A Fixed Rate Production Problem with Poisson Demand and Lost Sales Penalties
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Bibliographic record
Abstract
We solve a variation of a classic make‐to‐stock inventory problem introduced by Gavish and Graves. A machine is dedicated to a single product whose demand follows a stationary Poisson distribution. When the machine is on, items are produced one at a time at a fixed rate and placed into finished‐goods inventory until they are sold. In addition, there is an expense for setting up the machine to begin a production run. Our departure from Gavish and Graves involves the handling of unsatisfied demand. Gavish and Graves assumed it is backordered, while we assume it is lost, with a unit penalty for each lost sale. We obtain an optimal solution, which involves a produce‐up‐to policy, and prove that the expected time‐average cost function, which we derive explicitly, is quasi‐convex separately in both the produce‐up‐to inventory level Q and the trigger level R that signals a setup for production. Our search over the ( Q, R) array begins by finding Q 0 , the minimizing value of Q for R = 0. Total computation to solve the overall problem, measured in arithmetic operations, is quadratic in Q 0 . At most 3 Q 0 cost function evaluations are required. In addition, we derive closed‐form expressions for the objective function of two related problems: one involving make‐to‐order production and another for control of an N‐policy M/ D/1 finite queue. Finally, we explore the possibility of solving the lost sales problem by applying the Gavish and Graves algorithm for the backorder problem.
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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 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| 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 it