M 16888 esd ESD.J7 Goal Programming and Isoperformance March 29. 2004 Lecture 15 Olivier de Weck o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
1 Goal Programming and Goal Programming and Isoperformance Isoperformance March 29, 2004 Lecture 15 Olivier de Weck © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
Mlesd Why not performance-optimal 35.9 The experience of the 1960s has shown that for military aircraft the cost of the final increment of performance usually is excessive in terms of other characteristics and that the overall system must be optimized, not just performance Ref: Current State of the art of Multidisciplinary design Optimization (MDO TC)-AlAA White Paper, Jan 15, 1991 TRW Experience Industry designs not for optimal performance, but according to targets specified by a requirements document or contract -thus, optimize design for a set of GOALs o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
2 Why not performance Why not performance-optimal ? optimal ? “The experience of the 1960’s has shown that for military aircraft the cost of the final increment of performance usually is excessive in terms of other characteristics and that the overall system must be optimized, not just performance” Ref: Current State of the Art of Multidisciplinary Design Optimization (MDO TC) - AIAA White Paper, Jan 15, 1991 TRW Experience Industry designs not for optimal performance, but according to targets specified by a requirements document or contract - thus, optimize design for a set of GOALS. © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
16888 esd Lecture Outline ESD.J7 Motivation -why goal programming? EXample: Goal Seeking in EXcel Case 1: Target vector T in Range Isoperformance Case 2: Target vector T out of Range Goal Programming Application to Spacecraft Design Stochastic Example: baseball Many-To-One Forward Perspective Choose x What isJ? Reverse Perspective Choose J What x satisfy this? o Massachusetts Institute of Technology-Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
3 Lecture Outline Lecture Outline • Motivation - why goal programming ? • Example: Goal Seeking in Excel • Case 1: Target vector T in Range = Isoperformance • Case 2: Target vector T out of Range = Goal Programming • Application to Spacecraft Design • Stochastic Example: Baseball Forward Perspective Choose x What is J ? Reverse Perspective Choose J What x satisfy this? © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics A Domain Range B T f a x J Target Vector Many-To-One
M 16888 esd Goal Seeking ESD.J7 max(s) L.ISO mIne B X max min UB o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
Goal Seeking Goal Seeking max(J) T=Jreq J x * min(J) * i iso , x x i LB x , , xmin xi UB i max 4 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
Mles Excel: Tools-Goal Seek 16888 ESD.J7 Excel-example sin(x)/x-example single variable x no solution if T is out of range For information about ' Goal seek consult Microsoft Excel help files o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
5 Excel: Tools Excel: Tools – Goal Seek Goal Seek Excel - example J=sin(x)/x -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -6 -5.2 -4.4 -3.6 -2.8 -2 -1.2 -0.4 0.4 1.2 2 2.8 3.6 4.4 5.2 6 x J sin(x)/x - example • single variable x • no solution if T is out of range For information about 'Goal Seek', consult Microsoft Excel help files. © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
M esd Goal Seeking and Equality Constraints ESD.11 Goal Seeking- is essentially the same as finding the set of points x that will satisfy the following"soft equality constraint on the objective Find all x such that 网≤E req Example 1000kg I arget mass Target Jre (x)= Rara=1.5 Mbps Target data rate ector 15M$ Target Cost o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
6 Goal Seeking and Equality Constraints Goal Seeking and Equality Constraints • Goal Seeking – is essentially the same as finding the set of points x that will satisfy the following “soft” equality constraint on the objective: J ( ) x − Jreq Find all x such that ≤ ε Jreq Target mass Example ª msat º ª 1000kg º « » Target J ( ) x = R Target data rate data » » ≡ « 1.5Mbps req «« » Vector: Target Cost ¬ » « 15M $ » ¼ « Csc ¼ ¬ © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
M esd Goal Programming vS Isoperformance ESD. 77 Criterion space Decision space (Design Space) (Objective Space) 2 X 2 Case 1: The target (goal) vector is in Z- usually get non-unique solutions Isoperformance Case 2: The target (goal)vector is not in Z-don't get a solution -find closest Goal Programming o Massachusetts Institute of Technology-Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
7 Goal Programming vs. Isoperformance Goal Programming vs. Isoperformance Criterion Space Decision Space (Objective Space) (Design Space) J2 is not in Z - don’t get a solution - find closest x2 J1 c S Z 2 x1 x4 x3 x2 J1 J3 J2 J2 The target (goal) vector = Isoperformance T2 T1 The target (goal) vector Case 1: is in Z - usually get non-unique solutions Case 2: = Goal Programming © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
M 168g esd Isoperformance Analogy ESD.J7 Non-Uniqueness of Design if n>z Analogy: Sea Level Pressure [mbar Chart: 1600 Z, Tue 9 May 2000 Performance: Buckling load Constants: 1=15 m), c=2.05 P- CTT EI Isobars= Contours of Equal Pressure Parameters Longitude and latitude Variable Parameters: E,I(r) Requirement L E, REQ 1000 metric tons L Solution 1: V2 ,[=10cm,E=19.1e+10 008 Solution 2 ,r=128cm,E=7.1e+10 012 d 2r EI Isoperformance Contours= Locus of constant system performance Bridge- Column多 Parameters=e.g Wheel Imbalance Us Support Beam Ix, Control Bandwidth o o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
Isoperformance Analogy Isoperformance Analogy Non-Uniqueness of Design if n > z Analogy: Sea Level Pressure [mbar] Chart: 1600 Z, Tue 9 May 2000 Performance: Buckling Load 2 c EI π P = Isobars = Contours of Equal Pressure Constants: l=15 [m], c=2.05 E l 2 Parameters = Longitude and Latitude Variable Parameters: E, I(r) Requirement: PE REQ = 1000 metric tons , Solution 1: V2A steel, r=10 cm , E=19.1e+10 Solution 2: Al(99.9%), r=12.8 cm, E=7.1e+10 L L L H 1008 1008 1012 1008 1008 1008 1012 1016 1012 1012 1012 1016 1012 1004 1016 1012 1012 l 2r PE E,I c Bridge-Column Isoperformance Contours = Locus of constant system performance Parameters = e.g. Wheel Imbalance Us, Support Beam Ixx, Control Bandwidth ω c 8 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
M 168g esd Isoperformance and LP ESD.J7 In LP the isoperformance surfaces are hyperplanes min C x Let c'x be performance objective and kx a cost objective S.t. XLB Sxsxur B(primal feasibility) 1. Optimize for performance C X 2. Decide on acceptable performance penalty e 3. Search for solution on isoperformance Efficient hyperplane that Solution minimizes cost kx C X Performance c xtE= C Optimal Solution o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
Isoperformance and LP Isoperformance and LP T • In LP the isoperformance surfaces are hyperplanes min c x • Let cTx be performance objective and kTx a cost objective s. . t x ≤ ≤x x LB UB 1. Optimize for performance cTx* 2. Decide on acceptable performance penalty ε 3. Search for solution on isoperformance hyperplane that minimizes cost kTx* cTx* = cT k c Efficient Is Solution operformance hyperplane x** B (primal feasibility) Performance cTx*+ Optimal Solution ε xiso 9 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
M esd Isoperformance Approaches 16888 ESD.J7 (a)deterministic I so performance Approach Deterministic Isoperformance Model Algorithms Jz, reg- (b)sto cha stic Iso performance Approach Design Space Ind x ●80% k Isoperformance Empirical 10.7592117.34 Algorithms System Model 20913118343 50% Statistical data o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
Isoperformance Algorithms Empirical System Model 10 Isoperformance Approaches Isoperformance Approaches (a) deterministic I soperformance Approach Jz,req Deterministic System Model Isoperformance Algorithms Design A Design B Design C Jz,req Design Space (b) stocha stic I soperformance Approach Ind x y Jz 1 0.75 9.21 17.34 2 0.91 3.11 8.343 3 ...... ...... ...... Statistical Data Design A Design B 50% 80% 90% Jz,req Empirical System Model Isoperformance Algorithms Jz,req P(Jz) © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
