A Short Overview of Designed Experiments for Process Optimization

Process development and optimization for prototyping, R&D or production is often a challenging task, as it consumes considerable project time and can lead to development delays. Effective and careful planning and conducting of the necessary experiments are crucial to mitigate these challenges and ultimately gain process understanding and knowledge about the optimal process conditions. As a result, there can be considerable time and cost savings as well as dramatically improved product quality. ?

Finding the important factors in early-stage development

In early-stage development many factors can be considered as influential on the experimental outcome. Optimization designs with many factors (e.g., n > 5) tend to get too large for most situations. Read here on how to screen through the factors and find the important ones for follow up optimization designs.

Favorable Design Properties for Process Optimization

Most experimental designs aim to fit a refined statistical model to as few experimental runs as possible. If this model is to be used to optimize the process it should predict the outcome as accurately as possible. To achieve this, the design’s underlying model needs to account for common process properties such as interacting factors or nonlinear behavior. These types of designs are often referred to as response surface designs (RSM).

Figure 1 below shows a hypothetical simple linear model compared to a model containing a combination of linear effects and the interaction as well as a full RSM containing the interaction and quadratic effect, all based on the same true response function. It is easy to see that the models predicted optimal areas of the surface can differ substantially. More refined models containing interaction and quadratic effects usually better mimic the true process behavior. The following sections will summarize some classical and more modern versions of these optimization designs.

Historical RSM Designs – CCDs and Box Behnken Designs

One of the oldest and most common RSM designs is the Central Composite Design (CCD). These are constructed by using (fractional) factorial design augmented with center points and so-called star or axial points to account for curvilinear behavior. There are some different categories of CCDs, which mainly differ by the position of the star points compared to the design points of the underlying fractional factorial design. Another common historical design is the Box Behnken Design which is very similar to CCDs but with a different way of distributing the additional runs to account for possible nonlinear behavior. The best way to see these differences is to look for figure 2 below and compare the exemplary design spaces. All these designs have in common that they only work with continuous factors, have fixed numbers of runs for different number of factors and cannot account for any additional needs of the experimenter, such as constrains of the design space, categorical factors, or pre-experimental knowledge. ?

The Bridge between Screening and RSM: OMARS Designs

Classical RSM designs have favorable design properties like high power to estimate effects as well as enabling independent estimation of effects, but they tend to require a larger number of runs. On the other hand RSM enabling definitive screening designs (DSDs) with fewer runs might not be solid enough to model a RSM for a greater number of factors. To close the gap between the small screening DSDs and the rather large classical RSM designs like CCDs a new family of designs, the orthogonal minimal aliased response surface designs (OMARSs) were introduced a couple of years ago. They give much more flexibility regarding the number of runs compared to DSDs while ensuring that the linear main effects can be estimated without being correlated with higher order effects.

Optimal Designs

Both, classical RSM designs and OMARS designs have in common that they are not very flexible when it comes to specific needs of the experimenter, which are quite common in real world experimentation. These can range from hard to design factor type combinations like continuous and multilevel categorical factors in combination with different types of blocks, constrained designs spaces (see figure 2), mixture factors, incorporation of additional data with covariates or hard to change factors. Optimal designs can dynamically calculate the best possible designs for these problems and tailor them to the experimenters needs, such as the number of runs or canceling out specific model effects we know are unlikely to occur. They might not be the best designs for the specific use cases of classical RSM or OMARS designs but fill the white space these more static designs cannot cover. Figure 2 below shows a comparison of the design space between different designs for three factors to illustrate some of the differences.

Design Augmentation

Designed experiments do not need to be a one-shot-approach. In fact, in many cases the experimenters are starting with a plain design and augment it in different steps after an initial successful set of runs. This is mostly done by using the algorithms used for optimal design generation to augment already existing (non) DoE data. Approaches might range from adding optimal distributed runs to observational data, extend a simple screening design into a RSM optimization design, change the factor ranges or even investigate specific additional model terms, such as cubic effects or higher order interactions after initial experiments to further refine the model for optimization.

Space Filling Designs and Machine Learning Modelling

In recent years space filling designs have gained increased attention especially in the context of mixture components factors in the design. While originally intended for deterministic systems like computer experiments, their property of distributing design points in the design space in a uniform way, using different distribution algorithms have proven beneficial for highly nonlinear system behaviours. Since space filling designs do not assume a classical least squares statistical model with polynomial terms for interactions and quadratic effects in the background to distribute the design points in the design space the focus for modelling the system behaviour changed to other machine learning approaches. An overview about recent often used design choices and machine learning models can be found here.

Summary

Designing experiments for process optimization requires models which go beyond fitting linear hyperplanes, ?taking interactions and curvilinear behaviour into account. This can be achieved by using classical designs like CCDs and or Box Behnken Designs in case of continuous factors only or OMARS designs in case the experimenter wants to move in the white space between screening designs and full RSM designs. Other options are optimal designs, which tend to be the most flexible family of designs, or space filling designs in combination with modern machine learning modelling to account for highly nonlinear behavior.

Despite the differences, all of theses approaches should yield much better results than traditional ways of experimenting like one factor at a time or relying on gut feeling. They enable scientists and engineers to gain insights into their processes, avoid firefighting, save time and effort and bring their products to market faster and more reliable.

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