Mathematical models problems involving restrictions

For the first problem, one will usually write a mathematical model of how insulation of varying thicknesses restricts the loss of heat from a pipe. Evaluation requires that one develop a cost model for the insulation (a capital cost in dollars) and the heat that is lost (an operating cost in dollars/year). Some method is required to permit these two costs to be compared, such as a present worth analysis. Finally, if the model is simple enough, the method one can use is to set the derivative of the evaluation function to zero with respect to wall thickness to find candidate points for its optimal thickness. For the second problem, selecting a best operating schedule involves discrete decisions, which will generally require models that have integer variables. 

Its objective is to select tiie best possible decision for a given set of circumstances without having to enumerate all of the possibilities and involves maximization or minimization as desired. In optimization decision variables are variables in the model which you have control over. Objective function is a fimction (mathematical model) that quantifies the quality of a solution in an optimization problem. Constraints must be considered, conditions that a solution to an optimization problem must satisfy and restrict decision variables are determined by defining relationships among them. It must be found the values of die decision variables that maximize (minimize) the objective function value, while staying widiin the constraints. The objective function and all constraints are linear functions (no squared terms, trigonometric functions, ratios of variables) of the deeision variables [59, 60]. 

As mentioned earlier, there have been many attempts to develop mathematical models that would accurately represent the nonlinear stress-strain behavior of viscoelastic materials. This section will review a few of these but it is appropriate to note that those discussed are not all inclusive. For example, numerical approaches are most often the method of choice for all nonlinear problems involving viscoelastic materials but these are beyond the scope of this text. In addition, this chapter does not include circumstances of nonlinear behavior involving gross yielding such as the Luder s bands seen in polycarbonate in Fig. 3.7. An effort is made in Chapter 11 to discuss such cases in connection with viscoelastic-plasticity and/or viscoplasticity effects. The nonlinear models discussed here are restricted to a subset of small strain approaches, with an emphasis on the single integral approach developed by Schapery. 

The supplier selection problem involves decisions that need to be made by an oiganization that would not only minimize total purchasing cost but also minimize rejects, the lead time of the products, and VaR risk. We consider the least restrictive case for modeling as a situation where the buyers can acquire one or more products from any of the suppliers. The mathematical model for the problem is discussed next. 

Detailed modeling study of practical sprays has a fairly short history due to the complexity of the physical processes involved. As reviewed by O Rourke and Amsden, 3l() two primary approaches have been developed and applied to modeling of physical phenomena in sprays (a) spray equation approach and (b) stochastic particle approach. The first step toward modeling sprays was taken when a statistical formulation was proposed for spray analysis. 541 Even with this simplification, however, the mathematical problem was formidable and could be analyzed only when very restrictive assumptions were made. This is because the statistical formulation required the solution of the spray equation determining the evolution of the probability distribution function of droplet locations, sizes, velocities, and temperatures. The spray equation resembles the Boltzmann equation of gas dynamics[542] but has more independent variables and more complex terms on its right-hand side representing the effects of nucleations, collisions, and breakups of droplets. 

The important problem of stability of viscoelastic flows is far more involved than for Newtonian flows, and many problems still remain open. However, it is clearly established that the difficulty lies in the relationship between various mathematical notions of stability. Some results have been obtained in this direction for restricted classes of flows and/or of models. Moreover several important studies of spectral stability have been performed. 

An example of such an approximation may be found in the applied mathematical field of quantum mechanics, by which the behavior of electrons in molecules is modeled. The classic quantum mechanical model of the behavior of an electron bound to an atomic nucleus is the so-called particle-in-a-box model. In this model, the particle (the electron) can exist only within the confines of the box (the atomic orbital), and because the electron has the properties of an electromagnetic wave as well as those of a physical particle, there are certain restrictions placed on the behavior of the particle. For example, the value of the wave function describing the motion of the electron must be zero at the boundaries of the box. This requires that the motion of the particle can be described only by certain wave functions that, in turn, depend on the dimensions of the box. The problem can be solved mathematically with precision only for the case involving a single electron and a single nuclear proton that defines the box in which the electron is found. The calculated results agree extremely well with observed measurements of electron energy. 

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