Optimization Algorithm for Inventory Allocation in Gravity-Flow Racks with Classical and Quantum-Hybrid Computing

TL;DR

The paper tackles the problem of inventory allocation in gravity-flow racks under FIFO constraints, where frequent reinsertions inflate costs. It formulates a joint, multi-item allocation as a $QUBO$ problem, enabling execution on classical, quantum, and quantum-hybrid hardware via a Problem Hamiltonian $H_P = H_A + H_B + H_C$ that encodes assignment, affinity, and capacity constraints with a binary-expanded capacity term. Empirical results compare two Simulated Annealing variants, Gurobi, and D-Wave's Constrained Quadratic Model hybrid solver; across small to large configurations, the $CQM$ solver consistently delivers superior solution quality and speed, while INT-SA provides a strong classical baseline and Gurobi struggles at scale. A factory-scale simulation using real operational data shows the proposed method can reduce reinsertion events by more than an order of magnitude relative to factory logs, underscoring substantial practical impact for industrial logistics and highlighting the potential of quantum-hybrid optimization in real-world supply chains.

Abstract

Warehouses play a central role in industrial logistics, functioning as critical hubs for storing and organizing inventory to support efficient production. Optimizing item allocation within these facilities is essential for reducing operational costs and improving delivery times. In this work, we address the optimization of inventory allocation in warehouses equipped with gravity-flow racks, which are designed for First In, First Out (FIFO) logistics, a configuration that inherently requires item reinsertions during retrieval operations to maintain flow continuity. These reinsertions, however, are time-consuming and costly, so minimizing their occurrence is crucial for operational efficiency. We propose an optimization strategy that simultaneously allocates multiple items, determining their placement across available shelves in a single decision step, explicitly accounting for every item and every shelf in the warehouse. By jointly evaluating multiple items, our approach enables globally optimized placement decisions, minimizing conflicts that arise in sequential methods. The problem is formulated as a QUBO, allowing implementation on both classical metaheuristics and quantum-hybrid solvers. We assess performance by comparing three classical optimization approaches - two variants of Simulated Annealing and the commercial solver Gurobi - with D-Wave's hybrid solver, which uniquely combines quantum annealing with classical metaheuristics. Complementing these benchmarks, a factory-scale simulation based on real operational data shows that considering larger batches of items in the allocation step can significantly reduce reinsertions, highlighting the practical potential of the proposed approach for industrial logistics.

Figures (4)

• Figure 1: Schematic representation of a gravity-flow rack. The rack has inclined shelves with rolling rails, allowing items to move forward under gravity. Items are represented by colored pallet boxes, and the rack comprises three gravity-flow shelves. On the left, at the loading side, a forklift places a red box onto the middle shelf. On the right, at the retrieval side, another forklift removes a green box from the top shelf. To reach an item located in the middle of a shelf (e.g., the dark green box on the top shelf), the forklift on the right must first remove all pallet boxes in front of it (e.g., the red and blue boxes), illustrating the operational constraints imposed by the FIFO method.
• Figure 2: Average energy obtained by the CQM Solver, the Gurobi Optimizer, and the two Simulated Annealing variants (RS-SA and INT-SA) for the $10 \times 10$ and $15 \times 15$ warehouse configurations, shown in panels (a) and (b), respectively. Error bars represent the standard deviation across 100 runs for each SA variant and Gurobi, and 50 runs for the CQM Solver. The inset reports the relative error of each method with respect to the average energy achieved by the CQM Solver. The time limit for each method was set to 5 seconds, corresponding to the minimum runtime required by the CQM Solver across all instances.
• Figure 3: Average energy obtained by the CQM Solver, the Gurobi Optimizer, and the two Simulated Annealing variants (RS-SA and INT-SA) for the $20 \times 20$ and $25 \times 25$ warehouse configurations, shown in panels (a) and (b), respectively. Error bars represent the standard deviation across 100 runs for each SA variant and Gurobi, and 50 runs for the CQM Solver. The inset reports the relative error of each method with respect to the average energy achieved by the CQM Solver. The time limit for each SA variant was set to 5 seconds, equal to the minimum runtime required by the CQM Solver for all instances, except for the task of allocating 312 and 375 items in the $25\times25$ configuration, which required 7.04 and 9.79 seconds, respectively.
• Figure 4: Average number of reinsertions obtained in the factory-scale simulation for the three allocation strategies. For the Factory’s Method, the reported value corresponds directly to the number of reinsertions recorded in the real factory logs during the three-month period reproduced in the simulation; therefore, no standard deviation is shown. The Recommendation Method is deterministic by construction. The Proposed Method yielded the lowest number of reinsertions, with $200 \pm 0$ for $N=1$, $148.2 \pm 9.8$ for $N=5$, and $138 \pm 8.4$ for $N=10$, averaged over eight independent runs. Error bars indicate standard deviations.