Fully Packed and Ready to Go: High-Density, Rearrangement-Free, Grid-Based Storage and Retrieval

TL;DR

The paper addresses high-density, grid-based storage and retrieval with uniform loads, aiming to maximize space utilization while minimizing rearrangements. It presents exact characterizations and constructive algorithms: in the offline case, a rearrangement-free solution exists for any n whenever the open-side width c ≥ 3, with an O(n) placement algorithm; in online settings, a lookahead of ℓ = 3r − 1 suffices for zero relocations when c ≥ 3, and a 1.125-approximation is achievable under fully online constraints through L-paths; a density–action trade-off is established for the fully online case, with aisle-based layouts achieving optimal density bounds 2k/(2k+1) for depth k. The approach emphasizes column-adjacent and L-shaped paths, enabling efficient, parallelizable multi-robot execution, and experiments show substantial improvements over baselines. The results have practical relevance for warehouses, ports, and automated parking, guiding design choices between density, access patterns, and relocation costs.

Abstract

Grid-based storage systems with uniformly shaped loads (e.g., containers, pallets, totes) are commonplace in logistics, industrial, and transportation domains. A key performance metric for such systems is the maximization of space utilization, which requires some loads to be placed behind or below others, preventing direct access to them. Consequently, dense storage settings bring up the challenge of determining how to place loads while minimizing costly rearrangement efforts necessary during retrieval. This paper considers the setting involving an inbound phase, during which loads arrive, followed by an outbound phase, during which loads depart. The setting is prevalent in distribution centers, automated parking garages, and container ports. In both phases, minimizing the number of rearrangement actions results in more optimal (e.g., fast, energy-efficient, etc.) operations. In contrast to previous work focusing on stack-based systems, this effort examines the case where loads can be freely moved along the grid, e.g., by a mobile robot, expanding the range of possible motions. We establish that for a range of scenarios, such as having limited prior knowledge of the loads' arrival sequences or grids with a narrow opening, a (best possible) rearrangement-free solution always exists, including when the loads fill the grid to its capacity. In particular, when the sequences are fully known, we establish an intriguing characterization showing that rearrangement can always be avoided if and only if the open side of the grid (used to access the storage) is at least 3 cells wide. We further discuss useful practical implications of our solutions.

Figures (9)

• Figure 1: Application examples. Left: Grid-based storage using robotic vehicles (AGVs) for transferring loads yalcin2017multi. The AGV can go beneath a load and can move in all four grid directions. Right: Illustration of an automated parking garage where vehicles are the load to be autonomously placed and retrieved auto-garage. Similar to the first case, AGVs can go under vehicles to transport them.
• Figure 2: A solution without relocations for an input arrival sequence $A\xspace = (9,4,7,3,6,2,1,8,5)$ and departure sequence $D\xspace = (1,2,3,4,5,6,7,8,9)$ for a $(3 \times 3)$ grid accessible only from the bottom. Snapshots are illustrated from left to right. (a) The first arriving (load) 9 can be directly stored at the top using a (straight) upward path empty. (b) Next, 4 arrives and can be stored in front of 9, leaving the space in between. (c) 7 is stored using an upward path with a single turn, i.e., a column-adjacent path. (d) 3, 6, 2, can be stored as shown using upward paths. (e) 1, 8, 5 can be stored similarly. At this stage, all loads have arrived. (f) 1, 2, 3, 4, and 5 can be retrieved sequentially directly using downward paths. Then, 6 can be retrieved using a downward path with a turn, another column-adjacent path. (g) 7, 8, and 9 can be retrieved using downward paths.
• Figure 3: Consider an instance with $A = (1, 4, 2, 3)$ and ${D = (1, 2, 3, 4)}$. Given that load $1$ must depart first, it has to be stored in the front to avoid relocations. This forces the above-shown storage sequence (or its vertical mirror, where 1 is placed on the bottom right). This leaves load 2 buried behind loads 3 and 4. This means it cannot be retrieved without a rearrangement.
• Figure 4: Running Algorithm 1 on the inputs $D\xspace' = (\underset{1}{12}, \underset{2}{7}, \underset{3}{3}, \underset{4}{1}, \underset{5}{10}, \underset{6}{8}, \underset{7}{9}, \underset{8}{11}, \underset{9}{6}, \underset{10}{4}, \underset{11}{2}, \underset{12}{5})$ (top row) and ${D\xspace' = (\underset{1}{8}, \underset{2}{3}, \underset{3}{4}, \underset{4}{5}, \underset{5}{6}, \underset{6}{2}, \underset{7}{7}, \underset{8}{12}, \underset{9}{1}, \underset{10}{10}, \underset{11}{9}, \underset{12}{11})}$ (bottom row). Top: (a)(b) Partial solutions after the first two iterations, where non-equal matching loads of $D$ and $D'$ are put in the left two columns $C_1$ and $C_2$. (c) In the third interaction, $D$ and $D'$ have an equal matching load 3, which is put in $C_3$. (d) The next two iterations fill up columns $C_1$ and $C_2$. This leads to the solution following case 1 from here on. (e) The leftover loads in $C_3$ are filled up to complete the arrangement. The solution with arrows showing the guaranteed local adjacencies for $[12]$ and $D\xspace'$ is shown in (f) and (g), respectively. Bottom: (h)(i) A partial solution after two and three iterations. (j) The next three iterations fill up $C_3$, leading the algorithm into case 2. (k) The next two iterations fill up $C_2$ with loads 10 and 12. (l) $C_1$ is filled up to complete the arrangement. (m) and (n) show the solution with local adjacency as in the top row. Notice that the arrows indicate guaranteed paths for retrieval (in (f) and (m)) and for storage (in (g) and (n), when reversed. Better paths requiring fewer sideway-move actions may exist.
• Figure 5: An example execution of the algorithm described in Theorem \ref{['thm:general-full']}, which also works for proving Theorem \ref{['thm:char']}. $A = (4, 10, 6, 12, 2, 3, 9, 15, 1, 14, 13, 7, 5, 11, 8)$ and $D=[15]$. (a) Knowing the first three loads to arrive are $4, 10, 6$, we store them in the order they depart, which is $4, 6, 10$ from bottom to top. These go to $C_1$. Similarly, the next three arrivals are stored in $C_2$ as $2, 3, 12$, from bottom to top. (b) For the next $9$ arrivals, we run the algorithm from Theorem \ref{['thm:3col']}. For this, we turn the remaining loads of $A$ backward to get $D'=(8, 11, 5, 7, 13, 14, 1, 15, 9)$ and the corresponding portion of $D$ is $(1, 5, 7, 8, 9, 11, 13, 14, 15)$. (c)(d) Arrows showing how departures and arrivals can be handled without rearrangements, respectively. Note that the arrows for arrivals are drawn backwards to be consistent with Fig. \ref{['fig:alg_ex']}.

Theorems & Definitions (18)

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• Theorem 1
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• Proposition 1
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• Theorem 4