Equilibrium in Solid Bodies” • While working in Florence as a Civil Engineer from 1870- 1893, Pareto takes up the study of philosophy and politics and is one of the first to analyze economic problems with mathematical tools. • In 1893, Pareto becomes the Chair of Political Economy at the University of Lausanne in Switzerland, where he creates his two most famous theories: – Circulation of the Elites – The Pareto Optimum • “The optimum allocation of the resources of a society is not attained so long as it is possible to make at least one individual better off in his own estimation while keeping others as well off as before in their own estimation.” • Reference: Pareto, V., Manuale di Economia Politica, Societa Editrice Libraria, Milano, Italy, 1906. Translated into English by A.S. Schwier as Manual of Political Economy, Macmillan, New York, 1971