Benders Decomposition for Production Routing Under Demand Uncertainty
Bibliographic record
Abstract
The production routing problem (PRP) is a generalization of the inventory routing problem and concerns the production and distribution of a single product from a production plant to multiple customers using capacitated vehicles in a discrete- and finite-time horizon. In this study, we consider the stochastic PRP with demand uncertainty in two-stage and multistage decision processes. The decisions in the first stage include production setups and customer visit schedules, while the production and delivery quantities are determined in the subsequent stages. We introduce formulations for the two problems, which can be solved by a branch-and-cut algorithm. To handle a large number of scenarios, we propose a Benders decomposition approach, which is implemented in a single branch-and-bound tree and enhanced through lower-bound lifting inequalities, scenario group cuts, and Pareto-optimal cuts. For the multistage problem, we also use a warm start procedure that relies on the solution of the simpler two-stage problem. Finally, we exploit the reoptimization capabilities of Benders decomposition in a sample average approximation method for the two-stage problem and in a rollout algorithm for the multistage problem. Computational experiments show that instances of realistic size can be solved to optimality for the two-stage and multistage problems, and that Benders decomposition provides significant speedups compared to a classical branch-and-cut algorithm.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 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".