Issue 100, June 2009

Using DOE++ For Product and Process Optimization

In last month’s Reliability Basics article, we discussed reliability allocation. The purpose of reliability allocation is to meet the system reliability requirement and, in the meantime, to keep the cost as low as possible. High reliability usually means high cost. A desirable design is one that can provide satisfactory reliability with a minimal cost. This is an optimization problem with two responses: reliability and cost. In this article, we will illustrate how to use DOE++ to optimize several responses simultaneously.

Example:
A chemical engineer is interested in determining the operating conditions that maximize the yield of a process [1]. At the same time, she also wants the viscosity of the product to be within the range of 62 to 68, with the target value of 65; and she wants the molecular weight of the product to be less than 3400, with the target value of 3000.

Two controllable variables influence process yield: reaction time and reaction temperature. From previous experimentation, the engineeer has determined that the ranges of settings that she wants to examine are given in the table below.

A central composite design [2, 3] with 5 center points and alpha = 1.414 is used to conduct the experiment. (Central composite design usually is used to identify a second order model between input variables and output response.) Once the model has been identified, it can be used for optimization.

A quadratic function is used to establish the relationship between A, B and each response. The model is:

Since we have three responses (yield, viscosity and weight), the optimal setting of A and B for one of the responses may not be good for the other two. Therefore, a compromise should be made. A balanced setting that can optimize the overall performance should be found. The desirability function approach is used to come up with a balanced solution. This solution tries to satisfy the requirements for each of the responses as much as possible without compromising on any of the requirements too much.

Under the desirability function approach, each response is assigned a desirability function d_i. The value of d_i varies between 0 and 1, with 0 representing that the worst acceptable value and 1 representing the response that is the target value. The overall desirability function is defined as:

where r_i represents the relative importance of each response. The bigger the r_i value, the more influence the corresponding response has on the overall desirability function. Usually, r_i is assigned a value from 0.1 to 10.

Depending on the objective for the response (maximization, minimization or target value), the definition of d_i is different [2, 3].

For maximization:

For minimization:

For target value:

where:

• is the predicted value for the i^th response.
• T is the target value for a given response.
• L is the acceptable lower limit for a response.
• U is the acceptable upper limit for a response.
• ω is the weight for a response.

The weight, ω, determines how the desirability value changes for a response. A value of ω that is less than 1 is equivalent to saying that any response value between the limit and the target is desirable; a value of ω that is greater than 1 is equivalent to saying that it is very important that the target is met.

The following steps illustrate how to use DOE++ for this chemical process example. The data set for the example is provided in the Multiple Response Optimization Folio in the Multiple Response Central Composite Design.rdoe file, located in the Examples\Response Surface Method folder within the application directory (e.g. C:\Program Files\ReliaSoft\DOE\Examples\Response Surface Method).

Create the Design
Step 1: Choose Project > Add Standard Folio.

Step 2: On the first page of the Design Wizard, select to create a Response Surface Method Design and click Next to proceed to the next step.

Step 3: On the second page of the Design Wizard, select Central Composite Design and click Next to proceed to the next step.

Step 4: In the third step of the Design Wizard, use the following settings:

Step 5: Click the Factor Properties button. In the Factor Properties window that appears, enter the time and temperature values, as shown next, then click OK to close the window and return to the Design Wizard.

Step 6: Click the Response Properties button. In the Response Properties window that appears, define the settings, as shown next, then click OK to close the window and return to the Design Wizard.

Step 7: Click Finish to create the new Folio.

NOTE: The run orders are randomly generated when you create the design, therefore your Folio may be different from the Folio in this example.

Step 8: Conduct the experiment according to the run order in the design matrix and record all the response values. NOTE: For this example, we will use the data that is already entered in the Multiple Response Optimization Folio, as shown next.

Analysis and Results
Once the response values have been entered in the Data Sheet, you can analyze it. The data set is already entered for you in the example file.

Step 1: Open the Multiple Response Optimization Folio.

Step 2: Choose Data > Select Effects and use the following settings:

Step 3: Click the Apply to All Responses button to apply the selected effects to all three responses, then click OK to close the window.

Step 4: Click the Calculate icon on the Control Panel to calculate and display the results.

The Analysis tab provides the ANOVA table and the Regression Information for each response. You can switch the displayed results by choosing a different response on the Control Panel, as shown next.

Step 5: Choose Plot > Contour/Surface Plot to look at the surface plot to view where the optimal solution is. For example, for the Yield, the surface plot is:

From the surface plot, the approximate location for the mountain peak is at Time = 87 and Temperature = 177.

In fact, DOE++ has an optimization tool to help you find the global optimal solution. The steps for using this tool are presented next.

Optimization
Step 1: Choose Data > Optimization.

Step 2: On the Response Settings page of the Optimization Settings window, use the following settings:

Step 3: On the Factor Settings page of the Optimization Settings window, set the range of each factor. These default ranges are the limits of each factor that you used in the experiment.

Step 4: Click OK to get the optimal solution.

The plot shows where the optimal solution is and what the desirability values are. This solution also can be viewed by clicking the View All Solutions icon.

The Optimization Solutions window will appear, as shown next.

The optimal settings are Time = 83.92 minutes and Temperature = 170.42F. At this setting, the predicted yield is 78.3951 with a desirability value of 0.8395; predicted weight is about 3200 with a desirability value of 0.1; predicted viscosity is 65 and the desirability is 1. The achieved overall desirability value is 0.9434.

A follow-up experiment using these settings should be conducted to determine whether the predicted values are accurate. If the response values from the follow-up experiment are significantly different from the predicted values, you need to check whether the follow-up experiment was set up correctly. Otherwise, more experiments should be added in order to determine the true relationship between factors and responses.

Conclusions
In this article, we illustrated how to use DOE++ to optimize a chemical process. We also briefly explained the theory of multiple response optimization and we used the quadratic model for each response. It is worth mentioning that this model is validated in the low and high range of each factor. If the prediction is beyond the range, caution should be taken because the predicted results may no longer be accurate.

References:
[1] Montgomery, D. C. Design and Analysis of Experiments, 5th edition, New York: John Wiley & Sons, 2001.
[2] Myers, R. H. Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 2nd edition, New York: John Wiley & Sons, 2002.
[3] ReliaSoft Corporation, Design of Experiment and Analysis Reference, Tucson, AZ: ReliaSoft Publishing, 2008.

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