RSM Designs: Central Composite & Box-Behnken

Topic - When a process nears an optimum, first-order planar models exhibit significant lack-of-fit due to physical curvature. To fit a full second-order polynomial response surface with quadratic and interaction terms, experimenters transition to two specialized experimental designs: the Central Composite Design (CCD) and the Box-Behnken Design (BBD).

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1. Second-Order Modeling & The Need for Curvature Designs

A standard two-level factorial ($2^k$) design only samples the boundary corners of the design space. It provides information on main effects and linear interactions, but cannot estimate pure quadratic curvature ($\beta_{ii} X_i^2$).

To capture curvature, each factor must be evaluated at a minimum of three distinct levels. Response Surface Methodology addresses this requirement through quadratic polynomial models:

$$Y = \beta_0 + \sum_{i=1}^k \beta_i X_i + \sum_{i=1}^k \beta_{ii} X_i^2 + \sum_{i \lt j} \beta_{ij} X_i X_j + \epsilon$$

For a two-factor system ($k = 2$):

$$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_{12} X_1 X_2 + \beta_{11} X_1^2 + \beta_{22} X_2^2 + \epsilon$$

Where:

• $\beta_0$ = Intercept (predicted baseline at center conditions).
• $\beta_1, \beta_2$ = Linear main effects.
• $\beta_{12}$ = Cross-product interaction effect.
• $\beta_{11}, \beta_{22}$ = Quadratic curvature effects.

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2. Central Composite Design (CCD)

Core Definition & Purpose

The Central Composite Design (CCD) is the most widely utilized experimental design in Response Surface Methodology. It is specifically structured to evaluate:

• Linear effects
• Interaction effects
• Quadratic (curvature) effects
• Optimum operating setpoints

A CCD is created by augmenting an existing $2^k$ factorial design with axial (star) points and replicated center points.

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The Three Components of CCD Points

A CCD is partitioned into three distinct geometric categories:

A. Factorial Points ($2^k$)

Vertices of the $k$-dimensional hypercube evaluated at coded coordinates $(\pm 1, \pm 1, \dots)$.

• For two factors ($k = 2$): $2^2 = 4$ corner runs:
  • Run 1: $(-1, -1)$
  • Run 2: $(-1, +1)$
  • Run 3: $(+1, -1)$
  • Run 4: $(+1, +1)$
• Estimation Role: Main effects and interaction effects.

B. Axial / Star Points ($2k$)

Points placed symmetrically along each factor axis at distance $\pm \alpha$ from the center, with all other factor coordinates set to zero:

• For two factors ($k = 2$): $2(2) = 4$ axial runs:
  • Run 5: $(-\alpha, 0)$
  • Run 6: $(+\alpha, 0)$
  • Run 7: $(0, -\alpha)$
  • Run 8: $(0, +\alpha)$
• Estimation Role: Pure quadratic curvature effects ($\beta_{11}, \beta_{22}$).

C. Center Points ($n_c$)

Replicated trials conducted at the origin $(0, 0, \dots, 0)$.

• For two factors with $n_c = 5$: Runs 9, 10, 11, 12, 13 at $(0, 0)$.
• Estimation Role: Estimate pure experimental error variance ($\sigma^2$), detect global curvature, and assess model adequacy.

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Derivation of Total Number of Experiments

For $k$ factors, the total number of experimental runs in a standard CCD is given by:

$$\boxed{N = 2^k + 2k + n_c}$$

Where:

• $2^k$ = Number of factorial cube points.
• $2k$ = Number of axial (star) points.
• $n_c$ = Number of center point replications.

Example Calculation ($k = 2$ factors, $n_c = 5$ center points):

$$\begin{aligned} N &= 2^2 + 2(2) + 5 \\ N &= 4 + 4 + 5 \\ \mathbf{N} &= \mathbf{13 \text{ experimental runs}} \end{aligned}$$

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The Meaning and Derivation of $\alpha$ (Star Distance)

The parameter $\alpha$ designates the radial distance of the axial points from the design center.

To establish rotatability (meaning the prediction variance $\text{Var}[\hat{y}(\mathbf{x})]$ depends solely on the distance from the design center and is invariant to direction), $\alpha$ is derived as:

$$\boxed{\alpha = (2^k)^{\frac{1}{4}}}$$

For $k = 2$ Factors:

$$\alpha = (2^2)^{\frac{1}{4}} = 4^{\frac{1}{4}} = \sqrt{2} \approx \mathbf{1.414}$$

Thus, the axial coordinates for a rotatable two-factor CCD are:

$$(\pm 1.414, 0) \quad \text{and} \quad (0, \pm 1.414)$$

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Converting Axial Points into Physical Engineering Settings

Consider the baseline industrial ranges:

• Temperature ($X_1$): Center = $120^\circ\text{C}$, Half-range $\Delta X_1 = 20^\circ\text{C}$ (Low $100^\circ\text{C}$, High $140^\circ\text{C}$).
• Pressure ($X_2$): Center = $15\,\text{psi}$, Half-range $\Delta X_2 = 5\,\text{psi}$ (Low $10\,\text{psi}$, High $20\,\text{psi}$).

Converting the coded axial coordinates ($X = \pm 1.414$) into physical units:

Temperature Axial Conversions ($X_1$)

$$T_{\text{high}} = 120 + 20(1.414) = 120 + 28.28 = \mathbf{148.28^\circ\text{C}}$$ $$T_{\text{low}} = 120 - 20(1.414) = 120 - 28.28 = \mathbf{91.72^\circ\text{C}}$$

Pressure Axial Conversions ($X_2$)

$$P_{\text{high}} = 15 + 5(1.414) = 15 + 7.07 = \mathbf{22.07\,\text{psi}}$$ $$P_{\text{low}} = 15 - 5(1.414) = 15 - 7.07 = \mathbf{7.93\,\text{psi}}$$

Coded vs. Physical Design Matrix

Coded Point	Actual Physical Setting	Operational Classification
$(-1, 0)$	$100^\circ\text{C}, 15.0\,\text{psi}$	Factorial Boundary Run
$(+1, 0)$	$140^\circ\text{C}, 15.0\,\text{psi}$	Factorial Boundary Run
$(-1.414, 0)$	$\mathbf{91.72^\circ\text{C}}, 15.0\,\text{psi}$	Axial Curvature Run
$(+1.414, 0)$	$\mathbf{148.28^\circ\text{C}}, 15.0\,\text{psi}$	Axial Curvature Run
$(0, -1.414)$	$120^\circ\text{C}, \mathbf{7.93\,\text{psi}}$	Axial Curvature Run
$(0, +1.414)$	$120^\circ\text{C}, \mathbf{22.07\,\text{psi}}$	Axial Curvature Run
$(0, 0)$	$120^\circ\text{C}, 15.0\,\text{psi}$	Center Point Replicate

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Engineering Interpretation of Axial Coordinates

The calculated values $91.72^\circ\text{C}, 148.28^\circ\text{C}, 7.93\,\text{psi},$ and $22.07\,\text{psi}$ are not final optimum values. They are deliberate experimental probe settings extending beyond the initial factor boundaries to observe whether the surface continues to rise, flattens, or curves downward into a peak.

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The 11 Steps in CCD Execution

1. Define the problem and response: Formulate the objective (e.g., maximize yield).
2. Select important factors: Identify the continuous inputs ($X_1, X_2$).
3. Choose factor ranges: Determine realistic upper and lower operational limits.
4. Code the factor values: Normalize physical scales to $(-1, 0, +1)$.
5. Construct the CCD matrix: Assemble Factorial points ($2^k$) + Axial points ($2k$) + Center points ($n_c$).
6. Conduct experiments and record responses: Execute trials in randomized order.
7. Fit the second-order model: Estimate coefficients ($\beta_0, \beta_i, \beta_{ii}, \beta_{ij}$).
8. Perform ANOVA diagnostics: Evaluate $F$-value, $p$-value, $R^2$, Adjusted $R^2$, and Lack-of-Fit.
9. Generate response surfaces: Produce 2D contour maps and 3D topographical surface plots.
10. Determine optimum factor settings: Calculate the stationary point coordinates ($\mathbf{x}_0$).
11. Conduct a confirmation experiment: Physically test the predicted optimum setting.

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3. Box-Behnken Design (BBD)

Core Definition & Purpose

The Box-Behnken Design (BBD) is an independent, spherical three-level design ($-1, 0, +1$) used in Response Surface Methodology to fit second-order models without requiring axial star points.

It is particularly advantageous when:

• There are $3$ or more factors ($k \ge 3$).
• Factors are evaluated at three discrete settings: Low ($-1$), Middle ($0$), and High ($+1$).
• Experiments at extreme corner combinations are undesirable or hazardous (e.g., combining high temperature, high pressure, and high catalyst concentration might cause an explosion or equipment burnout).

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Geometric Architecture: Edge Midpoints

Unlike CCD, Box-Behnken does not use the corner points of the cubic domain (e.g., $(+1, +1, +1)$ is excluded) and contains no star points outside the cube ($\alpha$ points).

Instead, experimental points are located exclusively at the midpoints of the edges of the design space and at the center:

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Derivation of Number of Experiments in BBD

For $k$ factors, the total number of experimental trials in a standard Box-Behnken Design is:

$$\boxed{N = 2k(k - 1) + n_c}$$

Where:

• $k$ = Number of factors ($k \ge 3$).
• $2k(k - 1)$ = Number of edge midpoint points.
• $n_c$ = Number of center points.

Example Calculation ($k = 3$ factors, $n_c = 3$ center points):

$$\begin{aligned} N &= 2(3)(3 - 1) + 3 \\ N &= 2(3)(2) + 3 \\ N &= 12 + 3 \\ \mathbf{N} &= \mathbf{15 \text{ experimental runs}} \end{aligned}$$

(By comparison, a 3-factor CCD requires $2^3 + 2(3) + 3 = 17$ runs minimum, and up to 20 runs with recommended center replicates).

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Complete Standard 3-Factor BBD Matrix ($N = 15$)

Run	$X_1$	$X_2$	$X_3$	Point Category
1	$-1$	$-1$	$0$	Edge Midpoint
2	$-1$	$+1$	$0$	Edge Midpoint
3	$+1$	$-1$	$0$	Edge Midpoint
4	$+1$	$+1$	$0$	Edge Midpoint
5	$-1$	$0$	$-1$	Edge Midpoint
6	$-1$	$0$	$+1$	Edge Midpoint
7	$+1$	$0$	$-1$	Edge Midpoint
8	$+1$	$0$	$+1$	Edge Midpoint
9	$0$	$-1$	$-1$	Edge Midpoint
10	$0$	$-1$	$+1$	Edge Midpoint
11	$0$	$+1$	$-1$	Edge Midpoint
12	$0$	$+1$	$+1$	Edge Midpoint
13	$0$	$0$	$0$	Center Point
14	$0$	$0$	$0$	Center Point
15	$0$	$0$	$0$	Center Point

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Factor Coding in BBD

The standard coding transformation equation is:

$$X_i = \frac{x_i - x_{i, c}}{\Delta x_i}$$

Where $x_i$ is actual value, $x_{i, c}$ is center value, and $\Delta x_i$ is half-range.

For Temperature ($100^\circ\text{C}$ to $140^\circ\text{C}$):

$$X_1 = \frac{T - 120}{20} \implies \begin{cases} X_1 = -1 \to 100^\circ\text{C} \\ X_1 = 0 \to 120^\circ\text{C} \\ X_1 = +1 \to 140^\circ\text{C} \end{cases}$$

Critical Property: Unlike CCD (which pushes star points outside to $\pm 1.414$), BBD keeps all experimental runs strictly within the specified factor range.

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The 12 Steps in BBD Execution

1. Define the problem: Clarify the process objective.
2. Select the response: Specify output $Y$ (e.g., $Y = \text{Yield}$).
3. Identify important factors: Select continuous parameters (e.g., Temperature, Pressure, Time).
4. Select three levels: Establish $(-1, 0, +1)$ values.
5. Construct the BBD matrix: Assemble $2k(k-1)$ edge points + $n_c$ center points.
6. Conduct the experiments: Run trials in randomized sequence.
7. Record the responses: Collect quantitative performance metrics.
8. Fit the second-order model: Estimate full quadratic coefficients.
9. Perform ANOVA: Verify $F$-value, $p$-value, $R^2$, Adjusted $R^2$, and Lack-of-Fit.
10. Generate response surface & contour plots: Inspect 2D/3D interaction topologies.
11. Determine optimum factor settings: Solve for optimal coordinate vector.
12. Conduct a confirmation experiment: Verify mathematical predictions empirically.

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4. Comprehensive Comparison: BBD vs. CCD

Feature	Box-Behnken Design (BBD)	Central Composite Design (CCD)
Full Name	Box-Behnken Design	Central Composite Design
Main Purpose	Fit second-order quadratic model	Fit second-order quadratic model
Factor Levels	Exactly 3 levels ($-1, 0, +1$)	5 levels ($- \alpha, -1, 0, +1, + \alpha$) in rotatable CCD
Corner Points	Not used (all corners excluded)	Factorial corner points are used
Axial Points	No separate star points	Uses $2k$ axial / star points at distance $\pm \alpha$
Extreme Combinations	Avoided (protects hazardous systems)	Included through factorial and axial layout
Operational Scope	Experimental domain restricted strictly within factor range	Axial points extend beyond original factor range
Useful When	Extreme parameter settings are undesirable or unsafe	Wider exploration and rotatable curvature estimation is desired
Basic Points ($k = 3$)	$12$ edge points $+ n_c$ center points ($N = 15$)	$8$ factorial $+ 6$ axial $+ n_c$ center points ($N = 17\text{--}20$)

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5. Exam Traps & Operational Nuances

1. The 2-Factor BBD Fallacy

• The Trap: Attempting to build a Box-Behnken design for $k = 2$ factors.
• The Reality: Box-Behnken designs do not exist for $k = 2$. By definition, an edge midpoint requires at least 3 dimensions to leave one factor at center while varying the other two ($2k(k-1) = 2(2)(1) = 4$, which degenerates into an unrotated square). For $k=2$, always use a Central Composite Design (CCD).

2. The Safety Factor in Chemical Synthesis

• The Trap: Choosing a CCD when factor extremes can cause thermal runaway or mechanical breakdown.
• The Reality: In a rotatable CCD, axial points push the factors to $\pm 1.414$ times the nominal range, and corner points test all high extremes simultaneously. If exceeding $140^\circ\text{C}$ or combining high pressure with high temperature is dangerous, BBD is the mandatory industry choice.

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6. Summary & Cheatsheet

CCD Run Formulas

• Total Runs: $N = 2^k + 2k + n_c$.
• Rotatability: $\alpha = (2^k)^{1/4}$ ($\sqrt{2} \approx 1.414$ for $k=2$).
• Points: Cube ($2^k$) + Star ($2k$) + Center ($n_c$).

BBD Run Formulas

• Total Runs: $N = 2k(k - 1) + n_c$ ($k \ge 3$).
• Levels: Strictly 3 levels ($-1, 0, +1$).
• Points: Edge midpoints only; zero corners, zero stars.

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Key Takeaways

• Curvature Necessity: Estimating quadratic curvature parameters ($\beta_{ii}$) strictly requires testing at three or more factor levels.
• CCD Rotatability: Central Composite Designs provide rotatable variance contours by extending axial star points to distance $\alpha = (2^k)^{1/4}$.
• BBD Operational Safety: Box-Behnken designs protect fragile or hazardous experimental units by omitting extreme factorial corners and eliminating out-of-range star points.

Next Section: RSM: Multi-Response Optimization (MRO) - Balancing competing responses, linear and exponential desirability functions, and complete multi-criteria trade-off walkthroughs.