Module 4 Exam Prep Checklist & Formula Guide

Quick Summary - High-yield exam revision package for Module 4: Design of Experiments and Optimization Methods. Use this checklist and master formula sheet for rapid self-assessment, memory consolidation, and formula verification prior to university examinations.

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1. High-Yield Conceptual Checklist

Review each syllabus topic below. Click the topic titles to jump directly to the detailed theoretical lectures and derivations.

Section A: DOE Principles & Standard Designs

• Foundations of Experimental Design:
  • Define Factor, Level, Response, Treatment, and Experimental Unit.
  • State and explain Fisher's 3 Fundamental Principles: Replication, Randomization, and Local Control (Blocking).
  • Differentiate between true replication and repeated measurements.
  • Memorize the 7-step sequence of designing a valid scientific experiment.
• Standard Experimental Designs & ANOVA:
  • Understand conditions for Completely Randomized Design (CRD) (homogeneous units, 1-way ANOVA).
  • Understand conditions for Randomized Block Design (RBD) (1 nuisance gradient, 2-way ANOVA).
  • Understand conditions for Latin Square Design (LSD) (2 orthogonal nuisance sources, $p \times p$ matrix, 3-way ANOVA without interactions).
  • Construct ANOVA tables from scratch (Degrees of freedom, Sum of Squares, Mean Squares, and $F_{\text{cal}}$).
  • Master the ANOVA Decision Rule: If $F_{\text{cal}} > F_{\text{tab}}(\alpha, \text{df}_1, \text{df}_2)$, reject $H_0$.
  • Full Factorial $2^k$ designs: Compute contrasts ($C$), estimated effects ($C / 2^{k-1}r$), and sum of squares ($C^2 / 2^k r$).

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Section B: Taguchi Methodology & Robust Parameter Design

• Fractional Factorials & Taguchi Robust Design:
  • Explain drawbacks of Full Factorials ($N = L^k$ run explosion, sparsity of effects).
  • Define Fractional Factorial designs ($2^{k-p}$) and confounding/aliasing.
  • State Dr. Genichi Taguchi's definition of Quality ("Loss imparted to society from shipment").
  • Write and calculate Taguchi's Quality Loss Function (QLF): $L(y) = k(y - m)^2$.
  • Contrast traditional "goalpost" tolerances with Taguchi continuous quadratic loss.
  • Explain the 3-step design method: System Design, Parameter Design, Tolerance Design.
  • Differentiate Control Factors, Noise Factors, and Signal Factors.
  • Define Orthogonal Arrays (OAs) ($L_4, L_8, L_9, L_{16}, L_{27}$) and identify why factor level balances eliminate correlation.
  • Memorize the 3 Signal-to-Noise ($S/N$) ratio formulas:
    • Larger-the-better (LTB)
    • Smaller-the-better (STB)
    • Nominal-the-best (NTB)
  • Remember: Always MAXIMIZE the $S/N$ ratio, regardless of objective.
  • Execute an $L_9$ response table, identify optimal factor combinations, and project predicted optimum performance: $\overline{S/N}_{\text{pred}} = \overline{S/N} + \sum (\overline{S/N}_{\text{opt}} - \overline{S/N})$

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Section C: Response Surface Methodology & Optimization

• RSM Fundamentals & First-Order Modeling:
  • Memorize the Canonical 10-Step Methodological Framework of RSM:
    1. Define the Problem (Objective: Maximize, Minimize, or Target).
    2. Select Response Variable(s) ($Y$).
    3. Identify Factors and Levels (Coded levels: Low $-1$, Center $0$, High $+1$).
    4. Select Experimental Design (CCD / BBD).
    5. Conduct Experiments according to design matrix and record responses.
    6. Fit the Mathematical Model ($Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2$).
    7. Analyze the Model using statistical tests (ANOVA, $F$-value, $p$-value, $R^2$, Adjusted $R^2$, Lack-of-Fit).
    8. Study the Response Surface (Contour plots, 3D surface plots).
    9. Determine the Optimum Conditions (Steepest ascent path / stationary point $\mathbf{x}_0$).
    10. Conduct Confirmation Experiments (Validate predicted vs. actual responses).
  • Calculate 1st-order regression coefficients ($\beta_0 = \frac{\sum Y}{N}, \beta_j = \frac{\sum X_j Y}{N}$).
  • Execute the Method of Steepest Ascent along gradient $\nabla Y = [\beta_1, \beta_2]^T$ with proportional step conversions.
• RSM Designs: Central Composite & Box-Behnken:
  • Understand Central Composite Design (CCD) point structure: Factorial ($2^k$) + Star ($2k$) + Center ($n_c$).
  • Calculate total runs in CCD: $N = 2^k + 2k + n_c$ (e.g., $k=2, n_c=5 \implies N = 13$).
  • Derive star distance for rotatability: $\alpha = (2^k)^{1/4}$ ($\sqrt{2} \approx 1.414$ for $k=2$).
  • Convert coded star coordinates $(\pm \alpha)$ into physical engineering settings.
  • Understand Box-Behnken Design (BBD): Edge midpoints with strictly 3 levels ($-1, 0, +1$).
  • Calculate total runs in BBD: $N = 2k(k - 1) + n_c$ (e.g., $k=3, n_c=3 \implies N = 15$).
  • Explain why BBD is safer for hazardous processes (omits extreme factorial corners and out-of-range star points).
  • Reproduce the comparison matrix between CCD and BBD.
• RSM: Multi-Response Optimization (MRO):
  • Formulate linear desirability functions for Larger-the-better ($d_Y = \frac{Y - Y_{\min}}{Y_{\max} - Y_{\min}}$) and Smaller-the-better ($d_R = \frac{R_{\max} - R}{R_{\max} - R_{\min}}$).
  • Compute overall composite desirability via geometric mean: $D = (d_1 \times \dots \times d_k)^{1/k}$.
  • Explain the Zero-Product Rule: if any $d_i = 0$, $D = 0$.
  • Solve multi-response trade-off problems balancing competing criteria (e.g., Yield vs. Roughness).
• Global Optimization & Metaheuristics Taxonomy:
  • Define the 5 components of an optimization problem: Decision Variables, Objective Function, Constraints, Parameters, and Algorithm.
  • Differentiate between a Local Optimum and a Global Optimum.
  • Define Heuristic vs. Metaheuristic; explain the balance between Exploration (Diversification) and Exploitation (Intensification).
  • Reproduce the 5-Branch Taxonomy Tree:
    1. Evolutionary: GA, DE, GP, ES.
    2. Physics-Based: SA, GSA, HS, MA.
    3. Swarm-Based: PSO, ACO, ABC, FSA.
    4. Bio-Inspired: AIS, BFO, DCA, KHA.
    5. Nature-Inspired: CS, FA, BA, IWO.
• Genetic Algorithms (GA):
  • Explain Darwin's "Survival of the Fittest", chromosomes, genes, and population.
  • Detail the 7 execution stages: Chromosome formation, population init, fitness evaluation, selection, crossover, mutation, and elitist replacement.
  • Perform a manual 1-generation GA calculation maximizing $f(x) = x^2$ with 5-bit strings.
• Physics & Swarm Algorithms: SA & PSO:
  • Simulated Annealing (SA): Physical metallurgy analogy (Heating, Isothermal, Cooling), Metropolis acceptance probability: $P = \exp\left(-\frac{\Delta E}{T}\right)$ Geometric cooling schedule ($T \leftarrow \alpha T$).
  • Particle Swarm Optimization (PSO): Swarm dynamics, personal best ($\mathbf{pbest}_i$), global best ($\mathbf{gbest}$), velocity update equation ($w$, $c_1$, $c_2$), position update equation.

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2. Master Formula Cheatsheet

1. Analysis of Variance (ANOVA) Layouts

$\text{Correction Factor: } CF = \frac{G^2}{N}, \quad \text{Total SS: } \text{SST} = \sum y^2 - CF$

Experimental Design	Treatment SS ($\text{SSTreat}$)	Blocking / Gradient SS	Error SS ($\text{SSE}$)	Error Degrees of Freedom ($\text{df}_{\text{Error}}$)
CRD ($t$ treat, $r$ rep)	$\sum \frac{T_i^2}{r} - CF$	None	$\text{SST} - \text{SSTreat}$	$N - t = t(r - 1)$
RBD ($t$ treat, $r$ blocks)	$\sum \frac{T_i^2}{r} - CF$	$\text{SSBlock} = \sum \frac{B_j^2}{t} - CF$	$\text{SST} - \text{SSTreat} - \text{SSBlock}$	$(t - 1)(r - 1)$
LSD ($p \times p$ square)	$\sum \frac{T_i^2}{p} - CF$	$\text{SSRow} = \sum \frac{R_j^2}{p} - CF$ $\text{SSCol} = \sum \frac{C_k^2}{p} - CF$	$\text{SST} - \text{SSTreat} - \text{SSRow} - \text{SSCol}$	$(p - 1)(p - 2)$

Test Statistic: $F_{\text{cal}} = \frac{\text{MSTreat}}{\text{MSE}} = \frac{\text{SSTreat} / \text{df}_{\text{Treat}}}{\text{SSE} / \text{df}_{\text{Error}}}$

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2. $2^k$ Full Factorial Design Formulas

$\text{Contrast: } C = \sum_{i=1}^{2^k} \pm y_{i+}, \quad \text{Estimated Effect: } \hat{E} = \frac{C}{2^{k-1} \cdot r}, \quad \text{Sum of Squares: } \text{SS}_E = \frac{C^2}{2^k \cdot r}$

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3. Taguchi Robust Design Formulas

• Quality Loss Function: $L(y) = k(y - m)^2, \quad \text{where } k = \frac{\text{Cost at Specification Limit}}{\Delta^2}$
• Signal-to-Noise Ratios (dB):
  • Larger-the-Better (LTB): $\eta = -10 \log_{10}\left(\frac{1}{n}\sum_{i=1}^n \frac{1}{y_i^2}\right) \quad \xrightarrow{n=1} \quad 20 \log_{10}(y)$
  • Smaller-the-Better (STB): $\eta = -10 \log_{10}\left(\frac{1}{n}\sum_{i=1}^n y_i^2\right)$
  • Nominal-the-Best (NTB): $\eta = 10 \log_{10}\left(\frac{\bar{y}^2}{s^2}\right)$
• Predicted Optimum Performance: $\overline{S/N}_{\text{pred}} = \overline{S/N}_{\text{overall}} + \sum_{j=1}^k \left(\overline{S/N}_{j, \text{opt}} - \overline{S/N}_{\text{overall}}\right)$

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4. Response Surface & Desirability Formulas

• First-Order Planar Model & Orthogonal Coefficient Estimation: $Y = \beta_0 + \sum_{i=1}^k \beta_i X_i, \quad \beta_0 = \frac{\sum Y_i}{N}, \quad \beta_j = \frac{\sum X_{ji} Y_i}{N}$
• Steepest Ascent Gradient Direction: $\nabla Y = \begin{bmatrix} \beta_1 & \beta_2 & \dots & \beta_k \end{bmatrix}^T$
• Second-Order Quadratic Model: $y = \beta_0 + \sum_{i=1}^k \beta_i x_i + \sum_{i=1}^k \beta_{ii} x_i^2 + \sum_{i < j} \beta_{ij} x_i x_j + \epsilon$
• Derringer-Suich Overall Desirability: $D = \left(d_1 \times d_2 \times \dots \times d_m\right)^{\frac{1}{m}} = \left(\prod_{i=1}^m d_i\right)^{\frac{1}{m}}$

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5. Metaheuristic Governing Formulas

• Simulated Annealing Metropolis Criterion: $\Delta E = E_{\text{new}} - E_{\text{current}}, \quad P = \begin{cases} 1 & \text{if } \Delta E \le 0 \\ \exp\left(-\frac{\Delta E}{T}\right) & \text{if } \Delta E > 0 \end{cases}$ $\text{Cooling Schedule: } T_{k+1} = \alpha T_k \quad (0.8 \le \alpha < 1)$
• Particle Swarm Optimization (PSO) Updates: $\mathbf{V}_i^{t+1} = w \mathbf{V}_i^t + c_1 r_1 \left(\mathbf{pbest}_i - \mathbf{X}_i^t\right) + c_2 r_2 \left(\mathbf{gbest} - \mathbf{X}_i^t\right)$ $\mathbf{X}_i^{t+1} = \mathbf{X}_i^t + \mathbf{V}_i^{t+1}$

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tip

Next Step for Practice

Proceed to the Module 4 Solved Practice Problems to test your calculation speed on complete numerical examination questions.