Efficient Solution Procedures for Multistage Stochastic Formulations of Two Problem Classes

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Solak, Senay
Johnson, Ellis L.
Clarke, John-Paul B.
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We consider two classes of stochastic programming models which are motivated by two applications related to the field of aviation. The first problem we consider is the network capacity planning problem, which arises in capacity planning of systems with network structures, such as transportation terminals, roadways and telecommunication networks. We study this problem in the context of airport terminal capacity planning. In this problem, the objective is to determine the optimal design and expansion capacities for different areas of the terminal in the presence of uncertainty in future demand levels and expansion costs, such that overall passenger delay is minimized. We model this problem as a nonlinear multistage stochastic integer program with a multicommodity network flow structure. The formulation requires the use of time functions for maximum delays in passageways and processing stations, for which we derive approximations that account for the transient behavior of flow. The deterministic equivalent of the developed model is solved via a branch and bound procedure, in which a bounding heuristic is used at the nodes of the branch and bound tree to obtain integer solutions. In the second study, we consider the project portfolio optimization problem. This problem falls in the class of stochastic programs in which times of uncertainty realizations are dependent on the decisions made. The project portfolio optimization problem deals with the selection of research and development (R&D) projects and determination of optimal resource allocations for the current planning period such that the expected total discounted return or a function of this expectation for all projects over an infinite time horizon is maximized, given the uncertainties and resource limitations over a planning horizon. Accounting for endogeneity in some parameters, we propose efficient modeling and solution approaches for the resulting multistage stochastic integer programming model. We first develop a formulation that is amenable to scenario decomposition, and is applicable to the general class of stochastic problems with endogenous uncertainty. We then demonstrate the use of the sample average approximation method in solving large scale problems of this class, where the sample problems are solved through Lagrangian relaxation and lower bounding heuristics.
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