Introduction
Welcome to the next lesson in our journey through the Analyze and Improve phases of DMAIC. In our previous session, we learned how to use efficient screening experiments, like Plackett-Burman designs, to filter a long list of potential factors down to the "vital few" that have a significant impact on our process output.
Now that we have this manageable list of 2 to 5 key factors, we need to understand their behavior in more detail. It's no longer enough to know that a factor is important; we need to know how it behaves and, critically, how it interacts with other important factors. For instance, increasing temperature might improve a result, but only when a specific type of material is used. This interplay is called an interaction effect, and understanding it is often the key to true process optimization.
This lesson will equip you with the tools to do just that. Our learning outcome is to design a full or fractional factorial experiment (e.g., 2^k) to study main effects and interactions. These designs are the gold standard for characterizing, or deeply understanding, the vital few factors that drive your manufacturing processes.
1. Full Factorial Designs: The Complete Picture
The most straightforward way to study your vital few factors is with a full factorial design. The name says it all: you test the full set of all possible combinations of the factor levels.
In Design of Experiments (DOE), we typically study factors at two levels: a "low" setting (coded as -1) and a "high" setting (coded as +1). A 2^k full factorial design refers to an experiment with:
- k factors
- 2 levels for each factor
- 2^k total experimental runs (or unique combinations)
For example, a 2^3 design involves 3 factors, requiring 2 x 2 x 2 = 8 runs to test every combination. This allows us to estimate:
- 3 main effects (one for each factor)
- 3 two-factor interactions (A*B, A*C, B*C)
- 1 three-factor interaction (A*B*C)
The key advantage of a full factorial design is its clarity. Because you run every combination, the effect of each factor and each interaction can be estimated independently, without being mixed up or "confounded."
2. A Practical Example: Designing a 2^4 Full Factorial
Let's walk through designing a full factorial experiment using a realistic manufacturing scenario. This example involves optimizing the viscosity of glue used in plywood manufacturing.
12. Full Factorial Design - Part 1
This video by Wan Fazli provides an excellent, step-by-step walkthrough of designing a full factorial experiment. It starts by introducing a case study and then uses Design-Expert software (though the principles apply to any statistical package like Minitab) to build the experimental plan.
Please watch the following segments to understand the process: Case Study Background (00:00 - 10:14): Understand the problem: reducing glue viscosity (measured by filtration rate) while trying to decrease the concentration of a hazardous chemical (formaldehyde). The team identifies four key factors: Formaldehyde concentration, Pressure, Temperature, and Agitation (RPM). Defining Factors and Levels (10:14 - 13:12): See how the high and low levels are defined for each of the four factors. This transforms the process variables into a structured experimental design. Creating the Design in Software (13:12 - 15:47): Observe how a 2^4 (2 levels, 4 factors) full factorial design is selected. Notice that this will require 2^4 = 16 runs. Entering Factor & Response Details (17:41 - 20:50): Watch how the factor names, units, levels, and the response variable ('Filtration Rate') are entered into the software to generate the final experimental plan.
After completing these steps, the software generates a randomized run sheet with 16 distinct combinations of the factor settings. Your job as the engineer would be to run these 16 experiments and record the filtration rate for each one. The final step, which we'll cover in a future lesson, is to analyze that data.
3. The Need for Efficiency: Fractional Factorial Designs
A full factorial design is powerful, but it has a significant drawback: the number of runs grows exponentially.
- 4 factors (2^4) = 16 runs (as in our example)
- 5 factors (2^5) = 32 runs
- 6 factors (2^6) = 64 runs
- 7 factors (2^7) = 128 runs
In many industrial situations, running 64 or 128 experiments is too costly and time-consuming. This is where fractional factorial designs become essential.
A fractional factorial design, denoted as 2^(k-p), is a clever way to study k factors by running only a fraction (1/2^p) of the full factorial runs.
- A half-fraction (2^(k-1)) uses half the runs. For 6 factors, a 2^(6-1) design requires 32 runs instead of 64.
- A quarter-fraction (2^(k-2)) uses one-fourth of the runs. For 6 factors, a 2^(6-2) design requires only 16 runs.
This efficiency is based on the sparsity of effects principle, which you'll recognize from our last lesson: in most systems, the main effects and low-order interactions (e.g., two-factor) are much more significant than high-order interactions (three-factor and above). Fractional designs leverage this by sacrificing our ability to estimate those high-order interactions in exchange for fewer runs.
The Trade-Off: Confounding and Aliasing
This efficiency comes at a price: confounding, also called aliasing. When you don't run all combinations, effects get tangled together. Unlike in screening designs where we just accepted that main effects were mixed with interactions, here we need to understand the confounding pattern precisely.
To understand this, let's learn a few key terms.
Two Level Fractional Factorial Designs
This article from ReliaSoft's documentation provides a clear, foundational explanation of fractional factorial designs.
Please read the sections 'Half-fraction Designs' and its subsection 'Calculation of Aliases'. Focus on understanding these concepts: Generator: The rule used to create the fraction. For example, in a 2^(3-1) design, you might set Factor C equal to the interaction of A*B (written C = AB). This means the 'C' column in your design matrix is just the 'A' column multiplied by the 'B' column. Defining Relation: An equation that defines the design and its confounding. If the generator is C = AB, the defining relation is I = ABC. Alias Structure: The complete pattern of confounding. Using the defining relation I = ABC, we can find that the main effect of A is aliased with the BC interaction (A = BC), B is aliased with AC, and C is aliased with AB. This means the number you calculate for the effect of 'A' is actually the combined effect of (A + BC).
Understanding the alias structure is the most critical part of designing a fractional factorial experiment. You must know which effects are confounded to ensure you can answer your experimental questions.
4. Designing a Practical Fractional Factorial Experiment
The alias structure is determined by the design's Resolution. The resolution is a Roman numeral (III, IV, V, etc.) that tells you the "cleanliness" of the estimates. For characterization experiments, we are typically interested in Resolution IV or V designs.
- Resolution III: Main effects are aliased with two-factor interactions. (These are screening designs, like Plackett-Burman. We generally avoid them for characterization).
- Resolution IV: No main effects are aliased with two-factor interactions, but two-factor interactions are aliased with other two-factor interactions. (e.g., AB is aliased with CD). This is often a very good compromise between efficiency and clarity.
- Resolution V: No main effects are aliased with two-factor interactions, and no two-factor interactions are aliased with each other. This is a great design, but requires more runs than a Resolution IV.
Let's see how this works in a practical manufacturing scenario.
Two Level Fractional Factorial Designs
Let's return to the ReliaSoft article and look at a detailed manufacturing example. This involves a company trying to understand six factors in their injection molding process for an automobile fuel cone.
Please read the example in the section titled 'Example' under 'Quarter and Smaller Fraction Designs'. Pay attention to the following: The Problem: They have 6 factors, so a full factorial would be 2^6 = 64 runs. This is too many. The Design Choice: They choose a 2^(6-2) Resolution IV design, which requires only 16 runs. This is a quarter-fraction. The Alias Structure: Look at the alias list provided. Notice that main effects (like A) are only aliased with three-factor interactions or higher, which are assumed to be negligible. However, two-factor interactions are aliased with each other (e.g., AB = CE, AE = BC = DF). This is the signature of a Resolution IV design.
In this fuel cone example, the team successfully reduced the experiment from 64 runs to a manageable 16. The design is powerful enough to give clean estimates of all the main effects. The trade-off is that if they see a significant effect for the AE term, they don't know if it's the AE, BC, or DF interaction that's truly active. In practice, using process knowledge and other principles like effect heredity (the idea that interactions are more likely between factors that have strong main effects) often helps resolve this ambiguity. If not, a follow-up experiment might be needed.
Conclusion
In this lesson, we have moved from screening for important factors to designing experiments that characterize their behavior in detail. You now have a framework for choosing between a comprehensive full factorial design and a more efficient fractional factorial design based on your resources and experimental goals.
Key Takeaways:
- Full factorial (2^k) designs are the most thorough, testing every combination of factor levels. They estimate all main effects and interactions without confounding.
- The number of runs in a full factorial grows exponentially, often making them impractical for more than 4 or 5 factors.
- Fractional factorial (2^(k-p)) designs offer an efficient alternative by running a fraction of the full experiment.
- This efficiency comes at the cost of aliasing, where effects are confounded. Understanding the alias structure is crucial.
- Design Resolution (e.g., III, IV, V) is a key metric that describes the confounding pattern and helps you select an appropriate design for your objectives. A Resolution IV design is often a great choice for characterization.
Preview of the Next Lesson:
We've focused today on the design of these experiments. But what do you do once you've chosen a design? How do you fully understand its strengths and weaknesses before you run it? Our next lesson, "Evaluate the aliasing structure and resolution of a fractional factorial design to understand its limitations," will be a deeper dive into interpreting the alias reports from statistical software so you can be confident that your experiment will answer the right questions.