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MIT 2 810 - Quality and Variation I

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2.810 Quality and Variation I2.810 Quality and Variation I2.810 Quality and Variation I2.810 Quality and Variation IT. Gutowski1Ref: 1. Random Variables, Lipschutz2. Hogg & Ledolter (Control Charts)pp 50-56, 339-342 + p386 Table C.1 (also see Kalpakjian)3. A Brief Intro to Designed Experiments, Taken from Quality Engineering using Robust Design by Madhav S. Phadke,OutlineI. Looking for problemsTypes of ProblemsMinimizing Random EffectsMinimizing Assignable Causes2Minimizing Assignable CausesII. Prof Dan FreyDesigned ExperimentsLooking for Problems• Where you find problems makes a difference, the closer to home the better.3• Make problems obvious, don’t hide them.Where you find the problem matters• Customer complaints• Warranty claims• Dealer returns•In-house inspectionIncreasingIncreasingIncreasingIncreasingCostCostCostCost4•In-house inspection• Assembly• Observed at the processCostCostCostCostTaguchi Quality Loss Function5Deviation, δL+′′+′+=2!2)0()0()0()(δδδfffQLQL = k δ2Fix problems close to home!Repair and upgrade of the Hubble TelescopeQuality Systems• Make problems obvious– Poke yoke at the process level– Clear flow paths and responsibility– Andon board–Simplify the system6–Simplify the system• Stop operations to attend to quality problems– Stop line – Direct attention to problem– Involve Team• Reduce inventory– Reduces rework, shorter trail to the problemTypes of Problems7Mean on target, but large variation due tomany random effectsMean drift hasassignable cause,tight groupingmeans small variationReducing Randomness8Goal: 6σ < (USL-LSL)To do this…• reduce variation– How does variation propagate?– Choose the right processes–Control actions9–Control actions• make product robust- insensitive to variation– Design for mfg– Design for functionRandom VariablesRandom VariablesRandom VariablesRandom Variables1. Expectation = mean = average2. Variance = (std deviation)23. Properties of E() and Var()4.Propagation of Errors104.Propagation of ErrorsRef. LipshutzProcess VariationProcess measurement reveals a distribution in output values.05101520253035123456789101112131112345678910111213Discrete 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)If 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 12both 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)Calculation of E(X) = µ and Var (X) = σ2x13Properties of the Expectation1. If Y = aX + b;where Y, X are random variables; a, b are constants,E(Y) = aE(X) + b(4)14E(Y) = aE(X) + b(4)2. If X1,…Xnare random variables, E(X1+ … + Xn) = E(X1) +…+ E(Xn) (5)Properties of the Variance1. For a and b constants, Var(aX + b) = a2Var(X) (6)2.If X1,…..Xnare independentrandom variables152.If X1,…..Xnare independentrandom variablesVar(X1+…+ Xn) = Var(X1)+ Var(X2)+ + Var(Xn) (7)θyxPropagation of errors• Dimensional Effects– Abbe error: y ≈ θ x –thermal expansion: δL = L α∆T16–thermal expansion: δL = L α∆T– Mean E(y) = E(θ) E(x), if independent, but– Var (y) = ?LPropagation of errors])[()())((2δδθδθδδδθθδyEyVarxxyxxyyy=+≅++=+=17)()()()()( recall)(2)(22222θθδθδθδθδθδVarxxVaryVarxpxxExxxxyii+≅=+⋅+≅∑Propagation of errors• this result is called “quadrature”, in general, if y=θx, with θ, x independent random variables with small variation, thenwith Var (x) = σ218with Var (x) = σx2222+=xyxyσθσσθPropagation of errors• A more general result is for any relationship like y=zαxβ, with z, x independent random variables with small variation, then19variation, then22222+=xzyxzyσβσασexample: errors due to thermal example: errors due to thermal example: errors due to thermal example: errors due to thermal expansionexpansionexpansionexpansion• 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, 20size (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.The out of specification parts are 2(0.5-φ(2σ))= 2(0.5 -0.4772) = 0.0456 or 4.56%21Upper Specification LimitLower Specification LimitTarget22Some 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−=C% out23Upper Specification LimitLower Specification LimitTargetCp% out⅔ 4.551 0.271 ⅓ .0063Kotz & JohnsonIn 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?0.135%13.6+2.1+0.13515.835+0.13524Upper Specification LimitLower Specification LimitTarget+0.13515.97In this case an alternative process capability can be used called the Cpkµµ),min(LSLUSL−−=25σµµ3),min(LSLUSLCpk−−=Upper Specification LimitLower Specification LimitTargetComparisonCase 1 (µ on target)Cp= 4σ/6σ = 2/3Cpk=Case 2 (µ drift)Cp= 4σ/6σ = 2/3Cpk=26Cpk=Min(2σ/3σ,2σ/3σ)=2/3Out of Spec = 4.55%Cpk=Min(1σ/3σ,3σ/3σ)=1/3Out of Spec = 15.835%Why the two different distributions at Sony?20% Likelihood set will be returned27“Tolerance Stack up”, really about variance,recall thatX1Xn28recall thatE(X1+ … + Xn) = E(X1) +…+ E(Xn)but how aboutVar(X1+…+ Xn) = ?If 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” as29(9)where L = X1+ X2σL2=σ12+σ22+2σ1σ2ρIf X1and X2are correlated (ρ = 1), then(14)for X1= X2= X0(15)σL2=σ12+σ22+2σ1σ2=(σ1+σ2)2224σσ=30(15)for N (16)or (17)σL2=N2σ020σσNL=2024σσ=LNow, if X1and X2are uncorrelated (ρ = 0) we get the result as in eq’n (7) or,(10)and for N


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MIT 2 810 - Quality and Variation I

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