Partial‐Outsourcing Strategy for the Vehicle Routing Problem With Stochastic Demands
Bibliographic record
Abstract
ABSTRACT This paper studies a combined delivery strategy involving a private vehicle and external carriers under stochastic customer demands. The routing problem focuses on a single private vehicle, while external carriers are allowed to determine their own routes independently and are compensated with a fixed price per unit demand served. A strategy incorporating routing re‐optimization is proposed, along with a new recourse mechanism that leverages outsourcing through external carriers. To enable routing re‐optimization, a novel approximate linear programming (ALP) approach is introduced. This offers a new pathway for addressing vehicle routing problems (VRPs) under stochastic demand considerations. The ALP approach is adapted to the specific structure of routing under stochastic demands, leading to the development of a decomposition‐based ALP solution framework. This adaptation arises from changes in the decision sequence of routing and restocking at each step of the Markov decision process (MDP), which differs from previous formulations of vehicle routing under stochastic demands. Additionally, further adaptations are made to facilitate the computation of the proposed strategy by exploring the relationships among variables and constraints specific to the problem context, as well as by developing a constraint sampling procedure designed to mimic the near‐optimal heuristic policy. Our numerical results show that the proposed outsourcing‐based policy yields notable operating‐cost savings, with an average improvement of 4.06% over the traditional recourse strategy in midpoint‐depot instances. Moreover, in small instances where the optimal policy within the traditional partial re‐optimization framework can be computed, the proposed price‐directed (PD) policy still provides cost advantages over this re‐optimization scheme, demonstrating the value of our ALP‐based framework.
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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 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".