Applications of optimization

Formulation of the Objective Function The formulation of objective functions is one of the crucial steps in the application of optimization to a practical problem. You must be able to translate the desired objective into mathematical terms. In the chemical process industries, the obective function often is expressed in units of currency (e.g., U.S. dollars) because the normal industrial goal is to minimize costs or maximize profits subject to a variety of constraints. 

G. 147 Swan, Applications of Optimal Control Theory in Biomedicine (1984)... 

This chapter has provided a brief overview of the application of optimal control theory to the control of molecular processes. It has addressed only the theoretical aspects and approaches to the topic and has not covered the many successful experimental applications [33, 37, 164-183], arising especially from the closed-loop approach of Rabitz [32]. The basic formulae have been presented and carefully derived in Section II and Appendix A, respectively. The theory required for application to photodissociation and unimolecular dissociation processes is also discussed in Section II, while the new equations needed in this connection are derived in Appendix B. An exciting related area of coherent control which has not been treated in this review is that of the control of bimolecular chemical reactions, in which both initial and final states are continuum scattering states [7, 14, 27-29, 184-188]. 

In the last few years, optimization techniques have become more widely used in the pharmaceutical industry. Some of these have appeared in the literature, but a far greater number remain as in-house information, using the same techniques indicated in this chapter, but with modifications and computer programs specific to the particular company. An excellent review of the application of optimization techniques in the pharmaceutical sciences was published in 1981 [20]. This covers not only formulation and processing, but also analysis, clinical chemistry, and medicinal chemistry. 

The following attributes of processes affecting costs or profits make them attractive for the application of optimization ... 

To compensate for the errors involved in experimental data, the number of data sets should be greater than the number of coefficients p in the model. Least squares is just the application of optimization to obtain the best solution of the equations, meaning that the sum of the squares of the errors between the predicted and the experimental values of the dependent variable y for each data point x is minimized. Consider a general algebraic model that is linear in the coefficients. 

The formulation of objective functions is one of the crucial steps in the application of optimization to a practical problem. As discussed in Chapter 1, you must be able to translate a verbal statement or concept of the desired objective into mathematical terms. In the chemical industries, the objective function often is expressed in units of currency (e.g., U.S. dollars) because the goal of the enterprise is to minimize costs or maximize profits subject to a variety of constraints. In other cases the problem to be solved is the maximization of the yield of a component in a reactor, or minimization of the use of utilities in a heat exchanger network, or minimization of the volume of a packed column, or minimizing the differences between a model and some data, and so on. Keep in mind that when formulating the mathematical statement of the objective, functions that are more complex or more nonlinear are more difficult to solve in optimization. Fortunately, modem optimization software has improved to the point that problems involving many highly nonlinear functions can be solved. 

This chapter includes a discussion of how to formulate objective functions involved in economic analysis, an explanation of the important concept of the time value of money, and an examination of the various ways of carrying out a profitability analysis. In Appendix B we cover, in more detail, ways of estimating the capital and operating costs in the process industries, components that are included in the objective function. For examples of objective functions other than economic ones, refer to the applications of optimization in Chapters 11 to 16. 

The concept of convexity is useful both in the theory and applications of optimization. We first define a convex set, then a convex function, and lastly look at the role played by convexity in optimization. 

Martin and coworkers described an application of optimization to an existing tower separating propane and propylene. The lighter component (propylene) is more valuable than propane. For example, propylene and propane in the overhead product were both valued at 0.20/lb (a small amount of propane was allowable in the overhead), but propane in the bottoms was worth 0.12/lb and propylene 0.09/lb. The overhead stream had to be at least 95 percent propylene. Based on the data in Table E12.4A, we will determine the optimum reflux ratio for this column using derivations provided by McAvoy (personal communication, 1985). He employed correlations for column performance (operating equations) developed by Eduljee (1975). 

Ramirez, W. F. Application of Optimal Control to Enhanced Oil Recovery. Elsevier, Amsterdam (1998). 

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