Mathematical modeling approach, applications

Throughout the data collection phase unexpected complications were encountered in the pilot application because calculations obtained from the ERP system were often not comparable across plants. Even if the same ERP system was used (which was not the case for all sites considered), different conventions were used for distributing costs across products. Additionally, recipes pulled from the ERP system often did not reflect the actual recipes employed in production. For example, significant differences in absolute raw material quantities required and relative relationship between the different raw materials were found. As similar problems are also reported in the literature discussing practical applications of mathematical modeling approaches (e.g., Kallrath 2000, p. 817 Lee and Billington 1995, p. 46), this appears to be the norm rather than a company-specific exception. 

One of the major uses of molecular simulation is to provide useful theoretical interpretation of experimental data. Before the advent of simulation this had to be done by directly comparing experiment with analytical (mathematical) models. The analytical approach has the advantage of simplicity, in that the models are derived from first principles with only a few, if any, adjustable parameters. However, the chemical complexity of biological systems often precludes the direct application of meaningful analytical models or leads to the situation where more than one model can be invoked to explain the same experimental data. 

Mathews and Rawlings (1998) successfully applied model-based control using solids hold-up and liquid density measurements to control the filtrability of a photochemical product. Togkalidou etal. (2001) report results of a factorial design approach to investigate relative effects of operating conditions on the filtration resistance of slurry produced in a semi-continuous batch crystallizer using various empirical chemometric methods. This method is proposed as an alternative approach to the development of first principle mathematical models of crystallization for application to non-ideal crystals shapes such as needles found in many pharmaceutical crystals. 

There are two statistical assumptions made regarding the valid application of mathematical models used to describe data. The first assumption is that row and column effects are additive. The first assumption is met by the nature of the smdy design, since the regression is a series of X, Y pairs distributed through time. The second assumption is that residuals are independent, random variables, and that they are normally distributed about the mean. Based on the literature, the second assumption is typically ignored when researchers apply equations to describe data. Rather, the correlation coefficient (r) is typically used to determine goodness of fit. However, this approach is not valid for determining whether the function or model properly described the data. 

Thiel et al. (1983) and Scott et al. (1992) have proposed more quantitative approaches, based on mathematical modelling of instantaneous and/or continuous U gain and loss. These models are applicable to weathering and account for the position of the data in the forbidden zones of the ( " U/ U) and ( °Th/ U) diagram. Several studies used this approach to interpret data in soils and weathering profiles (e.g.,... 

Model selection, application and validation are issues of major concern in mathematical soil and groundwater quality modeling. For the model selection, issues of importance are the features (physics, chemistry) of the model its temporal (steady state, dynamic) and spatial (e.g., compartmental approach resolution) the model input data requirements the mathematical techniques employed (finite difference, analytic) monitoring data availability and cost (professional time, computer time). For the model application, issues of importance are the availability of realistic input data (e.g., field hydraulic conductivity, adsorption coefficient) and the existence of monitoring data to verify model predictions. Some of these issues are briefly discussed below. 

The National Academy of Sciences of Ukraine suggested that an economic-environmental-social model be devised and employed for the purposes of the country s sustainable development. This is a very complex and time-consuming approach that may not be usable at this time of industrial restructuring, privatization and other involved processes occurring in a collapsed national economy. An alternative tactics is put forward, which is applicable at both national and regional level. Instead of mathematical modeling and optimization, it uses systems approach and decision theory techniques. 

Fora recent survey of reactive and stochastic chemical batch scheduling approaches, the reader is referred to Floudas and Lin [2], For a survey of the different types of probabilistic mathematical models that explicitly take uncertainties into account, see Sahinidis [12]. For detailed information about stochastic programming and its applications, the reader is referred to the books of Birge and Louveaux [9], Ruszczynski and Shapiro [10], or Wallace and Ziemba [26]. 

Simplified mathematical models These models typically begin with the basic conservation equations of the first principle models but make simplifying assumptions (typically related to similarity theory) to reduce the problem to the solution of (simultaneous) ordinary differential equations. In the verification process, such models must also address the relevant physical phenomenon as well as be validated for the application being considered. Such models are typically easily solved on a computer with typically less user interaction than required for the solution of PDEs. Simplified mathematical models may also be used as screening tools to identify the most important release scenarios however, other modeling approaches should be considered only if they address and have been validated for the important aspects of the scenario under consideration. 

Despite the broad definition of chemometrics, the most important part of it is the application of multivariate data analysis to chemistry-relevant data. Chemistry deals with compounds, their properties, and their transformations into other compounds. Major tasks of chemists are the analysis of complex mixtures, the synthesis of compounds with desired properties, and the construction and operation of chemical technological plants. However, chemical/physical systems of practical interest are often very complicated and cannot be described sufficiently by theory. Actually, a typical chemometrics approach is not based on first principles—that means scientific laws and mles of nature—but is data driven. Multivariate statistical data analysis is a powerful tool for analyzing and structuring data sets that have been obtained from such systems, and for making empirical mathematical models that are for instance capable to predict the values of important properties not directly measurable (Figure 1.1). 

In terms of computer-based and chemometric approach, additional improvements were also needed in mathematical models for chromatography and in method development, in order to help identifying the correct type of model and the adequate experimental parameters then, application to high volume of generated data is possible. 

An interesting application of this approach in another field has been described by Keller (K2). In the design of steam turbines rather complicated heat-balance calculations are required. While each particular installation is different, and therefore requires a different mathematical model, the components of each turbine are always similar. A large-scale computer program was developed, therefore, which would through suitable instructions combine the calculations required for each component into an over-all heat balance for the turbine. 

Kousa et al. [20] classified exposure models as statistical, mathematical and mathematical-stochastic models. Statistical models are based on the historical data and capture the past statistical trend of pollutants [21]. The mathematical modelling, also called deterministic modelling, involves application of emission inventories, combined with air quality and population activity modelling. The stochastic approach attempts to include a treatment of the inherent uncertainties of the model [22],... 

Based on a comparison of alternative proposals from literature, in this chapter modeling approaches are developed that cover the specific requirements of specialty chemicals production networks. The focus is not so much on mathematical aspects but on applicability to problems from industry. 

During the approval process, the regulatory agencies can use models to check that the tests are adequately performed. The predictive modeling approach is strongly recommended by the American Food and Drug Administration that already uses mathematical models in its evaluation of applications for drug approval. 

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