Route optimization for open-close multiple travelling salesman problem with load-balancing constraint: A multi-chromosome based genetic algorithm
Bibliographic record
Abstract
The Multiple Travelling Salesman Problem (MTSP) is one of the prominent combinatorial optimization problems with both theoretical interest and practical applications. However, its less-explored variants, such as the Open-Close Multiple Travelling Salesman Problem (OCMTSP), have received comparatively limited attention. In the OCMTSP, all salesmen commence their routes from a central depot, but unlike the classical MTSP, not all are required to return to the starting point upon completing their deliveries. Additionally, allowing any salesman to visit the maximum number of cities can lead to an imbalanced workload distribution among the salesmen. To address this imbalance, the current study incorporates a load balancing constraint into the OCMTSP framework, ensuring a fair distribution of cities among all salesmen. This extended problem variant is termed as Open-Close Multiple Travelling Salesman Problem with Load Balancing (OCMTSPLB). The primary objective of the OCMTSPLB is to minimize the total travel distance or cost incurred by the combined open and closed tours while maintaining balanced workloads. To solve this variant, the study proposes two distinct crossover based multi-chromosome Genetic Algorithm (GA) frameworks. Given the novelty of this problem, the algorithms are assessed using standardized benchmark instances from the TSPLIB. Experimental findings indicate that one of the proposed GA variants consistently achieves superior solution quality, a result further validated through non-parametric statistical tests.
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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 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".