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**Executing Robust Design**

This module will present the analysis techniques to optimize product/process performance via both the mean and the variation, as discussed in the Introduction module. Generally takes 6 hours to complete material. Catapult exercise probably adds another hour.

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**Definition of Robust Design**

Robustness is defined as a condition in which the product or process will be minimally affected by sources of variation. A product can be robust: Against variation in raw materials Against variation in manufacturing conditions Against variation in manufacturing personnel Against variation in the end use environment ` Against variation in end-users Against wear-out or deterioration

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**Low Variation; Minimum Cost High Variation; High Cost**

Why We Need to Reduce Variation Cost Low Variation; Minimum Cost LSL USL Nom Cost High Variation; High Cost LSL USL Nom

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Purpose of this Module To introduce a variation improvement investigation strategy Can noise factors be manipulated? To provide the MINITAB steps to design, execute, and analyze a variability response experiment To provide the MINITAB steps to optimize a design for both mean and variation effects Note that all of the “variance” language from previous versions of this module were changed to “variation” and “variability”, mainly in response to MINITAB modeling the standard deviation, rather than the variance. This change doesn’t affect the approach or philosophy in any way.

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**Objectives of this Module**

At the end of this module, participants will be able to : Identify possible variation effects from residual plots Create a variability response from replicates Identify possible mean and variance adjustment factors from noise-factor interaction plots Use the MINITAB Response Optimizer to achieve a process on target with minimum variation

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**Strategies to Detect Variation Effects**

Passive Approach Noise factors are NOT included, manipulated or controlled in the experimental design Possible variation effects are identified through analysis of the variability of replicates from an experimental design Active Approach Noise factors ARE included in the experimental design in order to force variability to occur Analysis is similar to the passive approach The passive approach “hopes” for noise to happen during the time period the data collection consumes. Have no guarantee that it will happen or how extensive it will be. In the end, harder to model but you could include the noise as a covariate in the analysis. The active approach “makes” the noise happen along with the control factors changing. This enables a clear estimate of their effect, just like a factor within a DOE. Also allows for direct modeling of the variance. Analysis is both cases is nearly identical.

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The Passive Approach A factorial experiment is performed using Control factors. Noise factors are not explicitly manipulated nor is an attempt made to control them during the course of the experiment. Pros Simple extension of standard experimental techniques Does not require explicit identification of noise factors Cons Requires larger number of replicates than would typically be required to determine mean effects Requires “true” randomization and replication Requires that noise factors be “noisy” during the execution of the experiment Pro – you already know how to do it! Con – no guarantee of success, in terms of “noise happens”

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**How to ensure that noise is noisy?**

Let excluded factors vary Compare noise factor variations prior to and within DOE Monitor noise factor levels during normal process conditions Monitor noise factor variation during course of experiment Compare before/during levels Run DOE over a longer period of time with : More replicates Full randomization First – identify the noise variables! Then, try to do everything to get the noise to be active. Let ALL other variables vary. Extend the range of the data collection period, if possible. To assess how much the noise varied : take measurements of the noise variables during normal operation prior to the DOE, take measurements of the noise variables during the DOE, compare the two. You want to ensure how the noise usually varies is how it will vary in the environment you want to optimize in. Example, the end-user customer of an engine may see the full range of ambient temp when they use the engine (driving in summer arizona and winter michigan), but development testing might span one change of the seasons (fall to winter), rarely all four – therefore the noise during the test is not “total”.

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A Passive Example A B A and B are control factors. Within each treatment combination, noise factors are allowed to naturally fluctuate. Within treatment variation is largely driven by this background noise. This 2x2 table represents a DOE where replications were run. This replication results in the distribution of Y within each of the four boxes. The spread is “assumed” to be due to the noise happening. The vertical dotted line represents a target value. If you look at moving Factor A from its low setting(-) to its high setting(+), what happens to the distributions? Look across the B(-) row. Going from low to high causes the mean to shift above the target. Look across the B(+) row. Going from low to high causes the mean to shift to above the target. This means Factor A has an effect on the mean. The spread is the same, from low to high. (we call this a mean adjustment factor, MAF) If you look at moving Factor B from its low setting(-) to its high setting(+), what happens to the distributions? Look down the A(-) column. Going from low to high causes the variation to decrease considerably. Look down the A(+) column. Going from low to high causes the variation to decrease considerably. This means Factor B has an effect on the variance. The mean is the same, from low to high. (we call this a variance adjustment factor, VAF)

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**Example output from a Passive Design**

Mean Y Variation Y The graphs at right illustrate the type of output which might be obtained from a Robust Parameter Design Experiment. Both are Main Effects plots with the top row showing the main effects of factors A and B on the mean and the bottom row showing the main effects of factors A and B on the variation. Note that in this example the mean and variation can be adjusted independently of each other! These are main effect plots from MINITAB. They represent what the data on the previous slide would look like, in this format. Observe that Factor A only affects the mean while Factor B affects only the variation. This is the “best case” scenario. You want to have a factor which solely affects the variation so that you can set it to its optimum value without moving the mean. If its setting causes the mean to be off target, then you want to have a factor which solely affects the mean so that you can use it to get on target without affecting the variation at all. Perfect! Would doubt that this happens much in practice.

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The Models Our objective in performing a designed experiment is to develop a transfer function between the factors (X’s) and the Y. Thus far, we have only addressed the mean of Y. Now we must also consider the variability of Y If our experiments are successful at identifying a variation effect, we now have an opportunity to simultaneously optimize both equations! The mean equation is what we are already familiar with. What we have been using all throughout week 3 & RSM. Now we are trying to write out an equation for the variation.

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**Example: A Passive Noise Experiment**

A design engineer has evaluated the output performance of a circuit design and performed an initial capability analysis of this design to determine if there is a problem with the mean and/or the variability. Stat > Quality Tools > Capability Analysis > Normal Y = Y1Initial; Lower Spec = 58; Upper Spec = 62 Get the lower and upper spec limits, They are seen here.

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**Design Capability Analysis**

Is there a problem? Cp = 0.5 means that the spec range is ½ of the 6s spread of the data. Based on the evaluation of the capability analysis, the engineer established the need for further design optimization through a robust design approach. The process has a problem with excessive variability and it needs to be centered as well.

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**24 Full Factorial Experiment**

A 24 full factorial has been designed to determine if four factors have an effect on the mean and/or variability of voltage drop (Y1). There are five replicates for a total of 80 runs, with no center points or blocks. Resistor R33 (A) Inductor L3 (B) Capacitor C23 (E) Capacitor C29 (F) Worksheet: “Passive Design” The selection of these factors was based on theory, preliminary analysis, and experience. Notice the number of replicates!! This is because we are looking for noise effects. Would you want to run this many experiments? This spreadsheet is NOT supplied to the students.

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**Passive Analysis Roadmap - Part 1 (for Mean Only)**

Analyze the response of interest Factorial Plots (Main Effects, Interaction) Statistical Results (ANOVA table and p-values) Residual Plots by factor Reduce model using statistical results Use the residuals plot to evaluate potential existence of variation effects If residuals plot indicates a possible variation effect, go to Passive Analysis Roadmap - Part 2 This is the roadmap for the Passive Analysis, Part 1 which is for the mean model only. The purpose of this Part 1 map is to determine if there is a variation effect worth moving on to Part 2.

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Interaction Plot Based on the interaction plot, a few of the interactions may be significant. Check the statistical output for verification. Create the plots first to get a rough visual idea. Make both the main effects & interaction plot at the same time. But always look at the interaction plot first. Options: Draw full interaction plot matrix Remember : looking for non-parallel lines Stat > DOE > Factorial > Factorial Plots > Interaction Plot

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Main Effects Plot The main effects plot indicates that factors B and E have the largest effects. Factor A also has a moderate positive effect. Factor F does not seem to be important. Let’s look at the results. Stat > DOE > Factorial > Factorial Plots > Main Effects Plot

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Factorial Analysis A preliminary look at the statistical output of the experiment indicates factor F may not be significant. Did we make a mistake by including it in the experimental design? * Note that the 3-way and 4-way interactions are still in the model but not presented in the output above Estimated Effects and Coefficients for Y1 (coded units) Term Effect Coef SE Coef T P Constant A B E F A*B A*E A*F B*E B*F E*F This is fitting the full model. Pareto of effects charts are not necessary since the DOE was replicated. We will reduce the model down according to a p-value cut-off = 0.2, just like in week 3. Stat > DOE > Factorial > Analyze Factorial Design

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**Reduce Model to Significant Terms**

Our final model indicates that factors A, B, and E are significant, along with interactions BE, BF, ABF, and BEF, using a p-value cut-off of 0.2 Estimated Effects and Coefficients for Y1 (coded units) Term Effect Coef SE Coef T P Constant A B E F B*E B*F A*B*F B*E*F S = R-Sq = 73.11% R-Sq(adj) = 70.08% Refresher on choosing p-value = 0.2 vs For purposes of model building, it’s OK to use a higher cut-off for interactions in the model. Keeping them in the model may allow a model with a lower mse (better predictions) and higher r-sq(adj). Stat > DOE > Factorial > Analyze Factorial Design

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**The Role of Residual Plots in RD**

In Robust Parameter Design, the residual plots can show the possibility for a variation effect Remember from ANOVA and Regression, we stated one of the assumptions on the residuals was constant variance and we checked this via plots Stat > DOE > Factorial > Analyze Factorial Design Choose Graphs > Residuals vs Variables > A B E F Generally use standardized residuals.

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What Next? After reducing the model, the “Residuals versus Factor F” plot still indicates that F contributes to a variation effect. This finding should encourage us to move further in the analysis of this data to create a variability response and analyze the data. Thus we move on to Part 2 of the roadmap. Stat > DOE > Factorial > Analyze Factorial Design Choose Graphs > Residuals vs Variables > F We face a statistical paradox when this situation occurs! The ANOVA procedure requires that the variances be equal across all treatment combinations. Yet in this test we have discovered that there is a violation of this assumption. Thus, the statistical validity of the ANOVA output may be questionable. But this is GOOD news for Robust Parameter Design optimization.

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**Passive Analysis Roadmap - Part 2 (if Variation effect present)**

Create a Variability Response Analyze Variability Factorial Plots (Main Effects, Interaction) Statistical Results (ANOVA table and p-values) Reduce model using statistical results Compare main effects plots for mean and variability to determine which are Mean Adjustment Factors and which are Variance Adjustment Factors (or both) Use the Multiple Response Optimizer to find optimal settings of the factors Mean on target Minimum variability Perform a capability study / analysis on the resulting factor settings Once a potential variation effect is observed via the residual plots, proceed to Part 2 of the roadmap and actually model the variability.

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**Create a Variability Response**

We are now going to use the replications to make a new response in order to model the variability. Once we have modeled the variability, we can use the MINITAB Response Optimizer to find the settings of the control factors that will put Y on target with minimum variation. MINITAB makes this easy with a pre-processing of the responses in preparation for a variability analysis You will see that MINITAB will use the standard deviation as the measure of variability, rather than the variance the results are equivalent MINITAB 14 has added a capability to calculate the variability for each treatment combination. Those using MTB14 for the first time won’t have full appreciation for this! Doesn’t matter whether you use st. dev. or variance. The results are the same.

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**Uses Natural Log (Standard Dev of Y)**

All of the statistical techniques that we are using to analyze this DOE assume that the data is symmetric (because we are testing for mean differences) Unfortunately, when we use a calculated standard deviation as a response, we do not meet this assumption because the sampling distribution of variances is expected to be skewed, hence the distribution of standard deviations would also be skewed Raw St Dev Ln (St Dev) A discussion of this response transformation can be found in “Response Surface Methodology” by Myers & Montgomery, p. 579 (although they work with the variance). In addition to the achievement of symmetry gained through a log transformation, we also have the added benefit of minimizing the effects of statistical outliers on our analysis. See graphs below. MINITAB takes the natural log automatically, behind the scenes. There is no way to turn it off. But we don’t have the extra step of taking the ln either.

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**Create a Variability Response**

Stat > DOE > Factorial > Pre-Process Responses for Analyze Variability Type whatever you want to name the column containing the standard deviations into the middle column. The last column is a MINITAB needed column. Don’t move, remove, or change this at all!

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**Create a Variability Response**

Worksheet should now contain the following new columns Have one standard deviation for each treatment combination.

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**Analyze the Variability**

Analysis of the variability will be essentially identical to the analysis for the mean Will select the “Terms” to estimate in the model Will use the Pareto of Effects “Graph” in order to facilitate the first model reduction Stat > DOE > Factorial > Analyze Variability Now, this data looks like a single replicate of a 24 to MINITAB so we can’t fit the full model.

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**Analyze the Variability**

TERMS GRAPHS Same as with the mean.

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**Pareto Chart of the Effects**

Because we don’t have any degrees of freedom for error, we must look at the Pareto of effects to decide which term to drop into the error and begin to reduce the model Drop ABD first, then reduce the model down according the p-value = 0.2 cut-off. Stop there. Drop ABD interaction first

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**Final Model for ln StDevY1**

Once the insignificant terms have been eliminated using a p-value cut-off of 0.2, the reduced model is shown below Regression Estimated Effects and Coefficients for Natural Log of StDevY1 (coded units) Ratio Term Effect Effect Coef SE Coef T P Constant A B E F A*B A*E A*F B*F A*B*E B*E*F A*B*E*F R-Sq = 99.93% R-Sq(adj) = 99.75% Remember that this model is for the natural log of st dev.

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**Interaction Plot for StDevY1**

The interaction plot indicates a moderately strong interaction between factors A & E and A & F Where should factors A, B, E and F be set in order to minimize the variability in voltage drop, Y1? Same steps as in the mean analysis. Make both main effects & interaction plots at the same time but always look at interactions first. Options: Draw full interaction plot matrix Answer: set A = 10, B = doesn’t matter, E = 13.5, F = 69 Stat > DOE > Factorial > Factorial Plots > Interaction Plot

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**Main Effects Plot for StDevY1**

Factor F has the largest effect on the variability. Increasing F should reduce variability. But what did the interaction plot show? Factor A is the next strongest. Set A = 10. What did the interaction plot show? Factors B and E are weak but what did the interaction plot show? Answer from interaction plot: set A = 10, B = doesn’t matter, E = 13.5, F = 69 F is the same, A is the same, E is a weak main effect but it’s interaction with A calls for setting it to 13.5, B is weak but statistically significant so set to 5. Stat > DOE > Factorial > Factorial Plots > Main Effects Plot

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**Determine Mean & Variation Effects**

The graphs at right allow us to directly compare each factor’s singular effect on both the mean and variation Based on these graphs, Factors B and E are Mean Adjustment Factors since they affect the mean with little or no effect on the variability Factor F is a Variance Adjustment Factor since it affects the variability with little or no effect on the mean Factor A appears to affect both mean and variability Affects Both Affects Mean Affects Variation There were significant interactions!! So watch out basing conclusions on main effect plots only. These plots are ideal to use for this purpose if no interactions are present. For the students, it may be easier just to tile these 8 graphs but then there is no guarantee that the plots will be arranged as above. The graph above is quite involved. When making the main effects plots from the “Factorial Plots” menu, select both Y1 and StDevY1 as responses. Then select only 2 factors at a time, A & B first. Makes 2 plots. Then ctrl-E and select E & F. Makes 2 plots. Will need to make sure the scales are the same. If not, adjust BEFORE using the layout tool. Then highlight these 4 graphs and right-click to use the layout tool from the Graph Manager window. Arrange both Y1 graphs in the top row, arrange both StDevY1 graphs in the bottom row. Then choose “Finish”.

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**Quality Check: Status of Your Models**

Use the “Show Design” icon to check on the status of the analysis. You should make sure that the correct model has been fit for each response that you intend to specify in the response optimizer. As shown in this window, models have been fit for both the Y1 and StDevY1 responses Just to make sure both the mean and variability models are fit.

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**Multiple Response Optimizer**

Set the Weight for Y1 to 10 to ensure hitting 60, tight lower & upper range Read “first-guess” target & upper values for StDevY1 from interaction plots Note that StDevY1 is in regular units here, NOT logged units! For StDevY1, set weight low to protect against a bad first guess Set up the optimizer to hit the mean target of 60 and minimize the st dev. Stress that st dev is in the regular units!! NOT ln stdev! Look at the interaction plots for StDevY1 to get an idea of what the StDevY1 target/upper value should be. Same comments about the optimizer holds true again. One way to “help” it is to put in tight limits on the mean and set a weight to 10. This makes the desirability drop down significantly as soon as it’s off the target value. And put in a wide range on the stdev and set the weight to 0.1. This makes the desirability drop very slowly as you move away from the target. It drops off as you get above the upper limit. Stat > DOE > Factorial > Response Optimizer

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**Multiple Response Optimizer**

We use the multiple response optimizer to provide a stacked main effects plot. This plot allows us to interactively manipulate the values of each factor in the model and see the effect on both the mean and the variation. You can use the red sliders to tune each of the factors

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**Use the Equations to Confirm Y1**

Let’s use the model coefficients to predict and see that it matches (make sure to use un-coded)! From the optimized solution, A = , B = 1, E = 13.5, F = 69 Y1 = *A *B *E *F *B*E *B*F *A*B*F *B*E*F Y1 = * * * * *1* *1* * *1* *1*13.5*69 Y1 = , as seen in the optimizer window There were model coefficients for both the mean model and the variability model. Make sure to use the un-coded coefficients printed out at the bottom of the analysis. This exercise is to confirm that the optimizer is using the coefficients from the fitted model as well as to show how to use the model coefficients to predict at ANY setting of the factors. This is for the mean model. Show where these coefficients are in the session window.

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**Use the Equations to Confirm StDevY1**

Again, make sure to use the un-coded coefficients! Again, A = , B = 1, E = 13.5, F = 69 lnStDevY1 = *A *B *E *F *A*B *A*E *A*F *B*F *A*B*E *B*E*F *A*B*E*F lnStDevY1 = * * * * * * * * * * *1* * *1* *1*13.5* * *1*13.5*69 lnStDevY1 = StDevY1 = e = , as seen in the optimizer There were model coefficients for both the mean model and the variability model. Make sure to use the un-coded coefficients printed out at the bottom of the analysis. This exercise is to confirm that the optimizer is using the coefficients from the fitted model as well as to show how to use the model coefficients to predict at ANY setting of the factors. This is for the variability model, ln(StDev). After predicting using the equation, must raise e to that power in order to get StDev in regular units. Show where these coefficients are in the session window.

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**Final Design Capability Analysis**

Did we achieve our objectives? worksheet “passive capability” Stat > Quality Tools > Capability Analysis > Normal Y = Y1Final; Lower Spec = 58; Upper Spec = 62 If the students ask, we can talk about this being one of the engineering situations where we could go with something that is perhaps lower cost and not improve it as much as it possibly could be. Maybe opening the tolerances up would result in a process cost savings or being able to go with a less-expensive supplier, while still remaining capable. This is overkill. Instructors only : reference that the blackbox was used to obtain this data. Factors were set to the levels given in the optimizer and a capa data set was generated. The one key thing to note here, if anyone tries to match this, is that F is coded backwards in the blackbox, i.e. if you want F = 69, the slider MUST be set to 63 … AND … if you want F = 63, the slider MUST be set to 69. This is model set B.

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**Remember the Two Strategies?**

We just reviewed the Passive Approach Noise factors are NOT included, manipulated or controlled in the experimental design We analyzed the variability of replicates from an experimental design Now we will look at the Active Approach Noise factors ARE included in the experimental design in order to force variability to occur We will see that the analysis is similar to the passive approach This should be for Tuesday

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The Active Approach A factorial experiment is performed using Control AND Noise factors in the same experiment. Analysis can be performed by characterizing Control*Noise interactions only or by moving forward to analyze the variability by dropping the noise factors into the error term. Pros Simple extension of standard experimental techniques Guarantees noise in the Noise factors Provides for flexibility in analysis methods Can allow for reduced replication Cons Requires ability to manipulate and control Noise factors Optimal designs for minimization of unneeded effects (noise by noise interactions) can be difficult to create

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**Example: An Active Noise Experiment**

3 control factors 1 noise factor 24 full factorial design Two Approaches to Analysis Use only interpretation of interaction plots to choose settings of the control factors to minimize effect of noise Model the variability by dropping the noise factors into the error and analyze like the passive approach

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**Active Analysis Roadmap – Plots Only**

Create and execute with noise included as a factor Analyze the response of interest Factorial Plots (Main Effects, Interaction) Statistical Results (ANOVA table and p-values) Reduce model using statistical results Review Interactions Plot Interpret the interaction plots to look for evidence of variation effects Review Main Effects Plot (if applicable) Use the Multiple Response Optimizer to find the optimal settings of the factors such that the mean is on target Will force in settings obtained from the interaction plots Perform a capability study / analysis on the resulting factor settings

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**Example: An Active Noise Experiment**

An engineer is interested in improving the stability and robustness of a filtration product Review the capability of the current performance to determine the opportunity to apply robust design techniques Stat > Quality Tools > Capability Analysis > Normal Y = Y4Initial; Lower Spec = 60; Upper Spec = 80 Y could in theory be efficiency or flow or something you want to target at 70.

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**Design Capability Analysis**

What conclusions can you draw from this graph? This study indicates that the design is on target, but is marginal with respect to variation The objective is to reduce this variation through design modifications

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**Example: An Active Noise Experiment**

The device contains several control factors from which three were identified as DOE candidates Pressure (A) Concentration (B) Stir Rate (C) Ambient Temperature was identified as being significant, but not economically controllable Temperature would not change appreciably during the time in which it would take to execute a three factor experiment Decided to include it as a factor in the design to force it to change Call this Factor G

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**The Experimental Design**

A 24-1 fractional design (Res IV) was rejected because the 2-way interactions are of great interest in this experiment A 24 full factorial design was used Because of time constraints, only 1 replicate was performed The variables are listed below: A = Pressure B = Concentration C = Stir Rate G = Temperature The data is in worksheet “Active Design” Open worksheet “Active Design” within “Robust Design.mpj” The crux of Robust Design is the control*noise factor interactions so you need a design which will estimate the 2-way interactions. Certain resolution IV designs may be OK but it depends on the number of factors and alias structure.

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Factorial Analysis This DOE is an unreplicated, 4-factor full factorial We need to create the “Pareto of Effects” chart Have the class reduce the model down using this chart and a p-value cut-off of 0.2. Stat > DOE > Factorial > Analyze Factorial Design

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Fit the Reduced Model Based on the p-values, the A*G and B*G and C*G interactions are important. Since we have identified some important control*noise interactions, the next step is to examine the interaction plots. Estimated Effects and Coefficients for Y4 (coded units) Term Effect Coef SE Coef T P Constant A B C G A*B A*G B*C B*G C*G A*B*C A*B*G A*B*C*G You will notice highly significant 3-way and the 4-way interaction. How would you study those? Even though they are statistically significant, do you think they are practically significant, i.e. worth the grand effort to plot that 4-way interaction? You remember how involved plotting up a 3-way interaction was from the Multi-Level example in week 3? Stat > DOE > Factorial > Analyze Factorial Design

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Interaction Plot The interaction plot show the valuable interactions available to the designer. Let’s take a closer look at the strongest ones. Yeah, the interactions we need! Options: Draw full interaction plot matrix If we are lucky, B can be set at 5, C can be set at 150 and then A can be usefd to set the mean Stat > DOE > Factorial > Factorial Plots > Interaction Plot

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**Interaction Plot – A Closer Look**

The two interaction plots at right indicate that both B and C can be exploited to desensitize Y4 to the noise variable G If B is set at its high level, the slope of the G effect line is minimized If C is set at its low level, the slope of the G effect line is also minimized To minimize output variation due to noise in ambient temperature (G), the above two settings should be controlled in the design Choosing B = 5 makes the response insensitive to the noise distribution of G. Choosing C = 150 makes the response insensitive to the noise distribution of G. The graph above was obtained by creating the B*G interaction plot, then the C*G interaction plot, then using the “Layout Tool” within the MINITAB graph manager to stack them on top of each other.

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Main Effects Plot Since factor A is not involved in a significant interaction and its main effect is significant, we should take a look at its main effect plot to see if there is some potential value in controlling A This plot indicates that factor A has about +/- 3 units of control over the nominal value of Y4. Thus, if manipulating one of the other factors takes the mean value off target, this factor could be used to exert some control over the mean value of Y. A was not involved in any strong interactions so can use it to get back on target. Leave it at nominal, according to this plot. Stat > DOE > Factorial > Factorial Plots > Main Effects Plot

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**New Settings : Capability Analysis**

B was set to 5, C was set to 150, A was left at nominal (15) It appears that the changes to B & C were successful in reducing variation in Y4 but the mean is now off target Use factor A to adjust back to target! This data was obtained from the Blackbox with the settings given above. Now off target so where should A be set? Use the optimizer and the mean model.

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**How much should we shift factor A?**

Set up the response optimizer to target 70. The lower and upper limits are not important since we will manually manipulate this. Set factor B=5 and factor C=150. The optimizer indicates a nominal Y4=67.7, very close to that observed in the validation study. Finally, slowly slide the bar for factor A to the right while observing the predicted value of Y4. This indicates that a nominal setting of A=18.8 should achieve Y4=70. For instructors only – can see that the sensitivity to G remains about the same, whether A is set at 15 or To see this, take the top optimization and move the slider on G from -1 to +1 and observe that the predicted Y4 changes from – Doing the same with the bottom optimization, move the slider on G from -1 to +1 and observe that the predicted Y4 changes from 63.3 – This is nearly the same delta but now Y4 is on target of 70. Also note that the s overall from the previous capa study is 2.15 while the s overall from the next capa study is 2.20, confirming that moving A doesn’t really affect the variation. If someone points out that the slope of the G line should be flat, and that as above, it implies that we’re still not insensitive to the noise, they would be right but you don’t want to go there. Simply state that it is improved from the current process before the optimization. Note that in the initial capa study, s overall was 3.98. Worksheet “active design” Stat > DOE > Factorial > Response Optimizer

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**New Setting for A : Capability Analysis**

When we shift factor A to this new nominal value, we succeed at shifting the response to put it on target without degrading the variation Shows results when run with the values from the optimizer.

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**Summary Variation improvement strategies can take two forms:**

Passive Approach Active Approach The choice of which strategy to use depends on the ability to control or manipulate noise factors (at least for the duration of the experiment) Standard full and fractional designs can be used Variation effects must be calculated using replications A log transform of the variability response is automatically used to minimize the effects of asymmetry in the variance distribution The response optimizer can be used to simultaneously optimize both mean and variability responses A validation study must be made at the end of a Robust Parameter Design study

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Objectives Revisited At the end of this module, participants should be able to : Identify possible variation effects from residual plots Create a variability response from replicates Identify possible mean and variance adjustment factors from noise-factor interaction plots Use the MINITAB Response Optimizer to achieve a process on target with minimum variation Complete validation capability studies

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Chapter 13 Multiple Regression

Chapter 13 Multiple Regression

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