A heuristic algorithm to solve the one‐warehouse multiretailer problem with an emission constraint
Bibliographic record
Abstract
Abstract In this paper, we consider the one‐warehouse multiretailer problem with a global carbon emission cap constraint (OWMR‐EC). This constraint aims at limiting the carbon emissions related to the production, setup, and inventory‐holding operations. We develop a penalized relaxation (PR) method to heuristically solve the considered problem, both with and without the possibility of having initial inventory. This heuristic uses in itself another heuristic that we propose to solve the standard one‐warehouse multiretailer problem (OWMR). Our PR method is tested on numerous instances adapted from the literature. Our results indicate that the penalized method is able to find between 87.4% and 89.8% of feasible solutions for this NP‐hard problem, with an average optimality gap of 2.1% and 2.2% depending on the algorithms we use to solve the different subproblems involved in the method. The results show that our method is highly effective in terms of run‐time and solution quality, when a feasible solution is found. Furthermore, the results indicate that the heuristic for the standard OWMR is also very effective. We further perform a sensitivity analysis on the optimal solutions of the OWMR‐EC to better understand the implications of the carbon emission cap constraint. The sensitivity analysis indicates that the marginal cost of reducing carbon emissions increases as the emission cap decreases. The analysis also shows that the correlation between the cost and emission parameters has an important impact on the potential to further lower the emissions, compared to the emission of the minimum cost solution.
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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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".