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We consider a risk-averse stochastic capacity planning problem under uncertain demand in each period. Using a scenario tree representation of the uncertainty, we formulate a multistage stochastic integer program to adjust the capacity expansion plan dynamically as more information on the uncertainty is revealed. Specifically, in each stage, a decision maker optimizes capacity acquisition and resource allocation to minimize certain risk measures of maintenance and operational cost. We compare it with a two-stage approach that determines the capacity acquisition for all the periods up front. Using expected conditional risk measures, we derive a tight lower bound and an upper bound for the gaps between the optimal objective values of risk-averse multistage models and their two-stage counterparts. Based on these derived bounds, we present general guidelines on when to solve risk-averse two-stage or multistage models. Furthermore, we propose approximation algorithms to solve the two models more efficiently, which are asymptotically optimal under an expanding market assumption. We conduct numerical studies using randomly generated and real-world instances with diverse sizes, to demonstrate the tightness of the analytical bounds and efficacy of the approximation algorithms. We find that the gaps between risk-averse multistage and two-stage models increase as the variability of the uncertain parameters increases and decrease as the decision maker becomes more risk averse. Moreover, a stagewise-dependent scenario tree attains much higher gaps than a stagewise-independent counterpart, whereas the latter produces tighter analytical bounds. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work of Dr. X. Yu was partially supported by the U.S. National Science Foundation Division of Information and Intelligent Systems [Grant 2331782]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0396 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0396 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .
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This content will become publicly available on December 13, 2025
A Bilevel Optimization Approach for a Class of Combinatorial Problems with Disruptions and Probing
We consider linear combinatorial optimization problems under uncertain disruptions that increase the cost coefficients of the objective function. A decision maker, or planner, can invest resources to probe the components (i.e., the coefficients) in order to learn their disruption status. In the proposed probing optimization problem, the planner, knowing just the disruptions’ probabilities, selects which components to probe subject to a probing budget in a first decision stage. Then, the uncertainty realizes, and the planner observes the disruption status of the probed components, after which the planner solves the combinatorial problem in the second stage. In contrast to standard two-stage stochastic optimization, the planner does not have access to the full uncertainty realization in the second stage. Consequently, the planner cannot directly optimize the second-stage objective function, which is given by the actual cost after disruptions, and the decisions have to be made based on an estimate of the cost. By assuming that the estimate is given by the conditional expected cost given the information revealed by probing, we reformulate the probing optimization problem as a bilevel problem with multiple followers and propose an exact algorithm based on a value function reformulation and three heuristic algorithms. We derive theoretical results that bound the value of information and the price of not having full information and a bound on the required probing budget that attains the same performance as full information. Our extensive computational experiments suggest that probing a fraction of the components is sufficient to yield large improvements in the optimal value, that our exact algorithm is competitive for small- to medium-scale instances, and that the proposed heuristics find high-quality solutions in large-scale instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0236] and the Division of Civil, Mechanical and Manufacturing Innovation [Grant CMMI 2145553]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0629 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0629 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .
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- Award ID(s):
- 2145553
- PAR ID:
- 10560981
- Publisher / Repository:
- INFORMS
- Date Published:
- Journal Name:
- INFORMS Journal on Computing
- ISSN:
- 1091-9856
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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