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Robust Design 16.888 – Multidisciplinary System Design Optimization Robust Design Response Control Factor Prof. Dan Frey Mechanical Engineering and Engineering Systems Plan for the Session • Basic concepts in probability and statistics • Review design of experiments • Basics of Robust Design • Research topics – Model-based assessment of RD methods – Faster computer-based robust design – Robust invention Ball and Ramp Ball Response = Ramp the time the Funnel ball remains in the funnel Causes of experimental error = ? Probability Measure • Axioms – For any event A, P(A) t 0 – P(U)=1 – If the intersection of A and B=I, then P(A+B)=P(A)+P(B) Continuous Random Variables • Can take values anywhere within continuous ranges • Probability density function b – P a x d b f (x)dx ^`³ x a f (x) – 0 d f x (x) for all x x f – f (x)dx 1 ³ x f a b x Histograms • A graph of continuous data • Approximates a pdf in the limit of large n Histogram of Crankpin Diameters 5 Frequency 0 Diameter, Pin #1 Measures of Central Tendency • Expected value E(g(x)) g(x) f (x)dx ³ x S • Mean P= E(x) • Arithmetic average 1 n ¦ xi n i 1 Measures of Dispersion •Variance VAR(x) V 2 E((x E(x))2 ) • Standard deviation V E((x E(x))2 ) 1 n • Sample variance 2 2 S ¦(xi x) n 1 i1 • nth central moment E((x E(x))n ) th • n moment about m E((x m)n ) Sums of Random Variables • Average of the sum is the sum of the average (regardless of distribution and independence) E(x y) E(x) E( y) • Variance also sums iff independent V 2 (x y) V (x)2 V ( y)2 • This is the origin of the RSS rule – Beware of the independence restriction! Concept Test • A bracket holds a component as shown. The dimensions are independent random variables with standard deviations as noted. Approximately what is the standard deviation of the gap? A) 0.011” V 0.01" B) 0.01” V 0.001" C) 0.001” gap Expectation Shift Under utility theory, S=E(y(x))- y(E(x)) S is the only difference between probabilistic and deterministic design S y(E(x)) y(x) E(y(x)) fy(y(x)) fx(x) x E(x) Probability Distribution of Sums • If z is the sum of two random variables x and y z x y • Then the probability density function of z can be computed by convolution z p (z) x(z ] )y(] )d] z ³ f Convolution z p (z) x(z ] )y(] )d] z ³ f Convolution z p (z) x(z ] )y(] )d] z ³ f Central Limit Theorem The mean of a sequence of n iid random variables with – Finite P 2G – E xi E(xi ) < f G ! 0 approximates a normal distribution in the limit of a large n. Normal Distribution ( xP )2 1 2V 2 f x (x) e V 2S -6V -3V -1V P +1V +3V +6V 68.3% 99.7% 1-2ppb Engineering Tolerances • Tolerance --The total amount by which a specified dimension is permitted to vary (ANSI Y14.5M) pp(qy)) • Every component within spec adds to the yield (Y) Y L U qy Process Capability Indices UL /2 • Process Capability Index Cp { 3V UL P 2 • Bias factor k { ()/UL 2 CC()1 k • Performance Index pk{ p p(q) UL 2 L UL U q 2 18 Concept Test • Motorola’s “6 sigma” programs suggest that we should strive for a Cp of 2.0. If this is achieved but the mean is off target so that k=0.5, estimate the process yield. Plan for the Session • Basic concepts in probability and statistics • Review design of experiments • Basics of Robust Design • Research topics – Model-based assessment of RD methods – Faster computer-based robust design – Robust invention Pop Quiz • Assume we wish to estimate the effect of ball position on the ramp on swirl time. The experimental error causes V = 1 sec in the response. We run the experiment 4 times. What is the error our estimate of swirl time? A) V = 1 sec B) V = 1/2 sec Ball C) V = 1/4 sec Ramp Funnel History of DoE • 1926 – R. A. Fisher introduced the idea of factorial design • 1950-70 – Response surface methods • 1987 – G. Taguchi, System of Experimental Design Full Factorial Design 4 A B C D Response • This is the 2 +1 +1 +1 +1 +1 +1 +1 -1 • All main effects +1 +1 -1 +1 and interactions +1 +1 -1 -1 +1 -1 +1 +1 can be resolved +1 -1 +1 -1 +1 -1 -1 +1 • Scales very +1 -1 -1 -1 poorly with -1 +1 +1 +1 -1 +1 +1 -1 number of factors -1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 -1 -1 +1 -1 -1 -1 -1 Replication and Precision A B C D Response +1 +1 +1 +1 +1 +1 +1 -1 The average of +1 +1 -1 +1 trials 1 through 8 +1 +1 -1 -1 has a of 1/8 +1 -1 +1 +1 V +1 -1 +1 -1 that of each trial +1 -1 -1 +1 +1 -1 -1 -1 -1 +1 +1 +1 -1 +1 +1 -1 “the same precision as if -1 +1 -1 +1 the whole … -1 +1 -1 -1 had been devoted to one -1 -1 +1 +1 single component” -1 -1 +1 -1 -1 -1 -1 +1 – Fisher -1 -1 -1 -1 Resolution and Aliasing Trial A B C D E F G FG=-A 1 -1 -1 -1 -1 -1 -1 -1 +1 2 -1 -1 -1 +1 +1 +1 +1 +1 3 -1 +1 +1 -1 -1 +1 +1 +1 4 -1 +1 +1 +1 +1 -1 -1 +1 5 +1 -1 +1 -1 +1 -1 +1 -1 6 +1 -1 +1 +1 -1 +1 -1 -1 7 +1 +1 -1 -1 +1 +1 -1 -1 8 +1 +1 -1 +1 -1 -1 +1 -1 27-4 Design (aka “orthogonal array L8”) Resolution III. Projective Property + B + - - C - A + Considered important for exploiting sparsity of effects. DOE – Key Assumptions • Pure experimental error error in observations is random & independent • Hierarchy lower order effects are more likely to be significant than higher order effects • Sparsity of effects there are few important effects • Effect heredity for an interaction to be significant, at least one parent should be significant Sparsity of Effects • An experimenter may list several factors 1.2 • They usually affect the 1 response to greatly 0.8 varying degrees 0.6 0.4 • The drop off is 0.2 surprisingly steep Factor effects 0 2 (~1/n ) 1234567 • Not sparse if prior Pareto ordered factors knowledge is used or if factors are screened Hierarchy • Main effects are usually more important than two- factor interactions A B C D • Two-way interactions are usually more important than three-factor interactions AB AC AD BC BD CD •And so on • Taylor’s series seems to support the idea ABC ABD ACD BCD f f (n) (a) ¦(x a)n ABCD n 0 n! Inheritance • Two-factor interactions are most likely when A B C D both participating factors (parents?) are strong AB AC AD BC BD CD • Two-way interactions are least likely when ACD ABC ABD BCD neither parent is strong • And so on ABCD Resolution • II Main effects are aliased with main effects • III Main effects are clear of other main effects but aliased with two-factor interactions • IV Main effects are clear of other main effects and clear of two-factor interactions but main effects are aliased with three-factor interactions and two-factor interactions are aliased with other two-factor interactions • V Two-factor interactions are clear of other two-factor interactions but are aliased with three factor interactions… Discussion Point • What are the four most important factors affecting swirl time? • If you want to have sparsity of effects and hierarchy, how would you formulate the variables? Important Concepts in DOE • Resolution – the ability of an experiment to provide estimates of effects that are clear of other effects • Sparsity of Effects – factor effects are few • Hierarchy – interactions are generally less significant than main effects • Inheritance – if an interaction is significant, at least one of its “parents” is usually significant • Efficiency – ability of an experiment to estimate effects with small error variance Plan for the Session • Basic concepts in probability and statistics • Review design of experiments • Basics of Robust Design • Research topics – Model-based assessment of RD methods – Faster computer-based robust design – Robust invention Major Concepts of Taguchi Method • Variation causes quality loss • Two-step optimization • Parameter design via orthogonal arrays • Inducing noise (outer arrays) • Interactions and confirmation Loss Function Concept • Quantify the economic consequences of performance degradation due to variation L(y) What should the function be? y Fraction Defective Fallacy • ANSI seems to imply a “goalpost” L(y) mentality • But, what is the A difference between o – 1 and 2? 1 2 3 – 2 and 3? m y m-' m+'R Isn’t a continuous function R more appropriate? A Generic Loss Function • Desired properties L(y) – Zero at nominal value – Equal to cost at A specification limit o – C1 continuous • Taylor series m m+' m-'R R y f 1 f (x) | ¦ (x a)n f (n) (a) n 0 n! Nominal-the-best • Defined as L(y) Ao 2 L( y) 2 ( y m) 'o A • Average loss is o proportional to the 2nd moment y m-' m m+' about m R R quadratic quality loss function "goal post" loss function Average Quality Loss P L(y) A Ao o 2 2 V E[L( y)] 2 >@V (P m) 'o y m m+ m-'R 'R quadratic quality loss function probability density function Other Loss Functions Ao 2 • Smaller the better L( y) 2 y 'o 2 1 • Larger the better L( y) Ao'o 2 y • Asymmetric Ao 2 2 ( y m) if y ! m 'Upper L( y) Ao 2 2 ( y m) if y d m 'Lower Who is the better target shooter? Sam John Who is the better target shooter? Sam John Sam can just John requires adjust his sights lengthy training The “P” Diagram Noise Factors Product / Process Response Control Factors There are usually more control factors than responses Exploiting Non-linearity Response Use your extra “degrees of freedom” and search for robust set points.
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