Objective Functions and Factors

Simultaneous methods are based on a mathematical model for the response area, whereas sequential approaches represent search methods. 

A prerequisite for any optimization is the definition of one or several objective criteria (figures of merit). For computer-aided or automatic optimizations, the objective functions need to 

The optimum sought of an objective function is either the minimum, for example, minimum time demand, or the maximum, for example, the yield of a chemical reaction. 

In analytics, the analytical performance characteristics constitute ever-important objective criteria that must often be considered in combination. 

In connection with the calibration function performance, several characteristics are combined, such as the sensitivity of an analytical method and its working range. 

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Typical ctors in analytical chemistry are the pH value, reagent concentration, temperature, flow rate, solvent, elution strength, mixture components, irradiation, atomization time, or sputtering rate. Typical responses are the analytical figures of merit and objective functions that consist of combinations of different quality criteria. Objective functions and factors are considered in Section 4.2 in detail. 

To do this, the product of the mass flowrate and the specific enthalpy was substituted by the corresponding enthalpy flow. Results of the reconciliation procedure using the Q-R factorization are given in Table 7. Table 8 compares the residuum of the balance equations, the value of the objective function, and the computing time of the MATLAB implementation for both approaches (Q-R factorization and use of SQP with the reduced set of balance equations). These results show the improvement and the efficiency achieved using Q-R decomposition when the system can be represented as bilinear. 

The simplest device that requires no factorizations, involves checking whether some variables that are not included in the equality constraints have the same sign in the objective function and in all the satisfied inequality constraints (including the lower and upper bound constraints). This alternative is always adopted in the initial... 

As objective function for fitting the required g -model parameters different types of objective functions and experimental or derived properties X, for example, vapor phase mole fraction, pressure, temperature, K-factor K , separation factor au, and so on, can be selected, where either the relative or the absolute deviation of the experimental and correlated values (pressure, temperature, vapor phase composition, etc.) can be minimized. 

Optimization problems can be classified as unconstrained, where no limitations are imposed on the range of possible values of independent factors, and constrained, where additional conditions (constraints) define the range of admissible values of the factors. The objective function and the... 

Table 6.4 lists the 31 optional objective functions, and Table 6.5 shows the 48 reaction activity factors for selection. Aspen H YSYS Petroleum Refining combines the input plant product distribution to construct the reactor effluent and partition the reactor effluent into Cl, C2, C3, C4, C5, and four square cuts , namely, naphtha (C6 to 430 °F cut), diesel (430-700 °F cut), bottom (700-1000 °F) cut and residue (1000 °F-i- cut) which are shown in Table 6.4. AU of the objective functions listed in Table 6.4 are either the prediction errors of crucial operations or important product yields for the HCR process. Aspen HYSYS Petroleum Refining allows us to select the desired objective functions during calibration. After selecting the objective functions, we choose appropriate activity factors to calibrate the reactor model. Figure 6.12 illustrates the relationships among the activity factor, catalyst bed, and reactor type, and Table 6.5 shows the major effect of each activity factor on the model performance, such as global activity (Rgiobai) on the bed temperature profile to help the selection of activity factors.

Uncertainties in amounts of products to be manufactured Qi, processing times %, and size factors Sij will influence the production time tp, whose uncertainty reflects the individual uncertainties that can be presented as probability distributions. The distributions for shortterm uncertainties (processing times and size factors) can be evaluated based on knowledge of probability distributions for the uncertain parameters, i.e. kinetic parameters and other variables used for the design of equipment units. The probability of not being able to meet the total demand is the probability that the production time is larger than the available production time H. Hence, the objective function used for deterministic design takes the form ... 

Froa the above discussion it should be obvious that the selection of an appropriate objective function is a difficult task. The choice of the objective function is a critical factor in automated aethods development, since it is used to define the response surface. It is highly likely that different objective functions will result in the production of different response surfaces and the location of different optimum ejqaerimental conditions for the separation. Yet, it is not possible to set hard guidelines for the selection of the objective function, which must be chosen by practical experience keeping the objectives of the separation in mind. 

The first factor k. 1 = 35, is expected to be temperature dependent via an Arrhenius fjfpe relationship the second factor defines functionality dependence on molecular size the third factor indicates that smaller molecules are more likely to react than larger species, perhaps due to steric hindrance potentials and molecular mobility. The last term expresses a bulk diffusional effect on the inherent reactivity of all polymeric species. The specific constants were obtained by reducing a least squares objective function for the cure at 60°C. Representative data are presented by Figure 5. The fit was good. 

In Eqs. (7)—(10), 5(A) is the spectral power distribution of the illuminant, and R A) is the spectral reflectance factor of the object. Jc(A), y(A), and 5(A) are the color-matching functions of the observer. In the usual practice, k is defined so that the tristimulus value, Y, for a perfect reflecting diffusor (the reference for R A)) equals 100. Using the functions proposed by the CIE in 1931, y(A) was made identical to the spectral photopic luminous efficiency function, and consequently its tristimulus value, Y, is a measure of the brightness of objects. The X and Z values describe aspects of color that permit identification with various spectral regions. 

With a proper balancing of C and Ci this turned out to be a valid strategy, which of course is very sensitive to changes in the weighing factors of other terms in the objective function components. The makespan can be simply obtained by introducing an additional inequality that describes that the makespan should be greater than the end-time of casting for the last - or all - batches. 

The objective to be minimized is a weighted sum of deviations of the produced amounts from the demanded amounts d at the due dates im. Overproduction and underproduction, i.e., positive differences — d% and d — p respectively, are weighted by the nonnegative factors am and fim. If the value of the objective function is represented by z e R the objective can be stated as follows ... 

The objective function is the net present value, NPV (sum of the discounted cash flows), using a discount rate of 10 percent. All flows except capital were assumed to be uniformly distributed over the year working capital was added or subtracted instantaneously at the beginning of each year, and fixed capital was added only in the zero year. The continuous discounting factors were... 

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