IE 361 Module 39. "Statistical" (Probabilistic) Tolerancing Part 1 (Ideas) Reading: Section 4.4 Statistical Methods for Quality Assurance

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1 IE 361 Module 39 "Statistical" (Probabilistic) Tolerancing Part 1 (Ideas) Reading: Section 4.4 Statistical Methods for Quality Assurance ISU and Analytics Iowa LLC (ISU and Analytics Iowa LLC) IE 361 Module 39 1 / 11

2 Combining Variation on Inputs In this module we consider simple probability-based methods of combining measures of input or component variability to predict output/system/overall variability. The basic tools available in this effort are exact formulas for means and variances of linear functions of several random variables, approximations for means and variances of general functions of several random variables (based on linear approximations to the functions and the above), and simple probabilistic simulations, easily done in any decent statistical package (or, heaven forbid, EXCEL). (ISU and Analytics Iowa LLC) IE 361 Module 39 2 / 11

3 The Problem General Statement/Context Sometimes geometry or physical theory gives one an equation for a variable of interest in terms of more basic variables X, Y,..., Z U = g (X, Y,..., Z ) and an issue of interest is how one might infer the level of variation to be seen in U from given information on the levels of variation in the inputs and the form of g. The context of this kind of study is usually design, where allowable levels of variation and the details that lead to a particular g are being decided in a "What if?" mode of thinking. (ISU and Analytics Iowa LLC) IE 361 Module 39 3 / 11

4 Example Example 39-1 Box Packing The Box Size example in Section 4.4 of SMQA is a nice "tolerance stack-up" example. A company wished to pack 4 units of product in a carton and was experiencing diffi culty in packaging. The linear size of the carton interior (Y ) and exterior linear sizes of four packages (X s), together with the carton "head space" (U) are illustrated in the figure below. In this problem, simple geometry reveals that U = Y X 1 X 2 X 3 X 4 Figure: Cartoon of a Box Size/Packaging Problem (ISU and Analytics Iowa LLC) IE 361 Module 39 4 / 11

5 Example Example 39-2 Resistor Asssembly The Assembly Resistance example in Section 4.4 of SMQA is a simple circuit example involving 3 resistors. The figure below illustrates this situation. In this problem, elementary physics produce an equation for assembly resistance, R, R = R 1 + R 2R 3 R 2 + R 3 Figure: A Schematic for the Assembly Resistance Example of SMQA (ISU and Analytics Iowa LLC) IE 361 Module 39 5 / 11

6 Example Example 39-3 Auto Door Tolerancing The figure below concerns a problem faced by an engineer who must set tolerances on various geometric features of a car door assembly, with the end goal of creating a uniform gap between the door and the body of an automobile on which it is to be hung. Figure: Cartoon of a Door Tolerancing Problem (ISU and Analytics Iowa LLC) IE 361 Module 39 6 / 11

7 Example Example 39-3 continued Some plane geometry and trigonometry applied to this situation produce the following set of equations that in the end express the gaps g 1 and g 2 at the elevation of the top hinge and a distance d below that hinge in terms of x, w, θ 1, y, θ 2, and φ (which are all quantities on which a design engineer would need to set tolerances). p = ( x sin φ, x cos φ) q = p + ( y cos ( φ + ( θ 1 π )) ( ( 2, y sin φ + θ1 π ))) 2 s = (q 1 + q 2 tan(φ + θ 1 + θ 2 π), 0) u = (q 1 + (q 2 + d) tan(φ + θ 1 + θ 2 π), d) g 1 = w s 1 g 2 = w u 1 That is, though we have not written them out explicitly here, there are 2 functions of the inputs x, w, θ 1, y, θ 2, and φ that produce the 2 gap values. (ISU and Analytics Iowa LLC) IE 361 Module 39 7 / 11

8 Tools Exact Formulas for Linear g In the three examples, then, what are tools for predicting how variation in the inputs will be reflected in the outputs? If one is willing to model inputs X, Y,..., Z as independent random variables equations (4.23) and (4.24) of SMQA are straight from basic probability and say that for g linear, i.e. where for constants a 0, a 1,..., a k, U has mean U = a 0 + a 1 X + a 2 Y + + a k Z µ U = a 0 + a 1 µ X + a 2 µ Y + + a k µ Z and variance σ 2 U = a 2 1σ 2 X + a 2 2σ 2 Y + + a 2 k σ2 Z (ISU and Analytics Iowa LLC) IE 361 Module 39 8 / 11

9 Tools Approximate Methods for Nonlinear g Still modeling inputs X, Y,..., Z as independent random variables equations (4.26) and (4.27) of SMQA are based on a (Taylor theorem) "linearization" of a general g and the above relationship and say that roughly and σ 2 U µ U g(µ X, µ Y,..., µ Z ) ( ) g 2 ( ) g 2 ( ) g 2 σ 2 X + σ 2 Y + + σ 2 Z x y z where the partial derivatives are evaluated at the point (µ X, µ Y,..., µ Z ), and simple probabilistic simulations can be used to approximate the distribution of U based on some choice of distributions for the inputs in very straightforward fashion for general g. (ISU and Analytics Iowa LLC) IE 361 Module 39 9 / 11

10 Perspective Before proceeding to illustrate these 3 methods, there are several points to be made. In the first place, notice that since in the case of a linear g, the a s are exactly the partial derivatives of U with respect to the input variables, the general approximation produces the exact result in case g is exactly linear. Second, note that while we will see that simulation is completely straightforward and indeed almost mindless to carry out, there will be occasions where it is desirable to use the formulas. In particular, it s possible to look at a term like ( ) g 2 σ 2 X x (in an approximation for σ 2 U ) as the part of the variance of U traceable to variation in the input X. Finally, we observe that the approximation for σ 2 U is "qualitatively right." The variability in U must depend upon both 1) how variable the inputs are and 2) what the rates of change of output with respect to inputs are. (These two are measured respectively by the variances and the derivatives.) (ISU and Analytics Iowa LLC) IE 361 Module / 11

11 Perspective The figure below illustrates the importance of the derivatives in determining how variance is transmitted through g. Figure: Cartoon Illustrating the Importance of Rates of Change in Determining Variance Transmission (g a Function of One Input) (ISU and Analytics Iowa LLC) IE 361 Module / 11