Discrete-valued objectives

Objective functions that allow only discrete values of the independent variable ) occur frequently in process design because the process variables assume only specific values rather than continuous ones. Examples are the cost per unit diameter of pipe, the cost per unit area for heat exchanger surface, or the insulation cost considered in Example 1.1. For a pipe, we might represent the installed cost as a function of the pipe diameter as shown in Figure 4.2 [see also Noltie (1978)]. For... 

Conclusions on pipe-fitting equations. Each of the ft values given in Table 6.1 is the experimental friction factor for that pipe size. This friction factor ft is not to be mistaken for the friction factor for straight pipe f. They are different discrete values to be applied per Eq. (6.11). Ku K2, K3, and KA each represent a different pipe valve or fitting. One could also be a control valve, item 10 in the pipe fitting list. The objective here is to add up all of the fittings expressed as K values and execute Eq. (6.11) to solve the pipe run pressure drop. In summary, this tells how much head hL the pump must put out in order to do the job. 

Mathematics plays a less dominant role in chemistry. Mostly mathematics has entered chemistry via physics. Physics-based quantum chemistry is so widely accepted as a theoretical device in chemistry that the term quantum chemistry is almost a synonym for theoretical chemistry. In general, quantum chemistry is used to generate numbers that can be compared with numerical values of experimental results concerning discrete molecular objects. Like the other physics-oriented theories in chemistry, quantum chemistry is not a true chemical theory, since it can not be used for the direct solution of chemical problems, but it provides physical data of molecules that may be used indirectly to interpret and to predict the chemical properties and behaviour of molecular systems. 

We will see later that energy transfer into macroscopic objects is also quantized. However, the discrete values are so closely spaced that they appear continuous. 

High-resolution transmission electron microscopy can be understood as a general information-transfer process. The incident electron wave, which for HRTEM is ideally a plane wave with its wave vector parallel to a zone axis of the crystal, is diffracted by the crystal and transferred to the exit plane of the specimen. The electron wave at the exit plane contains the structure information of the illuminated specimen area in both the phase and the amplitude.. This exit-plane wave is transferred, however affected by the objective lens, to the recording device. To describe this information transfer in the microscope, it is advantageous to work in Fourier space with the spatial frequency of the electron wave as the relevant variable. For a crystal, the frequency spectrum of the exit-plane wave is dominated by a few discrete values, which are given by the most strongly excited Bloch states, respectively, by the Bragg-diffracted beams. 

The attribute with the minimum entropy value will be selected as a node in the decision tree. The arcs out of this node represent different values of this attribute. If all the objects in an arc belong to one class, the partition process stops. Otherwise, another attribute will be identified using entropy values to partition further the objects that belong to this arc. This partition process continues until all the objects in an arc are in the same class. Before this algorithm is applied, aU attributes that have continuous values need to be transformed to discrete values. 

When there are only a few discrete-valued decision variables in a model, the most effective method of analysis is usually the most direct one total enumeration of all the possibilities. For example, a model with only eight 0-1 variables could be enumerated by trying all 2 = 256 combinations of values for the different variables. If the model is pure discrete, it is only necessary to check whether each possible assignment of values to discrete variables is feasible and to keep track of the feasible solution with best objective function value. For mixed models the process is more complicated because each choice of discrete values yields a residual optimization problem over the continuous variables. Each such continuous problem must be solved or shown infeasible to establish an optimtil solution for the full mixed problem. 

An ensemble of identical dipoles but with various values of energy-per-entity (i.e., a distribution of values over a range of discrete values) constitutes a Formal Object called dipole distribution. The energy-per-entity node of this Formal Object is drawn with a flattened pentagon (i.e., a regular pentagon deformed by moving its horizontal side toward the (tenter). 

GRAPH A3.4 Various shapes for the nodes used for representing energies-per-entity powered by external sources (left) or distributed over discrete values (center). On the right are shown the simplest Formal Objects (on the complexity scale), the inductive and the capacitive singletons. 

The rotation of micro-objects around a motionless axis was analyzed in Section 7.5.3. The orbital motion of an electron was used as an example. It was found that in this case the rotational energy can accept only discrete values defined by eq. (7.5.26). Since potential energy in free rotation is accepted to be equal to zero, the total energy is kinetic. One of the important characteristics of such movement is the rotational constant B ... 

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. 

In Section 42.2 we have discussed that queuing theory may provide a good qualitative picture of the behaviour of queues in an analytical laboratory. However the analytical process is too complex to obtain good quantitative predictions. As this was also true for queuing problems in other fields, another branch of Operations Research, called Discrete Event Simulation emerged. The basic principle of discrete event simulation is to generate sample arrivals. Each sample is characterized by a number of descriptors, e.g. one of those descriptors is the analysis time. In the jargon of simulation software, a sample is an object, with a number of attributes (e.g. analysis time) and associated values (e.g. 30 min). Other objects are e.g. instruments and analysts. A possible attribute is a list of the analytical... 

More significantly, a suboptimal objective value of 2944.1 rcu was reported as an optimal solution. Using the uneven discretization of time formulation proposed in this chapter, a globally optimal value of 3081.8 rcu was obtained in 24.5 CPU s. Only 72 binary variables were necessary and the model solution was based on... 

The case study was solved using the uneven discretization of time formulation presented in this chapter. The mathematical model for the scenario without heat integration (standalone mode) involved 88 binary variables and gave an objective value of 1060 rcu. This value corresponds to the production of 14 t of product and external utility consumption of 12 energy units of steam and 20 energy units... 

The decision-rule approach tried by Frieden (1974, 1975) will serve to introduce the concept of an object built up from grains. In this approach, both 6 and x are taken to be discrete. That is, at a particular value of independent variable xm, we permit 6 to be only an integral multiple of Ad, the grain size. We may consider the use of random numbers to select locations for grain placement. A decision rule provides the basis for acceptance or rejection... 

In this section we consider problems in which there is convective and diffusive transport in one spatial dimension, as well as elementary chemical reaction. The computational solution of such problems requires attention to discretization on a mesh network and solution algorithms. For steady-state situations the computational problem is one of solving a boundary-value problem. In chemically reacting flow problems it is not uncommon to have steep reaction fronts, such as in a flame. In such a case it is important to provide adequate mesh resolution within the front. Adaptive mesh schemes are used to accomplish this objective. 

In a retrofit batch design, we optimize the batch plant profitability defined as the total production value minus the cost of any new equipment. The objective is to obtain a modified batch plant structure, an operating strategy, the equipment sizes, and the batch processing parameters. Discrete decisions correspond to the selection of new units to add to each stage of the plant and their type of operation. Continuous decisions are represented by the volume of each new unit and the batch processing variables which are allowed to vary within certain bounds. 

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