Surface response designs

What is a Surface Response Design?

The family of surface response design is used for modeling and analysis of problems in which a response of interest is influenced by several variables and the objective is to optimize this response.

Remark: In contrast to this, screening designs aim to study the input factors, not the response value.

For example, suppose that an engineer wants to find the optimal levels of the pressure (x₁) and the temperature (x₂) of an industrial process to produce concrete, which should have a maximum hardness y.

y = x₁ + x₂ + ε_i (1)

Options for the Surface Response Design function in XLSTAT

The tool offers the following design approaches for surface modeling:

Full factorial design with 2 levels: All combinations of 2 values for each factor (minimum and maximum) are generated in the design. 

Full factorial design with 3 levels: All combinations of 3 values for each factor (minimum, mean and maximum) are generated in the design.

Central composite design: Proposed by Box G.E.P. and Wilson K.B. (1951), the points of experiments are generated on a sphere around the center point. The number of different factor levels is minimized. The center point is repeated in order to maximize the prediction precision around the supposed optimum. 

Box-Behnken: This design was proposed by Box G.E.P. and Behnken D.W (1960) and is based on the same principles as the central composite design, but with a smaller number of experiments. 

Doehlert: This design was proposed by Doehlert D.H. (1970) and is based on the same principles as the central composite and Box-Behnken design, but with a smaller number of experiments. This design has a larger amount of different factor levels for several factors of the design and might therefore be difficult to use. 

Results for the Surface Response Design function in XLSTAT

Variables information: This table shows the information about the factors. For each factor the short name, long name, unit and physical unit are displayed.

Experimental design: This table displays the complete experimental design. Additional columns include information on the factors and on the responses, a label for each experiment, the sort order, the run order and the repetition.

Responses optimization: The responses optimization table of is displayed after the experimental design. You must then select the following parameters:

• Objective: Choose the objective of the optimization. You have the choice between minimum, optimum and maximum.

If the selected objective is the optimum or the maximum, the following fields are activated:

• Lower: Enter for each answer the value of the lower bound below which the desirability is 0.

• Target (left): Enter the value of the lower bound above which desirability is 1 for each response.

If the selected objective is the optimum or the minimum, the following fields are activated:

• Target (right): Enter for each response the value of the upper bound below which the desirability is equal to 1.

• Lower: Enter for each answer the value of the upper limit above which the desirability is 0.

• s: Activate this option if the increasing desirability function must be non-linear. Then enter the value of the shape parameter which must be between 0.01 and 100.

• t: Activate this option if the decreasing desirability function must be non-linear. Then enter the value of the shape parameter which must be between 0.01 and 100.

• Weight: Activate this option if the answers must have an exponential value different from 1 when calculating desirability. Then enter the value of the shape parameter which must be between 0.01 and 100.

For more details on responses optimization, you can refer to the analysis of a factor effect design.

Endoded design: This table shows the encoded experimental design. This table is only displayed in the case of a d-optimal experimental design.