Value of the stochastic solution

The concept of the value of the stochastic solution (VSS) measures the advantage of using a two-stage stochastic program over using a deterministic one, in other words, it measures the cost of ignoring the uncertainty. 

The advantage of using a 2S-MILP instead of the corresponding deterministic approach is measured by the value of the stochastic solution (VSS) which is the difference of the respective optimal objective values ... 

The best 2S-MILP solution found for this example was —17.74, thus the value of the stochastic solution (see Section 9.3.2) is... 

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. 

The value of the stochastic solution can also be evaluated as the cost of ignoring uncertainty in the problem. These concepts will be evaluated in our case study. 

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. 

To facilitate the discussion it is helpful to specify three of the numerous meanings of the word state . We shall call a site any value of the stochastic variable X or n. We shall call a macrostate any value of the macroscopic variable . A time-dependent macrostate is a solution of the macroscopic equation (X.3.1), a stationary macrostate is a solution of (X.3.3). We shall call a mesostate any probability distribution P. A time-dependent meso-state is a solution of the master equation, the stationary mesostate is the time-independent solution PS(X). 

If the hazard rate of any single particle out of a compartment depends on the state of the system, the equations of the probabilistic transfer model are still linear, but we have nonlinear rate laws for the transfer processes involved and such systems are the stochastic analogues of nonlinear compartmental systems. For such systems, the solutions for the deterministic model are not the same as the solutions for the mean values of the stochastic model. 

V = V2 + Vji. Indeed, the mean value of the stochastic process from relation (4.131), noted as e j, becomes v, which is the solution of the differential equation dv ... 

The SC network configurations obtained by the stochastic and deterministic formulations are summarized in Figs. 7.7 and 7.8. Numerical results show that the solution computed by the stochastic formulation has higher performance than the optimal deterministic solution in terms of expected corporate value. Certainly, the optimal expeeted corporate value from the stochastic solution is 12% greater than the one computed by utilizing the deterministic approach. 

Let u be a vector valued stochastic variable with dimension D x 1 and with covariance matrix Ru of size D x D. The key idea is to linearly transform all observation vectors, u , to new variables, z = W Uy, and then solve the optimization problem (1) where we replace u, by z . We choose the transformation so that the covariance matrix of z is diagonal and (more importantly) none if its eigenvalues are too close to zero. (Loosely speaking, the eigenvalues close to zero are those that are responsible for the large variance of the OLS-solution). In order to liiid the desired transformation, a singular value decomposition of /f is performed yielding... 

Plugging the first-stage solution of the EV problem xEV into the stochastic program (2S-MILP) gives the expected result of using the EV solution (EEV problem). The solution of the EEV problem is not necessarily optimal for the original 2S-MILP. Consequently, the optimal objective value of the EEV problem is always greater than (or at least equal to) the optimal objective value of the 2S-MILP, such that the objective of EEV is an upper bound for the optimal solution of the 2S-MILP ... 

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). 

The possible states in each compartment are n0i + n02 + 1. Therefore R is a 256-dimensional matrix. The initial condition for the master equation is Pio,5 (0) = 1. Figures 9.25 and 9.26 show the associated probabilities for each state as functions of time for the central and peripheral compartments, respectively. In these figures the disk area is proportional to the associated probability, the full markers are the expected values, and the solid lines the solution of the deterministic model. As already mentioned, we note that the expectation of the stochastic model follows the time profile of the deterministic system. 

The solution of this equation system gives expressions E (s), k = 1,N, which can be solved analytically by using an adequate inversion procedure. Indeed, the stochastic model has now an analytical solution but only vdth mean values. It is important to notice that when the analytical solution of a stochastic model pro-diuces only mean values it is important to make relationships between these results and the experimental work. This observation is significant because more of the experimental measurements allow the determination of the mean values of the variables of the process state, for the model validation or for the indentifica-tion of process parameters. 

In this stochastic model, the values of the frequencies skipping from one state to another characterize the common deep bed filtration. This observation allows the transformation of the above-presented hyperbolic model into the parabolic model, given by the partial differential equation (4.297). With the univocity conditions (4.295) and (4.296) this model [4.5] agrees with the analytical solution described by relations (4.298) and (4.299) ... 

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