A Unified Branch-Price-and-Cut Algorithm for Multicompartment Pickup and Delivery Problems
Bibliographic record
Abstract
In this paper, we study the pickup and delivery problem with time windows and multiple compartments (PDPTWMC). The PDPTWMC generalizes the pickup and delivery problem with time windows to vehicles with multiple compartments. In particular, we consider three compartment-related attributes: (1) compartment capacity flexibility that allows the capacities of the compartments to be fixed or flexible, (2) item-to-compartment flexibility that specifies which items are compatible with which compartments, and (3) item-to-item compatibility that considers that incompatible items cannot be simultaneously in the same compartment. To solve the PDPTWMC, we propose an exact branch-price-and-cut algorithm in which the pricing problem is solved by means of a unified bidirectional labeling algorithm. The labeling algorithm can tackle all possible combinations of the studied compartment-related attributes of the PDPTWMC. Furthermore, we implement several acceleration techniques that allow to, among others, reduce the symmetry in the label extensions with empty compartments, the symmetry in the dominance between compartments with similar attributes, and the complexity of the algorithm with fixed compartment capacity. Finally, we introduce benchmark instances for the PDPTWMC and conduct an extensive computational campaign to test the limits of our algorithm and to derive relevant managerial insights in order to highlight the applicability of considering the studied compartment-related attributes. Funding: This work was supported by Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant 439.18.459] and the Natural Sciences and Engineering Research Council of Canada [Grant 2017-06106]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.0252 .
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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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.008 | 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".