A Lagrangean Heuristic for Hub-and-Spoke System Design with Capacity Selection and Congestion
Bibliographic record
Abstract
Hub-and-spoke networks are widely applied in a variety of industries such as transportation, postal delivery, and telecommunications. Although they are designed to exploit economies of scale, hub-and-spoke networks are known to favour congestion, jeopardizing the performance of the entire system. This paper looks at incorporating congestion and capacity decisions in the design stage of such networks. The problem is formulated as a nonlinear mixed-integer program (NMIP) that explicitly minimizes congestion, capacity acquisition, and transportation costs. Congestion at hubs is modeled as the ratio of total flow to surplus capacity by viewing the hub-and-spoke system as a network of M/M/1 queues. To solve the NMIP, we propose a Lagrangean heuristic where the problem is decomposed into an easy subproblem and a more difficult nonlinear subproblem. The nonlinear subproblem is first linearized using piecewise functions and then solved to optimality using a cutting plane method. The Lagrangean lower bound is found using subgradient optimization. The solution from the subproblems is used to find a heuristic solution. Computational results indicate the efficiency of the methodology in providing a sharp bound and in generating high-quality feasible solutions in most cases.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 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".