Picker Routing Problem Overview

• Picker Routing Problem is the challenge of determining the minimum-length tour among required warehouse storage locations, modeled as a structured Traveling Salesman Problem.
• Exact algorithms leverage dynamic programming and graph formulations to solve one- and two-block instances efficiently, while multi-block layouts are strongly NP-hard.
• Recent research integrates PRP with order batching, storage assignment, and learning-based methods to enhance routing efficiency in modern warehouse systems.

The Picker Routing Problem (PRP) is the problem of finding a minimum-length tour between a set of storage locations in a warehouse. In the standard rectangular warehouse setting, a picker starts and ends at a single depot, moves through parallel picking aisles and cross aisles, and must visit all required storage locations; formally, the PRP can be represented on a complete undirected graph $$\mathcal G^{PRP}=(\mathcal V^{PRP},\mathcal E^{PRP})$$ whose edge costs are shortest walking distances in the warehouse. In this sense, the PRP is a special case of the TSP on a highly structured metric induced by warehouse walk paths, and earlier literature also interprets it as a Steiner TSP on the warehouse graph (Prunet et al., 2023).

1. Formal model and warehouse representation

In the standard multi-block rectangular model, a warehouse consists of blocks $\mathcal K=\{1,\dots,K\}$ and aisles $\mathcal A=\{1,\dots,A\}$. For each block $k$ and aisle $a$, the locations are denoted $\mathcal L^{ka}=\{l_i^{ka},\,1\le i\le L\}$, and the set of all locations is their union. A key geometric convention is that a picker may retrieve items from either side of an aisle while traversing it, so opposite shelf positions that are equivalent in travel distance are aggregated into a single location. The layout is parameterized by $D^{loc}$, $D^{aisle}$, and $D^{block}$, and the depot $v_0$ is a single start/end point in the first cross aisle (Prunet et al., 2023).

The canonical objective is stated directly: “The objective of the PRP is to find a tour of minimum weight in $\mathcal K=\{1,\dots,K\}$0 that visits all vertices exactly once.” With $\mathcal K=\{1,\dots,K\}$1 the required storage locations, $\mathcal K=\{1,\dots,K\}$2, and edge weights $\mathcal K=\{1,\dots,K\}$3 equal to shortest walking distance, the model is

$\mathcal K=\{1,\dots,K\}$4

over tours $\mathcal K=\{1,\dots,K\}$5 in $\mathcal K=\{1,\dots,K\}$6 that visit every vertex in $\mathcal K=\{1,\dots,K\}$7 exactly once (Prunet et al., 2023).

A complementary exact viewpoint models the warehouse as a graph $\mathcal K=\{1,\dots,K\}$8, where $\mathcal K=\{1,\dots,K\}$9 is the intersection of aisle $\mathcal A=\{1,\dots,A\}$0 and cross-aisle $\mathcal A=\{1,\dots,A\}$1, while required product locations lie inside subaisles. In that formulation, routing can be expressed through a tour subgraph $\mathcal A=\{1,\dots,A\}$2 rather than directly through a sequence. Ratliff and Rosenthal’s characterization, as restated in later structural work, is that $\mathcal A=\{1,\dots,A\}$3 is a tour subgraph if and only if it contains all required locations, is connected, and every vertex in $\mathcal A=\{1,\dots,A\}$4 has even degree (Dunn et al., 1 Aug 2025). This Eulerian connected-subgraph perspective underlies much of the exact dynamic programming literature.

2. Tractability landscape and complexity status

The central complexity distinction in the literature is between single-block, small-block-count, and unrestricted multi-block warehouses. The current status is as follows (Prunet et al., 2023).

Warehouse setting	Complexity status	Representative statement
Single-block PRP (PRP-1)	Polynomially solvable	Ratliff–Rosenthal dynamic programming
Two-block PRP (PRP-2)	Polynomially solvable	Dynamic programming extensions
Multi-block PRP with bounded $\mathcal A=\{1,\dots,A\}$5	Fixed-parameter tractable in $\mathcal A=\{1,\dots,A\}$6	$\mathcal A=\{1,\dots,A\}$7
Multi-block PRP with unbounded number of blocks	Strongly NP-hard	Theorem 1

The decisive result is Theorem 1: “The Picker Routing Problem (PRP) is NP-hard in the strong sense” (Prunet et al., 2023). The proof resolves the open case of conventional multi-block rectangular warehouses with uniform spacings. The decision version asks whether a PRP instance admits a solution of value lower than or equal to a given quantity $\mathcal A=\{1,\dots,A\}$8. The reduction is from the Hamiltonian Cycle Problem on Grid Graphs (HCPGG), which the paper also shows to be NP-complete in the strong sense, even when restricted to connected grid graphs (Prunet et al., 2023).

The hardness construction is notably geometric. From a connected grid graph on an $\mathcal A=\{1,\dots,A\}$9 bounding rectangle, the reduction builds a warehouse with $k$0 blocks, $k$1 aisles, exactly one location per aisle in each block, and parameters

$k$2

with depot distance $k$3. The threshold is

$k$4

Every grid edge becomes a PRP edge of weight $k$5, whereas every other PRP edge either visits the depot or has weight at least $k$6 (Prunet et al., 2023). Because the construction keeps all numerical values polynomially bounded, the result is strong NP-hardness in the standard, highly structured warehouse model rather than in a relaxed irregular-distance variant.

This yields the modern dichotomy: single- and two-block conventional rectangular warehouses are polynomially solvable, whereas general multi-block conventional rectangular warehouses are strongly NP-hard (Prunet et al., 2023).

3. Exact algorithms and structural theory in rectangular warehouses

Despite the hardness result for unrestricted multi-block layouts, exact methods remain effective when the geometry is strongly constrained. In one- and two-block rectangular warehouses, classical dynamic programs work with a small number of local edge-configuration states. In the one-block setting, the standard parity/connectivity states are

$k$7

and the valid horizontal additions are

$k$8

These states encode degree status at the current top and bottom frontier vertices together with the number of connected components (Dunn et al., 1 Aug 2025).

A major recent structural refinement is that, in a minimal tour subgraph, horizontal structure determines vertical structure. After merging consecutive subaisles whose intermediate intersection vertices have no horizontal incident edges, Proposition 1 states: if all horizontal edges incident to the vertices of an aisle are known for a minimal tour subgraph, then the vertical edge configurations within that aisle are uniquely determined (Dunn et al., 1 Aug 2025). The proof proceeds by parity repair, connectivity, and minimum-length arguments. The vertical pattern must be one of a few minimal possibilities: a single vertical path for paired odd endpoints, a largest-gap through-configuration when both ends are usable, or a one-sided return when only one end is usable (Dunn et al., 1 Aug 2025).

Algorithmically, this removes an entire vertical-decision stage from the classic dynamic programs. For one-block warehouses, the number of stages drops from $k$9 to $a$0; for two-block warehouses, it drops from $a$1 to $a$2 (Dunn et al., 1 Aug 2025). A closely related single-block reformulation reaches the same conclusion from an aisle-to-aisle transition perspective: instead of alternating between vertical and horizontal stages, only aisle-to-aisle transitions are needed, reducing the number of stages from $a$3 to $a$4 (Dunn et al., 2024).

A second structural line concerns double traversals. In multi-cross-aisle warehouses, double traversals can still be necessary, but not all types are essential. The key theorem is that there exists a minimum-length tour subgraph that does not contain orthogonal double edges (Dunn et al., 23 Jan 2025). In warehouse terms, a subaisle need not be traversed twice merely to connect horizontal travel at both its top and bottom ends. Any necessary double edge in a minimal tour subgraph must be bridging and non-orthogonal, with one endpoint at a required point and the other endpoint at a required point and/or a vertex with horizontal incident edges (Dunn et al., 23 Jan 2025). This sharpens the characterization of feasible minimal tours and further explains why horizontal structure can determine vertical behavior.

4. Compact single-block formulations and modern warehouse variants

Single-block warehouses remain the domain in which the strongest compact exact formulations are known. A compact MIP for the standard SPRP in a rectangular single-block warehouse exploits two structural properties from Ratliff and Rosenthal: only four feasible cross-aisle connection configurations are needed between consecutive aisles, and subtour elimination can be replaced by parity and connectivity tracking along the aisle sequence (Goeke et al., 2019). This avoids classical subtour elimination constraints and yields a formulation whose size is tied to warehouse geometry rather than to a complete routing graph.

That same formulation extends to three settings described as important in modern e-commerce warehouses: scattered storage, decoupling of picker and cart, and multiple end depots (Goeke et al., 2019). In the scattered-storage extension, a SKU may be available at several pick positions, so the optimization decides both where to pick and how to route. In the decoupling extension, the picker may leave the cart temporarily and move faster without it, subject to a picker-alone carrying capacity. In the multiple-end-depot extension, the route starts at a fixed depot but may terminate at one of several end depots (Goeke et al., 2019).

The computational results are unusually strong for exact single-picker routing. On the standard SPRP benchmark, the formulation solved all 900 Scholz instances, with average runtime $a$5 s versus $a$6 s for the compared SHSW formulation, and it solved instances with 1000 aisles, 1000 possible positions per aisle, and 1000 required positions in roughly two minutes on average (Goeke et al., 2019). For the modern variants, the same framework solved all tested scattered-storage, picker/cart-decoupling, and multiple-end-depot instances to optimality, with low computational effort (Goeke et al., 2019).

The managerial conclusions are equally specific. Decoupling picker and cart produced average savings ranging from $a$7 to $a$8, depending on picker-alone capacity and speed factor, whereas multiple end depots produced only $a$9, $\mathcal L^{ka}=\{l_i^{ka},\,1\le i\le L\}$0, and $\mathcal L^{ka}=\{l_i^{ka},\,1\le i\le L\}$1 average savings for end-depot densities $\mathcal L^{ka}=\{l_i^{ka},\,1\le i\le L\}$2 in a single-block warehouse (Goeke et al., 2019). This suggests that mobility-mode flexibility changes single-block routing more profoundly than simply enlarging the set of allowed route termini.

5. Integrated optimization: batching, storage assignment, sequencing, and AGV-assisted routing

In applied warehouse optimization, PRP is frequently embedded in a larger decision problem. The integrated joint order batching and picker routing problem (JOBPRP) treats routing as a warehouse-specific Steiner TSP subproblem inside a batch-assignment model. A branch-and-cut formulation with warehouse-structured valid inequalities solved instances involving up to 20 orders to proven optimality, while instances involving up to 5000 orders were handled by heuristic batching with optimal routing (Valle et al., 2017). The most effective valid inequalities were warehouse-specific, especially aisle cuts and subaisle cuts, which greatly improved computational results (Valle et al., 2017).

Subsequent exact work reconstructed the connectivity constraints on an auxiliary graph of artificial locations and introduced subaisle cuts together with two equivalent improved formulations: a reduced-graph non-compact model and a compact multicommodity flow model. The compact formulation $\mathcal L^{ka}=\{l_i^{ka},\,1\le i\le L\}$3 has the same LP relaxation as the reduced-graph formulation $\mathcal L^{ka}=\{l_i^{ka},\,1\le i\le L\}$4 after projection, while remaining polynomially sized (Kai et al., 2022). This line of work shows that the geometry of subaisles and aisle ends can be used not only in dynamic programming, but also in branch-and-cut formulations for integrated batching-and-routing.

Approximation-based integration has also been influential. For order batching with routing-aware cost estimation in rectangular warehouses, one model uses aisle-edge selection, first/last aisle indicators, west/east correction terms for higher blocks, and parity corrections. For single-block no-reversal routing, the approximation is exact: if $\mathcal L^{ka}=\{l_i^{ka},\,1\le i\le L\}$5 is the optimal approximation value, $\mathcal L^{ka}=\{l_i^{ka},\,1\le i\le L\}$6 the routed distance after exact routing of batches, and $\mathcal L^{ka}=\{l_i^{ka},\,1\le i\le L\}$7 the optimal integrated batching-routing value, then

$\mathcal L^{ka}=\{l_i^{ka},\,1\le i\le L\}$8

(Valle et al., 2018). For multiple-block no-reversal, the approximation is a lower bound and remains very tight in experiments (Valle et al., 2018).

Routing has also been integrated with storage assignment. In the Storage Location Assignment and Picker Routing Problem (SLAPRP), storage decisions become operational rather than purely tactical. A generic Branch-Cut-and-Price framework based on a route-based Dantzig–Wolfe formulation and strengthened by non-robust SL inequalities solved 57 of 72 integrated optimal-routing benchmark instances, compared with 40 solved by CPLEX on the authors’ compact model and 0 by CPLEX on the Silva et al. model (Prunet et al., 2024). The exact routing component is handled by ESPPRC pricing, and the framework covers optimal routing together with return, S-shape, midpoint, and largest-gap policies (Prunet et al., 2024).

Additional integrated extensions place PRP inside even broader warehouse-control models. In AGV-assisted mixed-shelves warehouses, the integrated order batching and routing problem is formulated as an extended multi-depot vehicle routing problem, and a variable neighborhood search reports savings up to $\mathcal L^{ka}=\{l_i^{ka},\,1\le i\le L\}$9 in AGV driving distances under mixed-shelves storage relative to dedicated storage (Xie et al., 2021). In the JOBPRSP-D, picker routing is embedded in a route-column model with deadlines, where the sequencing layer is reformulated as a bin-packing-style route assignment problem and solved by a column-generation heuristic that provides valid lower and upper bounds; instances with up to 18 orders are solved to proven optimality, and instances with 100 orders receive high-quality bounds (Briant et al., 2023).

6. Learning-based and dynamic formulations

Recent learning-based work treats PRP not only as a static shortest-tour problem but as a sequential decision process over warehouse-specific route states. For a rectangular two-cross-aisle warehouse, one deep reinforcement learning approach reformulates single-picker routing as an MDP over partial tour subgraphs. The action space is split into vertical actions

$D^{loc}$0

and horizontal actions

$D^{loc}$1

while the state space is the same seven parity/connectivity classes used in exact dynamic programming (Dunn et al., 2024). On 30 problem classes with 5 to 30 aisles and 30 to 90 picks, the standard model achieved average optimality gaps about $D^{loc}$2 to $D^{loc}$3 across aisle counts and about $D^{loc}$4 to $D^{loc}$5 across pick-list sizes, outperforming S-shape, Return, Composite, and Largest Gap heuristics (Dunn et al., 2024). A simplified version masks out the gap action to reduce perceived route complexity (Dunn et al., 2024).

A different extension is the Dynamic In-store Picker Routing Problem (diPRP), which moves the setting from warehouses to shared retail stores with stochastic customer traffic. The problem is formulated as a finite-horizon MDP with state

$D^{loc}$6

action set given by adjacent store-graph nodes, and reward

$D^{loc}$7

balancing travel, same-node encounters, nearby-customer encounters, and pick completion (Neves-Moreira et al., 2023). In a real retailer case study, the learned policy reduced customer encounters by more than $D^{loc}$8 relative to shortest-path-focused policies while preserving much of the picking productivity (Neves-Moreira et al., 2023). This shifts the routing objective from pure travel minimization to a stochastic multi-objective control criterion.

The most recent multi-agent extension is the min-max Mixed-Shelves Picker Routing Problem (MSPRP), in which several pickers must jointly fulfill demand while minimizing the maximum route length across pickers. A hierarchical and parallel decoding architecture, MAHAM, models a cooperative MMDP with composite actions $D^{loc}$9 for location and SKU, split orders and split deliveries, and picker capacity $D^{aisle}$0 per tour (Luttmann et al., 14 Feb 2025). The reward is

$D^{aisle}$1

Empirically, MAHAM achieved $D^{aisle}$2 gap to best-known solutions on MSPRP40 and $D^{aisle}$3 gap in all tested MSPRP50 settings, while also improving inference speed relative to autoregressive and conflict-prone parallel baselines (Luttmann et al., 14 Feb 2025). This suggests that, once mixed shelves and multiple pickers are introduced, PRP becomes a joint routing-and-allocation problem in which coordinated decoding is as important as route construction itself.