Coagulation process to remove microplastics from water

Factors: Coagulant dosage, pH, stirring speed

Response: Removal efficiency
(% microplastics)

How do process parameters affect microplastic removal efficiency?

Experimental Design: A structured approach to planning experiments

Goal: Maximize information while minimizing resources

Focus: Identify key factors and their interactions

Key Benefits:

Improves efficiency and reliability of results

Provides insights into cause-and-effect relationships

1: Method optimization

Focus: Identify the best combination of factors (dynamically exploring design space)

Goal: Maximize performance (e.g., sensitivity, efficiency)

Example: Optimizing pH, temperature, and time for a chemical reaction

Factor: A variable that influences the outcome of an experiment

Also referred to as a parameter

Examples:

pH value in a chemical reaction

Dosage of coagulant in water treatment

Key: Factors can be controlled or adjusted during the experiment

Response: The dependent variable in an experiment

Also referred to as target or target value

Numeric value: Allows comparison of factor combinations relative to the target

Determined through a model (function) of independent variables (factors)

Experimental Design Space: The range of all possible factor combinations in an experiment

An n+1 dimensional space, where:

n dimensions: Represent the factors (independent variables) as perpendicular axes

1 dimension: Represents the response (dependent variable)

For dimensionality of the design space, we have to distinguish between dimensions from factors and the response

The response can be seen as a hyper surface(volume) in the design space

I.e., the response is a function of the factors

This is called the response surface

Resolution of the Response Function: Determines the level of detail in the design space

Higher resolution requires more measurement points

For 1 parameter: Resolution increases linearly with the number of measurements

For 2 parameters: Resolution increases quadratically with the number of measurements

Key: Adequate resolution is critical to accurately capture the behavior of the response

Trade-off: High resolution improves detail but increases experimental cost and time

Resolution of the Response Function: Determines the level of detail in the design space

Higher resolution requires more measurement points

For 1 parameter: Resolution increases linearly with the number of measurements

For 2 parameters: Resolution increases quadratically with the number of measurements

Key: Adequate resolution is critical to accurately capture the behavior of the response

Trade-off: High resolution improves detail but increases experimental cost and time

Resolution of the Response Function: Determines the level of detail in the design space

Higher resolution requires more measurement points

For 1 parameter: Resolution increases linearly with the number of measurements

For 2 parameters: Resolution increases quadratically with the number of measurements

Key: Adequate resolution is critical to accurately capture the behavior of the response

Trade-off: High resolution improves detail but increases experimental cost and time

Downhill Simplex Method: A strategy for response optimization

Focus: Minimize the number of measurement points

Dynamic exploration of the design space

Starts from an initial point and iteratively moves towards the optimal response

Key Features:

Empirical step-by-step response improvement

Well-suited for nonlinear and multidimensional problems

Relies on a geometric approach using an n-simplex (triangle in 2D, tetrahedron in 3D, etc.)

n-Simplex: The smallest geometric unit in the design space

Defined by n+1 points in an n-dimensional space

Represents the simplest approximation of the response surface

Geometric Interpretation:

for 1 Factor (2D): A line (2 points)

for 2 Factors (3D): A triangle (3 points)

for 3 Factors (4D): A tetrahedron (4 points)

for n Factors: An n-simplex (n+1 points)

We consider the vertices of the n-simplex to estimate local trends of the response surface

Principle of the Downhill Simplex:

Iteratively removes the worst point (worst response)

Replaces it with a new point in an unexplored region

Key Step: Reflection

Reflects the worst point across the centroid of the remaining points

Direction of reflection approximates the local gradient

Principle of the Downhill Simplex:

New point = Reflection Point (R): \[ \mathbf{R} = \mathbf{C} + \alpha (\mathbf{C} - \mathbf{W}) \]

\(\mathbf{C}\): Centroid (mean) of the remaining points

\(\mathbf{W}\): Coordinates of the worst point

\(\alpha\): Reflection coefficient (typically \(\alpha = 1\))

Shrink: Reducing the simplex size when reflection fails

Occurs when the reflected point is still the worst point

Purpose: Reduce the step size and explore the region more cautiously

Mathematical Step:

New point = Shrink Point (S):

\[ \mathbf{S} = \mathbf{C} + \beta (\mathbf{C} - \mathbf{W}) \] where \(\beta\) is the shrink coefficient
(typically \(\beta = 0.5\)), \(\mathbf{W}\) is the worst point and \(\mathbf{C}\) is the centroid of the remaining points

Goal: Focus exploration on a smaller region

Response Function Across Optimization Steps:

Initial simplex: Individual responses (red bars)

Subsequent steps: Best response per simplex (blue bars)

Rapid improvement followed by saturation

Stopping Criterion:

Occurs when further improvement is negligible

Response convergence: Minimal change in best response (e.g., < 1%)

Simplex size: Simplex shrinks below a predefined threshold

Iteration limit: Maximum number of steps reached (e.g., 25)

What to do if the new point leaves the design space?

A new point can fall outside the defined boundaries of the design space

Approaches to handle this:

a) Clip the point: Restrict the new point to the nearest boundary of the design space

b) Modify the reflection or contraction step: Adjust the reflection or shrink coefficient to ensure the point stays within bounds

Example:

If the pH range is 4–9 and a new point suggests pH = 9.5, adjust the point to pH = 9

Alternatively, scale the reflection step to bring the point back within bounds (e.g., \(\alpha = -0.5\))

Simplex Movement: A "Tumbling" Path to the Optimum

The Simplex does not move directly toward the optimum

Instead, it "tumbles," exploring the response surface dynamically

Reason for Tumbling:

Simplex geometry defines the movement freedom

An irregular Simplex may cause inefficient or erratic motion

Importance of the Initial Simplex:

The simplex should be of suitable size, orientation, and shape

Shape: Regular (equilateral)

Size: edge should be like 10% of the design space

avoid edges parallel to the axes

The initial simplex should be placed in a promising region of the design space.

To construct a regular n-simplex we need to find n+1 vertices of n dimensions. E.g., for 2 factors (2D) we need 3 points that form a regular triangle.

Regular means that all edges have the same length and all angles are equal

For a starting, we can use the axes vectors for all but one vertex. The last vertex is then calculated using the rule of Pythagoras.

E.g., for a 2D simplex: \[ \mathbf{A} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \quad \mathbf{B} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \quad \mathbf{C} = \begin{pmatrix} ? \\ ? \end{pmatrix} \]

To construct a regular n-simplex we need to find n+1 vertices of n dimensions. E.g., for 2 factors (2D) we need 3 points that form a regular triangle.

Regular means that all edges have the same length and all angles are equal

Now we use that all edges have the same length to calculate the last vertex.

E.g., for a 2D simplex: \[ dist(\mathbf{A}, \mathbf{B}) = dist(\mathbf{B}, \mathbf{C}) = dist(\mathbf{C}, \mathbf{A}) \]

I.e., \[ dist(\mathbf{A}, \mathbf{B}) = \sqrt{(x_A - x_B)^2 + (y_A - y_B)^2} = \sqrt{2} \]

To construct a regular n-simplex we need to find n+1 vertices of n dimensions. E.g., for 2 factors (2D) we need 3 points that form a regular triangle.

Regular means that all edges have the same length and all angles are equal

Now we use that all edges have the same length to calculate the last vertex.

I.e., \[ dist(\mathbf{A}, \mathbf{C}) = \sqrt{(1 - x_C)^2 + (0 - y_C)^2} = \sqrt{2} \] \[ dist(\mathbf{B}, \mathbf{C}) = \sqrt{(0 - x_C)^2 + (1 - y_C)^2} = \sqrt{2} \]

To construct a regular n-simplex we need to find n+1 vertices of n dimensions. E.g., for 2 factors (2D) we need 3 points that form a regular triangle.

Regular means that all edges have the same length and all angles are equal

Now we use that all edges have the same length to calculate the last vertex.

I.e., \[ dist(\mathbf{A}, \mathbf{C}) = dist(\mathbf{B}, \mathbf{C}) \]

\[ \sqrt{(1 - x_C)^2 + (0 - y_C)^2} = \sqrt{(0 - x_C)^2 + (1 - y_C)^2} \]

To construct a regular n-simplex we need to find n+1 vertices of n dimensions. E.g., for 2 factors (2D) we need 3 points that form a regular triangle.

Regular means that all edges have the same length and all angles are equal

Now we use that all edges have the same length to calculate the last vertex.

I.e., \[ dist(\mathbf{A}, \mathbf{C}) = dist(\mathbf{B}, \mathbf{C}) \]

\[ (1 - x_C)^2 + (0 - y_C)^2 = (0 - x_C)^2 + (1 - y_C)^2 \]

To construct a regular n-simplex we need to find n+1 vertices of n dimensions. E.g., for 2 factors (2D) we need 3 points that form a regular triangle.

Regular means that all edges have the same length and all angles are equal

Now we use that all edges have the same length to calculate the last vertex.

I.e., \[ dist(\mathbf{A}, \mathbf{C}) = dist(\mathbf{B}, \mathbf{C}) \]

\[ (1 - x_C)^2 - (0 - x_C)^2 = (1 - y_C)^2 - (0 - y_C)^2 \]

To construct a regular n-simplex we need to find n+1 vertices of n dimensions. E.g., for 2 factors (2D) we need 3 points that form a regular triangle.

Regular means that all edges have the same length and all angles are equal

Now we use that all edges have the same length to calculate the last vertex.

I.e., \[ dist(\mathbf{A}, \mathbf{C}) = dist(\mathbf{B}, \mathbf{C}) \]

\[ x_C = y_C \]

To construct a regular n-simplex we need to find n+1 vertices of n dimensions. E.g., for 2 factors (2D) we need 3 points that form a regular triangle.

Regular means that all edges have the same length and all angles are equal

Now we use that all edges have the same length to calculate the last vertex.

I.e., \[ dist(\mathbf{A}, \mathbf{C}) = \sqrt{2} \]

\[ \sqrt{(1 - x_C)^2 + (0 - y_C)^2} = \sqrt{2} \]

To construct a regular n-simplex we need to find n+1 vertices of n dimensions. E.g., for 2 factors (2D) we need 3 points that form a regular triangle.

Regular means that all edges have the same length and all angles are equal

Now we use that all edges have the same length to calculate the last vertex.

I.e., \[ dist(\mathbf{A}, \mathbf{C}) = \sqrt{2} \]

\[ \sqrt{(1 - u)^2 + (0 - u)^2} = \sqrt{2} \] where \(u\) is \(x_C\) and \(y_C\)

To construct a regular n-simplex we need to find n+1 vertices of n dimensions. E.g., for 2 factors (2D) we need 3 points that form a regular triangle.

Regular means that all edges have the same length and all angles are equal

Now we use that all edges have the same length to calculate the last vertex.

I.e., \[ dist(\mathbf{A}, \mathbf{C}) = \sqrt{2} \]

\[ (1 - u)^2 + (0 - u)^2 = 2 \]

To construct a regular n-simplex we need to find n+1 vertices of n dimensions. E.g., for 2 factors (2D) we need 3 points that form a regular triangle.

Regular means that all edges have the same length and all angles are equal

Now we use that all edges have the same length to calculate the last vertex.

I.e., \[ dist(\mathbf{A}, \mathbf{C}) = \sqrt{2} \]

\[ (1 - u)^2 + u^2 - 2 = 0 \]

To construct a regular n-simplex we need to find n+1 vertices of n dimensions. E.g., for 2 factors (2D) we need 3 points that form a regular triangle.

Regular means that all edges have the same length and all angles are equal

Now we use that all edges have the same length to calculate the last vertex.

I.e., \[ dist(\mathbf{A}, \mathbf{C}) = \sqrt{2} \]

\[ 1 - 2u + u^2 + u^2 - 2 = 0 \]

To construct a regular n-simplex we need to find n+1 vertices of n dimensions. E.g., for 2 factors (2D) we need 3 points that form a regular triangle.

Regular means that all edges have the same length and all angles are equal

Now we use that all edges have the same length to calculate the last vertex.

I.e., \[ dist(\mathbf{A}, \mathbf{C}) = \sqrt{2} \]

\[ 2u^2 - 2u - 1 = 0 \]

To construct a regular n-simplex we need to find n+1 vertices of n dimensions. E.g., for 2 factors (2D) we need 3 points that form a regular triangle.

Regular means that all edges have the same length and all angles are equal

Now we use that all edges have the same length to calculate the last vertex.

I.e., \[ dist(\mathbf{A}, \mathbf{C}) = \sqrt{2} \]

\[ u = \frac{2\pm \sqrt{4 - 4(-1*2)}}{4} = \frac{2\pm \sqrt{12}}{4} = \frac{1\pm \sqrt{3}}{2} \]

The initial simplex should be scaled that it fits into a normalized design space
(e.g., 0-1)

To achieve this, we perform a two step approach:

multiply all vertices with the fixed scaling factor of \(1/(\sqrt{2})\) to get the edge length of 1

\[ \mathbf{A_{norm\_edge}} = \mathbf{A} \cdot \frac{1}{\sqrt{2}} \]

The initial simplex should be scaled that it fits into a normalized design space
(e.g., 0-1)

To achieve this, we perform a two step approach:

multiply all vertices with the fixed scaling factor of \(1/(\sqrt{2})\) to get the edge length of 1

\[ \mathbf{A_{norm\_edge}} = \mathbf{A} \cdot \frac{1}{\sqrt{2}} \]

multiply all vertices with the defined scaling factor of e.g. 0.1 or 0.2 to get the desired edge length of 10% or 20% of the design space

\[ \mathbf{A_{scaled}} = \mathbf{A_{norm\_edge}} \cdot 0.1 \]

For now, we have a simplex that fits into a normalized design space (e.g., 0-1). However, we want to optimize the parameters within their defined ranges.

To achieve this, we re-scale the simplex to the desired parameter ranges

\[ \mathbf{A_{scaled\_range}} = \mathbf{A_{scaled}} \cdot (\text{max} - \text{min}) + \text{min} \]

for a pH range of 5-8: \[ \mathbf{B_{scaled\_range}} = 0.14 \cdot (8 - 5) + 5 = 5.42 \] (assuming a scaling of 20% in this example)

When we construct the initial simplex like described before, it is not aligned with the axes which is already fine.

However, we can rotate the simplex for further improvement using a rotation matrix. E.g., for a 2D simplex:

\[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \]

This step is optional and may not be necessary in all cases.

The initial simplex should be placed in a promising region of the design space.

This region should be close to the optimum but not too close to avoid premature convergence.

To achieve this, we use a suitable setting to estimate an offset for the simplex.

\[ \mathbf{A_{offset}} = \mathbf{A_{scaled\_range}} + \text{offset} \]

E.g., for a pH range of 5-8 and an offset of 2: \[ \mathbf{B_{offset}} = 5.42 + 2 = 7.42 \]

In some cases, it might be beneficial to use multiple initial Simplexes to confirm the results.

The reason for this is that the optimum might be local and not global.

Confirmation can be achieved by using different starting points or different scaling factors.

Task: Create an initial simplex for the following parameters and their ranges:

pH: 4-6

Temperature T: 38-78°C

Time t: 10-30 min