A Branch-and-Price Algorithm for the Multidepot Vehicle Routing Problem with Interdepot Routes
Bibliographic record
Abstract
This paper proposes a column generation algorithm for the multidepot vehicle routing problem with interdepot routes. This problem is an extension of the multidepot vehicle routing problem in which the vehicles are allowed to stop at intermediate depots along their routes to replenish. The problem can be modeled as a set covering problem in which the variables are rotations corresponding to feasible combinations of routes. We consider two pricing subproblems to generate rotations. The first one generates rotations directly by solving an elementary shortest path problem with resource constraints on a modified version of the original customer-depot network. The second one exploits the relationship between the sets of routes and rotations but results in a model with many columns. We discuss some issues related to solving this second pricing subproblem by column generation and we introduce an alternate approach to alleviate these difficulties. We show through computational experiments that the second pricing mechanism performs better than the first to compute the linear programming relaxation lower bound. We then embed it within a branch-and-bound algorithm to compute optimal integer solutions. Moreover, we assess the benefits of allowing interdepot routes in multidepot vehicle routing.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.009 | 0.002 |
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".