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Advancing Column Generation by a Novel Variable Fixing Method In the paper titled “DeLuxing: Deep Lagrangian Underestimate Fixing for Column-Generation-Based Exact Methods,” Dr. Yu Yang introduces DeLuxing—an innovative variable-fixing technique that significantly advances column-generation-based exact methods for solving large-scale optimization problems, particularly vehicle routing problems (VRPs). DeLuxing leverages a novel linear programming formulation with a small subset of the enumerated variables, which is theoretically guaranteed to yield qualified dual solutions for computing Lagrangian underestimates. By eliminating over 75% of the unnecessary variables, DeLuxing drastically boosts computational efficiency, doubling the size of CMTVRPTW (capacitated multitrip vehicle routing problem with time windows) instances that can be solved optimally. Additionally, this breakthrough accelerates top VRP solvers like RouteOpt by up to 71%. The core concept underpinning DeLuxing extends to broader contexts such as variable type relaxation and cutting plane addition, achieving an additional 25% speedup for difficult instances.
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Improving Column Generation for Vehicle Routing Problems via Random Coloring and Parallelization
We consider a variant of the vehicle routing problem (VRP) where each customer has a unit demand and the goal is to minimize the total cost of routing a fleet of capacitated vehicles from one or multiple depots to visit all customers. We propose two parallel algorithms to efficiently solve the column-generation-based linear-programming relaxation for this VRP. Specifically, we focus on algorithms for the “pricing problem,” which corresponds to the resource-constrained elementary shortest path problem. The first algorithm extends the pulse algorithm for which we derive a new bounding scheme on the maximum load of any route. The second algorithm is based on random coloring from parameterized complexity which can be also combined with other techniques in the literature for improving VRPs, including cutting planes and column enumeration. We conduct numerical studies using VRP benchmarks (with 50–957 nodes) and instances of a medical home care delivery problem using census data in Wayne County, Michigan. Using parallel computing, both pulse and random coloring can significantly improve column generation for solving the linear programming relaxations and we can obtain heuristic integer solutions with small optimality gaps. Combining random coloring with column enumeration, we can obtain improved integer solutions having less than 2% optimality gaps for most VRP benchmark instances and less than 1% optimality gaps for the medical home care delivery instances, both under a 30-minute computational time limit. The use of cutting planes (e.g., robust cuts) can further reduce optimality gaps on some hard instances, without much increase in the run time. Summary of Contribution: The vehicle routing problem (VRP) is a fundamental combinatorial problem, and its variants have been studied extensively in the literature of operations research and computer science. In this paper, we consider general-purpose algorithms for solving VRPs, including the column-generation approach for the linear programming relaxations of the integer programs of VRPs and the column-enumeration approach for seeking improved integer solutions. We revise the pulse algorithm and also propose a random-coloring algorithm that can be used for solving the elementary shortest path problem that formulates the pricing problem in the column-generation approach. We show that the parallel implementation of both algorithms can significantly improve the performance of column generation and the random coloring algorithm can improve the solution time and quality of the VRP integer solutions produced by the column-enumeration approach. We focus on algorithmic design for VRPs and conduct extensive computational tests to demonstrate the performance of various approaches.
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- PAR ID:
- 10318932
- Date Published:
- Journal Name:
- INFORMS Journal on Computing
- ISSN:
- 1091-9856
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1287/ijoc.2021.1105
