Design of experiments (DOE) is a structured approach to modeling components, systems or processes using a test sequence called an experimental array. In the test sequence, changes are made to the input variables, which in-turn affects performance of one or more quality characteristics as captured in the output. Figure 1 illustrates a full-factorial array (2^k) with two variables (k) at two levels. In this four run array, each variable level is coded with a plus (+) and a minus (-) to represent the extreme values at which each variable will be tested. The array outlines all tests per-formed on glass substrates from two manufacturers (A- and A+) etched with two concentrations of acid (B- and B+).

The experimental array displays an equal number of pluses and minuses for each variable. Array balance en-sures that each input variable can be quantified independently. Once all the experiments are run, a response variable analysis correlates the input and response variables in a simplified model, y = b₀ + bX.  These predictive models assist in making knowledge based decisions by demonstrating the input (x) and output (y) relationship quantitatively (b). Understanding these relationships, an engineer can readily adjust an input variable to target a performance for a desirable quality characteristic.

DOE is an efficient and cost effective means for solving problems, as well as modeling products and processes. Yet, many organiz-ations continue to apply the one-factor-at-a-time (OFAT) linear test methodology. The OFAT method requires large numbers of experiments and is incapable of modeling interactions between system variables. On the other hand, design of experiments relies on a parallel testing procedure to model relationships, y = f (x), between main effects (A, B) and variable interactions (AB) relative to a quality characteristic (y). Parallel testing minimizes development time and costs, adding value and reducing time-to-market.

The ability to quantify variable interactions is a major benefit of DOE. Interactions occur when the effect of one factor depends on the level of a second factor. Anderson defines an interaction as “the combined change in two factors that produces an effect greater (or less) than that of the sum of effects expected from either factor alone.¹ Interaction effects reveal themselves in two possible modes, synergistic or antagonistic. Knowing the combination of factor levels that produces a desired effect could contribute to a significant improvement in performance.

Interaction and main effect values from the acid etch tests are tallied in Figure 1. Variable effects and betas (b) measure how main and interaction variables contribute to the output – in this case silica precipitates in an acid-etch bath. Effect values indicate a factor’s relative level of influence on the system under investigation. In this test array, all factors influenced the quality measure as reflected by their respective effects. Betas, which depend on the number of factor levels tested, quantifies each factor represented in the prediction equation. The resultant prediction equation for the acid-etch data is

y = 24.0 - 2.0 A + 10.5 B - 7.5 AB.

The acid-glass type interaction ² (AB) is illustrated graphically in Figure 2. The quality characteristic - precipitate volume - varies depending on acid concentration and to a lesser degree, glass type. Since the two main effects are involved in an interaction, selecting variable levels that achieve a desired performance – low precipitates – are determined from the interaction plot. The interaction is attributed to compositional differences in the glass substrates. However, rather than make specification changes for the substrates, the interaction plot reveals that the low acid concentration provides a sufficient etch for both glass types. Once lower acid levels were implemented on the manufacturing floor, rework was reduced, plus equipment maintenance costs were improved. This single change led to several hundred thousand dollars in annual savings.

Design of experiments offers advantages over other experimental methodologies, like OFAT. Parallel testing reduces development resources and the methodology allows for the quantification of factor interactions. The availability of software simplifies the data analysis. DOE software can readily analyze large volumes of data and quickly construct complicated prediction models from the results. DOE offers the opportu-nity to reduce development and manufacturing costs by improving system efficiency, while simultaneously improving product performance. There is one catch however – you must actually apply the method in order to reap its benefits!

1. Anderson, Whitcomb; DOE Simplified, 2000.
2. Interaction graphic generated using Design-Expert software.

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