The two-echelon location-routing problem: A comparative analysis of novel and existing compact formulations
Bibliographic record
Abstract
• Two novel compact formulations for the two-echelon location routing problem. • New valid inequalities for the proposed and existing formulations. • A theoretical comparison of the linear programming relaxations of the formulations. • Computational experiments to assess how the formulations perform in a MIP solver. • 125 new best known lower bounds for benchmark instances and 55 new optimal solutions. The two-echelon location-routing problem (2E-LRP) is a well-known problem in the literature that is commonly used to address applications in which deliveries occur at two levels. It concerns the location of facilities and the routing of vehicle fleets. Most studies addressing this problem and its variants rely on mixed-integer programming (MIP) formulations that are compact (i.e., have a polynomial number of variables and constraints). Although the formulations with two-index arc variables tend to perform better than those with vehicle index variables in vehicle routing problems, most of the literature on the 2E-LRP is based on the latter. In this paper, we present a comparative analysis of three compact formulations for the 2E-LRP: a literature-based formulation with vehicle index variables, and two novel formulations with two-index arc variables. Additionally, we propose enhancements for the literature-based formulation and polynomial valid inequalities for all of them. The linear programming relaxations of these formulations are thoroughly compared, showing that those of the two-index formulations are stronger. Extensive computational experiments evaluate the performance of the formulations on a general-purpose MIP solver. The results show that the formulations with vehicle index variables, despite being the standard approach in the literature, lead to poor solver performance, failing to find feasible solutions even for instances with only 50 customers. In fact, the best performance comes from the novel formulations, one of which leads to feasible solutions for all benchmark instances evaluated. Valid inequalities can be used to improve this performance even further. These experiments resulted in the discovery of 125 new best known lower bounds and 55 new optimal solutions (out of 131 benchmark instances evaluated).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.012 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".