Abstract
Abstract In this work, we address the modeling and solution of mixed-integer linear multistage stochastic programming problems involving both endogenous and exogenous uncertain parameters. We first propose a composite scenario tree that captures both types of uncertainty, and we exploit its unique structure to derive new theoretical properties that can drastically reduce the number of non-anticipativity constraints (NACs). Since the reduced model is often still intractable, we discuss two special solution approaches. The first is a sequential scenario decomposition heuristic in which we sequentially solve endogenous MILP subproblems to determine the binary investment decisions, fix these decisions to satisfy the first-period and exogenous NACs, and then solve the resulting model to obtain a feasible solution. The second is Lagrangean decomposition. We present numerical results for a process network and an oilfield development planning problem. The results clearly demonstrate the efficiency of the special solution methods over solving the reduced model directly.
Highlights
In the optimization of process systems, there is often some level of uncertainty in one or more of the input parameters
Note that in the reduced form of the model, first-period scenario-pair set SPF is defined in Equations (3.10) and (4.2), exogenous scenario-pair set SPX is defined in Equation (3.11), and endogenous scenario-pair set SPN is defined in Equations (3.22) and (3.23)
We have evaluated the performance of these approaches, as well as the impact of our theoretical reduction properties from Chapter 4, on a process network planning problem and an oilfield development planning problem
Summary
In the optimization of process systems, there is often some level of uncertainty in one or more of the input parameters. As we have shown, it is not uncommon for NACs to represent the majority of constraints in large problem instances It should not be surprising, that some authors have invested a considerable amount of effort into developing new formulations that eliminate redundant NACs. In the case that the uncertainty is purely exogenous, this process is fairly straightforward since the scenario tree is fixed. If we (or a colleague) can correctly formulate the mathematical optimization problem, we may be able to completely eliminate the guesswork from our decision making This is especially useful in cases with thousands of constraints, where it would be nearly impossible for a human being to manually arrive at the optimal solution. Subsequent recourse decisions allow operating conditions to be specified in response to this plan, based on the realizations observed for the exogenous-uncertain parameters (see, for instance, Liu and Sahinidis, 1996)
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