Order Retrieval in Compact Storage Systems

TL;DR

The paper tackles energy-efficient retrieval in compact storage systems by formalizing the Side-Access Compact Retrieval Problem (SACRP) for side-access storage slices. It develops dense and sparse encodings, derives structural properties of feasible retrieval batches, and proves SACRP to be strongly NP-complete. To solve SACRP, it introduces a MIP formulation and a dynamic programming algorithm, supplemented by a simple greedy heuristic; experiments reveal that DP consistently delivers optimal solutions for small to medium instances, while the MIP provides robust near-optimal performance with broader applicability, and the greedy method offers near-instant results with reasonable quality. The findings demonstrate that exploiting problem structure yields tractable, exact solutions for practical deployment in energy-conscious compact storage, and they propose future work on generalizing the model to multi-side lifts and benchmarking against industry systems like AutoStore.

Abstract

Growing demand for sustainable logistics and higher space utilization, driven by e-commerce and urbanization, increases the need for storage systems that are both energy- and space-efficient. Compact storage systems aim to maximize space utilization in limited storage areas and are therefore particularly suited in densely-populated urban areas where space is scarce. In this paper, we examine a recently introduced compact storage system in which uniformly shaped bins are stacked directly on top of each other, eliminating the need for aisles used to handle materials. Target bins are retrieved in a fully automated process by first lifting all other bins that block access and then accessing the target bin from the side of the system by a dedicated robot. Consequently, retrieving a bin can require substantial lifting effort, and thus energy. However, this energy can be reduced through smart retrieval strategies. From an operational perspective, we investigate how retrievals can be optimized with respect to energy consumption. We model the retrieval problem within a mathematical framework. We show that the problem is strongly NP-complete and derive structural insights. Building on these insights, we propose two exact methods: a mixed-integer programming (MIP) formulation and a dynamic programming algorithm, along with a simple, practitioner-oriented greedy algorithm that yields near-instant solutions. Numerical experiments reveal that dynamic programming consistently outperforms state-of-the-art MIP solvers in small to medium sized instances, while the greedy algorithm delivers satisfactory performance, especially when exact methods become computationally impractical.

Figures (4)

• Figure 1: Illustration of the entire retrieval process that is described in the example. Targets are highlighted in red. The total energy expenditure of the retrieval process is $4$. a) shows the initial configuration before any target is retrieved. b) depicts the configuration in the beginning of cycle 1, in which targets $b_1$, $b_3$, $b_4$, $b_5$ and $b_6$ are retrieved in a single cycle with an energy expenditure of $3$. In c) first target of cycle 1 ($b_1$) is retrieved. d) shows the configuration in the beginning of cycle 2, in which target $b_2$ is retrieved with an energy expenditure of $1$.
• Figure 2: Illustration of four examples of weakly triangular batches $Y$. Targets included in $Y$ are highlighted in red.
• Figure 3: Illustration of the construction for the following instance $\mathcal{I}_P$: $S=\{2,2,2,2,4,4\}$, thus $m=2$, and $T=8$. There is solution to $\mathcal{I}_P$ with $A_i = \{2,2,4\}$ and $A_2 = \{2,2,4\}$. The red ULs corresponds to the set $X$, the green UL corresponds to the set $Y$ and the blue ULs corresponds to the set $Z$.
• Figure 4: Model 1: Mathematical programming model for the SACRP.

Theorems & Definitions (25)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof