Note: Optimality Conditions for an (s, S) Policy with Proportional and Lump-Sum Penalty Costs
Bibliographic record
Abstract
We consider the optimality of the (s, S) policy for a periodic-review stochastic inventory problem with two types of shortage costs. The problem may arise in a rush-order application at a bank branch where the emergency provision costs during a foreign currency stockout are represented by proportional and lump-sum penalties. Aneja and Noori (1987) analyzed this problem and presented a set of conditions for the convexity of a particular function and made a claim about the K-convexity of another function to prove the optimality of the (s, S) policy. We show that because the function that is claimed to be K-convex is actually concave over a subset of its domain, Aneja and Noori's arguments cannot be used to prove the optimality of the (s, S) policy. However, we argue that Aneja and Noori's problem is equivalent to the typical lost-sales problem, and using this equivalence, we .nd a simple convexity condition that assures the optimality of the (s, S) policy.
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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 | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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".