Large Language Model for Operations Research Formulation Selection in Multi-Warehouse Inventory Allocation

Abstract

Multi-warehouse inventory allocation is typically formulated as a mixed-integer programming (MIP) problem, yet no single formulation consistently matches heterogeneous instance-level regimes induced by demand concentration, inventory imbalance, replenishment scale, service constraints, and forecast volatility. We study this issue as instance-wise operations research (OR) formulation selection, where each allocation instance is assigned to a solver-executable formulation from a candidate OR expert library. We propose a solver-guided large language model (LLM) framework for OR formulation selection, in which each OR expert corresponds to a MIP formulation encoding a distinct allocation priority. To train the selector, the framework first constructs balanced expert-conditioned supervised fine-tuning (SFT) records for schema learning, and then uses MIP solver evaluation on historical instances to convert solver-evaluated allocation-quality gaps into margin-weighted identity preference optimization (IPO) preferences and per-instance expert-score metadata for reward lookup during group relative policy optimization (GRPO) to assign rewards to sampled responses. Experiments on multi-warehouse inventory allocation instances from JD$\mathord{.}$com, one of China's largest e-retailers, demonstrate that GRPO substantially improves expert-selection accuracy relative to the SFT+IPO selector and, more importantly, produces higher realized allocation quality than both the preference-trained selector and the best fixed formulation. With GRPO, Hit Ratio@1 and Hit Ratio@2 increase from 21.45% to 50.42% and from 70.47% to 82.31%. The resulting selector achieves an allocation accuracy gain of 12.57 percentage points over the incumbent baseline, outperforming both the SFT+IPO selector and the best fixed OR expert, and reduces the gap to the ex-post oracle to 4.85 percentage points.