Two-Stage Robust Network Flow and Design Under Demand Uncertainty

We describe a two-stage robust optimization approach for solving network flow and design problems with uncertain demand. In two-stage network optimization, one defers a subset of the flow decisions until after the realization of the uncertain demand. Availability of such a recourse action allows one...

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Bibliographic Details
Published in:Operations research Vol. 55; no. 4; pp. 662 - 673
Main Authors: Atamturk, Alper, Zhang, Muhong
Format: Journal Article
Language:English
Published: Linthicum, MD INFORMS 01.07.2007
Institute for Operations Research and the Management Sciences
Subjects:
applications
Applied sciences
Commodities
Design optimization
Evaluation
Exact sciences and technology
Flows in networks. Combinatorial problems
integer
Integer programming
Integers
Inventory control, production control. Distribution
Linear programming
Logistics
Mathematical vectors
Network flow problem
Network management systems
network/graphs
Observational research
Operational research and scientific management
Operational research. Management science
Optimization
Optimization techniques
programming
Robust optimization
Stochastic analysis
Studies
Transportation
Transportation demand
Transportation problem (Operations research)
Uncertainty
United States
Availability
Script
Conservatism
Probabilistic approach
Network flow
Lot sizing
Random vector
Separation principle
Location problem
Optimization
Stochastic programming
Integer programming
Upper bound
Uncertain system
Transportation problem
Network topology
network/graphs: applications
Multicommodity flow problem
Minimax method
Budget
Bipartite graph
programming: integer
Infeasibility
ISSN:0030-364X, 1526-5463
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Summary:We describe a two-stage robust optimization approach for solving network flow and design problems with uncertain demand. In two-stage network optimization, one defers a subset of the flow decisions until after the realization of the uncertain demand. Availability of such a recourse action allows one to come up with less conservative solutions compared to single-stage optimization. However, this advantage often comes at a price: two-stage optimization is, in general, significantly harder than single-stage optimization. For network flow and design under demand uncertainty, we give a characterization of the first-stage robust decisions with an exponential number of constraints and prove that the corresponding separation problem is -hard even for a network flow problem on a bipartite graph. We show, however, that if the second-stage network topology is totally ordered or an arborescence, then the separation problem is tractable. Unlike single-stage robust optimization under demand uncertainty, two-stage robust optimization allows one to control conservatism of the solutions by means of an allowed "budget for demand uncertainty." Using a budget of uncertainty, we provide an upper bound on the probability of infeasibility of a robust solution for a random demand vector. We generalize the approach to multicommodity network flow and design, and give applications to lot-sizing and location-transportation problems. By projecting out second-stage flow variables, we define an upper bounding problem for the two-stage min-max-min optimization problem. Finally, we present computational results comparing the proposed two-stage robust optimization approach with single-stage robust optimization as well as scenario-based two-stage stochastic optimization.
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ISSN:0030-364X
1526-5463
DOI:10.1287/opre.1070.0428