Introduction

Imagine you are baking a cake, and the texture or taste is just slightly off. There are too many variables (baking temperature/time, butter %, sugar %, etc... to immediately know what combination works best without previous experience. Similarly in manufacturing, a machined part's surface roughness could be affected by feed rate, spindle speed, depth of cut, mill size/type... and much more. Proper Design of Experiments helps find the optimal settings without cycling through every combination of every variable (which is unfeasible in a timely manner).

There is a mathematical method to systematically determine cause-effect relationships, which might otherwise be difficult to see. This produces a "system" equation to relate input vs expected output. This is much more efficient than changing one variable at a time and uncovers other hidden relations between inputs.

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Statistical Analysis of Data provides the most unbiased and precise results at the cost of experiments in both time and money. It also allowed for

• more information per experiment compared to unplanned testing
• self-evident and clear conclusions
• credibility to experimental conclusions when variability and sources of error can be clarified
• ability to discover interactions and relations between variables

Key Definitions

An experiment is a process that generates data via controlled changes to inputs. It involves a test (or series of tests) where intentional changes are made to the inputs, so the output's response can be observed and identified. The experiment determines which inputs (x) are most influential on the output (y), and then determines what where to set x such that y is near the nominal requirement with minimal variability.

• Response: Measurable output (e.g., time, strength).
• Factors: Controlled input variables (e.g., blade angle).
• Levels: Specific values of factors (e.g., high/low).
• Treatment: A combination of different factors and levels.
• Noise: Uncontrolled variables that output or system performance.
• Replication: Repeating treatments to estimate precision and experimental error.

DOE involves choosing the relevant factors, selecting their appropriate levels, determining the treatment combinations for trial, and setting the number of times for replication to bound experimental error.

Types of Experiments

• Best Guess: Based on experience and expertise; not quantifiable!
• OFAT (One-Factor-at-a-Time): Sometimes associated with "scientific method". Inefficient; ignores interactions between factors.
• \( 2^k \) Factorial Design: The most efficient experiment type to deal with multiple factors; explores all combinations of \( k \) factors at 2 levels.

Principles of DOE

• Randomization: Randomize the order of testing to reduce variability and prevent bias.
• Replication: required to obtain error estimates and improves precision. The quantity depends on time \( \& \) cost of experiments.
• Blocking: Run experiments in "blocks" (or stages) to help control for nuisance variables. Ex: replicate 1 on day 1, replicate 2 on day 2

2^k Factorial Design

• \( k \) = number of factors, each at two levels for a total of \( 2^k \) runs.
• Example: \( k=2 \) → 4 runs; \( k=3 \) → 8 runs
• With two levels, we use Yates notation. Each factor is represented by a lowercase letter arranged in standard order. When a lowercase letter representing a factor is missing, it means the factor is at a lower level. Otherwise, it is at a higher level.
• (1) for all factors at lower levels
• ab for A and B at high levels
• a for A at high level, B at low level
• b for B at high level, A at low level
• \( m^k \) Factorial Designs also exist for m levels.
• With only 2 levels, you cannot distinguish between linear and higher-order effects.

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Effects and Interactions

• Main Effect (A): Change in response due to factor A = average(a, ab) - average(1, b)
• Main Effect (B): Change in response due to factor B = average(b, ab) - average(1, a)
• Interaction (AB): Combined effect of A and B beyond individual effects = average(1, ab) - average(a, b)

Characteristic Equation

Where \( x_A, x_B \in \{-1, +1\} \), \( \bar{y} \) is the average effect of all experiments, \( A \) is the effect of factor A, \( B \) is the effect of factor B, and \( AB \) is the interaction effect. The characteristic equation allows for estimation for new inputs in the range (-1, 1).

Statistical Significance

Use ANOVA- "analysis of variance" to determine if the effects are significant.

• Estimate the effects
• Compute standard deviation of the averages, or standard errors:
$$ SE = \sqrt{\frac{s^2}{n}} $$
• Estimate approximate 95% Confidence Interval:
$$ \text{Effect} \pm t_{\alpha/2} \cdot SE $$
• If the Confidence Interval includes, 0 then the factor is not statistically significant at that confidence level.

DOE Workflow

1. Design Experiment: choose factors, levels, and measurable output
2. Collect Data: randomize the order, replicate at least once.
3. Analyze Data: compute effects, interactions, significance.
4. Report Conclusions: reduced characteristic equation, find the optimal settings.

Example:

Factorial Design

Objective: Minimize surface roughness in machining.

Factors:

• A: Spindle Speed (Low = 1000 RPM, High = 2000 RPM)
• B: Feed Rate (Low = 0.1 mm/rev, High = 0.3 mm/rev)

Treatments and Responses:

Main Effects:

Interaction Effect:

Conclusion:

• Higher spindle speed reduces roughness.
• Higher feed rate increases roughness.
• Interaction is minor.
• Optimal setting: High spindle speed, Low feed rate.