Robust Out-of-Order Retrieval for Grid-Based Storage at Maximum Capacity

TL;DR

This work tackles dense, grid-based storage with one-side access under sequence uncertainty by introducing $k$-bounded perturbations to the departure order. It develops a two-part framework: a $k$-robust storage algorithm that achieves zero relocations using $\Theta(k)$ columns (with tight $2k+3$–$3k+3$ bounds) and a retrieval strategy that greedily relocates blocking loads to minimize future relocations. The authors provide a fast solver for constructing robust arrangements and an enhanced retrieval procedure, with extensive experiments showing up to 60--70% relocation reductions and near-elimination of relocations when $k$ is up to half the grid width. The results yield practical design guidelines for high-density warehouses, including column-width choices and strategies that confine relocations within the storage area, enabling scalable, robust automated storage systems.

Abstract

This paper proposes a framework for improving the operational efficiency of automated storage systems under uncertainty. It considers a 2D grid-based storage for uniform-sized loads (e.g., containers, pallets, or totes), which are moved by a robot (or other manipulator) along a collision-free path in the grid. The loads are labeled (i.e., unique) and must be stored in a given sequence, and later be retrieved in a different sequence -- an operational pattern that arises in logistics applications, such as last-mile distribution centers and shipyards. The objective is to minimize the load relocations to ensure efficient retrieval. A previous result guarantees a zero-relocation solution for known storage and retrieval sequences, even for storage at full capacity, provided that the side of the grid through which loads are stored/retrieved is at least 3 cells wide. However, in practice, the retrieval sequence can change after the storage phase. To address such uncertainty, this work investigates \emph{$k$-bounded perturbations} during retrieval, under which any two loads may depart out of order if they are originally at most $k$ positions apart. We prove that a $Θ(k)$ grid width is necessary and sufficient for eliminating relocations at maximum capacity. We also provide an efficient solver for computing a storage arrangement that is robust to such perturbations. To address the higher-uncertainty case where perturbations exceed $k$, a strategy is introduced to effectively minimize relocations. Extensive experiments show that, for $k$ up to half the grid width, the proposed storage-retrieval framework essentially eliminates relocations. For $k$ values up to the full grid width, relocations are reduced by $50\%+$.

Figures (8)

• Figure 1: Consider a $3 \times 3$ storage area $W\xspace$ accessible only from one side (bottom) that must store 9 loads arriving in the order $A\xspace = (4,1,7,6,3,2,9,8,5)$ and planned to depart in the order $D\xspace =(1,2,\ldots,9)$. Top row: a solution that avoids relocations if $D\xspace$ does not change. (a) The first three arriving loads, $4,1,7$, are stored. (b) Loads $6,3,2$ are stored. (c) Loads $9,8,5$ are stored. At this point, while the loads can depart according to $D\xspace$ (not shown), the actual retrieval sequence becomes $\tilde{D} = (2,1,3,5,4,7,6,9,8)$, a slight perturbation of $D\xspace$, requiring relocations: (d) Load 2 is blocked, so 1 is relocated outside $W\xspace$. (e) Load 2 is retrieved. (f) Load 1 is stored back until it is needed. (g) After $1,3,5,4$ are retrieved, load 7 is next but is blocked by 6, which is relocated within $W\xspace$. Bottom row: (h)–(j) Loads are stored according to $A\xspace$ but in a robust arrangement, as proposed in this work. (k)–(n) In this solution, loads can be retrieved without relocations, not only according to $D\xspace$, but also under $\tilde{D}$.
• Figure 2: Examples of the column bounds for $k = 1$. Upper bound (left): Consider a $2 \times 6$ grid with $A\xspace = (7, 3, 11, 1, 9, 4, 6, 12, 2, 10, 8, 5)$. Per \ref{['thm:UB']}, we partition $A\xspace$ into subsequences of odd and even loads: $A_0 = (7, 3, 11, 1, 9, 5)$ and $A_1 = (4, 6, 12, 2, 10, 8)$. We treat $A_0$ as a StoRMR instance on a $2 \times 3$ grid with $D = (1, 3, 5, 7, 9, 11)$ and use the solution to fill the leftmost 3 columns. Similarly, we treat $A_1$ as a separate instance for the rightmost 3 columns (shaded). The combined solution is a robust arrangement. Lower bound (right): Consider $A^R$ starting with 10, 3, 4. As $k = 1$, we must have 1 and 2 on the bottom row. Treating $A^R$ as a departure sequence to satisfy, 10 must also be on the bottom row. Next, to avoid placing 3 in the bottom row, we must store it in a cell adjacent to both 10 and 1. As no such cell exists, 3 must also be placed in the bottom row. Similarly, 4 must be adjacent to one of 1 and 2 and one of 10 and 3 to avoid the bottom row. Again, this is not possible, so 4 is also assigned to the bottom row. Thus, 5 columns are required.
• Figure 3: The storage algorithm for a ${3 \times 5}$ grid with $A^R = (11, 3, 15, 8, 2, 7, 13, 12, 1, 9, 10, 14, 6, 4, 5)$, $D=[15]$, and $k=2$. (a)(b) Snapshots showing partial arrangements as the algorithm runs. First column pair: 1 and 11 are placed per column initialization. Next, we set $x=4$ as the first load in $D$ that can be placed above 1. In the inner loop, $y = 3$ is discarded as it does not satisfy adjacencies for $D$; we proceed to $y = 15$, which is valid. The main loop then sets $x = 7$, but no matching valid $y$ is found until we reach $y = 7$, at which point 7 cannot be placed in $R$. Continue to $x = 8$, reaching the same conclusion. Then, for $x = 9$, a valid match is found with $y = 13$. (c)(d) Arrows showing valid adjacencies for departures and arrivals (treating arrivals in reverse).
• Figure 4: Example relocations: (a) Load $11$ is to be retrieved. $\pi$ is a straight downward path and $9$, $10$ are blockers (red). To relocate $9$, we compute the set of unblocked loads $U = \{5,6,8\}$ (green). Destination cell $c_3$ is not chosen because it would disconnect $10$ from empty cells and $c_2$ is also not chosen as placing $9$ there would block $5$. (b) $9$ is relocated to $c_1$ which keeps $6$ unblocked due to its adjacency to $5$. (c) $U$ is recomputed and $10$ is relocated to $c_3$, as it does not block any loads.
• Figure 5: Mean relocations $\pm$ st. dev. (top) and mean I/O row usage $\pm$ st. dev. (bottom) for varying grids and $k$ values, comparing storage and retrieval algorithms. The heatmaps are the same in the two leftmost columns since BaseR always uses the I/O row.

Theorems & Definitions (12)

• Definition 1
• Remark 1
• Definition 2
• Theorem 1
• Definition 3
• Proposition 1: (Robust adjacency conditions)
• proof
• Corollary 1
• Theorem 2: (Upper bound)
• proof