Figure 1. Strategy of Experimentation

What methods are currently used for process optimization? How does one find the ideal settings when there are competing criteria? Design of experiments (DOE) is a formalized method of data collection and analysis. Computer technology has turned what used to be a series of tedious mathematical calculations into fairly quick, yet sophisticated, analyses. From these analyses, a subject-matter expert can optimize process settings by making knowledgeable tradeoffs in the response criteria. DOE can be used both in laboratory research and on the manufacturing plant floor.

Researchers and engineers who are unfamiliar with DOE can be overwhelmed by the number of design options. Designed experiments can be run to accomplish many goals. The right design must be chosen to meet the objective of the problem. Figure 1 illustrates how the objective of the design determines the type of design chosen. The discovery phase uses screening designs to identify the primary factors that control the system. These designs are especially helpful to quickly sift through a large number of possible factors whose true contributions to the system are not understood. Typically, two-level fractional-factorial designs are used in this phase, preferably something called a Resolution IV design.¹ The breakthrough phase uses either full factorial or slightly fractional factorial designs (Resolution V or higher) to positively identify both main effects and interaction effects. At this point, center points may also be added to the design in order to determine if any quadratic effects might be present. The optimization phase is required when at least some of the factors have a curvilinear relationship with some of the responses. It uses response surface designs, such as central composite (CCD) and Box-Behnken (BB) designs.² Lastly, validation runs finalize the DOE process by confirming the results in a longer-term setting.

Case Study

Two standard response surface designs³ are central composite designs and Box-Behnken designs. This client chose the Box-Behnken design because each of his three factors could be easily set to the three levels required by the design. The total number of runs is 17, which was deemed reasonable by the manufacturing personnel. The design is shown in Table 1. The runs are physically completed in the random run order. Notice that the design includes five center points that are randomly scattered throughout the other runs.

Product samples were made at each of the 17 sets of conditions. The laboratory then measured seven key quality characteristics, including some viscosities, tensiles, and subtak and subadhesion properties. This data was entered into a statistical software package⁴ that can perform an analysis of variance in order to establish a polynomial prediction equation for each response.

Figure 2a. Contour Plots with X3 Set at Low Level

Sample Data Analysis

The p-value is the primary column of interest. It quantifies how significant each term is in the polynomial model. A smaller number is better, and as a general rule of thumb, terms are considered statistically significant when their p-value is 0.05 or lower. In this case all but one of these terms is substantially lower than 0.05. You might notice that the AC term is missing. This term had a p-value significantly higher than 0.05, so it was removed from the model.

Figure 2b. Contour Plots with X3 Set at High Level

In addition to the p-values, the R-squared values given in the lower part of the table are also of interest when a response surface design is run. The R-squared represents the amount of variation in the data that is explained by the model. The predicted R-squared represents the amount of variation in predictions that is explained by the model. When the objective of the experiment is to optimize, higher R-squared values are important, implying that the polynomial model is a very good predictor of the response. The higher the R-squared values are, the better the polynomial is at either describing the system or making predictions about the system. For this response, the R-squared values of approximately 98-99% indicate that this polynomial is a very good description of the relationship between these three factors and the Tensile at 0 day response.

Sample Graphical Analysis

These contour plots have X1 on the horizontal axis and X2 on the vertical axis, with X3 fixed at a specific level (in this case, the low and high levels of 5 and 20). Tensile at 0 day predicted contours fill the graphical area. According to the experimenter's subject matter knowledge, the optimal response value is 65. Notice that on the left-hand graph, there is a contour line labeled "65" that cuts through the middle of the graph area. When X3 is set at its low value of 5, any of the combinations of X1 and X2 settings that fall on that 65 line would be feasible. Now the X3 factor is moved to its high setting of 20 and the right-hand graph shows the line for "65" falling closer to the lower-left corner.

Figure 3. Multiple Response Optimization Sweet Spot

Multiple Response Optimization

Table 3 shows a summary of all seven responses, the terms that were significant in their models and summary statistics for each model. Typically, some of the responses will fit to polynomials better than others. In this case, Tensile at 0 day fits almost perfectly, while Subadhesion does not fit as well, but is still adequate.

After the analysis of each response is complete, these models are used in the numerical optimization routine. This case study uses the optimization criteria in Table 4. The goal is the targeted value, but it is surrounded by an acceptable range that is defined by the low and high values.

The "sweet spot" in processing conditions that achieves all the goals for this adhesive is shown in Figure 3, which displays factors X1 and X2, with X3 set at its low level of 5.

The client may have stumbled upon this set of operating conditions by chance, but it is unlikely that they would have had a clear picture of which responses define this region. It turns out that the left side of the operating window is defined by the initial viscosity limit of 55,000. The bottom is defined by the viscosity at 50%, and the right side is defined by both the subtak at 0 day and the tensile at 3 day. If the limits of any of these can be expanded, then the operating region will open up. This is a critical gain in process understanding, because it also indicates that the other responses do not control the operating conditions.

Summary

This was simply one illustration of the use of design of experiments. Two-level factorial DOE can be used to identify which variables in a process are critical to control. Mixture designs are used to optimize a chemical formulation to meet specific criteria. Robust design helps find factor settings that make a product or process robust to variations in uncontrollable variables.

Technical professionals will find many ways to apply design of experiments. It just takes a little education and a bit of practice to discover that experimentation can be done on a whole new level.