Mathematical Models of Response Surfaces

Earlier we noted that a response surface can be described mathematically by an equation relating the response to its factors. If a series of experiments is carried out in which we measure the response for several combinations of factor levels, then linear regression can be used to fit an equation describing the response surface to the data. The calculations for a linear regression when the system is first-order in one factor (a straight line) were described in Chapter 5. A complete mathematical treatment of linear regression for systems that are second-order or that contain more than one factor is beyond the scope of this text. Nevertheless, the computations for 

Progress of a variable-sized simplex optimization for the response surface of Example 14.1. The optimum response is at (3, 7). 

Theoretical Models of the Response Surface Mathematical models for response surfaces are divided into two categories those based on theory and those that are empirical. Theoretical models are derived from known chemical and physical relationships between the response and the factors. In spectrophotometry, for example, Beer s law is a theoretical model relating a substance s absorbance. A, to its concentration, Ca 

Empirical Models of the Response Surface In many cases the underlying theoretical relationship between the response and its factors is unknown, making impossible a theoretical model of the response surface. A model can still be developed if we make some reasonable assumptions about the equation describing the response surface. For example, a response surface for two factors, A and B, might be represented by an equation that is first-order in both factors 

A model describing a system s response that has a theoretical basis and can be derived from theoretical principles. 

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The RSM-based procedure consists of selection of proper design of experiments (DOE), development of an appropriate mathematical model of response surface with the best fittings of the experimental data and graphical representation of interaction effects of process parameters. The RSM has been applied for developing the mathematical models in the form of multiple regression equations. For the development of regression equations related to various quality characteristics of any process, the second order response surface has been assumed as (Montgomery, 2003) ... 

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