A new mathematical model and solution method for the asymmetric traveling salesman problem with replenishment arcs
Bibliographic record
Abstract
This paper presents a new mathematical model for the Asymmetric Traveling Salesman Problem with Replenishment Arcs, an extension of the Asymmetric Traveling Salesman Problem, incorporating constraints on subpaths within the tour. Many existing modeling approaches to this problem require the generation of replenishment feasible or replenishment violation paths as a parameter set, which may lead to computational difficulties. Our formulation addresses these difficulties and provides direct computation of an optimal tour without relying on the set of paths as a parameter set. In this paper, we also propose a Lagrangian relaxation-based solution method. Given that ordinary Lagrangian functions can encounter duality gap in nonconvex problems, we employ a special augmented Lagrangian function, which is proven to overcome the issue of duality gap for many classes of nonconvex problems, including ours. In this paper, we utilize a hybrid solution method by combining the F-MSG method with an ant colony optimization algorithm. A similar solution method was previously used in Bulbul and Kasimbeyli (2021) [13]. In this paper, the method used in the aforementioned paper is enhanced in terms of computational complexity and solution efficiency. We assess the proposed method on 180 randomly generated instances, demonstrating that it achieves optimal solutions for almost all cases. Additionally, we apply our methodology to the aircraft maintenance routing problem, testing it on 11 instances from the aforementioned study. The results highlight the effectiveness of our approach, with an average improvement of 48.6% in solution time and a 0.93% enhancement in solution quality.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 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".