Integration and Computational Issues in Stochastic Design and Planning Optimization Problems

Abstract

In stochastic process design and planning optimization problems, the expected value of the objective function in face of uncertainty is typically evaluated through an n-dimensional integral, where n is the number of uncertain parameters. In this paper, suitable integration techniques are presented and computational issues are discussed in relation to the number of uncertain parameters and the uncertainty model considered. A specialized cubature technique, suitable to integrate normally distributed uncertainties, is introduced, which for n < 10 can reduce significantly the computational effort required when compared to other strategies, such as product Gauss rules or efficient sampling techniques. The computational performance of the different integration techniques and their applicability are discussed through two process-engineering examples.

Supporting chemical process design under uncertainty

“…The number of simulations, that are performed sampling the uncertain parameters according to the above-described quasi-Monte Carlo technique, is increased until the results converge to 10 -1 . It is found that 1000 simulations are required, which is in agreement with results reported by Bernardo et al (1999). Overall, the simulations require computation time on the order of several hours.…”

“…In contrast to the random selection, a deterministic sequence of points can ensure that the most representative sample of the parameter space is obtained. It has been pointed out in the literature that these sampling methods only need to visit a small fraction of the space to obtain a representative answer (Bernardo et al, 1999). As Monte Carlo methods are often applied to problems where each function evaluation at a sample point is computationally expensive, as is the case in the problem studied here, simulations only on the order of hundreds to thousands can be performed in reasonable time.…”

“…Other authors have applied different numerical integration methods to the problem of design under uncertainty (e.g., Straub andGrossmann, 1990 andBernardo et al, 1999). As Bernardo et al remark, the number of required samples grows exponentially with the dimensionality of the problem when quadrature methods, e.g., Gaussian quadrature, are used.…”

“…Lastly, it should be noted that constraint relaxation in the context of this paper differs from the idea suggested by Bernardo et al (1999). They introduced a strategy to simplify the integration to determine the expected value of the objective function that they also call constraint relaxation.…”

A modern two-stage stochastic programming portfolio model for an oil refinery with financial risk management

“…The literature varies on the choice of risk measure, and how the decision-maker's preferences are taken into account. The prices in each scenario are independent by definition, but within each scenario the product prices are conspicuously correlated with each other, and not independent as stated for example in Bernardo et al (1999). Therefore, the prices are considered constant, and the amounts that go into each product as the random variables.…”