Optimizing Vehicle Scheduling Based on Variable Timetable by Benders-and-Price Approach
Bibliographic record
Abstract
In practice, vehicle scheduling is planned on a variable timetable so that the departure times of trips can be shifted in tolerable ranges, rather than on a fixed timetable, to decrease the required fleet size. This paper investigates the vehicle scheduling problem on a variable timetable with the constraint that each vehicle can perform limited trips. Since the connection-based model is difficult to solve by optimization software for a medium-scale or large-scale instance, a designed path-based model is developed. A Benders-and-Price algorithm by combining the Benders decomposition and column generation is proposed to solve the LP-relaxation of the path-based model, and a bespoke Branch-and-Price is used to obtain the integer solution. Numerical experiments indicate that a variable timetable approach can reduce the required fleet size with a tolerable timetable deviation in comparison with a fixed timetable approach. Moreover, the proposed algorithm is greatly superior to GUROBI in terms of computational efficiency and guarantees the quality of the solution.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".