Value to Information and Stochastic Solution

In order to understand the effect of each term on the overall objective function of the petrochemical network, different values of 0i and 02 should be evaluated, as will be shown in the illustrative case study. 

Since stochastic programming adds computational burden to practical problems, it is desirable to quantify the benefits of considering uncertainty. In order to address this point, there are generally two values of interest. One is the expected value of perfect information (EVPI) which measures the maximum amount the decision maker is willing to pay in order to get accurate information on the future. The second is the value of stochastic solution (VSS) which is the difference in the objective function between the solutions of the mean value problem (replacing random events with their means) and the stochastic solution (SS) (Birge, 1982). 

A solution based on perfect information would yield optimal first stage decisions for each realization of the random parameter Then the expected value of these decisions, known as wait-and-see (WS) can be written as (Madansky, 1960)  

However, since our objective is profit maximization, the EVPI can be calculated as  

The other quantity of interest is the VSS. In order to quantify it, we first need to solve the mean value problem, also referred to as the expected value (EV) problem. This can be defined as Min z(x, [) ]) where [ ] = f (Birge, 1982). The solution of the EV problem provides the first stage decisions variables evaluated at expectation of the random realizations. The expectation of the EV problem, evaluated at different realization of the random parameters, is then defined as (Birge, 1982)  

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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. 

This chapter addresses the planning, design and optimization of a network of petrochemical processes under uncertainty and robust considerations. Similar to the previous chapter, robustness is analyzed based on both model robustness and solution robustness. Parameter uncertainty includes process yield, raw material and product prices, and lower product market demand. The expected value of perfect information (EVPI) and the value of the stochastic solution (VSS) are also investigated to illustrate numerically the value of including the randomness of the different model parameters. 

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. 

Numerical integration (sometimes referred to as solving or simulation) of differential equations, ordinary or partial, involves using a computer to obtain an approximate and discrete (in time and/or space) solution. In chemical kinetics, these differential equations are typically the rate laws that describe the time evolution of the system. One obtains results for the mean concentrations, without any information about the (typically very small) fluctuations that are inevitably present. Continuation and sensitivity analysis techniques enable one to extrapolate from a numerically obtained solution at one set of parameters (e.g., rate constants or initial concentrations) to the behavior of the system at other parameter values, without having to carry out a full numerical integration each time the parameters are changed. Other approaches, sometimes referred to collectively as stochastic methods (Gardiner, 1990), can provide data about fluctuations, but these require considerably more computational labor and are often impractical for models that include more than a few variables. 

In this section, we will present results of microldnetics simulations based on elementary reaction energy schemes deduced from quantum chemical studies. We use an adapted scheme to enable analysis of the results in terms of the values of elementary rate constants selected. For the same reason, we ignore surface concentration dependence of adsorption energies, whereas this can be readily implemented in the simulations. We are interested in general trends and especially the temperature dependence of overall reaction rates. The simulations will also provide us with information on surface concentrations. In the simulations to be presented here, we exclude product readsorption effects. Microldnetics simulations are attractive since they do not require an assumption of rate-controlling steps or equilibration. Solutions for overall rates are found by solving the complete set of PDFs with proper initial conditions. While in kinetic Monte Carlo simulations these expressions are solved using stochastic techniques, which enable formation... 

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