L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 14 1 MER301: Engineering Reliability LECTURE 14: Chapter 7: Design of Engineering Experiments

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 14 2 Summary of Topics  Design of Engineering Experiments DOE and Engineering Design Coded Variables Optimization  Factorial Experiments Main Effects Interactions Statistical Analysis

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 14 3 Design of Experiments and Engineering Design  Applications of Designed Experiments Evaluation and comparison of design configurations Establish Production Process Parameters Evaluation of mechanical properties of materials/comparison of different materials Selection of ranges of values of independent variables in a design (Robust Design) Determination of Vital x’s (Significant Few versus the Trivial Many…..)

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 14 4 Design of Engineering Experiments  Objectives of engineering experiments include acquiring data that can be used to generate an analytical model for Y in terms of the dependent variables X i  The model may be linear or non-linear in the X i ’s and it defines a Response Surface of Y as a function of the X i ’s  The model can be used to generate statistical parameters(means, std dev) for use in product design, as in DFSS

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 14 5 Factorial Experiments  Factorial Experiments are used to establish Main Effects and Interactions  Levels of each factor are chosen to bound the expected range of each Xi

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 14 6 DOE Glossary  Model- quantitative relationship ; also called Transfer Function  DOE- systematic variation of X i ’s to acquire data to generate Transfer Function  Factorial Experiment-all possible combinations of X i ’s are tested Main Effects – change in Y due to change in X i. Interactions – joint effects of two or more X i ’s Replicates and Center Points  Response Surface- surface of Y generated by the Transfer Function Optimum Response- local max/min of Y  Partial Factorial Experiments- can run fewer points if can neglect higher order interactions

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 14 7 Coded Variables  Coded Variables.. Each x typically has some (dimensional) range in which it is expected to vary…. In Designed Experiments the lower value of each x is often assigned a value of –1 and the upper value of x a value of +1  Coded Variables have two advantages First, discrete variables( eg, “yes/no”, Operator A/ Operator B) can be included in the experiment Second, the magnitude of the regression coefficients is a direct measure of the importance of each x variable  Coded Variables have the disadvantage that an equation that can be directly used for engineering design is not specifically produced

L Berkley Davis Copyright 2009 DOE Process Map 8

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 14 9 Example Text Example 7-1  Two level factorial design applied to a process for integrated circuit manufacturing Y= epitaxial growth layer thickness A= deposition time( ),levels are long(+1) or short(-1) B= arsenic flow rate( ), levels are 59%(+1) or 55%(-1)  Experiment run with 4 replicates at each combination of A and B

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 14 10

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14.1( con’t)  What are the questions we need to answer? What is the quantitative effect of changes in A on the value of Y, ie the response of Y? What is the response of Y to changes in B? What is the interaction effect on Y when both A and B are changing, if any? Are there values of A and B such that Y is at an optimum level?

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14.1(con’t) Two level factorial design applied to integrated circuit manufacturing Y= epitaxial growth layer thickness A= deposition time( X1),levels are long(+1) or short(-1) B= arsenic flow rate( X2), levels are 59%(+1) or 55%(-1) Experiment run with 4 replicates at each combination of A and B How do we answer these Questions? What is the form of Experimental Design ? What is the quantitative effect of changes in A on the value of Y? What is the effect of changes in B on Y? What is the interaction effect on Y when both A and B are changing, if any? Are there values of A and B such that Y is at an optimum level?

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Factorial Design  factorial design is used when each factor has two levels Establish both main effects/interactions Assumes linearity of response Smallest number of runs to test all combinations of x’s  Factor(X i ) levels often described as “+ or –” Called geometric or coded notation 7-8 Factors A B

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Response of Y to A and B: Interaction/No Interaction  In Fig 7-1, response of Y to change in A is independent of B level; there is No Interaction between A and B  In Fig 7-2, the response of Y to change in A is shown with a different slope to illustrate an interaction between A and B

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Interaction/ No Interaction Interactions change the shape of the response surface significantly Experimental design must identify interactions and allow their impact to be quantified

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Main Effects  Main Effect term captures the change in the response variable due to change in level of a specific factor

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture AB Interactions  Interaction terms show the effects of changes in one variable at different levels of the other variables For Eq 7-3, this would give the effects of A at different levels of B(or vice versa)

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14.1: Data Set  Two level factorial design applied to integrated circuit manufacturing Y= epitaxial growth layer thickness A= deposition time(x 1 ),levels are long(+1) or short(-1) B= arsenic flow rate(x 2 ), levels are 59%(+1) or 55%(-1)  Experiment run with 4 replicates at each combination of A and B

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14.1: Effect Values

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Coded Least Squares Model  The Regression Equation is of the Form  The Coded Least Squares Model is of the Form X1 and X2 are coded variables and range from –1 to +1 is the average of all observations and the coefficients are

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Relationship between Regression Coefficients and the DOE Effects  The Regression Equation is  The Coded Least Squares Model is so that

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Relationship between Regression Coefficients and the DOE Effects  The Regression Equation is of the Form  The Coded Least Squares Model is of the Form For k=2

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14.1(con’t) Regression Analysis

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14.1 Excel Regression Analysis

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14.1:Regression Equation  Effects are calculated as A=0.836, B=-0.067, and AB=0.032 Large effect of deposition rate A Small effect of arsenic level B and interaction AB  Sample Variance(, pooled data set)=  Sample Mean( ) =

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14.1(con’t) Two level factorial design applied to integrated circuit manufacturing Y= epitaxial growth layer thickness A= deposition time( X1),levels are long(+1) or short(-1) B= arsenic flow rate( X2), levels are 59%(+1) or 55%(-1) Experiment run with 4 replicates at each combination of A and B How do we answer these Questions? What is the form of Experimental Design ? What is the quantitative effect of changes in A on the value of Y? What is the effect of changes in B on Y? What is the interaction effect on Y when both A and B are changing, if any? Are there values of A and B such that Y is at an optimum level?

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture One Factor at a Time Optimization  The One Factor at a Time method of conducting experiments is intuitively appealing to many engineers

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture One Factor at a Time Optimization  One Factor at a Time will frequently fail to identify effects of interactions Learning to use DOE factorial experiments often difficult for new engineers to accept DOE’s however are the most efficient and reliable method of experimentation 7-7

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Optimization  Screening tests establish factors (vital ’s) that affect Y  Range of ’s is a critical choice in the experimental design  Optimization will require multiple experiments 7-7a

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14.1:Regression Equation Optimization  Take partial derivatives wrt X1 and X2 Set equal to zero and solve for X1 and X2 X1~2 and X2~-26 No Optimum of Y in Range of X’s -1<=X,=1

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14-1:Statistical Analysis of the Regression Model-  There are three ways of conducting a statistical analysis of the regression model- and all will lead to the same conclusion Standard Error of the Effects calculated from Sample Data Sum of Squares based on Mean and Interaction Effects ANOVA/Significance Analysis of the Regression Equation  All of these methods are based on analysis of the single data set generated in the DOE.

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14-1:Statistical Analysis of the Regression Model-  There are three ways of conducting a statistical analysis of the regression model- and all will lead to the same conclusion Standard Error of the Effects calculated from Sample Data Sum of Squares based on Mean and Interaction Effects ANOVA/Significance Analysis of the Regression Equation  All of these methods are based on analysis of the single data set generated in the DOE.

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Standard Error of the Effects calculated from Sample Data  The magnitude/importance of each effect can be judged by comparing each effect to its Estimated Standard Error.  The first step in the analysis is to calculate the means and variances at each of the i factorial run conditions using data from the n replicates. For the variances  The second step is to calculate an overall(pooled) variance estimate for the factorial run conditions

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Calculation of Variance  The variances for the i factorial runs are  The overall(pooled) variance estimate for the 2 k =4 factorial run conditions is

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Standard Error of the Effects calculated from Sample Data (con’t)…..  Given the overall variance, the effect variance is calculated as follows The Effect Estimate is a difference between two means, each of which is calculated from half of the N measurements. Thus the Effect Variance is where  The Standard Error of each Effect is then

L Berkley Davis Copyright 2009

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Standard Error of the Effects calculated from Sample Data (con’t)…..  Because is the same for all of the effects and is used to calculate the Standard Error of any specific Effect( ie, A, B, AB,…) the value calculated will be the same for each one…  For the Epitaxial Process  The value of the Effect is twice that of the coefficient in the Regression Equation. Similarly, the Standard Error for the Coefficient is half that of the Effect

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Standard Error of the Effects calculated from Sample Data (con’t)…..  A Hypothesis Test is carried out on each of the Main Effects and the Interaction Effects. This is a t-test.  The A Effect is significant and the B and AB Effects are not 7-5

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14.1 Excel Regression Analysis

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Standard Error of the Effects calculated from Sample Data (con’t)…..

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14-1:Statistical Analysis of the Regression Model-  There are three ways of conducting a statistical analysis of the regression model- and all will lead to the same conclusion Standard Error of the Effects calculated from Sample Data Sum of Squares based on Mean and Interaction Effects ANOVA/Significance Analysis of the Regression Equation  All of these methods are based on analysis of the single data set generated in the DOE.

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Sum of Squares:Main Factors and Interaction  The Sum of Squares for the Effects can be expressed as  Sum of Squares from the Main and Interaction Effects can be used to assess the relative importance of each term  The Total Sum of Squares is obtained from and the Mean Square Error from

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Sum of Squares: Two Calculation Methods  The Total Sum of Squares,Effect Sum of Squares,and Mean Square Error are obtained from

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14-1:Statistical Analysis of the Regression Model-  There are three ways of conducting a statistical analysis of the regression model- and all will lead to the same conclusion Standard Error of the Effects calculated from Sample Data Sum of Squares based on Mean and Interaction Effects ANOVA/Significance Analysis of the Regression Equation  All of these methods are based on analysis of the single data set generated in the DOE.

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14.1 Regression Analysis and ANOVA (con’t) 7-6

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture ANOVA: for each term  The Significance of each term (A,B,AB) can be obtained from ANOVA A is Significant B and AB are not Significant

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Example 14.1 Regression Analysis and ANOVA(con’t) 7-6

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture Summary of Topics  Design of Engineering Experiments DOE and Engineering Design Coded Variables  Factorial Experiments Main Effects Interactions Optimization Statistical Analysis