Path inequalities for the vehicle routing problem with time windows
Bibliographic record
Abstract
Abstract In this paper we introduce a new formulation of the vehicle routing problem with time windows (VRPTW) involving only binary variables. The new formulation is based on the formulation of the asymmetric traveling salesman problem with time windows by Ascheuer et al. (Networks 36 (2000) 69–79) and has the advantage of avoiding additional variables and linking constraints. In the new formulation, time windows are modeled using path inequalities that eliminate time and capacity infeasible paths. We present a new class of strengthened path inequalities based on the polyhedral results obtained by Mak (Ph.D. Thesis, 2001) for a variant of the TSP. We study the VRPTW polytope and determine its dimension. We show that the lifted path inequalities are facet defining under certain assumptions. We also introduce precedence constraints in the context of the VRPTW. Computational experiments are performed with a branch and cut algorithm on the Solomon test problems with wide time windows. Based on results on 25‐node problems, the outcome is promising compared to leading algorithms in the literature. In particular, we report a solution to a previously unsolved 50‐node Solomon test problem R208. The conclusion is therefore that a polyhedral approach to the VRPTW is a viable alternative to the path formulation of Desrochers et al. (Oper Res 40 (1992), 342–354). © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 49(4), 273–293 2007
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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 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 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 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".