Stochastic Model Formulation

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. 

We demonstrate the implementation of the proposed stochastic model formulations on the refinery planning linear programming (LP) model explained in Chapter 2. The original single-objective LP model is first solved deterministically and is then reformulated with the addition of the stochastic dimension according to the four proposed formulations. The complete scenario representation of the prices, demands, and yields is provided in Table 6.2. 

The basic principles of modeling the physical, chemical and biological processes that determine pesticide fate in unsaturated soil are reviewed. The mathematical approaches taken to integrate diffusion, convection, sorption, degradation and volatilization are presented. Deterministic and stochastic models formulated to describe these processes in a soil-water pesticide system are contrasted and evaluated. The use of pesticide models for research or management purposes dictates the degree of resolution with thich these processes are modeled. 

In summary, models can be classified in general into deterministic, which describe the system as cause/effect relationships and stochastic, which incorporate the concept of risk, probability or other measures of uncertainty. Deterministic and stochastic models may be developed from observation, semi-empirical approaches, and theoretical approaches. In developing a model, scientists attempt to reach an optimal compromise among the above approaches, given the level of detail justified by both the data availability and the study objectives. Deterministic model formulations can be further classified into simulation models which employ a well accepted empirical equation, that is forced via calibration coefficients, to describe a system and analytic models in which the derived equation describes the physics/chemistry of a system. 

Stochastic modeling. Some researchers may categorize models differently as for example into numerical or analytic, but this categorization applies more to the techniques employed to solve the formulated model, rather than to the formulation per se. 

Despite the ability of the GLM to reproduce any realizable Reynolds-stress model, Pope (2002b) has shown that it is not consistent with DNS data for homogeneous turbulent shear flow. In order to overcome this problem, and to incorporate the Reynolds-number effects observed in DNS, a stochastic model for the acceleration can be formulated (Pope 2002a Pope 2003). However, it remains to be seen how well such models will perform for more complex inhomogeneous flows. In particular, further research is needed to determine the functional forms of the coefficient matrices in both homogeneous and inhomogeneous turbulent flows. 

The formulation outlined above allows for a simple stochastic implementation of the deterministic differential equation (35). Starting with an ensemble of trajectories on a given adiabatic PES W, at each time step At we (i) compute the transition probability pk k, (h) compare it to a random number ( e [0,1], and (iii) perform a hop if pt t > C- In Ih se of a pure A -level system (i.e., in the absence of nuclear dynamics), the assumption (37) holds in general, and the stochastic modeling of Eq. (35) is exact. Considering a vibronic problem with coordinate-dependent however, it can be shown that the electronic... 

The remainder of this Chapter is organized as follows. In Section 8.2 we will discuss model formulation for petrochemical network planning under uncertainty and with uncertainty and risk consideration, referred to as robust optimization. Then we will briefly explain the concept of value of information and stochastic solution, in Section 8.3. In Section 8.4, we will illustrate the performance of the model through an industrial case study. The chapter ends with concluding remarks in Section 8.5. 

We now undertake the formulation of a one-dimensional stochastic model describing a chain for which the directions of adjacent bonds are correlated. Since the chain is confined to one dimension, the bond vectors [Pg.311]

To formulate this stochastic model in terms of concentrations and joint correlation functions only, i.e., in a manner we used earlier in Chapters 2, 4 and 5, it is convenient to write down a master equation of the Markov process under study in a form of the infinite set of coupled equations for many-point densities. Let us write down the first equations for indices (m + m ) = 1 ... 

In Section 9.1.1 we have introduced a stochastic model for the description of surface reaction systems which takes correlations explicitly into account but neglects the energetic interactions between the adsorbed particles as well as between a particle and a metal surface. We have formulated this by master equations upon the assumption that the systems are of the Markovian type. In the model an infinite set of master equations for the distribution functions of the state of the surface and of pairs of surface sites (and so on) arise. This chain of equations cannot be solved analytically. To handle this problem practically this hierarchy was truncated at a certain level. The resulting equations can be solved numerically exactly in a small region and can be connected to a mean-field solution for large distances from a reference point. 

An important property of the stochastic version of compartmental models with linear rate laws is that the mean of the stochastic version follows the same time course as the solution of the corresponding deterministic model. That is not true for stochastic models with nonlinear rate laws, e.g., when the probability of transfer of a particle depends on the state of the system. However, under fairly general conditions the mean of the stochastic version approaches the solution of the deterministic model as the number of particles increases. It is important to emphasize for the nonlinear case that whereas the deterministic formulation leads to a finite set of nonlinear differential equations, the master equation... 

The stochastic formulation would be the most appropriate choice to capture the structural and functional heterogeneity in these biological media. When the process is heterogeneous, one frequently observes chaos-like behaviors. Heterogeneity is at the origin of fluctuations, and fluctuations are the prelude of instability and chaotic behavior. Stochastic modeling is able to ... 

This transform is widely used to formulate semi-Markov stochastic models, where t and / (t) are the random variable and its probability density function, respectively. In Table F.l, we briefly report some Laplace transform pairs. 

When matrix P is filled, vector E(0) is known, the calculation for E(l), E(2), E(3)... E(n)... can be easily carried out. At this time, we can formulate the following question which is also valid in almost all chemical engineering cases What information is produced with the assistance of this stochastic model The answer to this question shows that the model is frequently used for ... 

Roberts, S. M. 1960b. Stochastic models for the dynamic programming formulation of the catalyst replacement problem. Conference on Optimization Techniques in Chemical Engineering, New York University, May 18, 1960, 171-188. 

In the past 10 years or so, there have been a number of theoretical contributions to the fundamental problem of describing fluids in a mesoscopic context. If one wants to go beyond the usual Debye formulation, it is evident that the simplicity of one-body stochastic models must be abandoned. Stochastic models which are able to describe the dynamical behavior of a complex liquid (for instance, a highly viscous solution), exact their price in terms of a more involved formalism. One must be careful to achieve a balance between complexity in formulation and new information gained from the model. Often one can resort to a phenomenological model, which may or may not be the starting point for a more complete (and complicated) theoretical treatment. 

---