Course Introduction Probability, Statistics and Quality Loss HW#1 Presentations Robust System Design 16.881 Session #1 MIT Background ? SDM split-summer format – Heavily front-loaded – Systems perspective – Concern with scaling ? Product development track, CIPD – Place in wider context of product realization ? Joint 16 (Aero/Astro) and 2 (Mech E) Robust System Design 16.881 Session #1 MIT Course Learning Objectives ? Formulate measures of performance ? Synthesize and select design concepts ? Identify noise factors ? Estimate the robustness ? Reduce the effects of noise ? Select rational tolerances ? Understand the context of RD in the end-to-end business process of product realization. Robust System Design 16.881 Session #1 MIT Instructors ? Dan Frey, Aero/Astro ? Don Clausing, Xerox Fellow ?Joe Saleh, TA ? Skip Crevelling, Guest from RIT ? Dave Miller, Guest from MIT Aero/Astro ?Others ... Robust System Design 16.881 Session #1 MIT Primary Text Phadke, Madhav S., Quality Engineering Using Robust Design. Prentice Hall, 1989. Robust System Design 16.881 Session #1 MIT Computer Hardware & Software ? Required – Access to a PC running Windows 95 or NT – Office 95 or later – Reasonable proficiency with Excel ? Provided – MathCad 7 Professional (for duration of course only) Robust System Design 16.881 Session #1 MIT Learning Approach ? Constructivism (Jean Piaget) – Knowledge is not simply transmitted – Knowledge is actively constructed in the mind of the learner (critical thought is required) ? Constructionism (Seymour Papert) – People learn with particular effectiveness when they are engaged in building things, writing software, etc. – http://el.www.media.mit.edu/groups/el/ Robust System Design 16.881 Session #1 MIT Format of a Typical Session ? Lecture Well, almost ? Reading assignment ?Quiz ? Labs, case discussions, design projects ? Homework Robust System Design 16.881 Session #1 MIT Grading ? Breakdown – 40% Term project – 30% Final exam – 20% Homework (~15 assignments) – 10% Quizzes (~15 quizzes) ? No curve anticipated Robust System Design 16.881 Session #1 MIT Grading Standards Grade Range Letter Equivalent Meaning 97-100 A+ Exceptionally good performance demonstrating superior understanding of the subject matter. 94-96 A 90-93 A- 87-90 B+ Good performance demonstrating capacity to use appropriate concepts, a good understanding of the subject matter and ability to handle problems. 84-86 B 80-83 B- 77-80 C+ Adequate performance demonstrating an adequate understanding of the subject matter, an ability to handle relatively simple problems, and adequate preparation. 74-76 C 70-73 C- 67-70 D+ Minimally acceptable performance demonstrating at least partial familiarity with the subject matter and some capacity to deal with relatively simple problems. 64-66 D 60-63 D- <60 F Unacceptable performance. Robust System Design 16.881 Session #1 MIT Reading Assignment ? Taguchi and Clausing, “Robust Quality” ?Major Points – Quality loss functions (Lecture 1) – Overall context of RD (Lecture 2) – Orthogonal array based experiments (Lecture 3) – Two-step optimization for robustness ? Questions? Robust System Design 16.881 Session #1 MIT Learning Objectives ? Review some fundamentals of probability and statistics ? Introduce the quality loss function ? Tie the two together ? Discuss in the context of engineering problems Robust System Design 16.881 Session #1 MIT Probability Definitions ? Sample space - List all possible outcomes of an experiment – Finest grained – Mutually exclusive – Collectively exhaustive ? Event - A collection of points or areas in the sample space Robust System Design 16.881 Session #1 MIT Probability Measure ? Axioms – For any event A, P( A) ≥ 0 – P(U)=1 – If AB=φ, then P(A+B)=P(A)+P(B) Robust System Design 16.881 Session #1 MIT Discrete Random Variables ? A random variable that can assume any of a set of discrete values ? Probability mass function – p x (x o ) = probability that the random variable x will take the value x o – Let’s build a pmf for one of the examples ? Event probabilities computed by summation Robust System Design 16.881 Session #1 MIT Continuous Random Variables ? Can take values anywhere within continuous ranges ? Probability density function b – P{a < x ≤ b}= ∫ f x ( x)dx a – 0 ≤ f x ( x) for all x ∞ – ∫ f x ( x)dx = 1 ?∞ Robust System Design 16.881 Session #1 MIT 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 Robust System Design 16.881 Session #1 MIT Measures of Central Tendency b ? Expected value E( g( x)) = ∫ g( x) f x ( x)dx a ? Mean μ = E(x) n ? Arithmetic average 1 ∑ x i n i=1 Robust System Design 16.881 Session #1 MIT Measures of Dispersion ?Variance VAR( x) =σ 2 = E(( x ? E( x)) 2 ) ? Standard deviation σ = n ) ))( (( 2 xEx E ? ? Sample variance S 2 = 1 ∑ ( x i ? x ) 2 n ? 1 i?1 ? n th central moment E(( x ? E( x)) n ) ? n th moment about m E(( x ? m) n ) Robust System Design 16.881 Session #1 MIT 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 σ 2 ( x + y) =σ( x) 2 +σ( y) 2 ? This is the origin of the RSS rule – Beware of the independence restriction! Robust System Design 16.881 Session #1 MIT Central Limit Theorem The mean of a sequence of n iid random variables with – Finite μ – E ( x i ? E( x i ) 2+δ ) < ∞ δ > 0 approximates a normal distribution in the limit of a large n. Robust System Design 16.881 Session #1 MIT Normal Distribution 2 1 ( x ) e( x) σ π ? ?μ σ 2 f x 2 = 2 μ 99.7% 68.3% 1-2ppb Robust System Design 16.881 Session #1 MIT Engineering Tolerances ? Tolerance --The total amount by which a specified dimension is permitted to vary (ANSI Y14.5M) ? Every component p(y) within spec adds to the yield (Y) L U Y y Robust System Design 16.881 Session #1 MIT Crankshafts ? What does a crankshaft do? ? How would you define the tolerances? ? How does variation affect performance? Robust System Design 16.881 Session #1 MIT GD&T Symbols Robust System Design 16.881 Session #1 MIT Loss Function Concept ? Quantify the economic consequences of performance degradation due to variation L(y) What should the function be? y Robust System Design 16.881 Session #1 MIT Fraction Defective Fallacy ? ANSI seems to imply a “goalpost” L(y) mentality ? But, what is the difference between A o – 1 and 2? – 2 and 3? 3 2 1 Isn’t a continuous function m-? ο m m+? ο y more appropriate? Robust System Design 16.881 Session #1 MIT A Generic Loss Function ? Desired properties – Zero at nominal value – Equal to cost at specification limit – C1 continuous ? Taylor series ∞ f ( x) ≈ ∑ 1 ( x ? a) n f ( n ) (a L(y) A o m-? ο m m+? ο y n=0 n! ) Robust System Design 16.881 Session #1 MIT ? Defined as L(y) Nominal-the-best L( y) = A o 2 ( y ? m) 2 ? o A o ? Average loss is proportional to the 2 nd moment m-? ο m m+? ο y about m quadratic quality loss function (HW#2 prob. 1) "goal post" loss function Robust System Design 16.881 Session #1 MIT Average Quality Loss L(y) m m+? [L E σ μ A o ( y)] = A o 2 [σ 2 + (μ? m) 2 ] ? o m-? ο ο y quadratic quality loss function probability density function Robust System Design MIT16.881 Session #1 Other Loss Functions ? Smaller the better L( y) = A o 2 y 2 ? o (HW#3a) ? Larger-the better L( y) = A o ? o 2 1 2 y (HW#3b) ? Asymmetric A o 2 ( y ? m) 2 if y > m (HW#2f&#3c) L( y) = ? Upper A o 2 ( y ? m) 2 if y ≤ m ? Lower Robust System Design 16.881 Session #1 MIT Printed Wiring Boards ? What does the second level connection do? ? How would you define the tolerances? ? How does variation affect performance? Robust System Design 16.881 Session #1 MIT Next Steps ? Load Mathcad (if you wish) ? Optional Mathcad tutoring session – 1hour Session ? Complete Homework #2 ? Read Phadke ch. 1 & 2 and session #2 notes ? Next lecture – Don Clausing, Context of RD in PD Robust System Design 16.881 Session #1 MIT