Cross-Item Pooled Calibration of Safety Stocks: Impossibility, Finite-Sample Coverage, and the Price of Per-Item Validity
- 1Department of Data and Systems Engineering, The University of Hong Kong, Pokfulam, Hong Kong SAR, China
- 2School of Economics and Management, Shanghai Maritime University, Shanghai 201306, China
Abstract
An order-up-to level serving a share α of replenishment cycles is a quantile of an unknown demand law, and practice substitutes a normal quantile of the forecast error—an approximation hardest to defend for the intermittent, integer-valued demand of spare parts. We instead take the safety term to be an order statistic of standardized forecast errors. If those errors are exchangeable with the error of the cycle being decided, such a level attains α for every demand law at finite sample size—but only if the history holds α/(1−α) cycles: nineteen for 95%, ninety-nine for 99%. That ceiling belongs to the distribution-free criterion rather than to the construction, since no finite level computed in any way from the n cycle totals is distribution-free valid above n/(n+1). Pooling standardized residuals across the assortment restores the certificate, under exchangeability of the pooled scores with the score being decided, for an item drawn at random from the portfolio. Validity for an item named in advance is strictly stronger: assuming independence, it inflates the target by the item’s one-sided discrepancy ρ from the pool and then demands (α+ρ)/(1−α−ρ) pooled cycles, the same arithmetic one level up. On four public panels spanning 6215 series, eight of ten level setters sharing one forecast, the pooled level found the order statistic it requires at every decision and met its target in 27 of 36 configurations, against 10 for the better of the two normal variants by that count. Those configurations re-use four panels and are not independent replicates, so the paired service difference is reported per panel with an item-clustered interval: 1.09 to 3.06 percentage points, each excluding zero, at a median 1.252 times the normal rule’s stock. That difference is bought with stock and not with efficiency—matched on stock, it is −0.09 to +0.37 points—so what the construction supplies is the target reached without knowing in advance which inflated target would reach it.
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BibTeX
@article{zhang2026crossitem,
title = {{Cross-Item Pooled Calibration of Safety Stocks: Impossibility, Finite-Sample Coverage, and the Price of Per-Item Validity}},
author = {Jia’en Zhang and Zhongmiao Sun},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203701},
url = {https://www.mdpi.com/2227-7390/14/20/3701},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Cross-Item Pooled Calibration of Safety Stocks: Impossibility, Finite-Sample Coverage, and the Price of Per-Item Validity
AU - Jia’en Zhang
AU - Zhongmiao Sun
PY - 2026
DA - 2026/10/12
JO - Mathematics
DO - 10.3390/math14203701
UR - https://www.mdpi.com/2227-7390/14/20/3701
AB - An order-up-to level serving a share α of replenishment cycles is a quantile of an unknown demand law, and practice substitutes a normal quantile of the forecast error—an approximation hardest to defend for the intermittent, integer-valued demand of spare parts. We instead take the safety term to be an order statistic of standardized forecast errors. If those errors are exchangeable with the error of the cycle being decided, such a level attains α for every demand law at finite sample size—but only if the history holds α/(1−α) cycles: nineteen for 95%, ninety-nine for 99%. That ceiling belongs to the distribution-free criterion rather than to the construction, since no finite level computed in any way from the n cycle totals is distribution-free valid above n/(n+1). Pooling standardized residuals across the assortment restores the certificate, under exchangeability of the pooled scores with the score being decided, for an item drawn at random from the portfolio. Validity for an item named in advance is strictly stronger: assuming independence, it inflates the target by the item’s one-sided discrepancy ρ from the pool and then demands (α+ρ)/(1−α−ρ) pooled cycles, the same arithmetic one level up. On four public panels spanning 6215 series, eight of ten level setters sharing one forecast, the pooled level found the order statistic it requires at every decision and met its target in 27 of 36 configurations, against 10 for the better of the two normal variants by that count. Those configurations re-use four panels and are not independent replicates, so the paired service difference is reported per panel with an item-clustered interval: 1.09 to 3.06 percentage points, each excluding zero, at a median 1.252 times the normal rule’s stock. That difference is bought with stock and not with efficiency—matched on stock, it is −0.09 to +0.37 points—so what the construction supplies is the target reached without knowing in advance which inflated target would reach it.
SN - 2227-7390
ER -