Two-Level Capacitated Discrete Location with Concave Costs
Bibliographic record
Abstract
In this paper, we study a general class of two-level capacitated discrete location problems with concave costs. The concavity arises from the economies of scale in production, inventory, or handling at the facilities and from the consolidation of flows for transportation and transshipment on the links connecting the facilities. Given the discrete nature of the problem, it is naturally formulated as a mixed-integer nonlinear program that uses binary variables for locational decisions and continuous variables for routing flows. We present an alternative formulation that only uses continuous variables and discontinuous functions, resulting in a nonlinear program with a concave objective function. Our main goal is to computationally compare these two modeling approaches under the same solution framework. In particular, we present an exact branch-and-bound algorithm that uses (integer) linear relaxations of the proposed formulations to optimally solve large-scale instances. The algorithm is enhanced with a cost-dependent spatial branching strategy and preprocessing step to improve its convergence. Extensive computational experiments are performed to assess the performance of the exact algorithm. Based on real location data from 3,109 counties in the contiguous United States, we also present a sensitivity analysis to showcase the impact of considering concave costs in location and assignment decisions.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.003 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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 teacher head, 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".