Response surface models applications

Bokonjic, D., Stojiljkovic, M.P., Stulic, D., Kovacevic, V., Maksimovic, M. (1993). Application of response surface modeling for evaluation of the efficacy of a Hl-6/trimedoxime... 

There is, however, one application when the overall desirability D can be rather safely used, namely as a tool in conjunction with response surface modelling. In this context, it can be used to explore the joint modelling of several responses so that a near-optimum region can be located by simulations against the response surface models. The search for conditions which incrase D can be effected either by simplex techniques or by the method of steepest ascent. For the steepest ascent, a linear model for D is first determined from the experiments in the design used to establish the response surface models. The settings which increase D can be translated back into the individual responses by using the response surface models. Thus, it is possible to establish immediately whether the simulated reponse values correspond to suitable experimental conditions. Such results must, of course, be verified by experimental runs. 

S. Agatonovic-Kustrin, M. Zecevic, Lj. Zivanovic, and I. G. Tucker, Anal. Chim. Acta., 364, 265 (1998). Applications of Neural Networks for Response Surface Modelling in HPLC Optimization. 

Curved one-factor response surface showing (a) the limitation of a 2 factorial design for modeling second-order effects and (b) the application of a 3 factorial design for modeling second-order effects. 

One important use of experimental designs is to achieve optimum operating conditions of industrial processes. For a discussion of this application, see Box and Wilson (1951). This paper is extraordinarily rich in response surface concepts. What is the steepest ascent technique discussed in this paper What models are assumed, and what experimental designs are used ... 

The application of screening experiments is obligatory when operating with a relatively large number of factors (k>7), because in the first phase, it facilitates the inclusion of all those factors that do not affect the response greatly. Thus, they also considerably simplify the research of the factor space-domain and the modeling of the response surface. An active selective method, which may be applied in solving this problem is the analysis of variance. 

It first introduces the reader to the fundamentals of experimental design. Systems theory, response surface concepts, and basic statistics serve as a basis for the further development of matrix least squares and hypothesis testing. The effects of different experimental designs and different models on the variance-covariance matrix and on the analysis of variance (ANOVA) are extensively discussed. Applications and advanced topics such as confidence bands, rotatability, and confounding complete the text. Numerous worked examples are presented. 

The meaning of this complex definition is illustrated in figure 5.28. The procedure starts with a (small) set of initial experiments. The next step is the application of a model to the data. This model can be a graphical or a mathematical one, but may also be a simple linear interpolation between the individual data points. Typically, the model is applied to the retention surfaces of the individual solutes, and not to the response surface. Alternatively [537], it may describe relative retentions with respect to a reference component in the... 

Douglas C. Montgomery is Professor of Engineering andProfessor of Statistics at Arizona State University. His research interests are in response surface methodology, empirical modeling, applications of statistics in engineering, and the physical sciences. 

The approach of using a mathematical model to map responses predictively and then to use these models to optimize is limited to cases in which the relatively simple, normally quadratic model describes the phenomenon in the optimum region with sufficient accuracy. When this is not the case, one possibility is to reduce the size of the domain. Another is to use a more complex model or a non-polynomial model better suited to the phenomenon in question. The D-optimal designs and exchange algorithms are useful here as in all cases of change of experimental zone or mathematical model. In any case, response surface methodology in optimization is only applicable to continuous functions. 

TJ apid entrainment carbonization of powdered coal under pressure in a partial hydrogen atmosphere was investigated as a means of producing low sulfur char for use as a power plant fuel. Specific objectives of the research were to determine if an acceptable product could be made and to establish the relationship between yields and chemical properties of the char, with special emphasis on type and amount of sulfur compound in the product. The experiments were conducted with a 4-inch diameter by 18-inch high carbonizer according to a composite factorial design (1, 2). Results of the experiments are expressed by empirical mathematical models and are illustrated by the application of response surface analysis. 

Characterizing the overall uncertainties associated with the PBPK model estimates is also an important component of the PBPK model evaluation and application. This includes characterizing the uncertainties in model outputs resulting from the uncertainty in the PBPK model parameters. Traditionally, Monte Carlo has been employed for performing uncertainty analysis of PBPK models (39, 40). Some of the recent techniques that have been applied for the uncertainty analysis of PBPK models include the stochastic response surface method (SRSM) (38, 41) and the high-dimensional model reduction (HDMR) technique (42). 

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. 

There are yet further sophistications such as the supermodifled simplex, which allows mathematical modeling of the shape of the response surface to provide guidelines as to the choice of the next simplex. Simplex optimization is only one of several computational approaches to optimization, including evolutionary optimization, and steepest ascent methods, however, it is the most commonly used sequential method in analytical chemistry, with diverse applications ranging from autoshimming of instruments to chromatographic optimizations, and can easily be automated. 

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