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A Branch-and-Cut based Pricer for the Capacitated Vehicle Routing Problem

A Mixed Integer Programming (MIP) model and branch-and-cut framework algorithm for solving the CPTP. The CPTP appears as the pricing sub-problem in column generation schemes for the CVRP.

💡 About

The Capacitated Vehicle Routing Problem, CVRP for short, is a combinatorial optimization routing problem in which, a geographically distributed set of customers with associated known demands, must be served with a fleet of vehicles stationed at a central facility. In the last two decades, column generation techniques embedded inside branch-price-and-cut frameworks have been the de facto state-of-the-art dominant approach for building exact algorithms for the CVRP. The pricer, an important component in column generation, needs to solve the so-called Pricing Problem (PP) which asks for an Elementary Shortest Path Problem with Resource Constraints (ESPPRC) in a reduced cost network. Little scientific efforts have been dedicated in studying branch-and-cut based approaches for tackling the PP. The ESPPRC has been traditionally relaxed and solved through dynamic programming algorithms. This approach, however, has two main downsides. First, it leads to a worsening of the obtained dual bounds. Second, the running time deteriorates as the length of the generated paths increases.

Little scientific efforts were devoted at studying branch-and-cut approaches for tackling the PP. Jepsen et al. (2014) [1] studied the CPTP problem in the context of pricing. Almost ten years have passed from their work, and both, dynamic programming algorithms and MIP optimizers have improved a lot.

This work revisits the original efforts of Jepsen et al. (2014) [1] and verifies whether branch-and-cut algorithms may be effectively employed for tackling the PP.

🚀 Getting Started

⚓ Requirements

  • A working C compiler
  • cmake
  • git
  • CPLEX>=12.10: A free academic license can be downloaded at this URL

📥 Getting the source code

git clone --recursive https://github.com/dparo/master-thesis/

🔨 Building it

cd master-thesis
mkdir -p build
cd build
cmake -DCMAKE_BUILD_TYPE=Release ../
cmake --build ./ --config Release

🔧 Running the tests

ctest --progress --force-new-ctest-process --output-on-failure -C Release

🚗 Running the main solver

./src/cptp --help

📬 Contact

📑 References

  • [1] Jepsen, M. K., Petersen, B., Spoorendonk, S., & Pisinger, D. (2014). A branch-and-cut algorithm for the capacitated profitable tour problem. Discrete Optimization, 14, 78–96. https://doi.org/10.1016/j.disopt.2014.08.001
  • [2] Baldacci, R., Christofides, N., & Mingozzi, A. (2008). An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts. Mathematical Programming, 115(2), 351–385.
  • [3] Baldacci, R., Mingozzi, A., & Roberti, R. (2011). New route relaxation and pricing strategies for the vehicle routing problem. Operations Research, 59(5), 1269–1283.

♎ License

This project is licensed under the MIT license. Check the LICENSE file.

This project makes use of thirdy party projects, located under the deps folder, licensed respectively in their corresponding licenses:

  1. rxi/log.c MIT license
  2. argtable3/argtable3 BSD license
  3. silentbicycle/greatest/ MIT license
  4. DavGamble/cJSON MIT license
  5. scottt/debugbreak BSD2 license
  6. B-Con/crypto-algorithms Public domain
  7. johthepro/doxygen-awesome-css MIT license

💖 Funding

ko-fi