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MIT 2 810 - Quality and Variation

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12.810 Quality and Variation1. Design2. Process3. SystemT. GutowskiCharacterizethe ProcessVariationProblemsolving:1. Identify problem2. Identify cause3. Take corrective action2Technical Topics1. Random Variables• Expectation and Variance Operators• Propagation of errors• Process capability/Tolerance2. Control Actions• Statistical Process Control• Design of Experiments3Readings;• Hogg & Ledolter (Control Charts)pp 50-56 +p386 Table C.1 (also see Kalpakjian) • A Brief Intro to Designed ExperimentsTaken from Quality Engineering using Robust Design by Madhav S. Phadke, Prentice Hall, 1989• “Robust Quality” by Genichi Taguchi and Don Clausing• 5 homework Problems at end4Random Variables1. Expectation = mean = average2. Variance = (std deviation)23. Properties of E() and Var()4. Propagation of Errors5. Process Capability6. Quality Loss7. Tolerance Stack up5If the dimension “X” is a random variable, the mean is given byµ = E(X) (1)and the variation is given by Var(x) = E[(x - µ)2](2)both of these can be obtained from the probability density function p(x).For a discrete pdf, the expectation operation is:(3)E(X)=xii∑p(xi)6Calculation of E(X) = µ and Var (X) = σ2x7Mean and VarianceMean on target, but large variation due tomany random effectsMean drift hasassignable cause,tight groupingmeans small variation8Process VariationProcess measurement reveals a distribution in output values.0510152025303512345678910111213Discrete probability distribution based upon measurementsContinuous “Normal” distributionIn general if the randomness is due to many different factors, the distribution will tend toward a “normal” distribution.(Central Limit Theorem)9Properties of the Expectation1. If Y = aX + b; a, b are constants,E(Y) = aE(X) + b (4)2. If X1,…Xnare random variables, E(X1+ … + Xn) = E(X1) +…+ E(Xn)(5)10Properties of the Variance1. For a and b constantsVar(aX + b) = a2Var(X) (6)2. If X1,…..Xnare independent random variablesVar(X1+…+ Xn) = Var(X1)+ Var(X2)+ + Var(Xn) (7)11Propagation of errors• Size Effects– Abbe error: y ≈ θ x– thermal expansion: δL= L α∆T– Mean E(y) = E(θ) E(x), if independent, but– Var (y) = ?Lθyx12Propagation of errors)()()()()( recall)(2)(])[()())((222222θθδθδθδθδθδδδθδθδδδθθδVarxxVaryVarxpxxExxxxyyEyVarxxyxxyyyii+≅=+⋅+≅=+≅++=+=∑13Propagation of errors• this result is called “quadrature”, in general, if y=θx, with θ, x independent random variables with small variation, thenwith Var (x) = σx2222+=xyxyσθσσθ14example: errors due to thermal expansion• Say in a machine shop the conditions fluctuate such that the means and standard deviations (µ, σ) are for stock size (10, 1), linear expansion coef.(1000, 1) and temperature (20, 3).Quadature tells us that the variation in thermal expansion of the work pieces is dominated by the temperature and is approximately 0.39 times its mean value.15Tolerance is the specification given on the part drawing, and variationis the variability in the manufacturing process. This figure confuses the two by showing the process capabilities in terms of tolerance. Never the less, we can see that the general variability (expressed as tolerance over part dimension) one gets from conventional manufacturing processes is on the order of to210000,110−=41010001.−=Q. What products can not meet these ratios?16Part Tolerance17Product specifications are given as upper and lower limits, for example the dimensional tolerance +0.005 in.Upper Specification LimitLower Specification LimitTarget18Consider a process centered on the target, with ±2σ within the tolerances Upper Specification LimitLower Specification LimitTarget19The out of specification parts are 2(0.5-φ(2σ))= 2(0.5 -0.4772) = 0.0456 or 4.56%Upper Specification LimitLower Specification LimitTarget2021Some propose a process capability index Cpthat compares the tolerance interval USL-LSL vs the process variation 6σ. For the previous case this would be 4σ/6σ =0.67σ6LSLUSLCp−=Upper Specification LimitLower Specification LimitTarget22In general the mean and the target do not have to line up. In this case the Cpis misleading. A better question is how many parts are out of spec?Upper Specification LimitLower Specification LimitTarget23In this case an alternative process capability can be used called the Cpkσµµ3),min( LSLUSLCpk−−=Upper Specification LimitLower Specification LimitTarget24ComparisonCase 1 (µ on target)Cp= 4σ/6σ = 2/3Cpk=Min(2σ/3σ,2σ/3σ)=2/3Out of Spec = 4.6%Case 2 (µ drift)Cp= 4σ/6σ = 2/3Cpk=Min(1σ/3σ,3σ/3σ)=1/3Out of Spec = 16.1%Note; the out of Spec percentages are off slightly due to round off errors25Why the two different distributions at Sony?20% Likelihood set will be returned26Quality LossDeviation, δL+′′+′+=2!2)0()0()0()(δδδfffQLGoal Post QualityTaguchi Quality Loss FunctionQL = k δ227Tolerance Stack up, recall thatE(X1+ … + Xn) = E(X1) +…+ E(Xn)but how aboutVar(X1+…+ Xn) = ?X1Xn28If X1and X2are random variables and not necessarily independent, thenVar(X1 + X2) = Var(X1) + Var(X2) + 2Cov(X1Y) (8)this can be written using the standard deviation “σ”, and the correlation “ρ” as(9)where L = X1+ X2σL2=σ12+σ22+2σ1σ2ρ29If X1and X2are correlated (ρ = 1), then(14)for X1= X2= X0(15)for N (16)or (17)σL2=σ12+σ22+2σ1σ2=(σ1+σ2)2σL2=N2σ020σσNL=2024σσ=L30Now, if X1and X2are uncorrelated (ρ = 0) we get the result as in eq’n (7) or,(10)and for N (11)If X1=X=Xo(12)Or (13)σL2=σ12+σ22212iNiLσσ∑==σL2=Nσ02σL=Nσ031Tolerance Stack-upAs the number of variables grow so does the variation in the system;correlateduncorrelatedbut uncorrelated grows less fast, in fact the normalized value improves!σL= Nσ00σσNL=322. Control Actions1. Measure parameter2. Compare to desired level3. Take corrective action• Remove disturbances• Reduce sensitivity33How to control variationuuYYY ∆∂∂+∆∂∂=∆ααoutput variationdisturbancesensitivitycontrol actionref. paper by Dave Hardt34Statistical Process ControluuYYY ∆∂∂+∆∂∂=∆ααoutput variationdisturbancesensitivitycontrol action35Designed ExperimentsuuYYY ∆∂∂+∆∂∂=∆ααoutput variationdisturbancesensitivitycontrol action36Real Time ControluuYYY ∆∂∂+∆∂∂=∆ααoutput variationdisturbancesensitivitycontrol action37Statistical Process Control and Process


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MIT 2 810 - Quality and Variation

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