Probabilistic Analysis of Renewal Cycles: An Application to a Non-Markovian Inventory Problem with Multiple Objectives
Bibliographic record
Abstract
Many stochastic optimization problems are solved using the renewal reward theorem (RRT). Once a regenerative cycle is identified, the objective function is formed as the ratio of the expected cycle cost to the expected cycle time and optimized using the standard techniques. Application of the RRT requires only the first moments of the cycle-related random variables. However, if the start of a cycle corresponds to an important event, e.g., end of a period of shortages in an inventory problem, knowing only the expected time—and the cost—of the cycle may not give enough information on the functioning of the stochastic system. For example, it may be useful to know the probability that the cycle cost, or more importantly, the average cost per unit time will exceed predetermined levels. In this paper we provide a complete description of the cycle-related random variables for a stochastic inventory problem with supply interruptions. We assume a general phase-type distribution for the supplier's availability (ON) periods and an exponential distribution for the OFF periods. The first passage time of an embedded Markov chain of the ON/OFF process is used to develop the expressions for the exact distribution and the moments of the cycle time and cycle cost random variables. We then describe a method for computing the probability that the average cost per unit time will exceed a predetermined level. This method is used to construct an “efficient frontier” for the two criteria of (i) average cost and (ii) the probability of exceeding it. The efficient frontier is used to find a solution to the multiple-criteria optimization problem.
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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 | 0.006 | 0.014 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".