On the P-formulation and the Split-Fraction-Formulation for the Generalized Pooling Problem
Bibliographic record
Abstract
The generalized pooling problem (GPP) is a NP-hard problem for which the solution time for securing a global optimal solution heavily depends on the strength of the problem formulation. The existing GPP formulations use either quality variables (P-formulation and the variants) or split-fraction variables (SF-formulation and the variants) to model the material balance at the pools. This paper is the first attempt to develop theoretical results for comparing the strength of P-formulation and SF-formulation. It is found that, an enhanced version of P-formulation, called P + -formulation, is at least as strong as SF-formulation under mild conditions. Furthermore, P + -formulation becomes identical to P-formulation when the pooling network comprises only mixers and splitters. With additional conditions that are often satisfied at the root node, P-formulation is proved to be as least as strong as SF-formulation. The theoretical results are verified by the computational study of 23 problem instances.
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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 | 0.004 | 0.008 |
| Meta-epidemiology (narrow) | 0.003 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.007 | 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".