5.
Process Improvement
5.1. Introduction


Experimental Design (or DOE) economically maximizes information 
In an experiment, we deliberately change one or more process
variables (or factors) in order to observe the effect the changes have
on one or more response variables. The (statistical) design of experiments
(DOE) is an efficient procedure for planning experiments so that
the data obtained can be analyzed to yield valid and objective conclusions.
DOE begins with determining the objectives of an experiment and selecting the process factors for the study. An Experimental Design is the laying out of a detailed experimental plan in advance of doing the experiment. Well chosen experimental designs maximize the amount of "information" that can be obtained for a given amount of experimental effort. The statistical theory underlying DOE generally begins with the concept of process models. 

Process Models for DOE  
Black box process model 
It is common to begin with a process model
of the `black box' type, with several discrete or continuous input
factors that can be controlledthat
is, varied at will by the experimenterand
one or more measured output responses.
The output responses are assumed continuous. Experimental data are used
to derive an empirical (approximation) model linking the outputs and
inputs. These empirical models generally contain
first and secondorder terms.
Often the experiment has to account for a number of uncontrolled factors that may be discrete, such as different machines or operators, and/or continuous such as ambient temperature or humidity. Figure 1.1 illustrates this situation. 

Schematic for a typical process with controlled inputs, outputs, discrete uncontrolled factors and continuous uncontrolled factors 
FIGURE 1.1 A `Black Box' Process Model Schematic 

Models for DOE's  The most common empirical models fit to the experimental data take either a linear form or quadratic form.  
Linear model 
A linear model with two factors, X_{1} and
X_{2}, can be written as
For a more complicated example, a linear model with three factors X_{1}, X_{2}, X_{3} and one response, Y, would look like (if all possible terms were included in the model) \( Y = \beta_{0} + \beta_{1}X_{1} + \beta_{2}X_{2} + \beta_{3}X_{3} + \beta_{12}X_{1}X_{2} + \\ \beta_{13}X_{1}X_{3} + \beta_{23}X_{2}X_{3} + \beta_{123}X_{1}X_{2}X_{3} + \\ \mbox{experimental error} \)The three terms with single "X's" are the main effects terms. There are k(k1)/2 = 3*2/2 = 3 twoway interaction terms and 1 threeway interaction term (which is often omitted, for simplicity). When the experimental data are analyzed, all the unknown "\( \beta \)" parameters are estimated and the coefficients of the "X" terms are tested to see which ones are significantly different from 0. 

Quadratic model 
A secondorder (quadratic) model (typically used in
response surface
DOE's with suspected curvature) does not include the threeway
interaction term but adds three more terms to the linear model, namely
\( \beta_{11}X_{1}^{2} + \beta_{22}X_{2}^{2} + \beta_{33}X_{3}^{2} \)Note: Clearly, a full model could include many crossproduct (or interaction) terms involving squared X's. However, in general these terms are not needed and most DOE software defaults to leaving them out of the model. 