Reliability HotWire

Reliability HotWire

Issue 123, May 2011

Hot Topics

Taguchi Robust Design for Product Improvement: Part II

During the past several decades many methods on experimental design and data analysis for achieving product/process robustness have been proposed. In this article, we will use ReliaSoft’s DOE++ to illustrate the location and dispersion modeling method for doing so. Two models (transfer functions) will be built: one for the location (the mean of the observations) and the other for the dispersion (the standard deviation of the observations). The optimal setting of each control factor will be identified using these two transfer functions. This article provides a complement to Issue 122 of Reliability HotWire where we presented a case study on using the signal-to-noise ratio as the response for the regression model. The optimal setting of each control factor was identified using the transfer function between the control factors and the signal-to-noise ratio.

Philosophy

In Design of Experiments (DOE), the key to having a successful experiment is to carefully plan the design. In order to have a well designed experiment, engineering knowledge should be integrated into the design process from the very beginning, such as determining the number and values of the control factors and the noise factors. As pointed out by many researchers and engineers, the secret of robust design is the interactions between the control factors and the noise factors. If there are no interactions between these two types of factors, then it is very difficult to achieve a robust design. Let us use a simple example to illustrate this statement. Assume that there is only one control factor and one noise factor, and both are 2 level factors. The two possible scenarios of the interaction effect of these two factors are given below.

Factor interaction

The plot of (a) shows that there is no interaction between the two factors. The changes of the response y caused by the change of the noise factor are the same at the high and low levels of the control factor. In other words, it is impossible to improve the robustness to the noise of the product by adjusting the value of the control factor.

The plot of (b) shows that there is a strong interaction between the control factor and the noise factor. The changes of the response y caused by the change of the noise factor are different at the high and low levels of the control factor. In other words, it is possible to improve the robustness to the noise factor of the product by adjusting the setting of the control factor. For plot (b) we can see that the variation of y caused by the noise factor is smaller when the control factor is set to its low level. Therefore, setting the control factor to its low level will improve the robustness of the product to this noise factor.

From the above discussion we can see that reducing the response variation is one of the major goals of robust designs. Using the signal-to-noise ratio as the response can help in reaching this goal. Another widely used method in robust design is the so called location and dispersion modeling method, or the mean and standard deviation modeling method. This method assumes the mean and variation are independent. We will use the same example from the previous article to illustrate this method. Note that this example is only used for illustration purposes and the data presented here have been modified from a real application.

Example

An automobile manufacturer wants to improve the quality of the painted surface of its cars. The quality is measured by the gloss reading of the surface. The manufacturer wants the painted surface to have a higher gloss reading (i.e., maximize the response) and to be robust to the environment. Environmental factors, particularly temperature and humidity, are known to affect the painted surface. Thus, a robust design will be used here. For a detailed background on robust design, please see: http://reliawiki.org/index.php/Robust_Parameter_Design.

The control factors have been identified to be the flow rate of the paint, the pressure in the paint gun, the viscosity of the paint and the cure temperature (i.e., the surrounding air temperature).

The factors and levels are as follows:

Controllable Factors Name Low Level High Level
A Flow Rate 30 50
B Pressure 3 5
C Viscosity 10 15
D Cure Temperature 120 160

Two noise factors will be taken into account: the air temperature (ambient = room temperature) and humidity (ambient = room humidity).

Noise Factors Name Low Level High Level
A' Air Temperature 15 30
B' Humidity 30 90

The design matrix for the control factor in robust design is called the inner array, while the design matrix for the noise factor is called the outer array. For the inner (control) array, a 2 level fractional factorial design with D=ABC is used. For the outer (noise) array, a 2 level full factorial design is applied.

Assume that the experiment has been conducted properly and the data set was entered into DOE++, as shown below.

In the last issue, we used the signal-to-noise ratio method and found out that factors A and D are significant. In this article, we will build two models: one is for the Y mean and the other is for the standard deviation (Y Std) as shown in the above figure. The strategy is:

  • Build a model for the mean response (location) of each treatment (control factor combination).
  • Build a model for the standard deviation (dispersion) of each treatment.
  • Adjust the factors in these two models to optimize the response and reduce the variance.

The modeling process in DOE++ is given below.

Step 1: Click the Design tab.

Step 2: Include all two-way interactions in the model. To do this, click the Select Effects icon. In the Effects window, use the settings shown next, then click OK.

Select Effects

Click the Options page of the control panel and notice that Y Mean is showing in the Response drop-down list and that the Individual Terms check box is selected.

Step 3: In the Response drop-down list choose Y Std and select the Individual Terms check box.

Step 4: Return to the Main page of the control panel and click the Calculate icon.

Notice that the ANOVA table and the Regression Information table for the model are displayed on the Analysis tab, which is added to the folio upon calculation.

Step 5: Click the Plot icon to add the Plot sheet to the folio. From the normal probability plot for the Y Mean response (i.e., for the location model), we can see that effects A and D are significant, as shown next.

Normal Probability Plot of Effect

Step 6: To view the normal probability plot for the Y Std response (i.e., for the dispersion model), select Y Std from the Response field in the control panel. Effect A is identified as significant, as shown next.

Normal Probability Plot of Effect

From Step 5 and Step 6, the significant effects are identified for the Y Mean and the Y Std. Only these statistically significant effects are used to build the final models. The optimal values of the significant factors can be determined using the final models given in DOE++.

Optimization

Step 1: Click the Select Effects icon and select only the significant factors for the Y Mean response, the Y Std response and the Signal Noise Ratio response, as shown next, and then click OK.

Y Mean Effects

Y Std Effects

Signal Noise Ratio Effects

Step 2: Click Calculate.

The Regression Information table for the Y Mean response is shown next.

Y Mean Regression Information

Using the values found in the Coefficient column, the location model is found to be:

ymean = 79.3438 + 2.5312A + 2.6562D

Because factors A and D both have positive coefficients, both should be set to their high levels in order to maximize the mean.

The Regression Information table for the Y Std response is shown next.

Y Std Regression Information

Using the values found in the Coefficient column, the dispersion model is found to be:

ln(ystd ) = 0.7903 - 0.5425A

Because factor A has a negative coefficient, in order to minimize the variance, it should be set to its high level.

So the best settings for this simple example are: A = 50 and D = 160, with factors B and C at their most economical settings.

In cases where the models are complicated (e.g., a case where changing the value of a factor to maximize the mean will also maximize the standard deviation), the optimization tool in DOE++ can be used to find a compromise solution.

Step 3: Click the Optimization icon on the control panel. In the Optimization Settings window, use the settings shown next then click OK.

Optimization Settings

The optimization results are:

Optimization Results

The conclusion from the above modeling method is the same as the results we obtained through the signal-to-noise ratio method used in the last issue. Sometimes these two analysis methods will provide different results. If this occurs, use your engineering judgment on which results to use.

Conclusion

In this article, we introduced the location and dispersion modeling method for robust design and focused on presenting the statistical modeling procedure. In reality, when applying a DOE method, engineering knowledge must be integrated into the design and analysis process. Besides statistical modeling, you should also use other important factors such as design decomposition, measurement improvements, verification testing and the gains made by running a well-planned engineering experiment. The design layout used in this article and the article in the last issue is called the cross array design method. This is because the inner array for the control factors is crossed with the outer array for the noise factors. In other words, each combination of the control factors is tested under all the noise factor combinations, as seen in the data sheet in the Example section. Sometimes this will lead to a very large sample size. To overcome this drawback, you can also conduct robust design in a single array manner where noise factors are treated the same as control factors so the fractional factorial designs can be applied. You can include control factors only or you can include both noise factors and control factors in the transfer functions. If only the control factors are included in the transfer functions, the location and dispersion modeling method discussed in this article can still be used. Complicated statistical models also have been developed for the case when both types of factors are in the transfer function. For details, please see reference [1].

Acknowledgment: We would like to express our appreciation for the useful comments and suggestions from Louis Lavallee. He has many years of experience with Design of Experiments and is a senior reliability engineering consultant at Ops A La Carte.

References

[1] R. H. Myers and D. C. Montgomery, Response Surface Methodology: Process and Product Optimization Using Design Experiments, 2nd Edition, New York, NY: Wiley & Sons, 2002.