Chapter 5

1. Chapter 5 Six Sigma

2. Chapter 5 Contents • Design for Six Sigma • Input Parameter Selection • Input Parameter Variation • Six Sigma Analysis Procedure • Workshop 5-1

3. Design For Six Sigma • Real world engineering problems involve variation (scatter) in many forms. Scatter is inherent to some extent in material properties, component dimensions, loads, etc. • The input scatter will affect responses of the system to varying degrees. • Typical analyses assume a fixed value for each input quantity and assigns a safety factor to account for these assumptions (deterministic). • Design For Six Sigma (DFSS) provides a mechanism to include and account for scatter in input and provide insight into how they affect the system response (probabilistic).

4. Input Parameter Selection • Some basic questions DFSS addresses: • Given input scatter, how large is the scatter of the output parameters? • What is the probability that a response will fail to meet specific criteria? • Which input variable has the greatest impact on the scatter in the output? • Input parameters should be included in a DFSS study based on their impact on the overall response.

5. Input Parameter Variation • When an input quantity is specified as an “uncertainty variable”, a choice must be made as to its distribution. • DFSS allows the following distribution types: • Beta, Exponential, Gaussian, Lognormal, Uniform, Triangular, Truncated Gaussian and Weibull. • The choice of distribution function depends on the type, source and reliability of the data. For a full discussion of choosing distribution functions see “Guidelines for Selecting Probabilistic Design Variables”

6. Distribution Functions • Beta Distribution - Useful for random variables that are bounded at both sides. If linear operations are applied to random variables that are all subjected to a uniform distribution, then the results can usually be described by a Beta distribution. • Exponential Distribution - Useful in cases where there is a physical reason that the probability density function is strictly decreasing as the uncertainty variable value increases. • Gaussian (Normal) Distribution - Fundamental and commonly-used distribution for statistical matters. It is typically used to describe the scatter of the measurement data. • Lognormal Distribution -Typically used to describe the scatter of the measurement data of physical phenomena, where the logarithm of the data would follow a normal distribution. The lognormal distribution is suitable for phenomena that arise from the multiplication of a large number of error effects.

7. Distribution Functions • Uniform Distribution • For cases where the only information available is a lower and an upper limit. It is also useful to describe geometric tolerances. • Triangular Distribution • Helpful to model a random variable when actual data is not available. It is very often used to capture expert opinions. • Truncated Gaussian Distribution • Used where the physical phenomenon follows a Gaussian distribution, but the extreme ends are cut off are eliminated from the sample population by quality control measures. • Weibull Distribution • Most often used for strength or strength-related lifetime parameters, and is the standard distribution for material strength and lifetime parameters for very brittle materials (for these very brittle material the "weakest-link theory" is applicable).

8. Design For Six Sigma - Procedure • A Six Sigma study begins just like the other DX study types. • Following a Simulation analysis, Six Sigma Analysis is selected from the Project Page. • Geometry and analysis parameters are specified in CAD and Mechanical application as discussed in earlier chapters. • Since a DFSS study accounts for the random nature of the inputs they are treated differently than in the previous studies. First you will need to do DOE and to obtain Response Surface in order to do Six Sigma Analysis.

9. DFSS Procedure . . . Distribution attributes, mean and standard deviation are defined. Note: distribution attributes will vary depending on the distribution type chosen. DFSS inputs are changed to “Uncertainty Variables”. The distribution type is chosen from the drop down dialog.

10. DFSS Procedure . . . • Before obtaining DFSS data, a “Sample Set” must be generated. (This is simply a numerical evaluation of the existing response surface polynomials and is very fast.) • Sample sets are generated using a sampling technique known as the Latin Hypercube Sampling (a variation on the Monte Carlo simulation). • With a sample set generated, the inputs and responses for the study can be viewed. Expanding Sample Type you can choose LHS and WLHS (Weighted Latin Hypercube Sampling)

11. DFSS Procedure . . . • The procedure for a Six Sigma Analysis is: • Solve the Six Sigma Analysis by: • Updating each individual cell in the analysis either from the right menu in the Project Schematic or the Update button in the workspace toolbar while editing the cell. • Right clicking on the Six Sigma Analysis cell in the project schematic and selecting Update to update the entire analysis at once. • Clicking the Update Project button in the toolbar to update the entire project.

12. DFSS Procedure . . . • From the sample set various pieces of information can be gathered. • A synopsis for each parameter is given which includes mean, standard deviation, minimum and maximum etc. • Data is displayed in both histogram and various cumulative distribution function graphical formats.

13. DFSS Procedure . . . • By switching from the “Charts” view to the “Table” view, probability and inverse probability tables can be accessed.

14. DFSS Procedure . . . At the bottom of the tables new values for parameters, probabilities and sigma levels can be entered. RMB on the cell in the Table > Remove Lever to delete the cell

15. Postprocessing Six Sigma Results • Histogram • Most commonly used to visualize the scatter. A histogram is derived by dividing the range between the minimum value and the maximum value into equal size intervals. Six Sigma analysis determines how many samples fall within each interval. • Cumulative Distribution Function • Review tool, if you want to assess the reliability of the failure probability of your component of product. At the given point expresses the probability that the respective parameter value will remain below that point. • Probability and Inverse Probability Tables • Cumulative distribution function in tabular form. • Statistical Sensitivities in a Six Sigma Analysis • Global sensitivities based on correlation analysis using the generated sample points, which are located throughout the entire space of input parameters.

16. DFSS – Best Practice • Choosing and Defining Uncertainty Variables • Specify reasonable range and limits on the variability for each uncertainty variable. • Select the most important input variables, the ones you know have a significant impact on the result parameters. • Choosing a distribution for a random variable • Measured data • You must know how reliable the data is. In a mass production you will have a lot of data. In this case you could use a commercial statistical package to fit a statistical distribution function that best describes the scatter of the data. • No data • Geometric tolerances – Often Gaussian distribution is used • Material data – Often the scatter of material data is described by Gaussian distribution • Load data – You can use triangular distribution using the minimum, most likely, and maximum values obtained.