Choice of objective functions

Table 4 in Chapter 3.2.1 already characterizes the model developed in this work but does not provide the underlying rationale. Below, this rationale is given for all classification criteria except the choice of objective function to which Chapter 3.3.2 is dedicated. 

Obviously, for some choices of objective function, not all these constraints are needed. 

Company objectives. The notion of an optimal strategy is obviously sensitive to the choice of objective function. Like most economic models, the multi-channel literature has tended to assume that all firms seek to maximize some form of (expected) gross profit. While this may be the ultimate aim for most companies, certain legitimate channel strategies are not tied directly to this, at least not in the short run. 

Different approaches may be used, but it turns out that the models become very large with growing number of orders. The specific problem considered may be modeled by using a continuous time formulation and the effectively solved by using decomposition methods. Care must be taken when choosing decomposition strategy. Two different decomposition approaches are used, one product based and one time based. Real world problems can be solved using both decomposition schemes. The solution time also depend on the choice of objective function. 

No matter what the objectives and operations are, neither the functioning of an operation nor the choice of objectives is always decisively perfect at the start. There are situations in which uncertainty and lack of information are factors to reckon with. Sometimes it is necessary to make extrapolations and predictions without being able to study the actual environment of the operation. 

Here (j is the CG update parameter. In the above equations, e = e (tj) o vector notation for the discretized electric field strength, = g (fj) o objective functional J with respect to the field strength (evaluated at a field strength of e t) and dk = d (t ) o search direction at the feth iteration. The time has been discretized into N time steps, such as that tj=jx )t, where j = 0,1,2, , N. Different CG methods correspond to different choices for the scalar (j. ... 

We consider an nxn table D of distances between the n row-items of an nxp data table X. Distances can be derived from the data by means of various functions, depending upon the nature of the data and the objective of the analysis. Each of these functions defines a particular metric (or yardstick), and the graphical result of a multivariate analysis may largely depend on the particular choice of distance function. 

The objectives of the paper are the maximization of the system availability, the minimization of system cost, and achieving a reasonable ASIL using ASIL-decomposition. In order to make the best choice in the presence of reliahihty, safety, and cost challenges, an objective function and its constraints are defined by authors who involve the above mentioned interests. Subsequently, the optimmn value of objective function will be found using GA and pattern search methods. Adaptability of the optimization results will be finally verified by a practical approach which developed within an automotive project. 

The choice of activation function depends on its application object in the mixed intelligent algorithm. In Chap. 6 of the book, we use Sigmoid function ... 

A fault tree is viewed as a set of events connected AND and OR gates. In giving a formal semantics to the ult tree, we use actions to interpret the events in the tree and predicates to interpret the gates. The system being modelled is characterised by the choice of objects in the model EX. A-Time functions. The semantics of the fault tree is a set of boolean expressions which define a set of Time functions for the system. 

To verify the modelling of the data eolleetion process, calculations of SAT 4, in the entrance window of the XRII was compared to measurements of RNR p oj in stored data as function of tube potential. The images object was a steel cylinder 5-mm) with a glass rod 1-mm) as defect. X-ray spectra were filtered with 0.6-mm copper. Tube current and exposure time were varied so that the signal beside the object. So, was kept constant for all tube potentials. Figure 8 shows measured and simulated SNR oproj, where both point out 100 kV as the tube potential that gives a maximum. Due to overestimation of the noise in calculations the maximum in the simulated values are normalised to the maximum in the measured values. Once the model was verified it was used to calculate optimal choice of filter materials and tube potentials, see figure 9. 

The choice of the appropriate azeotropic distillation method and the resulting flowsheet for the separation of a particular mixture are strong functions of the separation objective. For example, it may be desirable to recover all constituents of the original feed mixture as pure components, or only some as pure components and some as azeotropic mixtures suitable for recycle. Not every objective may be obtainable by azeotropic distillation for a given mixture and portfolio of candidate entrainers. 

Although the minimization of the objective function might run to convergence problems for different NN structures (such as backpropagation for multilayer perceptrons), here we will assume that step 3 of the NN algorithm unambiguously produces the best, unique model, g(x). The question we would like to address is what properties this model inherits from the NN algorithm and the specific choices that are forced. 

There are different variants of the conjugate gradient method each of which corresponds to a different choice of the update parameter C - Some of these different methods and their convergence properties are discussed in Appendix D. The time has been discretized into N time steps (f, = / x 8f where i = 0,1, , N — 1) and the parameter space that is being searched in order to maximize the value of the objective functional is composed of the values of the electric field strength in each of the time intervals. 

Let II II denote the Euclidean norm and define = gk+i gk- Table I provides a chronological list of some choices for the CG update parameter. If the objective function is a strongly convex quadratic, then in theory, with an exact line search, all seven choices for the update parameter in Table I are equivalent. For a nonquadratic objective functional J (the ordinary situation in optimal control calculations), each choice for the update parameter leads to a different performance. A detailed discussion of the various CG methods is beyond the scope of this chapter. The reader is referred to Ref. [194] for a survey of CG methods. Here we only mention briefly that despite the strong convergence theory that has been developed for the Fletcher-Reeves, [195],... 

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 choice of the objective function is very important, as it dictates not only the values of the parameters but also their statistical properties. We may encounter two broad estimation cases. Explicit estimation refers to situations where the output vector is expressed as an explicit function of the input vector and the parameters. Implicit estimation refers to algebraic models in which output and input vector are related through an implicit function. 

For this last stage, the one-at-a-time procedure may be a very poor choice. At Union Carbide, use of the one-at-a-time method increased the yield in one plant from 80 to 83% in 3 years. When one of the techniques, to be discussed later, was used in just 15 runs the yield was increased to 94%. To see why this might happen, consider a plug flow reactor where the only variables that can be manipulated are temperature and pressure. A possible response surface for this reactor is given in Figure 14-1. The response is the yield, which is also the objective function. It is plotted as a function of the two independent variables, temperature and pressure. The designer does not know the response surface. Often all he knows is the yield at point A. He wants to determine the optimum yield. The only way he usually has to obtain more information is to pick some combinations of temperature and pressure and then have a laboratory or pilot plant experimentally determine the yields at those conditions. 

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