Objective function simplification

Finally, analysis of the objective function may permit some simplification of the problem. For example, if one product (A) from a plant is worth 30 per pound and all other products from the plant are worth 5 or less per pound, then we might initially decide to maximize the production of A only. 

Part I comprises three chapters that motivate the study of optimization by giving examples of different types of problems that may be encountered in chemical engineering. After discussing the three components in the previous list, we describe six steps that must be used in solving an optimization problem. A potential user of optimization must be able to translate a verbal description of the problem into the appropriate mathematical description. He or she should also understand how the problem formulation influences its solvability. We show how problem simplification, sensitivity analysis, and estimating the unknown parameters in models are important steps in model building. Chapter 3 discusses how the objective function should be developed. We focus on economic factors in this chapter and present several alternative methods of evaluating profitability. 

Remark 1 The main motivation behind the development of the simplified superstructure was to end up with a mathematical model that features only linear constraints while the nonlinearities appear only in the objective function. Yee et al. (1990a) identified the assumption of isothermal mixing which eliminates the need for the energy balances, which are the nonconvex, nonlinear equality constraints, and which at the same time reduces the size of the mathematical model. These benefits of the isothermal mixing assumption are, however, accompanied by the drawback of eliminating from consideration a number of HEN structures. Nevertheless, as has been illustrated by Yee and Grossmann (1990), despite this simplification, good HEN structures can be obtained. 

The objective function of a supply network design model can either minimize costs or maximize profits. In practice the production function is often required to assume that all forecasted demands have to be met. In this constellation cost minimization and profit maximization lead to identical results and consequently cost minimization models are used. From an economic perspective this simplification can be justified in cases where a high share of fixed costs allows the assumption that any product sale con-... 

A number of steps are involved in the solution of optimization problems, including analyzing the system to be optimized so that all variables are characterized. Next, the objective function and constraints are specified in terms of these variables, noting the independent variables (degrees of freedom). The complexity of the problem may necessitate the use of more advanced optimization techniques or problem simplification. The solution should be checked and the result examined for sensitivity to changes in the model parameters. 

Considering an SMB design problem with four objective functions was a novel approach because, previously, only two or three objective functions had been considered. This enabled full utilization of the properties of the SMB model without any unnecessary simplifications. In addition, the DM obtained more thorough understanding of the interrelationships of different objectives considered and, thus, learned more about the problem. 

If the problem considered has only two objective functions, methods generating a representation of the Pareto optimal set, like EMO approaches can be applied because it is simple to visualize the solutions on a plane. However, when the problem has more than two objectives, the visualization is no longer trivial and interactive approaches offer a viable alternative to solve the problem without artificial simplifications. Because interactive methods rely heavily on the preference information specified by the DM, it is important to select such a user-friendly method where the style of specif3ung preferences is convenient for the DM. In addition, the specific features of the problem to be solved must be taken into consideration. 

The variations of integral objective functionals can be further simplified at their optima by integrating by parts the terms involving derivative functions such as 6x and Srh in Equation (2.29). As a matter of fact, this simplification is an important step in deriving the optimality conditions. Therefore, it is very important to familiarize oneself with the following formula of integration by parts ... 

The objective function, that is, the target of the optimization activity, is always a simplification of reality. 

The main disadvantage of (15) is that the number of directed bonds 2B may be quite large, (in the example discussed presently, 2B = 30 while the matrix A has dimension 7. However, the tree structure of most of the isospectral graphs under consideration here can be used to obtain a much simplified, and almost explicit form of the secular function. This simplification is the object of the present appendix. 

If all possible long-range couplings are considered most of the molecules, which constitute interesting objects for investigation by dynamic NMR methods, represent quite complicated spin systems. We are usually compelled to make some simplifications in the model of exchange by considering lineshape functions for simpler spin systems. Otherwise the computational effort involved may go beyond reasonable limits. In this section we discuss such approximations in detail. 

However, the group gp is much too complicated for practical purposes of molecular shape characterization. Fortunately, the behavior of transformations t of family gp far away from the object p(r) is of little importance, and one can introduce some simplifications. Let us assume that the 3D function considered [e.g., an approximate electron density function p(r)], becomes identically zero outside a sphere S of a sufficiently large radius. As long as two symmorphy transformations tj and t2 have the same effect within this sphere, the differences between these transformations have no relevance to the shape of p(r), even if they have different effects in some domains outside the sphere. All such transformations t of equivalent effects within the relevant part of the 3D space can be collected into equivalence classes. In the symmorphy approach to the analysis of molecular shape, these classes are taken as the actual tools of shape characterization. 

A further simplification can-be made to the Bayes classifier if the covariance matrices for both groups are known to be or assumed to be similar. This condition implies that the correlations between variables are independent of the group to which the objects belong. Extreme examples are illustrated in Figure 5. In such cases the groups are linearly separable and a linear discriminant function can be evaluated. 

A concrete example of the variational principle is provided by the Hartree-Fock approximation. This method asserts that the electrons can be treated independently, and that the -electron wavefimction of the atom or molecule can be written as a Slater determinant made up of orbitals. These orbitals are defined to be those which minimize the expectation value of the energy. Since the general mathematical form of these orbitals is not known (especially in molecules), then the resulting problem is highly nonlinear and formidably difficult to solve. However, as mentioned in subsection (A 1.1.3.2). a common approach is to assume that the orbitals can be written as linear combinations of one-electron basis functions. If the basis functions are fixed, then the optimization problem reduces to that of finding the best set of coefficients for each orbital. This tremendous simplification provided a revolutionary advance for the application of the Hartree-Fock method to molecules, and was originally proposed by Roothaan in 1951. A similar form of the trial function occurs when it is assumed that the exact (as opposed to Hartree-Fock) wavefimction can be written as a linear combination of Slater determinants (see equation (A 1.1.104 ) ). In the conceptually simpler latter case, the objective is to minimize an expression of the form... 

The objective of the polarization model is to relate the material parameters, such as the dielectric properties of both the liquid and solid particles, the particle volume fraction, the electric field strength, etc., to the rheological properties of the whole suspension, in combination with other micro structure features such as fibrillatcd chains. A idealized physical model ER system—an uniform, hard dielectric sphere dispersed in a Newtonian continuous medium, is usually assumed for simplification reason, and this model is thus also called the idealized electrostatic polarization model. The hard sphere means that the particle is uncharged and there are no electrostatic and dispersion interactions between the particles and the dispersing medium before the application of an external electric field. For the idealized electrostatic polarization model, there are roughly two ways to deal with the suspensions One is to consider the Brownian motion of particle, and another is to ignore the Brownian motion and particle inertia. For both cases the anisotropic structure of such a hard sphere suspension is assumed to be represented by the pair correlation function g(r,0), derived by... 

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