{ "schemaVersion": "1.6", "slug": "warehouse-reoptimisation-tie-breaking", "title": "Stability and tie-breaking in online warehouse reoptimisation", "shortTitle": "Same three trips, different fate: tie-breaking decides warehouse stability", "url": "https://evidencepress.org/releases/warehouse-reoptimisation-tie-breaking/", "oneLine": "Five orders, two bins: plans (1,2,2) and (2,2,1) both take three trips, but a picker who reoptimises as orders arrive and always takes the remainder first is overwhelmed at about 76% of capacity, while filling the cart first is stable to full capacity; the paper also proves the Lorenz–Otto–Gendreau low-rate conjecture.", "abstract": "A warehouse picker who, whenever new orders arrive, recomputes the fastest way to finish the known orders is said to reoptimise. This candidate answers two questions about that policy for the cart models of Lorenz, Otto and Gendreau (Networks, 2025). First, solving every snapshot exactly does not determine stability. With five identical orders at one shelf and a two-bin cart, the plans (1,2,2) and (2,2,1) both need three trips, so a solver may return either. In the non-interventionist manual-cart model, always taking the remainder first is stable only below the root of E⌈Pois(ρ)/c⌉ = 1, about 76% of capacity with two bins, 49% with eight and tending to 1/e for very large carts, whereas always filling the cart first is stable up to full capacity. The two rules bracket every exact tie rule; the paper gives the exact long-run makespan ratio of remainder first at every load, shows that it approaches but never reaches two, and shows that the gap survives positive picking times (sharply with two bins) and two distinct shelf locations. Breaking ties by total order completion time selects full first. Second, the paper proves the authors' low-rate conjecture for all three of their cart models: if appending each new order on its own costs time B, Poisson arrivals with λE[B] < 1 keep the backlog logarithmic almost surely, with explicit finite-horizon bounds. That condition is deliberately conservative. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-10-31", "dateModified": "2026-10-31", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23845532", "doiUrl": "https://doi.org/10.5281/zenodo.23845532", "conceptDoi": "10.5281/zenodo.23845531", "pdfUrl": "https://github.com/ipitchford/warehouse-reoptimisation-tie-breaking/releases/download/v0.1.0-candidate/warehouse-reoptimisation-tie-breaking-0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23845532", "repoUrl": "https://github.com/ipitchford/warehouse-reoptimisation-tie-breaking", "releaseUrl": "https://github.com/ipitchford/warehouse-reoptimisation-tie-breaking/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/warehouse-reoptimisation-tie-breaking/index.md", "bibtexUrl": "https://evidencepress.org/releases/warehouse-reoptimisation-tie-breaking/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/warehouse-reoptimisation-tie-breaking.mp3", "imageUrl": "https://evidencepress.org/assets/og/warehouse-reoptimisation-tie-breaking.png?v=6ca117f045", "coverArtUrl": "https://evidencepress.org/assets/art/warehouse-reoptimisation-tie-breaking.svg?v=eeef348a96", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/warehouse-reoptimisation-tie-breaking.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/warehouse-reoptimisation-tie-breaking.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted operations-research candidate consolidating two independent derivations. The theorems have written proofs; thresholds are certified by exact rational brackets and high-precision evaluation; simulations are synthetic and no warehouse operating data were used. Internal AI-assisted review in five roles, a separate-model adversarial proof review and audit of new proofs, a clean replay of both source bundles, one revision round and a re-review whose minor items were applied without a further review round. External specialist review, independent reproduction and formal verification have not been established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23845532", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "From a fresh extraction of the frozen archive, all fourteen check scripts ran with Python -I -O -B and every regenerated JSON and CSV output matched the retained one within a relative tolerance of 1e-7, apart from recorded run times. GitHub CI repeats the replay on Linux." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "note": "Two independently developed derivations were consolidated and every overlapping constant is recomputed from the definitions at 60 digits; several quantities are checked by separately written scripts. This is producer-side diversity, not unaffiliated reimplementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Python 3.12 with NumPy, SciPy and Matplotlib versions pinned in requirements.txt; not a locked container. Simulations are seeded." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted operations-research mathematics (two independent derivations, consolidated)", "AI-assisted internal review, separate-model adversarial review and audit, re-review", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the final manuscript (revision of 31 October 2026 with DOI insertion, 35 pages), its verification scripts and the frozen release archive. The versioned GitHub and Zenodo research archives were checked against the frozen release assets on 31 October 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "online order picking", "reoptimisation", "warehouse", "stochastic stability", "bulk-service queue", "order batching", "tie-breaking", "makespan", "online algorithms" ], "keyResults": [ "Tie-breaking decides stability (Propositions 7–8, Theorem 9, proof): for identical orders at one shelf in Pcart-N with capacity c and ρ arrivals per tour, remainder first is positive recurrent exactly when E⌈Pois(ρ)/c⌉ < 1 (null recurrent at equality), full first is stable for ρ < c, and the two bracket every exact tie rule.", "Exact cost (Theorem 9, Theorem 13, proof): the long-run makespan ratio of remainder first is max{1, G_c}/max{1, ρ/c} at every load; it is always below two and approaches two along overloaded sequences.", "Where it breaks (Theorem 11, proof with rational root certificates): the remainder-first threshold is about 76% of capacity at c = 2 and 49% at c = 8, and ρ_c/c → 1/e with an explicit logarithmic correction; beyond it the drift can be exponentially small, so short simulations can miss the instability.", "A principled repair (Proposition 14, proof): with zero picking time, breaking ties by total order completion time selects full first.", "Robustness (Theorems 15–17, proof): the separation survives positive picking time at every c ≥ 2, has a unique sharp threshold at c = 2, and survives two distinct shelf locations, with a proved makespan loss of at least about 10.7% in an explicit instance.", "The conjecture (Proposition 1, Theorems 2–5, Corollary 6, proof): for Pcart-N, Pcart and Robot, Poisson arrivals with λE[B] < 1, where B is the cost of appending one order on its own, keep the backlog O(log n) almost surely and the makespan ratio 1 + O(log n / n), for every tie rule, with explicit completion, backlog and delay bounds." ], "reviews": [], "evidencePackage": "Written proofs of every theorem, proposition, lemma and corollary; fourteen verification scripts with retained outputs: a physical-state check of the clearance certificate on a finite weighted graph (48,912 value comparisons), recurrence checks (164,000 cases), the offline benchmark against all ordered partitions and an all-subset dynamic programme, exact rational root brackets for the thresholds (capacities 2 to 128), 60- and 90-digit recomputations, exact corrector identities, a recomputation of every constant derived by both independent derivations, and seeded synthetic simulations; replay with Python -O from a fresh extraction of the frozen archive.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Lorenz et al. (2025). Picking Operations in Warehouses With Dynamically Arriving Orders: How Good is Reoptimization?. Networks. The three cart models, the reoptimisation policy and the low-rate conjecture proved here.", "url": "https://doi.org/10.1002/net.22281", "doi": "10.1002/net.22281" }, { "citation": "Lorenz et al. (2024). Picking Operations in Warehouses with Dynamically Arriving Orders: How Good is Reoptimization?. arXiv:2409.12619. Earlier preprint of the source paper; the conjecture is quoted from this version.", "url": "https://arxiv.org/abs/2409.12619v1", "doi": "10.48550/arXiv.2409.12619" }, { "citation": "Christou et al. (2026). Online Batching with Lookahead: Adversarial Limits and Stochastic Gains. arXiv:2610.16013. Online batching with lookahead; its work-conserving definition excludes remainder first.", "url": "https://arxiv.org/abs/2610.16013v1", "doi": "10.48550/arXiv.2610.16013" }, { "citation": "Powell and Humblet (1986). 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