Response surface methodology example

Response-surface methodology has been used extensively for determining areas of process operation providing maximum profit. For example, the succinct representation of the rate surface of Eq. (114) indicates that increasing values of X3 will increase the rate r. If some response other than reaction rate is considered to be more indicative of process performance (such as cost, yield, or selectivity), the canonical analysis would be performed on this response to indicate areas of improved process performance. This information... 

In response surface methodology, it is frequently assumed that / can be approximated in some region of the design variables by a low-degree polynomial. For example, if p=2, and a first-order model is assumed appropriate then... 

In several cases, we highlighted theoretical discussions with supporting examples taken from the field of chemistry and process analytical chemistry. In particular, a simple calibration example using UV spectroscopy was selected to provide the reader with a familiar point of reference. In this way, the topics of experimental design and response-surface methodology were presented in a fashion that should help the nonexpert see the benefits of this approach prior to implementation. For the user who is unfamiliar with DOE methods, we hope our approach has provided a useful introduction. 

Response Surface Methodology (RSM) is a well-known statistical technique (1-3) used to define the relationships of one or more process output variables (responses) to one or more process input variables (factors) when the mechanism underlying the process is either not well understood or is too complicated to allow an exact predictive model to be formulated from theory. This is a necessity in process validation, where limits must be set on the input variables of a process to assure that the product will meet predetermined specifications and quality characteristics. Response data are collected from the process under designed operating conditions, or specified settings of one or more factors, and an empirical mathematical function (model) is fitted to the data to define the relationships between process inputs and outputs. This empirical model is then used to predict the optimum ranges of the response variables and to determine the set of operating conditions which will attain that optimum. Several examples listed in Table 1 exhibit the applications of RSM to processes, factors, and responses in process validation situations. 

Subsequently, we need to understand how the critical inputs affect the critical outputs (item 4). A second type of designed experiment, called response surface methodology (RSM), is used to accomplish this task. Sometimes we are fortunate and know the equation in advance. For example, the equation for dosage above is D = V x C. However, when the equation is not known, a DOE can be used to empirically fit a model. A response surface study is also presented as part of this case study. 

RSM yields the maximum amount of information from the minimum amount of work. For example, in the one-variable-at-a-time approach, shown in Fig, 1, ten experiments were run only to find the suboptimum conditions. However, using RSM and thirteen properly designed experiments not only would the true optimum have been found, but also the information necessary to design the process would have been made available. Secondly, since all of the experiments can be run simultaneously, the results could be obtained quickly. This is the power of response surface methodology. 

In the case of linear polynomial models that are used for example in response surface methodology [Box Draper 1987], there is a clear hierarchy between different models. Suppose that a yield y of a chemical reaction has to be modeled in terms of pressure (p) and temperature (T), then a first order response surface model for y is... 

P. Wehrle, Ph. Nobelis, A. Cuine, and A. Stamm, Response surface methodology an interesting tool for process optimization and validation example of wet granulation in a high shear mixer. Drug Dev. Ind. Pharm., 19, 1637-1653 (1993). 

The literature on the response surface methodology and the design of experiments is quite extensive and the reader is referred to the hterature for a more in-depth discussion. The intent here is to demonstrate how the previously developed software can be used in this type of analysis. This example has also illustrated the use of the actv[] table as input to nlstsqQ for eliminating flic use of particular parameters from a least squares data fitting analysis. 

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