An (<i>s</i>,<i>r</i>,<i>S</i>) Diffusion Inventory Model with Exponential Leadtimes and Order Cancellations
Bibliographic record
Abstract
We consider an inventory model of diffusion type for a single item, based on a so-called (s, r, S) policy, which is a refinement of the classical (s, S) policy. The content level process W = {W(t):t ≥ 0} behaves like a reflected Brownian motion with negative drift between jumps, at which replenishments are supplied which take the current content up to some prespecified level S. The process W starts at W(0) = S but is not bounded from above; the inventory is supposed to have infinite capacity. Whenever the content level drops to level s an order is placed to take the inventory back to level S, which the supplier will carry out after some random leadtime. However, if during a leadtime W reaches again a certain prespecified level r ∊ (s, S) (due to its intrinsic fluctuations), the order is cancelled and a penalty is paid. To assess the performance of this inventory system, one needs to compute several expected total discounted cost functionals of W : set-up cost (composed of the cost of actual replenishments and those of cancellations), variable delivery cost, holding cost, penalties on lost demands. All these functionals are derived in closed form as functions of the system primitives and in particular of the decision variables S, r and s. We also give some examples of numerical optimizations based on these results.
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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) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.003 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".