Discrete variables, examples

The last example differs from the previous examples in this section in that they involved discrete variables, while pressure and temperature are continuous functions. The same problem could also arise in the discrete case. For instance, although the initial design might favor crystallization over extraction, if the sequence of processing steps were changed the extractive process might be preferable. 

The results for this scenario were obtained using GAMS 2.5/CPLEX. The overall mathematical formulation entails 385 constraints, 175 continuous variables and 36 binary/discrete variables. Only 4 nodes were explored in the branch and bound algorithm leading to an optimal value of 215 t (fresh- and waste-water) in 0.17 CPU seconds. Figure 4.5 shows the water reuse/recycle network corresponding to fixed outlet concentration and variable water quantity for the literature example. It is worth noting that the quantity of water to processes 1 and 3 has been reduced by 5 and 12.5 t, respectively, from the specified quantity in order to maintain the outlet concentration at the maximum level. The overall water requirement has been reduced by almost 35% from the initial amount of 165 t. 

Continuous variables can assume any value within an interval discrete variables can take only distinct values. An example of a discrete variable is one that assumes integer values only. Often in chemical engineering discrete variables and continuous variables occur simultaneously in a problem. If you wish to optimize a compressor system, for example, you must select the number of compressor stages (an integer) in addition to the suction and production pressure of each stage (positive continuous variables). Optimization problems without discrete variables are far easier to solve than those with even one discrete variable. Refer to Chapter 9 for more information about the effect of discrete variables in optimization. 

In real life, other problems involving discrete variables may not be so nicely posed. For example, if cost is a function of the number of discrete pieces of equipment, such as compressors, the optimization procedure cannot ignore the integer character of the cost function because usually only a small number of pieces of equipment are involved. You cannot install 1.54 compressors, and rounding off to 1 or 2 compressors may be quite unsatisfactory. This subject will be discussed in more detail in Chapter 9. 

The analysis of polymer processing is reduced to the balance equations, mass or continuity, energy, momentum and species and to some constitutive equations such as viscosity models, thermal conductivity models, etc. Our main interest is to solve this coupled nonlinear system of equations as accurately as possible with the least amount of computational effort. In order to do this, we simplify the geometry, we apply boundary and initial conditions, we make some physical simplifications and finally we chose an appropriate constitutive equations for the problem. At the end, we will arrive at a mathematical formulation for the problem represented by a certain function, say / (x, T, p, u,...), valid for a domain V. Due to the fact that it is impossible to obtain an exact solution over the entire domain, we must introduce discretization, for example, a grid. The grid is just a domain partition, such as points for finite difference methods, or elements for finite elements. Independent of whether the domain is divided into elements or points, the solution of the problem is always reduced to a discreet solution of the problem variables at the points or nodal pointsinxxnodes. The choice of grid, i.e., type of element, number of points or nodes, directly affects the solution of the problem. 

Until now, the variable in every system has been discrete for example, the number obtained when a die is cast. As not only a discrete variable but also a continuous variable (for example, time) appears very often in chemical engineering, it is necessary to define the information entropy for continuous variable. Of course, it is possible to define the average amount of information for a system that is based on a continuous variable, for example, time. In the case based on a continuous variable t, let p tt) and At be the probability density at t and the very small change of continuous variable, respectively. The product of pif) and At corresponds to the probability P , in Eq. (1.3). When this method of... 

Numerical variables can either be continuous or discrete. Continuous variables are measured on a continuous, uninterrupted scale and can take any value on that scale. For example, height, weight, blood pressure, and heart rate are continuous variables. Depending on how accurately we want (or are able) to measure these variables, values containing one or more decimal points are certainly possible. In contrast, discrete variables can only take certain values, which are usually integers (whole numbers). The number of visits to an emergency room made by a person in one year is measured in whole numbers and is therefore a discrete variable. A subject s response to a questionnaire item that requires the choice of one of several specified levels (e.g., l=mild pain, 2=medium pain, 3=severe pain) yields a discrete variable. 

One s first reaction is to reject the adequacy of casting the problem as a linear one, but, as Grossmann and Santibanez show, the use of discrete (zero/one) decisions allow one to include to a very good approximation many of the nonlinearities. For example, a zero/one variable can be associated with the existence or non-existence of a unit. In the cost function that discrete variable can cause one to add in a fixed charge for the unit only if it exists. Also one can define a continuous "flow11 variable for the unit which can be forced to zero if the unit does not exist by the linear constraint ... 

We shall show that two of these parameters are enough to describe the H-bond dynamics ( Thb and pg, for example). We shaU be able to calculate these two parameters from experimental data (density and depolarized Rayleigh scattering at various temperatures) and to compare the predictions of our theory with the observed diffusional properties of water. Moreover, it will become clear that the assumption of a discrete variable i) essentially means... 

Many of the decisions faced in operations involve discrete variables. For example, if we need to ship 3.25 trucks of product from plant A to plant B each week, we could send 3 trucks for 3 weeks and then 4 trucks in the fourth week, or we could send 4 trucks each week, with the fourth truck only one-quarter filled, but we cannot send 3.25 trucks every week. Some common operational problems involving discrete variables include... 

Discrete variables are also sometimes used in process design, for example, the number of trays or the feed tray of a distillation column, and in process synthesis, to allow selection between flowsheet options, as described later. 

Data are usually unmanageable in the form in which they are collected. In this section, the graphical techniques of summarizing such data so that meaningful information can be extracted from it is considered. Basically, there are two kinds of variables to which data can be assigned continuous variables and discrete variables. A continuous variable is one that can assume any value in some interval of values. Examples of continuous variables are weight, volume, length, time, and temperature. Most environmental data are taken from continuous variables. Discrete variables, on... 

Examples of such variables are (Continuous variables) Reaction temperature, concentration of reagents, ratios of reagent concentrations, reaction times of intermediary steps, (Discrete variables) type of solvent, type of catalyst. 

Example Discrete variables which are difficult to change Assume that a reaction with the general formula... 

It is also worth mentioning that numerical solutions of the Schrodinger equation frequently enclose the atom in a spherical box of finite radius for example, discrete variable methods, finite elements methods and variational methods which employ expansions in terms of functions of finite support, such as -splines, all assume that the wave function vanishes for r > R, which is exactly the situation we deal with here. For such solutions to give an accurate description of the unconfined system it is, of course, necessary to choose R sufficiently large that there is negligible difference between the confined and unconfined atoms. 

Another approach is to treat the manufacturing scale as a normal qualitative variable, and optimize the process and/or formulation at each level. Alternatively it might also be treated as a quantitative discrete variable, as the volume of the apparatus, or possibly better, its logarithm. Literature examples of the use of experimental design in scale-up take this approach (14, 15, 16). 

A quantity x that can be determined quantitatively and that in successive but similar experiments can assume different values is called a random variable. Examples of random variables are the result of drawing one card from a deck of cards, the result of the throw of a die, the result of measuring the length of a nuclear fuel rod, and the result of counting the radioactivity of a sample. There are two types of random variables, discrete and continuous. 

Probability mass function. A function which assigns a probability to values of a random variable. Used for discrete variables as opposed to probability density function, which is used for continuous ones. An example is given in the entry under Poisson distribution. 

We present a new effective numerical method to compute resonances of simple but non-integrable quantum systems, based on a combination of complex coordinate rotations with the finite element and the discrete variable method. By using model potentials we were able to compute atomic data for alkali systems. As an example we show some results for the radial Stark and the Stark effect and compare our values with recent published ones. 

In all the preceding examples, the generalized time (or group parameter) has been a discrete variable. However, the concept of self-similarity is equally applicable to physical objects or processes described by a continu-... 

Conditional and Joint Pmbability Distributions In addition to its graphical structure, a Bayesian network needs to be speerfied by the conditional probability distribution of eaeh node given its parents. Let A and D be variables of interest with a direct causal (parental) relationship in Example 11.6. This relationship can be represented by a conditional probabUity distribution P D A) which represents the probabilistic distribution of child node D given the information of parent node A. When both child and parent nodes arc discrete variables, a contingeney table can summarize the conditional probabdities for aU possible states given each of its parent node states. For continuous variables, a eonditional probability density function needs to be defined. For the combination of continuous and discrete nodes, a mixture distribution, for example, mixture normal distribution, will be required (Imoto et al., 2002). 

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