Response Surface Models (RSM) are a set of statistical and mathematical techniques used for modeling and analysing problems where a response of interest is influenced by several variables.

The primary purpose of RSM is to optimise this response, often referred to as the output, by adjusting the input variables.

Here are the key aspects of RSM:

1. Objective: The main goal is to find the optimal settings of input variables that result in the best possible output (response). This could mean maximising, minimising, or achieving a target value of the response.
2. Types of Models:
   • First-order models: These are linear models that assume the relationship between the response and input variables is approximately linear.
   • Second-order models: These include quadratic terms and are used when the relationship is more complex and involves curvature.
3. Design of Experiments (DOE): RSM relies heavily on designed experiments to systematically explore the effects of input variables on the response. Common designs include:
   • Central Composite Design (CCD): This design includes a factorial or fractional factorial design with centre points and a group of “star points” that allow for estimation of curvature.
   • Box-Behnken Design: This design is a spherical design that does not contain an embedded factorial or fractional factorial design. It is useful for estimating the quadratic model without using an excessive number of experiments.
4. Response Surface Methodology Process:
   • Model fitting: Statistical techniques, like least squares regression, are used to fit a model to the experimental data.
   • Model adequacy checking: Diagnostic checks are performed to ensure the model is appropriate and meets the assumptions required for inference.
   • Optimization: Techniques such as gradient descent or ridge analysis are used to find the optimal conditions of the input variables.
5. Applications:
   • Product design and improvement: Identifying the best conditions for product features.
   • Process optimisation: Improving manufacturing processes to enhance yield or quality.
   • Engineering and quality control: Optimising the performance and reliability of systems and components.

Example

Consider a scenario where a chemical engineer wants to optimise the yield of a chemical reaction. The yield (response) depends on two factors: temperature () and pressure (). The steps would be:

1. Conduct experiments: Use DOE to systematically vary temperature and pressure.
2. Fit a model: Use regression to fit a quadratic model, such as:
3. Analyze the model: Check the statistical significance and adequacy of the model.
4. Optimize: Use the fitted model to find the temperature and pressure settings that maximise the yield.

RSM is a powerful tool in fields like engineering, manufacturing, pharmaceuticals, and many others where process optimisation is crucial.

Here are more detailed explanations of Central Composite Design (CCD) and Box-Behnken Design (BBD):

Central Composite Design (CCD)

Central Composite Design (CCD) is one of the most commonly used designs in Response Surface Methodology. It is particularly useful for fitting a second-order (quadratic) model without needing to use a complete three-level factorial experiment.

Key Features of CCD:

1. Components:
   • Factorial Points: These are the points from a full or fractional factorial design at all combinations of the high and low levels of the factors.
   • Centre Points: These are points at the center of the experimental region. Center points provide an estimate of the experimental error and check the adequacy of the model.
   • Axial (Star) Points: These points are added at a distance from the centre along each axis of the factors, allowing for the estimation of curvature.
2. Design:
   • The factorial portion can be a full factorial or a fractional factorial design.
   • The axial points can be placed at different distances from the center, controlled by the value of . The value of is often chosen to ensure rotatability, which means the prediction variance is the same at all points equidistant from the centre.
3. Rotatability: A design is rotatable if the variance of the predicted response is the same at all points that are equidistant from the center. For rotatability in CCD, is calculated as , where is the number of factors.

Example:

For a two-factor CCD with factors and :

• Factorial points: (-1, -1), (-1, 1), (1, -1), (1, 1)
• Centre point: (0, 0)
• Axial points: (-, 0), (, 0), (0, –), (0, )

Box-Behnken Design (BBD)

Box-Behnken Design (BBD) is another design used for fitting second-order models. It is more efficient than CCD in terms of the number of required runs when three or more factors are involved. BBD does not contain an embedded factorial design, and it does not require axial points, which makes it useful when experimental conditions cannot be easily extended to extreme levels.

Key Features of BBD:

1. Components:
   • Factorial Points: Unlike CCD, BBD does not include all combinations of high and low levels but rather uses points at the midpoints of the edges of the experimental space.
   • Centre Points: As with CCD, centre points provide an estimate of the experimental error and check the adequacy of the model.
2. Design:
   • Each factor is placed at three levels: low (-1), centre (0), and high (1).
   • The design consists of combinations where each factor is at the centre level, and one or two factors are at the high or low levels. This arrangement avoids extreme vertices of the experimental region.
3. Efficiency: BBD is more efficient for three or more factors because it requires fewer experimental runs compared to CCD. It is particularly useful when experimental regions are constrained by practical or physical limitations.

Example:

For a three-factor BBD with factors , , and :

• Factorial points: (-1, 0, 0), (1, 0, 0), (0, -1, 0), (0, 1, 0), (0, 0, -1), (0, 0, 1)
• Centre points: (0, 0, 0)
• Additional points to ensure the design’s balance: (-1, -1, 0), (-1, 1, 0), (1, -1, 0), (1, 1, 0), etc.

Comparison:

• CCD requires more runs than BBD for the same number of factors, especially when the number of factors increases.
• BBD avoids extreme points which may be outside the feasible experimental region but may not be rotatable.
• CCD can explore a wider range of the experimental space and is rotatable if is chosen correctly.

Both CCD and BBD are valuable designs in RSM, with the choice depending on the specific needs of the experiment and constraints of the system being studied.