A Branch-Price-and-Cut Algorithm for the Inventory-Routing Problem
Bibliographic record
Abstract
The inventory-routing problem (IRP) integrates two well-studied problems, namely, inventory management and vehicle routing. Given a set of customers to service over a multiperiod horizon, the IRP consists of determining when to visit each customer, which quantity to deliver in each visit, and how to combine the visits in each period into feasible routes such that the total routing and inventory costs are minimized. In this paper, we propose an innovative mathematical formulation for the IRP and develop a state-of-the-art branch-price-and-cut algorithm for solving it. This algorithm incorporates known and new families of valid inequalities, including an adaptation of the well-known capacity inequalities, as well as an ad hoc labeling algorithm for solving the column generation subproblems. Through extensive computational experiments on a widely used set of 640 benchmark instances involving between two and five vehicles, we show that our branch-price-and-cut algorithm clearly outperforms a state-of-the-art branch-and-cut algorithm on the instances with four and five vehicles. In this instance set, 238 were still open before this work and we proved optimality for 54 of them.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.002 | 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 | 0.003 |
| 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".