Probability and Combining Component VariationThe ProblemToolsIE 361 Module 19"Statistical" (Probabilistic) TolerancingReading: Section 5.4, Statistical Quality Assurance Methods forEngineersProf. Steve Vardeman and Prof. Max MorrisIow a State UniversityVardeman and Morris (I owa State University) IE 361 Module 19 1 / 18Combining Variation on InputsIn this module we consider simple probability-based methods of combiningmeasures of input or component variability to predictoutput/system/overall variability. The basic tools available in this e¤ortareexact formulas for means and variances of linear functions of severalrandom variables,approximations for means and variances of general functions ofseveral random variables (based on linear approximations to thefunctions and the above), andsimple probabilistic simulations, easily done in any decent statisticalpackage (or, heaven forbid, EXCEL).Vardeman and Morris (I owa State University) IE 361 Module 19 2 / 18The ProblemGene ral Statement /ContextSometimes geometry or physical theory gives one an equation for avariable of interest in terms of more basic variables X , Y , . . . , ZU = g(X , Y , . . . , Z)and an issue of interest is how one might infer the level of variation to beseen in U from given information on the levels of variation in the inputsand the form of g .The context of this kind of study is usually design, where allowable levelsof variation and the details that lead to a particular g are being decided ina "What if?" mode of thinking.Vardeman and Morris (I owa State University) IE 361 Module 19 3 / 18ExampleExampl e 19-1 Box Pack ingExample 5.9 of SQAME is a nice "tolerance stack-up" example. Acompany wished to pack 4 units of product in a carton and wasexperiencing di¢ culty in packaging. The linear size of the carton interior(Y ) and exterior linear sizes of four packages (X ’s), together with thecarton "head space" (U) are illustrated in the …gure below. In thisproblem, simple geometry reveals thatU = Y X1X2X3X4Figure: Cartoon of a Packaging Problem (Example 5.9 of SQAME )Vardeman and Morris (I owa State University) IE 361 Module 19 4 / 18ExampleExampl e 19-2 R esistor Assse mblyExample 5.8 of SQAME is a simple circuit example involving an assemblyof 3 resistors. The …gure below illustrates this situation. In this problem,elementary physics produce an equation for assembly resistance, R,R = R1+R2R3R2+ R3Figure: A Schematic for Example 5.8 of SQAMEVardeman and Morris (I owa State University) IE 361 Module 19 5 / 18ExampleExampl e 19-3 A uto Door Toler ancin gThe …gure below concerns a problem faced by an engineer who must settolerances on various geometric features of a car door assembly, with theend goal of creating a uniform gap between the door and the body of anautomobile on which it is to be hung.Figure: Cartoon of a Door Tolerancing ProblemVardeman and Morris (I owa State University) IE 361 Module 19 6 / 18ExampleExampl e 19-3 continuedSome plane geometry and trigonometry applied to this situation producethe following set of equations that in the end express the gaps g1and g2at the elevation of the top hinge and a distance d below that hinge interms of x, w , θ1, y, θ2, and φ (which are all quantities on which a designengineer would need to set tolerances).p = (x sin φ, x cos φ)q = p +y cosφ +θ1π2, y sinφ +θ1π2s =(q1+ q2tan(φ + θ1+ θ2 π) , 0)u = (q1+ (q2+ d ) tan(φ + θ1+ θ2 π) , d)g1= w s1g2= w u1That is, though we have not written them out explicitly here, there are 2functions of the inputs x, w, θ1, y, θ2, and φ that produce the 2 gap values.Vardeman and Morris (I owa State University) IE 361 Module 19 7 / 18ToolsExac t Formulas for Linear gIn the three examples, then, what are tools for predicting how variation inthe inputs will be re‡ected in th e outputs? If one is willing to modelinputs X , Y , . . . , Z as independent random variablesequations (5.23) and (5.24) of SQAME are straight from basicprobability and say that for g linear, i.e. where for constantsa0, a1, . . . , ak,U = a0+ a1X + a2Y +  + akZU has meanµU= a0+ a1µX+ a2µY+  + akµZand varianceσ2U= a21σ2X+ a22σ2Y+  + a2kσ2ZVardeman and Morris (I owa State University) IE 361 Module 19 8 / 18ToolsApproximate M ethods for Nonlinear gStill modeling inputs X , Y , . . . , Z as independent random variablesequations (5.26) and (5.27) of SQAME are based on a (Taylortheorem) "linearization" of a general g and the above relationshipand say that roughlyµU g (µX, µY, . . . , µZ)andσ2U∂g∂x2σ2X+∂g∂y2σ2Y+  +∂g∂z2σ2Zwhere the partial derivatives are evalua ted at the point( µX, µY, . . . , µZ), andsimple probabilistic simulations can be used to approximate thedistribution of U based on some choice of distributions for the inputsin very straightforward fashion for general g .Vardeman and Morris (I owa State University) IE 361 Module 19 9 / 18PerspectiveBefore proceeding to illustrate these 3 methods, there are several points tobe made. In the …rst place, notice that since in the case of a linear g , thea’s are exactly the partial derivatives of U with respect to the inputvariables, the general approximation produces the exact result in case g isexactly linear. Second, note that while we will see that simulation iscompletely straightforward and indeed almost mindless to carry out, therewill be occasions where it is desirable to use the formulas. In particular,it’s possible to look at a term like∂g∂x2σ2X(in an approximation for σ2U) as the part of the variance of U traceableto variation in the input X . Finally, we observe that the approximationfor σ2Uis "qualitatively right." The variability in U must depend uponboth 1) how variable the inputs are and 2) what the rates of change ofoutput with respect to inputs are. (These two are measured respectivelyby the variances and the derivative s.)Vardeman and Morris (I owa State University) IE 361 Module 19 10 / 18PerspectiveThe …gure below illustrates the importance of the derivatives indetermining how variance is transmitted through g .Figure: Cartoon Illustrating the Importance of Rates of Change in DeterminingVariance Transmission (g a Function of One Input)Vardeman and Morris (I owa State University) IE 361 Module 19 11 / 18ExampleExampl e 19-1 continuedReturning to box packing, the exact form for the variance of U