Alternative objective function

EXAMPLE 1 Optimizing for Cg using an alternative objective function... 

Combinatorial. Combinatorial methods express the synthesis problem as a traditional optimization problem which can only be solved using powerful techniques that have been known for some time. These may use total network cost direcdy as an objective function but do not exploit the special characteristics of heat-exchange networks in obtaining a solution. Much of the early work in heat-exchange network synthesis was based on exhaustive search or combinatorial development of networks. This work has not proven useful because for only a typical ten-process-stream example problem the alternative sets of feasible matches are cal.55 x 10 without stream spHtting. 

Optimization should be viewed as a tool to aid in decision making. Its purpose is to aid in the selection of better values for the decisions that can be made by a person in solving a problem. To formulate an optimization problem, one must resolve three issues. First, one must have a representation of the artifact that can be used to determine how the artifac t performs in response to the decisions one makes. This representation may be a mathematical model or the artifact itself. Second, one must have a way to evaluate the performance—an objective function—which is used to compare alternative solutions. Third, one must have a method to search for the improvement. This section concentrates on the third issue, the methods one might use. The first two items are difficult ones, but discussing them at length is outside the scope of this sec tion. 

For a circular aperture a typical set of basis functions are the Zemike polynomials (Noll, 1976), but for other geometries alternative basis functions may be more appropriate. The objective of most wavefront sensors is to produce a set of measurements, m, that can be related to the wavefront by a set of linear equations... 

Solution To determine the location of the azeotrope for a specified pressure, the liquid composition has to be varied and a bubble-point calculation performed at each liquid composition until a composition is identified, whereby X = y,-. Alternatively, the vapor composition could be varied and a dew-point calculation performed at each vapor composition. Either way, this requires iteration. Figure 4.5 shows the x—y diagram for the 2-propanol-water system. This was obtained by carrying out a bubble-point calculation at different values of the liquid composition. The point where the x—y plot crosses the diagonal line gives the azeotropic composition. A more direct search for the azeotropic composition can be carried out for such a binary system in a spreadsheet by varying T and x simultaneously and by solving the objective function (see Section 3.9) ... 

Optimization techniques are procedures to make something better. Some criteria must be established to determine whether something is better. The single criterion that determines the best among a number of alternatives is referred to as the performance index or the objective function. Economically, this is the expected profit for a plant design. It may be expressed as the net present value of the project. 

A computer can do only three things add, subtract, and decide whether some value is positive, negative, or zero. The last capacity allows the computer to decide which of two alternatives is best when some quantitative objective function has been selected. The ability to add and subtract permits multiplication and division, plus the approximation of integration and differentiation. 

The success of the dual approach and the form of the objective function and constraints suggest geometric (posynomial) programming as an alternative optimization technique. In the absence of the so-called reverse constraints, the posynomial program takes the following form ... 

The last entry in Table 1.1 involves checking the candidate solution to determine that it is indeed optimal. In some problems you can check that the sufficient conditions for an optimum are satisfied. More often, an optimal solution may exist, yet you cannot demonstrate that the sufficient conditions are satisfied. All you can do is show by repetitive numerical calculations that the value of the objective function is superior to all known alternatives. A second consideration is the sensitivity of the optimum to changes in parameters in the problem statement. A sensitivity analysis for the objective function value is important and is illustrated as part of the next example. 

In problems in which there are n variables and m equality constraints, we could attempt to eliminate m variables by direct substitution. If all equality constraints can be removed, and there are no inequality constraints, the objective function can then be differentiated with respect to each of the remaining (n — m) variables and the derivatives set equal to zero. Alternatively, a computer code for unconstrained optimization can be employed to obtain x. If the objective function is convex (as in the preceding example) and the constraints form a convex region, then any stationary point is a global minimum. Unfortunately, very few problems in practice assume this simple form or even permit the elimination of all equality constraints. 

Extended Kalman filtering has been a popular method used in the literature to solve the dynamic data reconciliation problem (Muske and Edgar, 1998). As an alternative, the nonlinear dynamic data reconciliation problem with a weighted least squares objective function can be expressed as a moving horizon problem (Liebman et al., 1992), similar to that used for model predictive control discussed earlier. 

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. 

Finally in this chapter, an alternative approach for nonlinear dynamic data reconciliation, using nonlinear programming techniques, is discussed. This formulation involves the optimization of an objective function through the adjustment of estimate functions constrained by differential and algebraic equalities and inequalities and thus requires efficient and novel solution techniques. 

In general, it is computationally expensive to find the solution to problem (11.12). An alternative approach based on a surrogate objective function was developed by Johnston and Kramer (1998). This approach, for the unconstrained and linearly constrained cases, has an analytical solution thus simplifying the calculations. The complete procedure can be found in the aforementioned publication. 

A mixture of acetone and chloroform is to be separated into pure products [Hostrup et al. (1999)]. Since they also form an azeotrope, one alternative to satisfy the separation objective is to find a suitable solvent for separation by extractive distillation. This type of problem in product design is usually encountered during the purification or recovery of products, by-products, reactants or removal of undesirable products from the process. Also, it can be noted that failure to find a suitable solvent may result in the discard of the product. Alternatively, a functional chemical product manufacturer may be interested to find, design and develop a new solvent. In this case, the solvent is the chemical product. 

Alternatively, the following objective function may be used, assuming that the errors in the measured concentrations are log-normaUy distributed ... 

In the field of selective hydrogenation two important properties are used to describe the catalytic performance the activity and the selectivity of the catalysts. Their values have to be optimized. The simplest approach is to fix the desired conversion level and ranking the catalysts according to their selectivity data. An alternative way for catalyst optimization is the use of the so called "desirability function" d. Upon using this function different optimization parameters can be combined in a common function Dj. In the combination different optimization parameters (often called as objective functions) can be taken into account with different weights [21]. The single desirability function for the conversion (a) can be described by the following formula ... 

Here x represents a vector of n continuous variables (e.g., flows, pressures, compositions, temperatures, sizes of units), and y is a vector of integer variables (e.g., alternative solvents or materials) h(x,y) = 0 denote the to equality constraints (e.g., mass, energy balances, equilibrium relationships) g(x,y) < 0 are the p inequality constraints (e.g., specifications on purity of distillation products, environmental regulations, feasibility constraints in heat recovery systems, logical constraints) f(x,y) is the objective function (e.g., annualized total cost, profit, thermodynamic criteria). 

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