Stability and tie-breaking in online warehouse reoptimisation

Unrefereed research candidate. A warehouse picker with a cart of c bins serves orders that keep arriving, and reoptimises: whenever the list of known orders changes, it recomputes the fastest way to finish them. Five orders, two bins: the plans (1,2,2) and (2,2,1) both need three trips, so a solver may return either. For identical orders at one shelf in the non-interventionist manual-cart model (Pcart-N) of Lorenz, Otto and Gendreau, always taking the remainder first (RF) is a reflected random walk with increments ceil(A/c)-1 and is stable only below the root of E ceil(Pois(rho)/c) = 1 (about 76% of capacity with two bins, 49% with eight, tending to 1/e), whereas filling the cart first (FF) is stable up to full capacity. RF and FF bracket every exact tie rule; the exact almost-sure makespan ratio of RF is given at every load, approaches but never reaches two, and the separation survives positive picking time (sharp threshold at c = 2) and two distinct shelf locations. Breaking ties by total order completion time selects FF. Separately, the paper proves the Lorenz-Otto-Gendreau low-rate conjecture (Networks, 2025) for all three of their cart models: Poisson arrivals with lambda E[B] < 1, where B is the solo append charge, keep the backlog O(log n) almost surely, with explicit finite-horizon completion, backlog and delay bounds. This condition is deliberately conservative.

The archive contains the manuscript (PDF, Markdown and generated LaTeX), proof notes, prior-art audits, the bibliography with its Crossref and arXiv records, fourteen verification scripts with retained outputs (rational root certificates, exhaustive finite checks, high-precision recomputations and seeded synthetic simulations), a one-command runner, CI and SHA-256 sums. Assurance: written proofs and finite computational checks; internal AI-assisted review only (not peer review, independent reproduction or formal verification). Creator: Anonymous. Text and data CC0-1.0; code MIT.