Stochastic objective function

Stochastic objective function. The preceding MPC formulation assumes that future process outputs are deterministic over the finite optimization horizon. For a more realistic representation of future process outputs, one may consider a probabilistic (stochastic) prediction for y[/c + i k] and formulate an objective function that contains the expectation of appropriate functionals. For example, if y[k + i k] is probabilistic, then the expectation of the functional in Eq. (4) could be used. This formulation, known as open-loop optimal feedback, does not take into account the fact that additional information would be available at future time points k + i and assumes that the system will essentially run in open-loop fashion over the optimization horizon. An alternative, producing a closed-loop optimal feedback law relies... 

Stochastic methods do not need auxiliary information, such as derivatives, in order to progress. They only require an objective function for the search. This means that stochastic methods can handle problems in which the calculation of the derivatives would be complex and cause deterministic methods to fail. 

Having evaluated the system performance for each setting of the six variables, the variables are optimized simultaneously in a multidimensional optimization, using for example SQP, to maximize or minimize an objective function evaluated at each setting of the variables. However, in practice, many models tend to be nonlinear and hence a stochastic method can be more effective. 

Various search strategies can be used to locate the optimum. Indirect search strategies do not use information on gradients, whereas direct search strategies require this information. These methods always seek to improve the objective function in each step in a search. On the other hand, stochastic search methods, such as simulated annealing and genetic algorithms, allow some deterioration... 

Consideration of the expected value of profit alone as the objective function, which is characteristic of the classical stochastic linear programs introduced by Dantzig (1955) and Beale (1955), is obviously inappropriate for moderate and high-risk decisions under uncertainty since most decision makers are risk averse in facing important decisions. The expected value objective ignores both the risk attribute of the decision maker and the distribution of the objective values. Hence, variance of each of the random price coefficients can be adopted as a viable risk measure of the objective function, which is the second major component of the MV approach adopted in Risk Model I. 

Compensating slack variables accounting for shortfall and/or surplus in production are introduced in the stochastic constraints with the following results (i) inequality constraints are replaced with equality constraints (ii) numerical feasibility of the stochastic constraints can be ensured for all events and (iii) penalties for feasibility violations can be added to the objective function. Since a probability can be assigned to each realization of the stochastic parameter vector (i.e., to each scenario), the probability of feasible operation can be measured. In this... 

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. 

The stochastic model with recourse in the previous section takes a decision merely based on first-stage and expected second-stage costs leading to an assumption that the decision-maker is risk-neutral (Sahinidis, 2004). In order to capture the concept of risk in stochastic programming, Mulvey, Vanderbei and Zenios (1995) proposed the following amendment to the objective function ... 

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). 

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. 

The results of the model considered in this Chapter under uncertainty and with risk consideration, as one can intuitively anticipate, yielded different petrochemical network configurations and plant capacities when compared to the deterministic model results. The concepts of EVPI and VSS were introduced and numerically illustrated. The stochastic model provided good results as the objective function value was not too far from the results obtained using the wait-and-see approach. Furthermore, the results in this Chapter showed that the final petrochemical network was more sensitive to variations in product prices than to variation in market demand and process yields when the values of 0i and 02 were selected to maintain the final petrochemical structure. 

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... 

It approximates the expectation of the stochastic formulation (usually called the true problem) and can be solved using deterministic algorithms. Problem (9.19) can be solved iteratively in order to provide statistical bounds on the optimality gap of the objective function value. The iterative SAA procedure steps are explained in Section 7.5 of Chapter 7. 

The importance of this result is that it leads to an overall objective criterion for sample size determination that averages criteria based on specific model assumptions. Thus it provides a solution that is robust to model uncertainty. Closed-form calculations of (8) are intractable, so we have developed numerical approximations to the conditional entropies Ent(6k n, yk, MLk) and Ent(9k n, yk, MGk). The computations of the expected Bayes risk are performed via stochastic simulations and the exact objective function is estimated by curve fitting as suggested by Miiller and Parmigiani (1995). These details are available on request from the authors. 

Figure 4.15 The summary description of a stochastic procedure used for the maximization of the objective function.

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