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SRM CT Examination: Module 4

Test Module: Module 4 (Design of Experiments & Optimization)
Max. Marks: 40 (Answer any two questions. Each question carries 20 marks, divided into four 5-mark subparts.)

Question 1: Standard Experimental Designs & ANOVA (20 Marks)​

  • A. Differentiate between the physical assumptions of a Completely Randomized Design (CRD) and a Randomized Block Design (RBD). Formulate the linear statistical models (yijy_{ij}) for both designs. (5 Marks)
  • B. Derive the degrees of freedom (df\text{df}) and construct the standard Analysis of Variance (ANOVA) table layout for a CRD comprising tt treatments and rr replications. Include the formulas for Mean Squares (MS\text{MS}) and the FF-ratio. (5 Marks)
  • C. An engineer tests 3 fertilizer compounds across 4 homogeneous plots (t=3,r=4,N=12t=3, r=4, N=12). Given the Total Sum of Squares (SST=110.92\text{SST} = 110.92) and Treatment Sum of Squares (SSTreat=92.17\text{SSTreat} = 92.17), mathematically calculate the Error Sum of Squares (SSE\text{SSE}), the Mean Squares, and the calculated FcalF_{\text{cal}} value. (5 Marks)
  • D. Explain the architectural constraints of a Latin Square Design (LSD). Why are the error degrees of freedom mathematically defined as (p−1)(p−2)(p-1)(p-2)? Based on this derivation, explain the statistical limitation of conducting a 2×22 \times 2 Latin Square. (5 Marks)
View Model Answer & Marking Scheme

### Part A (5 Marks)
*   **CRD Assumption (1.5 Marks):** Assumes all experimental units are completely homogeneous. No localized gradients exist.
*   **CRD Model (1 Mark):** $y_{ij} = \mu + \tau_i + \epsilon_{ij}$
*   **RBD Assumption (1.5 Marks):** Assumes units contain a known, one-directional nuisance gradient. Units are grouped into homogeneous blocks to control for this variance.
*   **RBD Model (1 Mark):** $y_{ij} = \mu + \tau_i + \beta_j + \epsilon_{ij}$

### Part B (5 Marks)
*   **Degrees of Freedom (2 Marks):** $\text{df}_{\text{Treat}} = t - 1$, $\text{df}_{\text{Error}} = N - t = t(r - 1)$, $\text{df}_{\text{Total}} = N - 1$.
*   **ANOVA Table Structure (3 Marks):**
    *   $\text{MS}_{\text{Treat}} = \text{SSTreat} / (t - 1)$
    *   $\text{MS}_{\text{Error}} = \text{SSE} / (N - t)$
    *   $F_{\text{cal}} = \text{MS}_{\text{Treat}} / \text{MS}_{\text{Error}}$

### Part C (5 Marks)
*   **Calculate SSE (1 Mark):** $\text{SSE} = \text{SST} - \text{SSTreat} = 110.92 - 92.17 = 18.75$
*   **Degrees of Freedom (1 Mark):** $\text{df}_{\text{Treat}} = 3 - 1 = 2$; $\text{df}_{\text{Error}} = 12 - 3 = 9$.
*   **Mean Squares (2 Marks):** $\text{MS}_{\text{Treat}} = 92.17 / 2 = 46.085$; $\text{MS}_{\text{Error}} = 18.75 / 9 = 2.083$.
*   **F-Ratio (1 Mark):** $F_{\text{cal}} = 46.085 / 2.083 = \mathbf{22.12}$

### Part D (5 Marks)
*   **LSD Architecture (2 Marks):** Controls two independent nuisance gradients. Number of treatments must equal rows and columns ($t = \text{row} = \text{col} = p$). Forms a $p \times p$ matrix.
*   **Error df Derivation (1 Mark):** $\text{df}_{\text{Total}} - \text{df}_{\text{Treat}} - \text{df}_{\text{Row}} - \text{df}_{\text{Col}} = (p^2 - 1) - (p - 1) - (p - 1) - (p - 1) = (p - 1)(p - 2)$.
*   **The 2x2 Limitation (2 Marks):** If $p = 2$, $\text{df}_{\text{Error}} = (2-1)(2-2) = (1)(0) = 0$. Since $\text{MSE} = \text{SSE} / 0$, the $F$-test denominator is undefined, making statistical inference impossible.

Question 2: Fractional Factorials & Taguchi Robust Design (20 Marks)​

  • A. Define Dr. Genichi Taguchi’s concept of Quality. Formulate the Taguchi Quality Loss Function (QLF) mathematically and contrast its behavior with traditional "goalpost" engineering tolerances. (5 Marks)
  • B. Write the exact mathematical formulations for the Taguchi Signal-to-Noise (S/NS/N) ratios for Larger-the-Better (LTB) and Smaller-the-Better (STB) responses. Explain the significance of the negative logarithmic transformation (−10log⁡10-10 \log_{10}). (5 Marks)
  • C. In a Taguchi L9L_9 robust parameter design, the calculated average S/NS/N ratios for the optimal factor levels are A3=35.84 dBA_3 = 35.84\,\text{dB}, B2=35.01 dBB_2 = 35.01\,\text{dB}, and C3=35.27 dBC_3 = 35.27\,\text{dB}. If the overall experimental mean S/N‾\overline{S/N} is 34.80 dB34.80\,\text{dB}, compute the predicted optimum performance (S/N‾pred\overline{S/N}_{\text{pred}}) using Taguchi's additive prediction model. (5 Marks)
  • D. Contrast 2k2^k Full Factorial Designs with 2k−p2^{k-p} Fractional Factorial Designs using the "Sparsity of Effects" principle. Define an Orthogonal Array (OA) and explain its mathematical necessity in industrial screening. (5 Marks)
View Model Answer & Marking Scheme

### Part A (5 Marks)
*   **Taguchi Quality Concept (1.5 Marks):** Quality is defined inversely as the "minimal loss imparted to society from the time the product is shipped."
*   **QLF Formula (1.5 Marks):** $L(y) = k(y - m)^2$, where $k$ is the cost constant and $m$ is the target.
*   **Contrast with Goalpost (2 Marks):** Traditional goalposts assume zero loss inside USL/LSL and 100% loss outside. Taguchi asserts loss grows quadratically as soon as the part deviates from the exact target $m$.

### Part B (5 Marks)
*   **LTB Formula (1.5 Marks):** $\eta_{\text{LTB}} = -10 \log_{10} \left( \frac{1}{n} \sum \frac{1}{y_i^2} \right)$
*   **STB Formula (1.5 Marks):** $\eta_{\text{STB}} = -10 \log_{10} \left( \frac{1}{n} \sum y_i^2 \right)$
*   **Negative Log Significance (2 Marks):** The negative sign mathematically inverts the ratio, ensuring that the golden rule of Taguchi Analysis—**always maximize the S/N ratio**—applies universally, whether minimizing defects or maximizing yield.

### Part C (5 Marks)
*   **Additive Formula (2 Marks):** $\overline{S/N}_{\text{pred}} = \overline{S/N}_{\text{overall}} + \sum (\text{Optimum Level} - \overline{S/N}_{\text{overall}})$
*   **Factor Deviations (2 Marks):**
    *   $A_3: 35.84 - 34.80 = +1.04$
    *   $B_2: 35.01 - 34.80 = +0.21$
    *   $C_3: 35.27 - 34.80 = +0.47$
*   **Final Prediction (1 Mark):** $34.80 + 1.04 + 0.21 + 0.47 = \mathbf{36.52\,\text{dB}}$

### Part D (5 Marks)
*   **Full vs. Fractional & Sparsity (3 Marks):** Full factorials require $2^k$ runs, which explodes exponentially. The Sparsity of Effects principle states that most system variation is driven by main effects and 2-way interactions, allowing us to run a $1/2^p$ fraction ($2^{k-p}$) and safely ignore negligible 3-way+ interactions.
*   **Orthogonal Array (2 Marks):** A balanced matrix (e.g., $L_9$) where every factor level appears an equal number of times across all columns. It is mathematically necessary to ensure factor effects are independently evaluated without confounding.

Question 3: Response Surface Methodology (RSM) (20 Marks)​

  • A. Outline the canonical 10-step sequential framework for executing Response Surface Methodology, detailing the transition from first-order screening to second-order quadratic modeling. (5 Marks)
  • B. Formulate the complete second-order polynomial model for a 2-factor system (X1,X2X_1, X_2). Explain the physical/geometric significance of the pure quadratic coefficients (β11,β22\beta_{11}, \beta_{22}) versus the cross-product interaction coefficient (β12\beta_{12}). (5 Marks)
  • C. Derive the formula for the total number of experimental runs (NN) in a Central Composite Design (CCD). If an engineer evaluates k=3k=3 factors with nc=3n_c=3 center points, calculate the required number of runs. How is the rotatability distance (α\alpha) mathematically derived for kk factors? (5 Marks)
  • D. Contrast the geometric architecture of a Box-Behnken Design (BBD) with a CCD. State the BBD run count formula (N=2k(k−1)+ncN = 2k(k-1) + n_c) and explain why BBD is mathematically preferred for hazardous chemical processes. (5 Marks)
View Model Answer & Marking Scheme

### Part A (5 Marks)
*   **Methodological Flow (5 Marks for defining at least 7 key steps logically):** 1) Define objective, 2) Select response, 3) Identify coded factors, 4) Select design (CCD/BBD), 5) Conduct experiments, 6) Fit regression model (1st or 2nd order), 7) ANOVA diagnostics, 8) Study 3D surface/contours, 9) Determine optimum (stationary point), 10) Physical confirmation.

### Part B (5 Marks)
*   **2nd Order Equation (2 Marks):** $Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_{12} X_1 X_2 + \beta_{11} X_1^2 + \beta_{22} X_2^2 + \epsilon$
*   **Pure Quadratic Significance (1.5 Marks):** $\beta_{11}, \beta_{22}$ dictate the non-linear curvature (peaks, valleys, mounds) along the principal factor axes.
*   **Interaction Significance (1.5 Marks):** $\beta_{12}$ captures synergistic or antagonistic twisting of the surface, establishing saddle points where the optimal setting of $X_1$ depends on the level of $X_2$.

### Part C (5 Marks)
*   **CCD Run Formula (1.5 Marks):** $N = (\text{Factorial } 2^k) + (\text{Axial } 2k) + (\text{Center } n_c)$
*   **Calculation (2 Marks):** For $k=3, n_c=3 \implies N = 2^3 + 2(3) + 3 = 8 + 6 + 3 = \mathbf{17 \text{ runs}}$.
*   **Rotatability Derivation (1.5 Marks):** $\alpha = (2^k)^{1/4}$. Ensures prediction variance depends only on radial distance from the design center.

### Part D (5 Marks)
*   **BBD Architecture (1.5 Marks):** Places points exclusively at the midpoints of the edges of the factor domain and the center. Uses 3 exact levels ($-1, 0, +1$).
*   **BBD Formula (1.5 Marks):** $N = 2k(k-1) + n_c$
*   **Hazardous Process Justification (2 Marks):** Unlike CCD, BBD deliberately excludes all extreme boundary corner points (e.g., $+1, +1, +1$) and out-of-bounds star points ($+\alpha$), making it physically safer for volatile systems where combined factor extremes cause thermal runaway or equipment failure.

Question 4: Multi-Response Optimization & Metaheuristics (20 Marks)​

  • A. Formulate the Derringer-Suich individual linear desirability function (did_i) for a Smaller-the-Better response constraint. If target T=1.0 μmT = 1.0\,\mu\text{m}, upper limit U=3.5 μmU = 3.5\,\mu\text{m}, and observed roughness y^=2.0 μm\hat{y} = 2.0\,\mu\text{m}, compute the individual desirability score. (5 Marks)
  • B. Given individual desirability scores d1=0.90d_1 = 0.90 (Yield) and d2=0.85d_2 = 0.85 (Roughness), calculate the Overall Composite Desirability (DD). State the mathematical formula and explain the operational importance of the "Zero-Product Rule". (5 Marks)
  • C. Formulate the Metropolis Acceptance Criterion probability equation used in Simulated Annealing (SA). If the change in system energy ΔE>0\Delta E \gt 0 (an uphill move), under what mathematical condition will the algorithm accept the inferior solution? (5 Marks)
  • D. Write the velocity vector (Vit+1\mathbf{V}_i^{t+1}) and position (Xit+1\mathbf{X}_i^{t+1}) update equations for Particle Swarm Optimization (PSO). Deconstruct the equation to define the mathematical roles of the Inertia (ww), Cognitive (c1c_1), and Social (c2c_2) components. (5 Marks)
View Model Answer & Marking Scheme

### Part A (5 Marks)
*   **STB Formula (2 Marks):** $d_i = \frac{U - \hat{y}}{U - T}$ (for $T \le \hat{y} \le U$)
*   **Calculation (3 Marks):**
    $$d_i = \frac{3.5 - 2.0}{3.5 - 1.0} = \frac{1.5}{2.5} = \mathbf{0.60}$$

### Part B (5 Marks)
*   **Composite Formula (1.5 Marks):** Geometric Mean $D = (d_1 \times d_2 \times \dots \times d_k)^{1/k}$
*   **Calculation (1.5 Marks):** $D = (0.90 \times 0.85)^{1/2} = \sqrt{0.765} \approx \mathbf{0.875}$
*   **Zero-Product Rule (2 Marks):** If any individual response fails specs ($d_k = 0$), the geometric mean forces overall $D = 0$. This guarantees an optimizer will not accept a system with catastrophic failure in one metric just because another metric is perfect.

### Part C (5 Marks)
*   **Metropolis Formula (2.5 Marks):** $P = \exp\left(-\frac{\Delta E}{T}\right)$
*   **Acceptance Condition (2.5 Marks):** A random number $r \sim \mathcal{U}(0, 1)$ is drawn. The algorithm accepts the inferior uphill move strictly if $r \lt P$. As Temperature $T$ drops over time, $P \to 0$, and uphill moves are systematically rejected.

### Part D (5 Marks)
*   **Update Equations (2 Marks):**
    *   $\mathbf{V}_i^{t+1} = w\mathbf{V}_i^t + c_1 r_1 (\mathbf{pbest}_i - \mathbf{X}_i^t) + c_2 r_2 (\mathbf{gbest} - \mathbf{X}_i^t)$
    *   $\mathbf{X}_i^{t+1} = \mathbf{X}_i^t + \mathbf{V}_i^{t+1}$
*   **Inertia ($w$) (1 Mark):** Preserves current flight momentum, dynamically decayed to transition from broad global exploration to focused local exploitation.
*   **Cognitive ($c_1$) (1 Mark):** Personal memory; pulls the particle toward its own historical best coordinate ($\mathbf{pbest}$).
*   **Social ($c_2$) (1 Mark):** Swarm consensus; pulls the particle toward the global highest fitness coordinate found by any member of the flock ($\mathbf{gbest}$).