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The Multi-AMR Buffer Storage, Retrieval, and Reshuffling Problem: Exact and Heuristic Approaches

Max Disselnmeyer, Thomas Bömer, Laura Dörr, Bastian Amberg, Anne Meyer

Abstract

Buffer zones are essential in production systems to decouple sequential processes. In dense floor storage environments, such as space-constrained brownfield facilities, manual operation is increasingly challenged by severe labor shortages and rising operational costs. Automating these zones requires solving the Buffer Storage, Retrieval, and Reshuffling Problem (BSRRP). While previous work has addressed scenarios where the focus is limited to reshuffling and retrieving a fixed set of items, real-world manufacturing necessitates an adaptive approach that also incorporates arriving unit loads. This paper introduces the Multi-AMR BSRRP, coordinating a robot fleet to manage concurrent reshuffling, alongside time-windowed storage and retrieval tasks, within a shared floor area. We formulate a Binary Integer Programming (IP) model to obtain exact solutions for benchmarking purposes. As the problem is NP-hard, rendering exact methods computationally intractable for industrial scales, we propose a hierarchical heuristic. This approach decomposes the problem into an A* search for task-level sequence planning of unit load placements, and a Constraint Programming (CP) approach for multi-robot coordination and scheduling. Experiments demonstrate orders-of-magnitude computation time reductions compared to the exact formulation. These results confirm the heuristic's viability as responsive control logic for high-density production environments.

The Multi-AMR Buffer Storage, Retrieval, and Reshuffling Problem: Exact and Heuristic Approaches

Abstract

Buffer zones are essential in production systems to decouple sequential processes. In dense floor storage environments, such as space-constrained brownfield facilities, manual operation is increasingly challenged by severe labor shortages and rising operational costs. Automating these zones requires solving the Buffer Storage, Retrieval, and Reshuffling Problem (BSRRP). While previous work has addressed scenarios where the focus is limited to reshuffling and retrieving a fixed set of items, real-world manufacturing necessitates an adaptive approach that also incorporates arriving unit loads. This paper introduces the Multi-AMR BSRRP, coordinating a robot fleet to manage concurrent reshuffling, alongside time-windowed storage and retrieval tasks, within a shared floor area. We formulate a Binary Integer Programming (IP) model to obtain exact solutions for benchmarking purposes. As the problem is NP-hard, rendering exact methods computationally intractable for industrial scales, we propose a hierarchical heuristic. This approach decomposes the problem into an A* search for task-level sequence planning of unit load placements, and a Constraint Programming (CP) approach for multi-robot coordination and scheduling. Experiments demonstrate orders-of-magnitude computation time reductions compared to the exact formulation. These results confirm the heuristic's viability as responsive control logic for high-density production environments.

Paper Structure

This paper contains 67 sections, 3 theorems, 21 equations, 8 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Foundations and Related Problems
  4. Extensions for Modeling the Buffer Storage, Retrival, and Reshuffling Problem
  5. Heuristic and Learning-Based Approaches
  6. Research Gap
  7. The Multi-AMR Buffer Storage, Retrieval, and Reshuffling Problem
  8. Static Lanes and Graph Overlay
  9. Model Simplifications
  10. Objective Function and Model Flexibility
  11. Exact Problem Formulation
  12. Notation
  13. Decision Variables
  14. Objective Function
  15. Constraints
  16. ...and 52 more sections

Key Result

Lemma B.1

The mapping of containers $c_i$ to unit loads $u_i$ and stacks $s_j$ to lanes $l_j$ defined by the transformation $T$ preserves all accessibility relationships between the elements.

Figures (8)

  • Figure 1: Real-world buffer scenario from the surface coating industry. Automating material flow in such space-constrained brownfield facilities requires solving the Buffer Storage, Retrieval, and Reshuffling Problem (BSRRP) to ensure continuous machine supply.
  • Figure 2: Conceptual comparison of buffer accessibility: a) Access from the Perimeter, as defined in the BSRRP. a) In-Grid Access, typical for Robotic Mobile Fulfillment Systems (RMFS) where agents navigate within the grid; In the former, AMRs are restricted to the exterior aisles, necessitating strategic reshuffling of obstructing unit loads to access deep LIFO slots.
  • Figure 3: Logical decomposition of the storage area. (a) The continuous floor is discretized into slots. (b) These slots are grouped into Static Lanes (colored), where each lane acts as a LIFO stack accessible from the perimeter. The grey numbered rectangles represent the unit loads
  • Figure 4: Overview of the hierarchical heuristic. The sequential stages transform the input from a linearized task sequence into strict precedence constraints, then into timed assignments, and finally into collision-free trajectories.
  • Figure 5: Performance comparison between the EF and the proposed Heuristic. Top panels: Runtime distributions (logarithmic scale) across $3\times3$ and $4\times4$ layouts. Bottom panels: Optimality gaps relative to the exact lower bound, categorized by unit-load-to-slot-ratio.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Definition B.1: BRP - Block Relocation Problem
  • Definition B.2: BSRRP - Buffer Storage, Retrieval, and Reshuffling Problem
  • Definition B.3: Transformation $T$
  • Lemma B.1: Correctness of Mapping $T$
  • Proof B.1
  • Lemma B.2: Time Window Correctness
  • Proof B.2
  • Theorem B.3: Solution Equivalence
  • Proof B.3