Design of Experiments

factors which, in comparison to the major factors, probably have only a low influence on the response and may unnecessarily increase the number of experiments.

Although the expected polynomial of a response surface is usually unknown, the strengths of factors are initially determined by level full factorial experimental designs or two-level fractional factorial experimental designs (Fig. 12) [197]. In these designs, all factors are varied on only two levels, which are referred to as low level (-1) or high level (+1).

These levels which circumscribe the design space, must be selected carefully, because if the high and the low level are too close to each other depending on the S/N-ratio, effects of these factors on the response might not be detected. Likewise, extreme levels should be avoided, as at these levels unrepresentative effects may occur.

Fig. 12: Examples of factorial designs with additional center points: a) two-level-one-factor (OFAT), b) two-level-two-two-level-one-factors, and c) two-level-three-two-level-one-factors. The blue dots represent the factorial points whereas the red dots represent the additional center points.

The choice of appropriate levels is crucial for the validity and the success of the experiments and require process understanding by the scientist. All other factors which are not included in the experimental design must be kept constant. If they cannot be kept

constant and are therefore disturbances, they have to be monitored because they may falsify the determination of the actual factor effects on the responses.

However, these experiments are only suited to estimate the factor influences within the range of the investigated factor levels if these influences are linear. To check the factor influences for linearity, two-level full and two-level fractional experimental designs can easily be augmented by center point experiments as long as all investigated factors of the screening remain in the experimental design. These experimental points are located at the center of the design space and may also be used to determine the S/N-ratio of a response which corresponds to the mean of measurements divided by its standard deviation. Therefore, these center point experiments are repeated several times, whereas the remaining factorial point experiments are only carried out ones. To estimate the significance of the factor influences within the entire design space, the standard deviation of the center point responses is applied to the factorial point responses. This approach might be criticized, as a process is often more stable at the center point than at the factorial points. If the deviation of the actual response values at the center point exceeds the standard deviation of the response surface model a lack of fit is significant. In this case it may be assumed, that the influence of the investigated factors on the response cannot be described by a linear model and additional terms have to be added to fit the model to the actual relationship between the investigated factors and the response. Therefore, additional experimental design points have to be added, to determine a response surface with a higher polynomial. For example, a suitable augmentation of an experimental design to determine quadratic terms of the response surface are star points, resulting in so-called central composite designs (CCDs) (Fig. 13). These designs are referred to as composite designs because they are built of three blocks, namely the already described factorial experiments (corners) of the two-level factorial design, replicated center points, and star

points which are symmetrically arrayed to the factor axis. These star points may be located at different distances to the center point resulting e.g. in central composite circumference designs (CCC) or central composite face centered designs (CCF).

Fig. 13: Examples of central composite designs (CCDs). a) two-factorial face centered design (CCF), b) two-factorial circumference design (2²-CCC), and c) three-factorial circumference design (2³-CCC).

In CCCs, the star point and corner experiments approximate the surface of a sphere, hence these design spaces are symmetrical. As the star points in CCCs are located outside the low and the high settings of the factors, the factors are investigated at five different levels which allows an estimation of quadratic terms with high rigor.

In CCF designs however, the star points are located plane to the factorial points whereby the star point and corner experiments approximate a surface of a cube which may also be considered as a symmetrical design space. Although these designs only result in three levels of each factor, they still support quadratic terms because they contain a sufficient number of experiments. On the one hand, CCF designs may be of advantage compared to CCCs, because factor levels that are plane to the factorial points are often easy to adjust in processes and the results of the experimental points may also be easily interpreted by the

scientists. On the other hand, CCF designs might be inferior to CCC designs as CCCs comprise a larger volume of the design space than CCFs, if the same settings of the low and the high level of the factors are applied in both experimental designs. Moreover, the investigation of five different factor levels also allows a better capture of strong curvatures by which even a cubic response behavior might be modeled. Therefore, CCCs should be preferred over CCFs, unless an extension of the design space beyond the factorial points is impossible.

However, CCDs are only suitable for a limited number of factors, as CCDs with too many factors result in an immense number of experiments. Therefore, these experimental designs should be avoided if more than approximately 5 factors (resulting in at least 26 experiments) have to be investigated. Moreover, these experimental designs are not always applicable, because the factors are not allowed to have constraints within the experimental design space. Alternatives to these symmetrical designs which allow the investigation of high numbers of factors are non-symmetrical optimal designs. Computer algorithms determine the optimal distribution of the design points within the design space, either to optimize the estimates of the specified model coefficients (D-optimal) or to optimize the prediction variance around the model (IV-optimal). These designs can also be augmented stepwise to any polynomial and allow constraints of the factors which are considered within the distribution of the design points within the design space. Because of the non-symmetric distribution of the factor levels, individual runs cannot be interpreted by the scientist, which is a disadvantage of these designs.

However, the planning and performing of experiments with the DoE approach offers many advantages compared to OFAT or unfocussed trial and error experiments. It is an organized approach with which the scientist is guided to perform a structured set of

experiments, which is adequate for the selected objective. Experience showed that DoE requires fewer experiments than any other approach. As all experiments belong to an experimental plan, they are mutually connected in a logical and theoretically favorable manner. Thus, with the implementation of the methodology of DoE more useful and precise information on those factors which significantly influence a response may be assessed by the investigation of the joint influence of all factors and by a defined number of experiments [126,131,191–195,198].