fractional factorial

**A fractional factorial design** is a DOE approach that tests only a **carefully selected subset** of the full factorial combinations, dramatically reducing the number of experimental runs while still extracting the most important information about main effects and key interactions. **Why Fractional Factorial?** - A full factorial with 7 factors at 2 levels requires $2^7 = 128$ runs — impractical in semiconductor manufacturing where each run costs wafers and fab time. - A **half-fraction** ($2^{7-1}$) requires only 64 runs. A **quarter-fraction** ($2^{7-2}$) needs only 32 runs. An **eighth-fraction** ($2^{7-3}$) needs just 16 runs. - The tradeoff: fewer runs means some effects become **aliased** (confounded) — you can't distinguish between certain main effects and interactions. **How It Works** - In a $2^{k-p}$ fractional factorial, $k$ is the number of factors and $p$ is the number of fractions (each $p$ halves the runs). - The arrangement is chosen using **generators** — mathematical relationships that define which combinations to include. - **Example**: $2^{4-1}$ = 8 runs for 4 factors (instead of 16). Factor D is defined as $D = A \times B \times C$. This means the main effect of D is **aliased** with the 3-way interaction $ABC$. **Resolution** - **Resolution III**: Main effects are aliased with 2-factor interactions. Useful for screening many factors but risky if interactions are large. - **Resolution IV**: Main effects are clear of 2-factor interactions, but 2-factor interactions are aliased with other 2-factor interactions. - **Resolution V**: Main effects and 2-factor interactions are clear of each other. 2-factor interactions are aliased with 3-factor interactions (usually negligible). - Higher resolution = better information but more runs. **Semiconductor Applications** - **Screening DOEs**: When 6–10+ factors need initial evaluation, use Resolution III or IV fractional factorials to identify the 3–4 most important factors. - **Follow-Up**: After screening, run a full factorial or RSM on only the important factors identified in the screening step. - **Process Transfer**: When transferring a process to a new tool or fab, screen for factors that need adjustment. **The Sparsity Principle** Fractional factorials work because of two empirical observations: - **Effect Sparsity**: In most real systems, only a few factors (and even fewer interactions) are important. - **Effect Hierarchy**: Main effects are generally larger than 2-factor interactions, which are larger than 3-factor interactions. - These principles mean that the information lost through aliasing usually involves effects that are negligibly small. Fractional factorial designs are the **workhorse of screening experiments** — they efficiently separate the vital few factors from the trivial many with minimal experimental cost.