The value of the stochastic solution in stochastic linear programs with fixed recourse

Stochastic linear programs have been rarely used in practical situations largely because of their complexity. In evaluating these problems without finding the exact solution, a common method has been to find bounds on the expected value of perfect information. In this paper, we consider a different method. We present bounds on the value of the stochastic solution, that is, the potential benefit from solving the stochastic program over solving a deterministic program in which expected values have replaced random parameters. These bounds are calculated by solving smaller programs related to the stochastic recourse problem.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47912/1/10107_2005_Article_BF01585113.pd

Stochastic Programming with Economic and Operational Risk Management in Petroleum Refinery Planning under Uncertainty

Rising crude oil price and global energy concerns have revived great interests in the oil and gas industry, including the optimization of oil refinery operations. However, the economic environment of the refining industry is typically one of low margins with intense competition. This state of the industry calls for a continuous improvement in operating efficiency by reducing costs through business-driven engineering strategies. These strategies are derived based on an acute understanding of the world energy market and business processes, with the incorporation of advanced financial modeling and computational tools. With regards to this present situation, this work proposes the application of the two-stage stochastic programming approach with fixed recourse to effectively account for both economic and operational risk management in the planning of oil refineries under uncertainty. The scenario analysis approach is adopted to consider uncertainty in three parameters: prices of crude oil and commercial products, market demand for products, and production yields. However, a large number of scenarios are required to capture the probabilistic nature of the problem. Therefore, to circumvent the problem posed by the resulting large-scale model, a Monte Carlo simulation approach is implemented based on the sample average approximation (SAA) technique. The SAA technique enables the determination of the minimum number of scenarios required yet still able to compute the true optimal solution of the problem for a desired level of accuracy within the specified confidence intervals. We consider Conditional Value-at-Risk (CVaR) as the risk metric to hedge against the three parameters of uncertainty, which affords a framework that also involves the use of the Value-at-Risk (VaR) measure. We adopt two approaches in formulating appropriate two-stage stochastic programs with mean–CVaR objective function. The first approach is by using the conventional definition of CVaR that leads to a linear optimization model approximation coupled with a graphical-based solution strategy to determine the value of VaR using SAA in order to arrive at the optimal solution. The second approach is to utilize auxiliary variables to formulate a suite of stochastic linear programs with CVaR-based constraints. We conduct computational studies on a representative refinery planning problem to investigate the various model formulations using GAMS/CPLEX and offer some remarks about the merits of these formulations

Chance Constrained Mixed Integer Program: Bilinear and Linear Formulations, and Benders Decomposition

In this paper, we study chance constrained mixed integer program with consideration of recourse decisions and their incurred cost, developed on a finite discrete scenario set. Through studying a non-traditional bilinear mixed integer formulation, we derive its linear counterparts and show that they could be stronger than existing linear formulations. We also develop a variant of Jensen's inequality that extends the one for stochastic program. To solve this challenging problem, we present a variant of Benders decomposition method in bilinear form, which actually provides an easy-to-use algorithm framework for further improvements, along with a few enhancement strategies based on structural properties or Jensen's inequality. Computational study shows that the presented Benders decomposition method, jointly with appropriate enhancement techniques, outperforms a commercial solver by an order of magnitude on solving chance constrained program or detecting its infeasibility