A Branch-Price-and-Cut Algorithm for the Two-Echelon Vehicle Routing Problem with Time Windows
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Bibliographic record
Abstract
In this paper, we propose an exact branch-price-and-cut (BPC) algorithm for the two-echelon vehicle routing problem with time windows. This problem arises in city logistics when high-capacity and low-capacity vehicles are used to transport items from depots to satellites (first echelon) and from satellites to customers (second echelon), respectively. The aim is to determine a set of least-cost first- and second-echelon routes such that the load on the routes respect the capacity of the vehicles, each second-echelon route is supplied by exactly one first-echelon route, and each customer is visited by exactly one second-echelon route within its time window. We model the problem with a route-based formulation where first-echelon routes are enumerated a priori, and second-echelon routes are generated using column generation. The problem is solved using BPC. To generate second-echelon routes, one pricing problem per satellite is solved using a labeling algorithm which keeps track of the first-echelon route associated with each (partial) second-echelon route considered. Furthermore, to speed up the solution process, we introduce effective deep dual-optimal inequalities and apply known valid inequalities. We perform extensive computational experiments on benchmark instances and show that our method outperforms a state-of-the-art algorithm. We also conduct sensitivity analyses on the different components of our algorithm and derive managerial insights related to the structure of the first-echelon routes.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it