Question

A chemical company is trying to determine an appropriate condition of a certain manufacturing process.

There are four factors can be controlled in this process, and each factor has two levels (please find the following table):

Factor number Notation Level
1 A "+", "-"
2 B "+", "-"
3 C "+", "-"
4 D "+", "-"


Please generate a fractional factorial design. ( D=ABC, the design with have 24−1=8 possible combinations/possibilities.).

Answer 1

In a fractional factorial design such as 2^4-1, only half of the 16 possible combinations of four binary factors are used, based on confounding the four-way interaction with a main effect or lower-order interaction. The selection of eight runs enables the estimation of main effects and lower-order interactions, making the process more efficient.

Step-by-step explanation:

A fractional factorial design allows you to evaluate the effects of multiple factors with fewer experiments, based on the principle of sparsity of effects, wherein not all interaction effects are expected to be significant. A very common fractional factorial design is the 2^4-1 design, where four factors are considered, each at two levels, but the design only uses half of the 16 possible combinations. This design assumes that one high-order interaction (the four-way interaction among A, B, C, and D) can be confounded with a smaller, potentially more significant interaction or a main effect.

In your case, the high-order interaction confounded is D=ABC. The eight experimental runs of the fractional factorial design can be represented as follows:

• -A -B -C +D
• +A -B -C -D
• -A +B -C -D
• +A +B -C +D
• -A -B +C -D
• +A -B +C +D
• -A +B +C +D
• +A +B +C -D

By carefully selecting and running these eight experimental configurations, engineers can estimate the main effects and lower-order interactions without having to run all 16 experiments of a full factorial design.

Using the factor-label method in the context of a fractional factorial design can be applied for unit conversions or more complex computations, ensuring that the units cancel or combine appropriately to provide a meaningful analysis in terms of the desired unit.

Answer 2

To generate a fractional factorial design for the given manufacturing process, consider the four factors and their levels (+ and -). D=ABC, resulting in 8 possible combinations. Each factor is coded as 1 for positive and -1 for negative.

Step-by-step explanation:

To generate a fractional factorial design for the given manufacturing process, we need to consider the four factors (A, B, C, and D) and their two levels (positive and negative). Since D=ABC, there are 8 possible combinations. We can use a 2^(4-1) design, where the main factor D is combined with each of the three other factors. Each factor will be coded as 1 for the positive level and -1 for the negative level. Here are the 8 combinations:

1. A: +, B: +, C: +
2. A: -, B: +, C: +
3. A: +, B: -, C: +
4. A: -, B: -, C: +
5. A: +, B: +, C: -
6. A: -, B: +, C: -
7. A: +, B: -, C: -
8. A: -, B: -, C: -