Linear stochastic model formulations

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. 

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... 

Chemical processes can be modelled in detail as a bunch of equations and differential equations based on chemical and physical laws. These laws rely on static assumptions about the environment where the corresponding processes take place. In practice, all components are infiuenced by stochastic factors that can influence the static process behaviour and/or the dynamic process characteristics. Incorporating continuous stochastic processes to a differential equation leads to stochastic differential equations. Linear stochastic differential equations (SDEs) are usually formulated in the following general form using the Wiener process W t) ... 

Stochastic models are also able to capture complicated pattern formation seen in chemically reacting media and can be used to study the effects of fluctuations on chemical patterns and wave propagation. The mesoscopic dynamics of the FHN model illustrates this point. In order to formulate a microscopically based stochastic model for this system, it is first necessary to provide a mechanism whose mass action law is the FHN kinetic equation. Some features of the FHN kinetics seem to preclude such a mechanistic description for example, the production of u is inhibited by a term linear in V, a contribution not usually encountered in mass action kinetics. However, if each local region of space is assumed to be able to accommodate only a maximum number m of each chemical species, then such a mechanism may be written. We assume that the chemical reactions depend on the local number of molecules of the species as well as the number of vacancies corresponding to each species, in analogy with the dependence of some surface reactions on the number of vacant surface sites or biochemical reactions involving complexes of allosteric enzymes that depend on the number of vacant active sites. 

The above formulation is an extension of the deterministic model explained in Chapter 5. We will mainly explain the stochastic part of the above formulation. The above formulation is a two-stage stochastic mixed-integer linear programming (MILP) model. Objective function (9.1) minimizes the first stage variables and the penalized second stage variables. The production over the target demand is penalized as an additional inventory cost per ton of refinery and petrochemical products. Similarly, shortfall in a certain product demand is assumed to be satisfied at the product spot market price. The recourse variables V [ +, , V e)+ and V e[ in... 

In this chapter, the Bayesian time-domain approach was introduced for identification of the model parameters and stochastic excitation parameters of linear multi-degree-of-freedom systems using noisy stationary or nonstationary response measurements. The direct exact formulation was presented but it turned out to be computationally prohibited for a large number of data points. Then, an approximated likelihood function expansion was proposed to resolve this obstacle. For a globally identifiable case with a large number of data points, the updated PDF... 

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