Solution of Stochastic Model

The two-stage mixed-integer stochastic program with recourse that includes a total number of200 scenarios for each random parameter is considered in this section. All random parameters were assumed to follow a normal distribution and the scenarios for all random parameters were generated simultaneously. Therefore, the recourse variables account for the deviation from a given scenario as opposed to the deviation from a particular random number realization. 

However, in order to properly evaluate the added-value of including uncertainty of the problem parameters, we will investigate both the EVPI and the VSS. 

In order to evaluate the VSS we first solved the deterministic problem, as illustrated in the previous section, and fixed the petrochemical network and the production rate of the processes. We then solved the EEV problem by allowing the optimization problem to choose second stage variables with respect to the realization of the uncertain parameters From (8.12), the VSS is  

This indicates that the benefit of incorporating uncertainty in the different model parameters for the petrochemical network investment is 513 622. On the other hand, the EVP I can be evaluated by first finding the wait-and-see (WS) solution. The latter can be obtained by taking the expectation for the optimal first stage decisions evaluated at each realization . From (8.11), the EVPI is  

This implies that if it were possible to know the future realization of the demand, prices and yield perfectly, the profit would have been 2 724 040 instead of 2 698 552, yielding savings of 25 488. However, since acquiring perfect information is not viable, we will merely consider the value of the stochastic solution as the best result. These results show that the stochastic model provided an excellent solution as the objective function value was not too far from the result obtained by the WS solution. 

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The SDE and transport equation can be used with the same univocity conditions. For simple univocity conditions and functions such as Di-a(Fa), the transport equations have analytical solutions. Comparison with the numerical solutions of stochastic models allows one to verify whether the stochastic model works properly. The numerical solution of SDE is carried out by space and time discretization into space subdivisions called bins. In the bins j of the space division i, the dimensionless concentration of the property (F = Fa/Faq) takes the Fj value. Taking into consideration these previous statements allows one to write the numerical version of relation (4.118) ... 

The Solution of Stochastic Models with Analytical Methods... 

We have presented applications of a parameter estimation technique based on Monte Carlo simulation to problems in polymer science involving sequence distribution data. In comparison to approaches involving analytic functions, Monte Carlo simulation often leads to a simpler solution of a model particularly when the process being modelled involves a prominent stochastic coit onent. 

The first approaches to compare stochastic models and chemical engineering were made in 1950, with the Higbie [4.1] and Dankwerts transfer models [4.2]. Until today, the development of stochastic modelling in chemical engineering has been remarkable. If we made an inventory of the chemical engineering modelling studies we could see that a stochastic solution exists or complements all the cases [4.3-4.8]. 

The change of a continuous poly stochastic model into its numerical form is carried out using the model described by Eq. (4.71) rewritten in Eq. (4.146). The solution of this model must cover the variable domain 0 z z, 0 t T. In accordance with the previous discussion, the following univocity conditions must be attached to this stochastic model. Here, fk(z),g (T) and h(t) are functions that must be specified. 

This mathematical model has to be completed with realistic univocity conditions. In the literature, a large group of stochastic models derived from the model described above (4.150), have already been solved analytically. So, when we have a new model, we must first compare it to a known model with an analytical solution... 

In this appendix, we show an alternative solution of Rouse model valid in the limit of long chains N—>>=o. It is less useful for comparison with stochastic models but is simpler and sometimes helpful to build up physical intuition. We start from eqn [21]... 

The general solution of the model can be obtained using kinetic Monte Carlo (kMC) simulations. This stochastic method has been successfully applied in the field of heterogeneous catalysis on nanosized catalyst particles (Zhdanov and Kasemo, 2000,2003). It describes the temporal evolution of the system as a Markovian random walk through configuration space. This approach reflects the probabilistic nature of many-particle effects on the catalyst surface. Since these simulations permit atomistic... 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

In this work, therefore we aim to combine the stochastic observer to input/output prediction model so that it can be robust against the influence of noise. We employ the modified I/O data-based prediction model [3] as a linear part of Wimra" model to design the WMPC and these controllers are applied to a continuous mefihyl methacrylate (MMA) solution polymerization reactor to examine the performance of controller. 

Stochastic or probabilistic techniques can be applied to either the moisture module, or the solution of equation (3) — or for example the models of Schwartz Crowe (13) and Tang et al. (16), or can lead to new conceptual model developments as for example the work of Jury (17). Stochastic or probabilistic modeling is mainly aimed at describing breakthrough times of overall concentration threshold levels, rather than individual processes or concentrations in individual soil compartments. Coefficients or response functions and these models have to be calibrated to field data since major processes are studied via a black-box or response function approach and not individually. Other modeling concepts may be related to soil models for solid waste sites and specialized pollutant leachate issues (18). 

The different theoretical models for analyzing particle deposition kinetics from suspensions can be classified as either deterministic or stochastic. The deterministic methods are based on the formulation and solution of the equations arising from the application of Newton s second law to a particle whose trajectory is followed in time, until it makes contact with the collector or leaves the system. In the stochastic methods, forces are freed of their classic duty of determining directly the motion of particles and instead the probability of finding a particle in a certain place at a certain time is determined. A more detailed classification scheme can be found in an overview article [72]. 

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 model equation for particle position, (7.27), is a stochastic differential equation (SDE). The numerical solution of SDEs is discussed in detail by Kloeden and Platen (1992).28 Using a fixed time step At, the most widely used numerical scheme for advancing the particle position is the Euler approximation ... 

The authors applied this model to the situation of dissolving and deposited interfaces, involving chemically interacting species, and included rate kinetics to model mass transfer as a result of chemical reactions [60]. The use of a stochastic weighting function, based on solutions of differential equations for particle motion, may be a useful method to model stochastic processes at solid-liquid interfaces, especially where chemical interactions between the surface and the liquid are involved. 

To interpret the solutions obtained from the stochastic models, we propose to investigate their corresponding coefficient of variation Cv. Cv for a set of values is defined as the ratio of the standard deviation to the expected value or mean and is usually expressed as a percentage. It is calculated as ... 

Computation of Cv is based on the objective function of the formulated model. Table 6.1 displays the expressions to compute Cv for the proposed stochastic model formulations. Note that Cv for the deterministic case of each stochastic model should be equal to zero, by virtue of its standard deviation assuming a value of zero since it is based on the expected value solution. 

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