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Bender Decomposition for LPs

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2024

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Bachelorarbeit
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Zusammenfassung

This thesis investigates the application of Benders decomposition to the Uncapacitated Facility Location Problem (UFLP), a pivotal optimization challenge in logistics and network design. Benders decomposition, introduced by J.F. Benders in 1962, is a technique that partitions large-scale optimization problems into a master problem and smaller, more manageable subproblems. This approach facilitates the solution of complex linear and mixed-integer programming problems by iteratively refining the master problem with feasibility and optimality cuts derived from subproblem solutions.

The UFLP focuses on determining the optimal placement of facilities to minimize total costs associated with facility establishment and customer allocation. This thesis provides a comprehensive exploration of the theoretical foundations of Benders decomposition, including key concepts in linear programming, dual problems, and the geometric properties of feasible regions. The core methodology involves the iterative solution of subproblems to generate constraints that progressively guide the master problem toward optimality.

A practical implementation of Benders decomposition is developed using Python and the Gurobi solver. The effectiveness of this implementation is evaluated by solving instances of the UFLP, with performance comparisons made against Gurobi’s direct solution method. The results reveal that while Benders decomposition exhibits superior scalability for larger problem instances, its computational efficiency may lag behind direct methods for smaller cases. This highlights the trade-off between decomposition methods and direct optimization techniques in terms of speed and scalability.

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Fachgebiet (DDC)
510 Mathematik

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Benders Decomposition, Linear Programs, Gurobi, Python

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ISO 690NGUYEN, Phi Long, 2024. Bender Decomposition for LPs [Bachelorarbeit]. Konstanz: Universität Konstanz
BibTex
@mastersthesis{Nguyen2024-12-16Bende-71776,
  year={2024},
  title={Bender Decomposition for LPs},
  address={Konstanz},
  school={Universität Konstanz},
  author={Nguyen, Phi Long}
}
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A practical implementation of Benders decomposition is developed using Python and the Gurobi solver. The effectiveness of this implementation is evaluated by solving instances of the UFLP, with performance comparisons made against Gurobi’s direct solution method. The results reveal that while Benders decomposition exhibits superior scalability for larger problem instances, its computational efficiency may lag behind direct methods for smaller cases. This highlights the trade-off between decomposition methods and direct optimization techniques in terms of speed and scalability.</dcterms:abstract>
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Konstanz, Universität Konstanz, Bachelorarbeit, 2024
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