Digital twin based sorting optimization for parcel distribution centers in logistics networks

Abstract

Parcel distribution centers in logistics networks aim to sort inbound parcels for downstream destinations through parcel-sorting systems, where parcels are diverted to assigned grids and packed into bins for outbound truck deliveries. Considering dynamic parcel movements, random packing behaviors, and conveyor congestion, maximizing sorting throughput by assigning grids to destinations presents significant computational challenges. This work establishes a high-fidelity parcel-sorting digital-twin system that models real-time interactions among parcels, sorters, and packers, and simulates throughput performance under specific sorting plans. To address the computational challenges and real-world operational constraints, a digital twin-based structured Monte Carlo tree search optimization framework, combining integer nonlinear programming and geographic destination graph networks, is proposed to optimize sorting plans. The proposed digital twin and optimization framework have been deployed in 146 parcel distribution centers (approximately 30% of SF Express parcel distribution centers in China) and demonstrated their effectiveness through both numerical and field experiments: the proposed framework increases average and peak sorting throughputs by 10.56% and 7.72%, respectively, and reduces recirculated parcels by 60.81%, compared to field-implemented method. Furthermore, it outperforms the basic digital twin-based Monte Carlo tree search baseline by 6.77% and 4.99% in average and peak throughputs, respectively, and reduces recirculated parcels by 54.57%.

Authors demonstrate a high-fidelity digital twin system for parcel sorting, integrating real parcel-level data and device mechanisms, and proposes a structured Monte Carlo tree search framework that combines integer nonlinear programming and geographic graph modules to optimize grid-to-destination assignments, improving throughput and reducing congestion in large-scale distribution centers.

Introduction

The logistics network is one of the most crucial and indispensable infrastructures for daily life and economic development, connecting industrial production, distribution, and consumption¹. In logistics networks, parcel distribution centers (PDCs) serve as the most pivotal hubs² to facilitate the consolidation, sorting, and distribution of vast volumes of parcels³. Notably, PDCs streamline the flow of parcels by efficiently routing them to their next destinations, ensuring continuous and timely parcel delivery^4,5, reducing transportation costs, and achieving economies of scale in logistics networks⁶.

Specifically, in PDCs, parcels are processed by three sequential procedures: first, parcels are unloaded from the inbound trucks⁷, then fed to the parcel-sorting system (PSS) and routed to the grids assigned to specific parcel destinations^8–11, and finally packed into bins for subsequent outbound truck deliveries^12–14. On the one hand, the random arrivals of the inbound trucks introduce stochasticity to parcel arrivals in PSS, posing great challenges for parcel-sorting. On the other hand, parcel-sorting in PSS significantly affects the timely departures of outbound trucks, as failure to complete parcel-sorting within designated time windows causes delayed outbound truck departures that propagate operational bottlenecks to downstream delivery procedures within the logistics network, resulting in substantial economic losses and deteriorated customer satisfaction. Therefore, improving parcel-sorting efficiency in PSS is the most important task to ensure the reliability and timeliness of parcel deliveries in PDCs.

The cross-belt sorter system (CBSS)¹⁵ is the predominant PSS used in PDCs, designed to efficiently sort each inbound parcel to one or more predefined grids according to its downstream destination. As depicted in Fig. 1, a CBSS comprises two sorting segments that share one central cross-belt conveyor. Each segment consists of induction stations, parcel scanners, grids, and label printers. Specifically, each parcel randomly arrives at an induction station and is conveyed onto the first passing empty tray on the cross-belt conveyor. The parcel moves along the conveyor and subsequently passes through a parcel scanner, which scans the barcode on the parcel and retrieves its destination. Then, the tray carrying this parcel is informed of the grid index for parcel unloading according to the sorting plan, i.e., the grid-to-destination assignments. When the tray reaches the designated grid, the parcel is unloaded if the grid’s bin is not full. Once the bin is full, the grid is locked, and the bin-packing process is initiated. During the bin-packing process, any parcel that reaches the locked grid will continue moving along the conveyor and attempt to be unloaded in the subsequent conveyor cycles. If a parcel is still not unloaded after the maximum allowable number of conveyor cycles, it is redirected to the “rejection grid” that recalls all these timed-out parcels. When the bin-packing process is initiated, a label with detailed dispatch information is generated by the label printer. Packers first retrieve and affix the printed label to the corresponding bin, and replace it with an empty bin.

Fig. 1. Layout of the cross-belt sorter system (CBSS) in parcel distribution centers (PDCs).

The CBSS consists of the following key sorting devices: 1. the induction station receives randomly arrived parcels and places each parcel onto the first empty tray passing by; 2. the cross-belt conveyor is equipped with hundreds of trays, each of which carries one parcel at a time. Trays move cyclically with the conveyor and unload parcels into the grids; 3. the parcel scanner scans the barcode on the parcel passing through; 4. the grid receives parcels into the bin with a specific downstream destination, and is locked once the bin reaches its maximum capacity; 5. label printer generates dispatch labels for the bins that reach their maximum capacity.

CBSS sorting optimization aims to maximize the sorting throughput, i.e., the number of sorted parcels per unit time, by assigning the grids to the parcel’s destinations. Specifically, for the parcel-sorting procedure, PDCs typically adopt a two-stage framework^10,16,17 to handle the increasing number of parcel destinations in recent years. In the first stage, PDCs perform destination clustering to aggregate the vast number of parcel final destinations into a manageable set of destination groups based on empirical parcel data, geographic proximity, and operational requirements. In the second stage, which is the focus of this paper, the sorting optimization determines the optimal assignment of sorting grids to these pre-clustered destination groups. To maintain consistency, the term “destination” in the paper refers specifically to the pre-clustered destination groups rather than individual delivery endpoints.

Given its central role in PDC operations, the sorting optimization problem considered in this work is important in two respects. First, it captures a real-world planning problem faced by large-scale CBSS in PDCs, where the grid-to-destination assignments must be determined in daily industrial operations. Second, high-quality solutions to this problem, i.e., well-designed mappings from grids to destinations, can deliver significant improvements in sorting throughput and other key performance indicators, which translate directly into economic benefits for logistics companies. Against this background, the sorting optimization problem has drawn considerable attention from both the academic and industrial spheres in recent years. The authors in ref. ¹⁸ developed a differentiated probabilistic queuing policy by assuming that customer order (a group of parcels belonging to the same destination) arrivals follow a Poisson distribution, and leveraged an alternating minimization algorithm to reduce the weighted average latency for the customer orders at PDCs. Similarly, the authors in refs. ^19,20 developed multi-class block-and-recirculate queueing models by assuming that customer order releases follow exponential distribution, and leveraged approximate aggregation methods to estimate and maximize the system throughput. However, existing works predominantly rely on idealized assumptions on the distributions of parcel arrivals¹⁸ and overly simplified representations for PSS (e.g., simplified tote (tray) release logic^19,20 and infinite queue capacities²¹), and thus are limited in capturing the precise behavior of complex, dynamic, and large-scale real-world parcel-sorting scenarios²².

To address these real-world limitations, simulation-based optimization has been widely adopted²³. By abstracting the essential dynamics of the system and replicating them in software, it enables the evaluation of existing solutions and the search for improved or even optimal ones without direct interaction with the physical system, thereby substantially reducing both the time and monetary expenditures. Such methods have been extensively applied in PDCs. For instance, the authors of^24–26 simulate PSS layout models to identify optimal configurations for general PSSs, including decisions on the number of induction stations and cross-belt conveyors and on the placement of additional conveyor routes. Software platforms, such as Plant Simulation²⁷, are widely utilized in such studies, with real-world PDCs (e.g., PDCs at Friedewald) serving as prototype systems for simulation-based layout optimization. In^28–30, truck scheduling simulation models were developed to solve the temporal mapping between inbound and outbound trucks and docking doors in PDCs, thereby facilitating the efficient arrival and departure of parcels. However, the above simulation models predominantly rely on fixed operational parameters and idealized assumptions, including parametric and stationary parcel arrivals, homogeneous or coarse-grained parcel characteristics that neglect the variability in parcel destinations and real-world handling constraints, and simplified routing rules without simulating congestion and equipment failures in real-world operations. Consequently, these models cannot faithfully reproduce the precise temporal and spatial variability of practical parcel-sorting operations, limiting their effectiveness and generalizability in solving practical sorting optimization problems.

To tackle the aforementioned challenges, we turn to the digital twin (DT) approach, which facilitates real-time seamless interactions with the physical environment^31–33. DT approaches have been increasingly adopted in logistics, with prior studies exploring DT deployments for various logistics systems, such as warehouse layout design³⁴, logistics transportation systems³⁵, and cold chain logistics platforms³⁶. However, these DT systems predominantly focus on architectural or platform-level designs that synchronize the physical and digital worlds to support real-time management or ensure safety guarantees. In contrast, our approach leverages a different insight by utilizing DT to achieve high-fidelity, parcel-level simulations of the sorting process, which allows us to address the sorting optimization problems within DT, without the need for costly real-world trial-and-error.

Specifically, compared to the conventional simulation-based approaches, our established parcel-sorting DT system (PSDTS) incorporates tens of thousands of real-world parcel data, including arrival timestamps with millisecond resolution, spatial coordinates in the CBSS, and downstream destinations, to replicate the exact parcel arrival time and position in the CBSS. Moreover, real-world parcel-sorting data are systematically sensed and stored via the established Industrial Internet of Things (IIoT) platform, enabling the PSDTS to precisely replicate the sorting operations and thus ensure high-fidelity parcel-level simulation of the considered CBSS. Leveraging these data, we adopt a joint model and data-driven approach that embeds the underlying physical mechanisms to construct DT models of key sorting devices in the CBSS. The established PSDTS integrates multi-scale simulations, including the detailed trajectory of each parcel, operational status of each sorting device, and throughput metrics of the CBSS, to comprehensively simulate the sorting performance of the CBSS with one particular sorting plan. It has been implemented in 146 PDCs with fidelity over 90%, accounting for approximately 30% of all SF Express PDCs in China, enabling data-driven design and rigorous validation of advanced sorting optimization algorithms.

Building on the developed PSDTS, we propose a DT-based structured Monte Carlo tree search (DT-SMCTS) framework to efficiently derive the sorting plan for CBSS while accommodating some practical operational constraints. Specifically, the sorting throughput maximization problem is formulated as a finite-horizon Markov decision process (MDP), which suffers from a vast solution space and computationally intensive PSDTS evaluations. To address these challenges, we adopt MCTS as the algorithmic backbone, which is widely adopted for solving large-scale combinatorial optimization problems with large branching factors, such as large language model reasoning and planning^37–40, robotics planning^41–43, multi-objective drug design^44,45, and retrosynthesis^46–50. Given the high complexity and strong realism requirements of CBSS, we embed a DT-based integer nonlinear programming (DT-INLP) module and a geographic destination graph network (GDGN) module to improve the expansion step of the MCTS, which significantly improves the convergence speed and sorting performance. We demonstrate the operational feasibility and effectiveness through both the DT numerical results and field experiments in real-world PDCs of SF Express. The proposed DT-SMCTS framework increases the average and peak sorting throughputs by 10.56% and 7.72%, respectively, and reduces the number of sorted parcels requiring multiple conveyor cycles by 60.81%, compared to the field-implemented method. Furthermore, it outperforms the basic DT-based MCTS baseline by 6.77% and 4.99% in average and peak throughputs, respectively, and reduces the recirculated parcels by 54.57%.

Results and discussion

IIoT within CBSS

As the foundational enabler of the PSDTS, the IIoT platform is implemented in CBSS to sense the parcel-sorting and bin-packing data and upload them to the cloud storage after they are acquired. Notably, the data acquisition and upload occur concurrently with the parcel-sorting operations. After each sorting shift and once all data from that shift have been uploaded to the cloud, the proposed DT-SMCTS framework retrieves these data and invokes the PSDTS to update the sorting plan. The updated plans are then deployed to the CBSS before the start of the upcoming sorting shifts. Consequently, both the plan update based on the acquired data and the deployment of the updated plan are performed in the interval between two successive sorting shifts, with one shift scheduled per day and lasting several hours (typically around 4 h). As a result, the sorting plan is always updated using data from completed parcel-sorting shifts and is applied only to subsequent sorting shifts.

Data acquisition and hardware configuration of the IIoT platform: specifically, the IIoT platform acquires both the parcel-sorting and bin-packing data. For a sorting shift containing N parcels, we use $w_n\triangleq \left\{T_n^{\mathrm{del}},K_n,S_n,E_n,V_n\right\}$ to record the parcel-sorting data associated with the n-th parcel: $T_n^{\mathrm{del}}$ denotes the exact delivery timestamp of the n-th parcel; K_n denotes the index of induction station receiving the n-th parcel, and $T_n^{\mathrm{del}}$ is accurately identified using the light curtain at the induction station; S_n denotes the index of parcel scanner, which inspects the n-th parcel’s barcode using a six-sided camera tunnel; E_n is the scanning error code, and it is logged as “error” if the barcode of the n-th parcel cannot be identified; $V_n\in \mathcal{D}$ represents the destination of the n-th parcel and is retrieved during the barcode scanning, with $\mathcal{D}=\left\{1,2,\cdots ,D\right\}$ being the destination set and D denoting the total number of destinations in the sorting shift. All parcel-related data are aggregated into the set W ≜ {w₁, w₂,…,w_N}.

Additionally, for a CBSS containing X grids, we use $g_x\triangleq \left\{{\left\{T_{x,l}^{\mathrm{pack}}\right\}}_{l=1}^{L_x},{\left\{C_{x,l}^{\mathrm{pack}}\right\}}_{l=1}^{L_x}\right\}$ to record the bin-packing data associated with the x-th grid: L_x is the total number of packed bins of the x-th grid throughout the sorting shift; $T_{x,l}^{\mathrm{pack}}$ and $C_{x,l}^{\mathrm{pack}}$ represent the overall bin-packing time and the number of collected parcels at the x-th grid during the l-th bin-packing process, respectively, which are measured based on the detection results from the light curtain at the bottom of the grid’s chute and the photoelectric sensor at the top of the grid’s bin. All the bin-packing data are aggregated into the set G ≜ {g₁, g₂,..., g_X}.

During the data acquisition, the adopted sensors in the IIoT platform, including light curtains, six-sided camera tunnels, and photoelectric sensors, are industrial-grade and ensure a millisecond-level sensing latency. These sensors perform data cleaning through built-in signal filtering and denoising to suppress transient noise and unintended sensor triggering, thereby ensuring that the acquired data are clean and reliable. Moreover, when a required data record cannot be successfully acquired, the corresponding sensor automatically triggers its predefined reacquisition procedure and records an error log for subsequent inspection.

Data-transmission protocols and latencies: the acquired parcel-sorting and bin-packing datasets, i.e., W and G, are immediately transmitted to the edge gateway of the IIoT platform using the MQTT⁵¹, HTTP⁵², and CoAP⁵³ protocols, depending on sensor configurations. At the edge gateway, the acquired data are further cleaned through automated consistency checks. In particular, the gateway verifies whether $T_n^{\mathrm{del}}$ is temporally consistent with the timestamps of adjacent parcel records. Duplicate parcel records and records that clearly violate the chronological order of the parcel-sorting process are removed. The cleaned data are subsequently uploaded to cloud storage via the HTTPS⁵⁴ protocol, resulting in an overall sensing-to-cloud latency on the order of several seconds. Once all acquired data are available in the cloud, the DT-SMCTS operates on the PSDTS to perform the sorting optimization. Importantly, this second-scale sensing-to-cloud latency does not affect either the sorting-plan update within the proposed DT-SMCTS framework or the subsequent deployment of the updated plan, as these steps are typically executed only once every 3 days, with nearly 1 day of available time in each update cycle.

DT of CBSS

As the simulation and evaluation backend for the DT-SMCTS, the PSDTS is designed to mimic the real-world parcel-sorting shift as faithfully as possible. Specifically, during each parcel-sorting shift, the IIoT platform identifies parcels, captures the associated parcel-sorting and bin-packing data, and uploads these data to cloud storage via the HTTPS⁵⁴ protocol. The DT-SMCTS framework subsequently retrieves the stored data from the cloud and invokes the PSDTS to compute an updated sorting plan. In each update cycle, DT-SMCTS iteratively generates candidate sorting plans based on the stored data and evaluates each candidate in the PSDTS through simulation; the candidate with the best simulated performance is selected as the updated sorting plan. This entire updating process is executed in an offline manner, i.e., during the intervals between successive parcel-sorting shifts. As a result, the PSDTS and the IIoT platform are functionally decoupled: the former is a software-based system that operates offline for modeling, simulation, and optimization, whereas the latter is primarily hardware-based and operates online together with the physical parcel-sorting shift. Notably, the online and offline phases of CBSS correspond, respectively, to the data-acquisition and plan-update phases of the DT-SMCTS framework. Accordingly, DT-SMCTS operates as a macroscopic online framework, in which data from the most recent 3 days are continually collected and leveraged to update the sorting plan.

As depicted in Fig. 2, the proposed PSDTS consists of four modules: the virtual parcel generation module leverages practical parcel-sorting data to generate virtual parcels; the CBSS mechanism module simulates the induction station, cross-belt conveyor, parcel scanner, and grid based on their respective operational mechanisms; the sorting simulation module simulates the entire life cycles of virtual parcels with the CBSS mechanism module and computes the sorting performance metrics under the given sorting plan, $\mathrm{\Pi }:\mathcal{X}\to \mathcal{D}$, which is defined as the mapping from the grid set $\mathcal{X}$ to the destination set $\mathcal{D}$; finally, the visualization platform provides a visual interface for comprehensive performance evaluation.

Fig. 2. Overview of the proposed parcel-sorting digital twin system (PSDTS) framework.

A The virtual parcel generation module takes the practical parcel-sorting dataset W as input and uses the extracted parcel attributes to generate the virtual parcels. B The cross-belt sorter system (CBSS) mechanism module replicates the operational mechanisms of the key sorting devices using the practical bin-packing dataset G and device-specific operational mechanisms. Specifically, in practical bin-packing dataset G, $T_{x,l}^{\mathrm{pack}}$ and $C_{x,l}^{\mathrm{pack}}$ denote the overall bin-packing time and the number of parcels collected at the x-th grid during the l-th bin-packing process, respectively. C The sorting simulation module evaluates a sorting plan Π, defined as the mapping from grids to destinations, by simulating the life cycles of all virtual parcels with the CBSS mechanism module and concurrently computing sorting performance metrics. D The visualization platform provides comprehensive visualization of the parcel-sorting performance.

Virtual parcel generation module

The virtual parcel generation module takes the parcel-sorting data from dataset W as its input and uses the extracted attributes, i.e., delivery time $T_n^{\mathrm{del}}$, induction station K_n, parcel scanner S_n, and scanning error E_n (if a scanning failure occurred), to generate the n-th virtual parcel, ensuring that it precisely replicates the arrival status and the subsequent sorting process of the n-th real-world parcel.

CBSS mechanism module

The CBSS mechanism module replicates four key sorting components of the CBSS based on their operational mechanisms and practical data.

• Virtual induction station module: in the physical CBSS, K induction stations are symmetrically located in each sorting segment. When the induction station K_n ∈ {1, 2,..., K} receives the n-th parcel, it records the current timestamp $T_n^{\mathrm{del}}$, and then conveys this parcel onto the first passing empty tray on the cross-belt conveyor. Correspondingly, the virtual induction station module in the PSDTS simulates K virtual induction stations to replicate their operational mechanisms by performing two functions. First, the virtual induction station K_n receives the n-th virtual parcel at time $T_n^{\mathrm{del}}$. Then, it conveys the n-th virtual parcel onto the first passing empty tray on the virtual cross-belt conveyor.

• Virtual cross-belt conveyor module: the cross-belt conveyor in the physical CBSS is equipped with hundreds of trays, each capable of carrying one parcel at a time. These trays move cyclically along with the conveyor, loading parcels from induction stations, conveying them to the parcel scanners, and unloading parcels at the grids according to the sorting plan. Correspondingly, the virtual cross-belt conveyor module in PSDTS replicates this operational mechanism by simulating a cyclically moving virtual cross-belt conveyor and a set of virtual trays. Specifically, the virtual cross-belt conveyor can be accelerated proportionally to shorten the simulation time. Each virtual tray moves with the virtual conveyor and is designed to perform three functions. First, if the virtual tray is empty, it loads a virtual parcel when passing the virtual induction stations. Second, it unloads the virtual parcel when it reaches the unlocked designated grid. Finally, it records the number of conveyor cycles that the carried parcel has traveled and unloads the parcel into the virtual rejection grid once the maximum allowable number of conveyor cycles is reached.

• Virtual parcel scanner module: in the physical CBSS, each sorting segment is equipped with one parcel scanner. As the n-th parcel moves along the cross-belt conveyor, it passes through a parcel scanner S_n ∈ {1, 2}, which scans the barcode on the n-th parcel to retrieve its destination V_n. If the scan is successful, the tray carrying the parcel is instantly informed of the grid index corresponding to its destination for parcel unloading; if the scan fails due to improper parcel orientation or a damaged barcode, an error code E_n is recorded. Correspondingly, the virtual parcel scanner module simulates two virtual parcel scanners, each of which is designed to perform two functions. First, when the n-th virtual parcel passes through, the virtual parcel scanner S_n reads its destination V_n, determines its designated grid according to the sorting plan Π, and notifies its virtual tray of the designated grid. Second, the virtual parcel scanner S_n checks whether a scanning error code E_n exists: if no error code is detected, the virtual parcel proceeds to its designated grid; if an error code is detected, the designated grid of the n-th virtual parcel is changed to the rejection grid, where it will be unloaded after the maximum number of conveyor cycles is reached.

• Virtual grid module: each grid in the physical CBSS is connected to a chute that directs the passing parcels from the tray to a bin placed below. When the grid’s bin is not full, all passing parcels with the assigned destination are unloaded; once the bin is full, the grid is locked and initiates the bin-packing process. The overall bin-packing time $T_{x,l}^{\mathrm{pack}}$ and the number of collected parcels $C_{x,l}^{\mathrm{pack}}$ at the x-th grid during the l-th bin-packing process are recorded. Notably, the overall bin-packing time $T_{x,l}^{\mathrm{pack}}$ is composed of two parts: a fixed $T_{x,l}^{\mathrm{base}}$ and a variable $T_{x,l}^{\mathrm{walk}}$, i.e., $T_{x,l}^{\mathrm{pack}}=T_{x,l}^{\mathrm{base}}+T_{x,l}^{\mathrm{walk}}$. Here, $T_{x,l}^{\mathrm{base}}$ accounts for the base packing time spent on replacing the bin, and $T_{x,l}^{\mathrm{walk}}$ is the label-fetching time for the packers, which is determined by the distance between the grid and the nearest label printer, and thus varies among the grids. Correspondingly, the virtual grid module simulates X virtual grids to replicate the parcel unloading and bin-packing process. Specifically, at each simulation time step, the virtual grids detect whether their bins have reached the maximum capacity. If so, the corresponding virtual grid transits to the locked status and initiates the bin-packing process. The maximum capacity and overall bin-packing time of the x-th virtual grid’s bin, denoted as ${\overline{C}}_x^{\mathrm{pack}}$ and ${\overline{T}}_x^{\mathrm{pack}}$, are modeled as the average number of the collected parcels $C_{x,l}^{\mathrm{pack}}$ and the average bin-packing time $T_{x,l}^{\mathrm{pack}}$ over all bin-packing processes, computed by using the grid-related dataset G. It is noted that the virtual rejection grid is modeled to remain unlocked throughout the sorting shift.

Sorting simulation module

This module takes sorting plan Π as the input and assesses its performance with the CBSS mechanism module by simulating the life cycles of all virtual parcels generated by the virtual parcel generation module at simulation time steps. Notably, we adopt a discrete event simulation approach for the PSDTS since the parcel-sorting shift itself is inherently characterized by discrete operational events (e.g., parcel arrivals, tray loading and unloading, scanning, and bin packing), and the system state remains piecewise constant between two neighboring events. Each simulation time step, denoted as Δt, corresponds to the real-world time required for a tray to advance one position on the conveyor of the physical CBSS. At each simulation time step, the sorting simulation module increments its virtual clock by Δt and faithfully replicates the life cycle of each virtual parcel in the sorting shift. The life cycle of the n-th virtual parcel begins with the virtual tray allocation by the virtual induction station K_n when the virtual clock reaches $T_n^{\mathrm{del}}$. It then proceeds to the determination of its designated grid at the specific virtual parcel scanner S_n, based on its destination V_n, scanning error code E_n (if it exists), and sorting plan Π. Finally, it is unloaded at its designated virtual grid.

Concurrently, the sorting simulation module computes the performance metrics, e.g., instant throughput, average throughput, and peak throughput, with respect to the real-world minute t ∈ {1, 2, ..., T_end}, where T_end denotes the real-world terminating time of the sorting shift. Instant throughput C_Π(t) quantifies the number of sorted parcels per minute as a moving average over a rolling window of the preceding T minutes, i.e.,

$$\begin{array}{c}\hfill C_{\mathrm{\Pi }}\left(t\right)\triangleq \frac{1}{t-\max \left\{t-T,0\right\}}{\sum }_{t_0=\max \left\{t-T,0\right\}+1}^t{N}_{t_0},t=1,...,T_{\mathrm{end}},\end{array}$$ 1

where $N_{t_0}$ represents the number of sorted parcels during the t₀-th minute. Based on the instant throughput defined in Eq. 1, the PSDTS computes average throughput C_avg(Π) and peak throughput C_peak(Π), which are defined as the average number of sorted parcels and the maximum instant throughput over the entire sorting shift, respectively, i.e.,

$$C_{\mathrm{avg}}\left(\mathrm{\Pi }\right)\triangleq \frac{1}{T_{\mathrm{end}}}{\sum }_{t_0=1}^{T_{\mathrm{end}}}{N}_{t_0},$$ 2

$$C_{\mathrm{peak}}\left(\mathrm{\Pi }\right)\triangleq {\max }_{t\in \left\{1,2,\dots ,T_{\mathrm{end}}\right\}}{C}_{\mathrm{\Pi }}\left(t\right).$$ 3

In practical CBSS, average throughput C_avg(Π) reflects the overall sorting performance, and peak throughput C_peak(Π) captures the stability during the most intense parcel arrival period within the sorting shift, thereby highlighting the potential performance bottleneck.

Visualization platform

The visualization platform is developed by integrating the real-time data interface of the sorting simulation module with dynamic sorting logic and virtual entities, including virtual parcels and four mechanism modules, implemented within the Unreal Engine environment⁵⁵, as detailed in Supplementary Movie 1. This integration facilitates continuous monitoring of the sorting performance and the operational status of the virtual mechanism modules.

Remark: to guarantee the effectiveness of the established PSDTS, fidelity is a critical metric to ensure that PSDTS precisely replicates the CBSS in real-world conditions. Considering that the number of sorted parcels per time interval directly quantifies the throughput of the CBSS, fidelity is defined as the weighted average accuracy between the simulated and actual recorded number of sorted parcels over fixed-length time intervals throughout the sorting shift, as detailed in Supplementary Method 1. To maintain high fidelity for the established PSDTS, a continuous calibration mechanism is incorporated that refines the key operational parameters, e.g., maximum capacity ${\overline{C}}_x^{\mathrm{pack}}$ and packing time ${\overline{T}}_x^{\mathrm{pack}}$ of virtual grids by utilizing practical data collected from the latest sorting shift.

To validate the accuracy and reliability of the established PSDTS, we compare the sorted parcel quantities between the practical sorting process in CBSS and the simulated sorting process in PSDTS across four typical PDCs using real-world parcel data with 17,394, 61,466, 41,643, and 30,207 parcels in each sorting shift, respectively. As observed in Fig. 3, the simulated sorted parcel quantities in each time interval align closely with the real-world parcel numbers in CBSS, underscoring the PSDTS’s capability to precisely replicate the real-world CBSS. Specifically, the PSDTS achieves fidelities of 96.13%, 95.27%, 97.23%, and 96.35% in the four PDCs according to Supplementary Method 1 under 1-min time intervals, underscoring its reliability and applicability as the performance evaluation system for sorting throughput maximization algorithms.

Fig. 3. Sorting time vs. quantities of sorted parcels per time interval for practical sorting process in cross-belt sorter system (CBSS) and simulated sorting process in parcel-sorting digital twin system (PSDTS) across four typical parcel distribution centers (PDCs).

a–d Represent four typical PDCs. The black square markers and black solid line denote the practical sorting process in CBSS, whereas the blue triangle markers and blue solid line denote the simulated sorting process in PSDTS. Source data are provided as a Source Data file.

Performance evaluation of DT-SMCTS across different PDCs

Empirical parcel data analysis in Supplementary Note 1 shows that the portion of parcels targeted to each destination in the sorting shift remains highly similar across consecutive days, and thus the sorting plan can be derived by inputting historical data into the DT-based optimization framework. Considering the deployment costs, sorting plans are typically re-optimized offline on a 3-day cycle in practical operations, using the real-world parcel-sorting dataset W and bin-packing dataset G collected from the latest sorting shift. Based on the established PSDTS, we propose the DT-SMCTS framework to derive the optimal sorting plan. Specifically, the DT-based MCTS is adopted as the algorithmic framework, which is detailed in the “Methods” section. Then, we improve the expansion step in the basic DT-based MCTS framework to accelerate the convergence of MCTS through two steps: first, the DT-INLP module is developed to determine the number of grids assigned to each destination; then, based on the obtained grid quantity assignment, the GDGN module is proposed to guide the tree expansion. These modules work collaboratively to prune inferior solutions, thereby accelerating the convergence of MCTS.

To evaluate the performance of the proposed DT-SMCTS, we compare it with the following baselines.

• Expert-Calibrated Rank Correlation Sorting (ECRCS) method⁵⁶: the field-implemented sorting plan provided by SF Express, which is first generated based on the rank correlation metric computed from the portion of parcels targeted to each destination, and is then slightly adjusted by SF experts in accordance with the PDC’s manual sorting experience. Notably, this approach operates independently of the PSDTS.

• Balanced-Assignment with the Adaptive Large Neighborhood Search (BA-ALNS): a DT-free baseline that equalizes parcel workloads across grids to prevent regional overloads, thereby reducing parcel recirculation and mitigating sorting throughput deterioration. Specifically, the BA-ALNS is derived by solving the following problem:

  $${\min }_{\begin{array}{c}\hfill \mathit{\alpha }\hfill \end{array}}\frac{1}{X}{\sum }_{d=1}^D{\alpha }_d{\left(\frac{P_d}{{\alpha }_d}-\frac{1}{X}\right)}^2,$$

  where α_d is the number of grids assigned to destination d, P_d denotes the proportion of parcels targeted to destination d in the sorting shift, and the objective is solved using the ALNS algorithm⁵⁷.

• Basic DT-based MCTS with GDGN: the basic DT-based MCTS algorithm, which directly derives the sorting plan with GDGN, without integrating the DT-INLP module.

• Simulation-based performance reference (SPR): we consider a relaxed CBSS setting in which bin capacity is unbounded so that grids never lock, while preserving the same parcel-arrival, parcel-sorting, and unloading logic as in the CBSS. Solving this relaxed problem with PSDTS yields a simulation-derived estimate of the maximal attainable C_avg and C_peak as defined in Eqs. 2 and 3.

The performance of each algorithm was assessed on four representative PDCs that differ in both the geometric layout and operational scale, chosen as representative case studies from 146 real-world datasets collected at distinct PDCs. Specifically, the real-world parcel data comprise 133 (A), 201 (B), 247 (C), and 376 (D) destinations in each sorting shift, respectively. Specifically, the four PDCs contain 177 (A), 270 (B), 354 (C), and 598 (D) grids, respectively. In addition, we include the parcel recirculation count (PRC) as the additional performance metric, which refers to the number of sorted parcels that require more than one conveyor cycle. This metric is crucial as it directly quantifies the sorting congestion and thus reflects the overall sorting performance. Experiments were conducted on a MacBook Pro with Apple M2 Max chip and 32 GB unified RAM, without relying on any GPU clusters or cloud computing infrastructure. Detailed analysis regarding the computational resources and operational feasibility is provided in Supplementary Note 2.

Table 1 presents a comprehensive evaluation of the proposed DT-SMCTS, ECRCS, and basic DT-MCTS across four PDCs. For the average throughput (AT) metric, DT-SMCTS achieves improvements ranging from 3.12 to 10.56% over ECRCS and from 0.29 to 6.77% over basic DT-MCTS, highlighting its effectiveness in maintaining consistently superior sorting performance throughout the sorting shift. For the peak throughput (PT) metric, DT-SMCTS outperforms ECRCS by 4.77–7.72% and basic DT-MCTS by 0.63–4.99%, indicating its superior capability to mitigate congestion and maintain operational stability during the most intense parcel arrival period. For the overall throughput (OT), which is the weighted sum of the average and peak sorting throughput, DT-SMCTS achieves improvements ranging from 3.79 to 9.62% compared to ECRCS and 0.40–6.18% improvements over basic DT-MCTS. Additionally, DT-SMCTS reduces the PRC by 32.13–60.81% relative to ECRCS, and 0.77–54.57% compared to basic DT-MCTS, underscoring its effectiveness in reducing parcel-sorting congestion throughout the entire sorting shift.

Table 1.

Performance comparison of various algorithms across PDCs

	A					B
	ECRCS	BA-ALNS	Basic	DT-SMCTS	SPR	ECRCS	BA-ALNS	Basic	DT-SMCTS	SPR
			DT-MCTS					DT-MCTS
AT (num/h)	14,566.23	14,359.49	15,005.89	15,049.75	15,815.42	13,941.93	14,354.87	14,436.65	15,413.94	15,975.82
Boost (%)	–	−1.42	3.02	3.32	8.58	–	2.96	3.55	10.56	14.59
PT (num/h)	16,356.00	16,164.00	17,028.00	17,136.00	17,664.00	16,164.00	16,548.00	16,584.00	17,412.00	17,832.00
Boost (%)	–	−1.17	4.11	4.77	8.00	–	2.38	2.60	7.72	10.32
OT (num/h)	15,103.16	14,900.84	15,612.52	15,675.62	16,369.99	14,608.55	15,012.81	15,080.86	16,013.36	16,532.68
Boost (%)	–	−1.34	3.37	3.79	8.39	–	2.77	3.23	9.62	13.17
PRC (num)	1136	1720	777	771	0	3631	3016	2228	1700	0
Boost (%)	–	−51.41	31.60	32.13	–	–	16.94	38.64	53.18	–

	C					D
	ECRCS	BA-ALNS	Basic	DT-SMCTS	SPR	ECRCS	BA-ALNS	Basic	DT-SMCTS	SPR
			DT-MCTS					DT-MCTS
AT (num/h)	15,800.35	15,234.67	15,912.99	16,293.26	16,638.29	13,872.49	14,070.13	14,239.91	14,686.67	14,820.00
Boost (%)	–	−3.58	0.71	3.12	5.30	–	1.42	2.65	5.87	6.83
PT (num/h)	17,064.00	17,112.00	17,760.00	18,252.00	18,360.00	15,912.00	15,948.00	16,488.00	17,052.00	17,148.00
Boost (%)	–	0.28	4.08	6.96	7.59	–	0.23	3.62	7.16	7.77
OT (num/h)	16,179.44	15,797.87	16,467.09	16,880.88	17,154.80	14,484.34	14,633.49	14,913.09	15,396.27	15,518.40
Boost (%)	–	−2.36	1.78	4.34	6.03	–	1.03	2.96	6.30	7.14
PRC (num)	1959	2648	1137	878	0	393	357	339	154	0
Boost (%)	–	−35.17	41.96	55.18	–	–	9.16	13.74	60.81	–

Moreover, as detailed in Table 1, relative to the BA-ALNS baseline that operates independently of the PSDTS, DT-SMCTS achieves AT gains of 4.38–7.38%, PT gains of 5.22–6.92%, and OT gains of 5.20–6.86% across PDCs, while reducing PRC by 43.63–66.84%. These results underscore the importance of the PSDTS, which not only precisely replicates the CBSS parcel-sorting mechanism and evaluates sorting throughput, but also provides adaptive capability for handling different CBSS layouts across PDCs. The integration of the PSDTS enables the DT-SMCTS framework to achieve superior parcel-sorting performance that baselines without PSDTS cannot match.

Figure 4 further illustrates the performance comparison of these algorithms across PDCs, showing the evolution of OT, AT, and PT over episodes. As observed in Fig. 4, DT-SMCTS not only exhibits a substantial initial advantage but also stabilizes at a higher throughput at earlier stages compared to basic DT-MCTS, demonstrating a more rapid convergence. This superior convergence is primarily attributed to the integration of the DT-INLP module, which provides better initial assignments. This allows the DT-SMCTS to concentrate computational efforts on the exploration of high-quality regions of the solution space early in the search process. In contrast, basic DT-MCTS relies heavily on extensive exploration to identify high-quality solutions, resulting in slower convergence, higher variance, and ultimately lower throughput within a limited time.

Fig. 4. Performance comparison among various algorithms across parcel distribution center (PDC) A, B, C, and D.

a The weighted sum of average throughput with ω₁ = 0.7 and peak throughput ω₂ = 0.3; b the average throughput; c the peak throughput. The purple dashed line with star markers denotes the simulation-based performance reference (SPR), the orange line with triangle markers denotes the digital twin-based structured Monte Carlo tree search (DT-SMCTS), the blue line with circle markers denotes the basic digital twin-based Monte Carlo tree search (DT-MCTS), the green line with square markers denotes the Expert-Calibrated Rank Correlation Sorting (ECRCS), and the gray line with inverted-triangle markers denotes the Balanced-Assignment with the Adaptive Large Neighborhood Search (BA-ALNS). The shaded areas represent the rollout throughputs at each episode, and the highlighted solid lines mark the throughputs of the best sorting plan selected up to that episode. Source data are provided as a Source Data file.

In conclusion, the evaluation results confirm that DT-SMCTS consistently achieves higher throughput and superior convergence. Additionally, it approaches the SPR for all throughput metrics across PDCs, with only 0.90–4.84% degradation in AT, 0.56–2.99% in PT, and 0.79–4.24% in OT, underscoring its strong adaptability and effectiveness for sorting throughput maximization across diverse PDCs.

Performance evaluation of DT-SMCTS over consecutive days

In practice, PDCs typically update the sorting plan on a 3-day cycle. Accordingly, we evaluate the effectiveness of the proposed DT-SMCTS over 3 consecutive days using an identical sorting plan. Additionally, we include the first segment sorted ratio (FSSR) as a complementary metric to evaluate the sorting performance of the same PDC across consecutive days. The FSSR quantifies the proportion of parcels that are inducted and successfully sorted within their first inducted sorting segment and is defined as

$$P_{\mathrm{FSSR}}\left(\mathrm{\Pi }\right)=\frac{{\sum }_{x=1}^X{\sum }_{n=1}^{N_x}\mathbf{1}\left(n,x\right)}{N},$$ 4

where the indicator function 1(n, x) is defined such that it equals 1 if parcel n in grid x is scanned only once and sorted within its first inducted sorting segment; and 0 otherwise. N_x denotes the number of sorted parcels in grid x throughout the sorting shift, and N represents the total number of parcels in the sorting shift. A higher FSSR indicates that more parcels can be successfully sorted in their first inducted sorting segment, which reduces congestion and enables more parcels to be inducted in the subsequent sorting segment, thereby improving the overall sorting performance.

To ensure temporal consistency, the first day of this consecutive-days evaluation aligns with the initial evaluation date of PDC B in the preceding experiment. As depicted in Fig. 5, over consecutive days, DT-SMCTS consistently outperforms the ECRCS, BA-ALNS, and basic DT-MCTS, and aligns closely with the SPR. Specifically, it achieves higher values in AT, PT, OT, and FSSR, along with a reduced PRC. Compared to ECRCS, it achieves average improvements of 7.83, 6.04, 7.24, and 45.72% in AT, PT, OT, and PRC, as detailed in Table 2, demonstrating its effectiveness in reducing sorting congestion and improving the overall sorting performance.

Fig. 5. Sorting time vs. throughput of various algorithms in each sorting shift over consecutive days.

The purple line with star markers denotes the simulation-based performance reference (SPR), the orange line with triangle markers denotes the digital twin-based structured Monte Carlo tree search (DT-SMCTS), the blue line with circle markers denotes the basic digital twin-based Monte Carlo tree search (DT-MCTS), the gray line with inverted-triangle markers denotes the Balanced-Assignment with Adaptive Large Neighborhood Search (BA-ALNS), and the green line with square markers denotes the Expert-Calibrated Rank Correlation Sorting (ECRCS). Source data are provided as a Source Data file.

Table 2.

Detailed performance comparison of various algorithms over consecutive days

	Day 1					Day 2					Day 3
	ECRCS	BA-ALNS	Basic	DT-SMCTS	SPR	ECRCS	BA-ALNS	Basic	DT-SMCTS	SPR	ECRCS	BA-ALNS	Basic	DT-SMCTS	SPR
			DT-MCTS					DT-MCTS					DT-MCTS
AT (num/h)	13,941.93	14,354.87	14,436.65	15,413.94	15,975.82	14,249.79	14,424.39	14,581.55	15,120.06	15,754.24	14,031.82	14,219.65	14,259.94	14,990.10	15,815.73
Boost (%)	–	2.96	3.55	10.56	14.59	–	1.23	2.33	6.11	10.56	–	1.34	1.63	6.83	12.71
PT (num/h)	16,164.00	16,548.00	16,584.00	17,412.00	17,832.00	16,104.00	16,356.00	16,368.00	16,992.00	17,784.00	16,236.00	16,320.00	16,320.00	17,028.00	17,820.00
Boost (%)	–	2.38	2.60	7.72	10.32	–	1.56	1.64	5.51	10.43	–	0.52	1.26	4.88	9.76
OT (num/h)	14,608.55	15,012.81	15,080.86	16,013.36	16,532.68	14,806.05	15,003.87	15,117.49	15,681.64	16,363.17	14,693.07	14,849.76	14,877.96	15,601.47	16,417.01
Boost (%)	–	2.77	3.23	9.62	13.17	–	1.34	2.10	5.91	10.52	–	1.07	1.29	6.18	11.73
PRC (num)	3631	3016	2228	1700	0	4333	5686	2738	2374	0	4293	5172	3206	2628	0
Boost (%)	–	16.94	38.64	53.18	–	–	−31.23	36.81	45.21	–	–	−20.48	25.32	38.78	–
FSSR (%)	72.20	75.65	74.17	80.63	84.64	70.98	74.45	73.36	79.35	83.31	71.53	74.48	73.70	79.65	83.69

Performance evaluation of DT-SMCTS in practical field tests

To assess the performance of DT-SMCTS in real-world parcel-sorting scenarios, we conducted consecutive practical field tests at a PDC in Shanghai from the 20th to the 25th of the month. During this period, ECRCS was employed from the 20th to the 22nd, while DT-SMCTS was implemented from the 23rd to the 25th. Considering the variability in parcel volumes and parcel arrivals in each sorting shift across consecutive days, we adopt the FSSR as the primary performance metric. As shown in Fig. 6, DT-SMCTS consistently outperformed ECRCS. Specifically, for the conventional ECRCS approach, the FSSR values were 78.14, 77.65, and 77.63% on days 20–22, with an average of 77.81%; DT-SMCTS achieved 81.15, 78.83, and 80.65% on days 23–25, with an average of 80.21%, achieving an average improvement of 2.4% in FSSR. This enhancement indicates more efficient sorting and a more continuous flow of parcels in practical implementations.

Fig. 6. Practical field test performance over consecutive days.

The orange line with square markers denotes the proportion of parcels sorted within 0–0.5 conveyor cycles in each sorting shift, the green line with circle markers denotes the proportion of parcels sorted within 0.5–1 conveyor cycle, and the blue line with triangle markers denotes the proportion of parcels sorted within 1–1.5 conveyor cycles. Expert-Calibrated Rank Correlation Sorting (ECRCS) was deployed from day 20 to 22, achieving first segment sorted ratio (FSSR) values of 78.14%, 77.65%, and 77.63%, respectively. Digital twin-based structured Monte Carlo tree search (DT-SMCTS) was implemented from days 23 to 25, yielding FSSR values of 81.15%, 78.83%, and 80.65%, corresponding to an average improvement of 2.4%. Source data are provided as a Source Data file.

In summary, we utilize the established PSDTS and real-world parcel data to conduct comprehensive evaluations across PDCs and over consecutive days, and practical field tests to underscore the effectiveness and operational feasibility of the proposed DT-SMCTS framework. By first evaluating its performance through detailed quantitative analyses, then confirming its consistency over consecutive days, and finally verifying its real-world applicability via practical implementations, we provide strong evidence that DT-SMCTS can significantly enhance sorting performance across diverse PDCs over consecutive days.

Methods

In a CBSS, inbound parcels are routed to their assigned grids according to a sorting plan based on their downstream destinations. For a given shift, the sorting plan is formally defined as a grid-to-destination mapping $\mathrm{\Pi }:\mathcal{X}\to \mathcal{D}$, where $\mathcal{X}$ is the set of grids and $\mathcal{D}$ is the set of destinations. Each grid $x\in \mathcal{X}$ is assigned to a unique destination $\mathrm{\Pi }\left(x\right)\in \mathcal{D}$, while multiple grids may be assigned to the same destination. Accordingly, the sorting optimization in a CBSS aims to determine an optimal sorting plan, i.e., an optimal mapping Π.

A direct approach to determine the optimal sorting plan is to generate several sorting plans and subsequently evaluate their performances through practical implementation in the CBSS. However, this approach is costly and risky, since an inappropriate trial deployed in the CBSS may result in substantial financial losses. Another intuitive approach is to model the bin-stacking process of each grid as a bulk-service queue with state-dependent arrival rates⁵⁸. However, the arrival and departure processes of each grid are determined by real-time tray occupancy and grid-locking statuses, which together induce a high-dimensional state space that is too complex to be precisely captured by tractable mathematical formulations, rendering the queueing theory-based methods inapplicable for the considered problem.

As such, we propose to adopt the DT of CBSS to evaluate the sorting performance of one candidate sorting plan, and further develop a DT-based optimization framework that facilitates simulation-driven refinement of sorting plans based on the collected real-world data. In particular, we begin by introducing three operational constraints derived from real-world operational requirements of PDCs that maintain load balance across different sorting segments⁵⁹, minimize additional transportation costs, and ensure equitable grid allocation.

• Balanced assignment constraint: for each destination $d\in \mathcal{D}$, let α_d denote the total number of grids assigned to d; and let ${\alpha }_d^{\left(1\right)}$ and ${\alpha }_d^{\left(2\right)}$ denote the numbers of assigned grids in sorting segments 1 and 2, respectively, i.e.,

  $${\alpha }_d\triangleq {\sum }_{x=1}^X\mathbb{I}\left\{\mathrm{\Pi }\left(x\right)=d\right\},{\alpha }_d^{\left(1\right)}\triangleq {\sum }_{x=1}^{X_0}\mathbb{I}\left\{\mathrm{\Pi }\left(x\right)=d\right\},{\alpha }_d^{\left(2\right)}\triangleq {\sum }_{x=X_0+1}^X\mathbb{I}\left\{\mathrm{\Pi }\left(x\right)=d\right\},$$ 5

  where X₀ denotes the number of grids in the first sorting segment, and the indicator function $\mathbb{I}\left\{\mathrm{\Pi }\left(x\right)=d\right\}$ equals 1 if Π(x) = d, and 0 otherwise. To ensure balanced assignment of grids across the CBSS, for any destination assigned multiple grids, these grids should be assigned proportionally to the number of grids in each sorting segment, i.e.,

  $$⌊{\alpha }_d\frac{X_0}{X}⌋\le {\alpha }_d^{\left(1\right)}\le ⌈{\alpha }_d\frac{X_0}{X}⌉,⌊{\alpha }_d\frac{X-X_0}{X}⌋\le {\alpha }_d^{\left(2\right)}\le ⌈{\alpha }_d\frac{X-X_0}{X}⌉,\forall d\in \mathcal{D},{\alpha }_d\ge 2,$$ 6

  where floor function ⌊⋅⌋ returns the greatest integer less than or equal to its argument and ceiling function ⌈⋅⌉ returns the smallest integer greater than or equal to its argument.
• Adjacent assignment constraint: during the practical sorting process, parcels might be erroneously diverted into the grids adjacent to their designated ones due to inertia or mechanical errors. These misdirected parcels can only be detected by operators at downstream PDCs, by which time they have already undergone line-haul transportation. The resulting economic and time costs of correcting such errors are strongly correlated with the geographical distance between the incorrectly assigned PDC and the intended destination, and they also degrade service quality and customer satisfaction. To minimize such costs, adjacent grids should be assigned to destinations that are geographically close to each other. Specifically, adjacent grids should be assigned primarily to destinations within the same or adjacent provinces. To rigorously formulate the above constraint, we define M_j,u = 1 if grids j and u are adjacent, and 0 otherwise. Let $V_{i,q}^{\mathrm{same}}=1$ if destinations i and q are in the same province, and 0 otherwise. Similarly, let $V_{i,q}^{\mathrm{adj}}=1$ if destinations i and q are in adjacent provinces, and 0 otherwise. Then, it follows

  $$\begin{array}{c}\hfill M_{j,u}\left(1-V_{\mathrm{\Pi }\left(j\right),\mathrm{\Pi }\left(u\right)}^{\mathrm{same}}-V_{\mathrm{\Pi }\left(j\right),\mathrm{\Pi }\left(u\right)}^{\mathrm{adj}}\right)=0,\forall j,u\in \mathcal{X}\backslash {\mathcal{X}}^{\mathrm{iso}},\end{array}$$ 7

  where ${\mathcal{X}}^{\mathrm{iso}}$ denotes the set of grids that PDCs reserve for geographically isolated destinations, i.e., those without any neighbors in the same or an adjacent province in the sorting shift. Let ${\mathcal{D}}^{\mathrm{iso}}$ denote the set of these isolated destinations, and for each grid $y\in {\mathcal{X}}^{\mathrm{iso}}$, the sorting plan Π predetermines its mapping to an isolated destination, i.e., $\mathrm{\Pi }\left(y\right)\in {\mathcal{D}}^{\mathrm{iso}},\forall y\in {\mathcal{X}}^{\mathrm{iso}}$.
• Bounded assignment constraint: each destination must be assigned at least one and no more than a given number of grids, i.e.,

  $$\begin{array}{c}\hfill 1\le {\alpha }_d\le {\gamma }_{\mathrm{th}},\forall d\in \mathcal{D},\end{array}$$ 8

  where γ_th denotes the maximum number of grids assigned to each destination.

Remark: while these constraints restrict the feasible solution space and could slightly reduce the attainable throughput under a fixed simulation budget, they yield operationally compliant and more robust sorting plans by ensuring balanced assignment across sorting segments, mitigating the additional transportation costs induced by unavoidable parcel misrouting, and avoiding extreme grid assignments. In particular, although relaxing the adjacent assignment constraint may increase sorting throughput, it can simultaneously incur substantial transportation and delay costs at downstream PDCs by amplifying the consequences of inevitable mechanical misrouting. Retaining this constraint therefore provides a more robust and economically sound design and constitutes a strict operational requirement in practice (see Supplementary Note 3).

Subject to the above constraints, the objective of the sorting throughput maximization problem in CBSS is to maximize the weighted sum of the average and peak sorting throughputs defined in Eqs. 2 and 3, respectively, thereby ensuring sustained overall sorting performance throughout the sorting shift and stability during the most intense parcel arrival period. Accordingly, the sorting throughput maximization problem for the CBSS is formulated as

$$\begin{array}{cc}\hfill \text{(P1)}& \hfill {\max }_{\mathrm{\Pi }}{\omega }_1{C}_{\mathrm{avg}}\left(\mathrm{\Pi }\right)+{\omega }_2{C}_{\mathrm{peak}}\left(\mathrm{\Pi }\right)\hfill \\ \hfill & \hfill \text{s.t.}\left(6\right),\left(7\right),\left(8\right),\hfill \end{array}$$ 9

where ω₁ and ω₂ ∈ (0, 1), satisfying ω₁ + ω₂ = 1, are the importance factors of the average and peak throughputs, respectively.

Remark: the sorting throughput maximization problem (P1) presents significant challenges primarily due to the following reasons.

• Problem (P1) involves assigning each of the X grids to one of the D destinations, thus it is a combinatorial problem and NP-hard⁶⁰, and the cardinality of its solution space is upper bounded by $D^X+{\sum }_{k=1}^{D-1}{\left(-1\right)}^k\left(\genfrac{}{}{0ex}{}{D}{k}\right){\left(D-k\right)}^X$, as shown in Supplementary Method 2. Conventional approaches to solve such problems, such as branch and bound⁶¹, dynamic programming (DP)⁶² (e.g., value iteration, policy iteration, limited lookahead, and approximate DP), integer programming⁶³, genetic algorithm⁶⁴, and Monte Carlo tree search (MCTS)⁶⁵, incur prohibitive computational costs, particularly when both the number of destinations D and grids X are large.

• The practical operational constraints in Eqs. 6, 7, and 8 pose additional challenges for conventional DP or MCTS methods, as the intrinsic trial-and-error mechanism of these methods makes it almost unavoidable to select infeasible actions, leading to increased evaluations and computational overhead, significantly reducing the convergence speed⁶⁶.

• Solving problem (P1) is inherently time-consuming, since its objective function Eq. 9 can only be evaluated by using the proposed PSDTS to simulate the sorting process over the entire sorting shift. Consequently, considering the time sensitivity of sorting optimization in practical deployment, the method for solving problem (P1) has to be data-efficient, aiming to find the best feasible solution within a limited time.

Based on the established PSDTS, we propose the DT-SMCTS to derive the optimal sorting plan as depicted in Fig. 7. Specifically, DT-SMCTS utilizes DT-based MCTS as its algorithmic framework, which is detailed in Fig. 7C. It then modifies the expansion step of the DT-based MCTS through two steps: first, the DT-INLP module is developed to determine the number of grids assigned to each destination using the parcel-sorting data (as described in Fig. 7A); second, based on the obtained grid quantity assignment, the GDGN module is proposed to guide the tree expansion (as outlined in Fig. 7B). These two steps work collaboratively to prune inferior solutions, thereby accelerating the convergence of MCTS.

Fig. 7. Framework of the digital twin-based structured Monte Carlo tree search (DT-SMCTS).

A Digital twin-based integer nonlinear programming (DT-INLP) module generates candidate grid quantity assignments for all destinations. P_d denotes the portion of parcels targeted to destination d during a sorting shift, and J_d(α_d) measures the discrepancy between the assigned grid quantity and the demand-proportional target. For each pair of sensitivity rank k and penalty factor β, DT-INLP constructs a candidate assignment α^{(k, β)} by sequentially assigning the remaining grids according to the largest discrepancy value. All candidates are subsequently evaluated by the parcel-sorting digital twin system (PSDTS), and the assignment that maximizes the sorting objective is selected as α^*. B The geographic destination graph network (GDGN) module guides tree expansion based on the grid quantity assignment α^* from the DT-INLP module. For each sorting segment e ∈ {1, 2}, graphs ${\mathcal{G}}_{e,1}$ and ${\mathcal{G}}_{e,2}$ are constructed to model the geographic adjacency among destinations in the same province and in adjacent provinces, respectively. X₀ denotes the boundary between the two sorting segments, and x denotes the current decision stage. C The framework of the DT-based Monte Carlo tree search (MCTS) that constructs a tree representation of the sorting plan. The destination assignment of the x-th grid is determined at the x-th stage. The action at the x-th stage is denoted as a_x and defined as the assigned destination of the x-th grid. The state s_x at the x-th stage is the ordered sequence of mappings from the first x − 1 grids to their assigned destinations.

Basic DT-based MCTS

To address these challenges, we first reformulate the sorting throughput maximization problem (P1) as an equivalent finite-horizon MDP, which consists of the following components.

• Stage: there are X + 1 stages in the formulated MDP. The destination assignment of the x-th grid is determined at the x-th stage, and stage X + 1 is the terminal stage.

• Action: the action at the x-th stage is denoted as a_x and defined as the assigned destination of the x-th grid.

• State: the state at the x-th stage is the ordered sequence of mappings from the first x − 1 grids to their assigned destinations, represented as a tuple with x − 1 elements: s_x ≜ (a₁,a₂,..., a_x − 1).

• Transition: The state at the (x + 1)-th stage is derived by appending action a_x taken at stage x to the previous state s_x, i.e., s_x+1 ≜ (s_x, a_x).

• Reward: for any intermediate stage x < X + 1, reward r_x is set as 0; for the terminal stage X + 1, reward r_X + 1 equals ω₁C_avg(Π) + ω₂C_peak(Π) as the objective function of problem (P1), where both C_avg and C_peak are obtained from the PSDTS.

Based on the above MDP formulation, we propose a DT-based MCTS framework to derive the optimal sorting plan, which adopts a tree structure of depth X to simulate the X decision stages of the MDP. In particular, nodes at depth x correspond to distinct values of s_x + 1, while the edges connecting nodes at depth x − 1 and x correspond to different values of a_x. Each node in the tree stores four attributes: s, N(s), N(s, a), and Q(s, a). s represents the state of the node; N(s) denotes the cumulative number of visits to the node s, which is initialized as 0 and increases as the tree grows; N(s, a) indicates the number of times action a has been selected at node s and also is initialized as 0; Q(s, a) represents the estimated expected reward of selecting action a at node s.

Specifically, at the initial stage, the tree contains only the root node and simulates the first stage of the MDP. Subsequently, it grows to a depth of X by recursively performing the following four steps, as depicted in Fig. 7.

Selection: if the current tree contains only the root node, we directly terminate the selection step; otherwise, we select a path from the root node to a leaf node (i.e., a node with no child nodes) based on the values N(s), N(s, a), and Q(s, a) stored in the nodes of the current tree. Specifically, we start from the root node, denoted as s₁, and determine the value of a₁ using upper confidence bound (UCB)⁶⁷ by

$$\begin{array}{c}\hfill a_x={\mathrm{argmax}}_{a\in \mathcal{A}\left(s_x\right)}\left(Q\left(s_x,a\right)+u\left(s_x,a\right)\right),\forall x\in \left\{1,2,...,X\right\},\end{array}$$ 10

where $\mathcal{A}\left(s_x\right)$ denotes the set of actions corresponding to all the existing edges between node s_x and the nodes at depth x of the current tree, and

$$\begin{array}{c}\hfill u\left(s_x,a\right)=c\frac{\sqrt{2ln\left(N\left(s_x\right)\right)}}{N\left(s_x,a\right)},\end{array}$$ 11

with c being the exploration coefficient. Once a₁ is determined, we reach the node at depth 1, denoted by s₂, and then a₂ can be determined similarly to Eqs. 10 and 11. This selection process is repeated until a leaf node is reached.

Expansion: suppose reaching the leaf node s_L + 1 at depth L, where L < X, the tree is expanded by adding a child node s_L + 2 to node s_L + 1, corresponding to a feasible action selected at random. After the expansion step, the attributes of s_L + 2 are initialized as N(s_L + 2) = 0, N(s_L + 2, a) = 0, and Q(s_L + 2, a) = 0.

Simulation: after the expansion, nodes from depth L + 2 to depth X are selected by using a trivial rollout policy⁶⁵ that randomly selects actions satisfying the constraints in Eqs. 6, 7, and 8. Then, the obtained sorting plan Π is evaluated through the PSDTS, which returns C_avg(Π) and C_peak(Π).

Backpropagation: based on C_avg(Π) and C_peak(Π), the attributes of each node are backwardly updated from the leaf node to the root node as

$$N\left(s_x\right)=N\left(s_x\right)+1,N\left(s_x,a_x\right)=N\left(s_x,a_x\right)+1,Q\left(s_x,a_x\right)=Q\left(s_x,a_x\right)+\frac{{\omega }_1{C}_{\mathrm{avg}}\left(\mathrm{\Pi }\right)+{\omega }_2{C}_{\mathrm{peak}}\left(\mathrm{\Pi }\right)-Q\left(s_x,a_x\right)}{N\left(s_x,a_x\right)},$$ 12

for all x ∈ {1, 2,..., L + 2}.

DT-based structured MCTS

Then, we improve the expansion step in the basic MCTS framework to accelerate the convergence of MCTS through two steps: first, the DT-INLP module is developed to determine the number of grids assigned to each destination; then, based on the obtained grid quantity assignment, the GDGN module is proposed to guide the tree expansion. These modules work collaboratively to prune inferior solutions, thereby accelerating the convergence of MCTS.

Grid quantity assignment

Let P_d denote the portion of parcels targeted to destination d in the sorting shift. To optimize the sorting performance, the grid quantity assigned to destination d should be proportional to P_d, i.e., the discrepancy between α_d and P_dX should be minimized. Based on this intuition, we propose the following discrepancy metric

$$J_d\left({\alpha }_d\right)=\left\{\begin{array}{cc}\beta {\alpha }_d{∣{\alpha }_d-P_{d}X∣}^k,\hfill & \text{if}{\alpha }_d=1,\hfill \\ {\alpha }_d{∣{\alpha }_d-P_{d}X∣}^k,\hfill & \text{if}2\le {\alpha }_d\le {\gamma }_{\mathrm{th}},\hfill \\ 0,\hfill & \text{if}{\alpha }_d>{\gamma }_{\mathrm{th}},\hfill \end{array}\right)$$ 13

where rank k adjusts the penalty sensitivity and β serves as the penalty factor.

Utilizing this metric, the optimal grid quantity assignment α ≜ [α₁,..., α_D] can be approximated by solving the following problem

$$\begin{array}{cc}\hfill \text{(P2)}& \hfill {\min }_{\begin{array}{c}\hfill \mathit{\alpha }\hfill \end{array}}{\sum }_{d=1}^D{J}_d\left({\alpha }_d\right)\hfill \\ \hfill & \hfill \text{s.t.}\left(6\right),\left(7\right),\left(8\right).\hfill \end{array}$$

Notably, problem (P2) is still NP-hard with the cardinality of the solution space bounded by $\frac{\left(X-1\right)!}{\left(D-1\right)!\left(X-D\right)!}$, as shown in Supplementary Method 2. To efficiently approximate its optimal solution, we introduce a heuristic DT-INLP algorithm summarized in Box 1. Specifically, given any k and β, we denote the solution to problem (P2) as ${\mathit{\alpha }}^{\left(k,\beta \right)}\triangleq \left[{\alpha }_1^{\left(k,\beta \right)},{\alpha }_2^{\left(k,\beta \right)},...,{\alpha }_D^{\left(k,\beta \right)}\right]$. To construct α^{(k, β)}, we adopt X − D + 1 rounds of grid quantity assignments. In the first round, each of the D destinations is initially assigned one grid. With a slight abuse of notation, we use α^{(k, β)} to denote the assigned grid quantities to D destinations up to the current round. Hence, after the first round, α^{(k, β)} = [1, 1,..., 1] holds. In each of the following X − D rounds, we first calculate $J_d\left({\alpha }_d^{\left(k,\beta \right)}\right)$ for each destination based on Eq. 13, and select the destination d^* with the largest $J_d\left({\alpha }_d^{\left(k,\beta \right)}\right)$ value. Then, α^{(k, β)} is updated by ${\alpha }_{d^{*}}^{\left(k,\beta \right)}={\alpha }_{d^{*}}^{\left(k,\beta \right)}+1$. In this paper, we restrict the values of k and β to be within the empirical sets $\mathcal{K}$ and $\mathcal{B}$. Therefore, we can derive at most $\mid \mathcal{K}\mid \mid \mathcal{B}\mid$ candidate solutions. Each candidate is then evaluated via PSDTS, with α^{(k, β)} replacing the grid quantity assignment in the reference sorting plan Π^ref (i.e., the latest sorting plan implemented in the PDC). Finally, α^* is selected as the one that maximizes the objective function of problem (P1).

Box 1 DT-INLP for grid quantity assignment.

Input: Portion of parcels targeted to each destination ${\left\{P_d\right\}}_{d=1}^D$, reference sorting plan Π^ref, number of grids X, importance factors ω₁, ω₂, and empirical sets $\mathcal{K},\mathcal{B}$.

Candidate generation via DT-INLP:

1: Initialize solution set $\mathcal{S}=\mathrm{\varnothing }$;

2: For each $\left(k,\beta \right)\in \mathcal{K}\times \mathcal{B}$

3: Initialize ${\alpha }_d^{\left(k,\beta \right)}=1$, $\forall d\in \mathcal{D}$;

4: For z = 1 to X − D

5: For d = 1 to D

6: Compute $J_d\left({\alpha }_d^{\left(k,\beta \right)}\right)$ using Eq. 13;

7: End for

8: Select $d^{*}=\arg {\max }_d{J}_d\left({\alpha }_d^{\left(k,\beta \right)}\right)$;

9: Update ${\alpha }_{d^{*}}^{\left(k,\beta \right)}={\alpha }_{d^{*}}^{\left(k,\beta \right)}+1$;

10: End for

11: $\mathcal{S}=\mathcal{S}\cup \left\{{\mathit{\alpha }}^{\left(k,\beta \right)}\right\}$;

12: End for

Evaluation via PSDTS:

13: For each candidate ${\mathit{\alpha }}^{\left(k,\beta \right)}\in \mathcal{S}$

14: Evaluate ${\mathrm{\Pi }}_{{\mathit{\alpha }}^{\left(k,\beta \right)}}^{\mathrm{ref}}$ through the PSDTS;

15: End for

16: ${\mathit{\alpha }}^{*}={\mathrm{argmax}}_{{\mathit{\alpha }}^{\left(k,\beta \right)}\in \mathcal{S}}\left\{{\omega }_1{C}_{\mathrm{avg}}\left({\mathrm{\Pi }}_{{\mathit{\alpha }}^{\left(k,\beta \right)}}^{\mathrm{ref}}\right)+{\omega }_2{C}_{\mathrm{peak}}\left({\mathrm{\Pi }}_{{\mathit{\alpha }}^{\left(k,\beta \right)}}^{\mathrm{ref}}\right)\right\}.$

Output: Grid quantity assignment α^*

Tree expansion

After obtaining the grid quantity assignment α^* from the proposed DT-INLP module, we introduce the GDGN module to specify the procedure for tree expansion while satisfying the constraints in Eqs. 6, 7, and 8. First, we construct and initialize the graphs to model the geographic adjacency among the destinations in the same province and in adjacent provinces, respectively. Then, these graphs are utilized to guide the tree expansion.

Graph initialization: for each sorting segment e ∈ {1, 2}, we construct graphs ${\mathcal{G}}_{e,1}$, ${\mathcal{G}}_{e,2}$. Each graph contains D vertices, each corresponding to one of the D destinations. In ${\mathcal{G}}_{e,1}$, edges are added between vertices i and q if $V_{i,q}^{\mathrm{same}}=1$; in ${\mathcal{G}}_{e,2}$, edges are added if $V_{i,q}^{\mathrm{adj}}=1$. Specifically, each vertex in ${\mathcal{G}}_{e,1}$ and ${\mathcal{G}}_{e,2}$ stores a grid quantity attribute, which is initialized to ${\alpha }_d^{\left(e\right)}$ as given by the DT-INLP module and consistent with the balanced assignment constraint Eq. 6.

Graph-guided tree expansion: for each stage x, we first check whether $x\in {\mathcal{X}}^{\mathrm{iso}}$. For such special cases, a_x is randomly assigned to a destination in ${\mathcal{D}}^{\mathrm{iso}}$. For all other cases, based on the constructed graphs ${\mathcal{G}}_{e,1}$ and ${\mathcal{G}}_{e,2}$, the tree expansion is summarized in Box 2 and specified as follows.

• If 1 ≤ x ≤ X₀, we input a_x − 1 to the GDGN and check whether the vertex corresponding to destination a_x − 1 has any incident edges in ${\mathcal{G}}_{1,1}$: if incident edges exist, we prioritize setting a_x = a_x − 1 in the presence of a self-loop; if no self-loop is present, we randomly select one of the incident edges connected to a_x − 1 and set the connected vertex as a_x. If vertex a_x − 1 has no incident edges in ${\mathcal{G}}_{1,1}$, we randomly select an edge incident to a_x − 1 in ${\mathcal{G}}_{1,2}$ and set its connected vertex as a_x. After the selection, we decrease the grid quantity attribute of vertex a_x in each graph by 1. Once the grid quantity attribute of a vertex reaches 0, all its incident edges are removed from both ${\mathcal{G}}_{1,1}$ and ${\mathcal{G}}_{1,2}$.

• If X₀ < x, we apply the same procedure, and replace ${\mathcal{G}}_{1,1}$ and ${\mathcal{G}}_{1,2}$ by ${\mathcal{G}}_{2,1}$ and ${\mathcal{G}}_{2,2}$.

By sequentially exploring the edges in the constructed graphs, the GDGN module ensures that the selected actions satisfy the constraints in Eqs. 6, 7, and 8, effectively pruning inferior solutions, thereby accelerating convergence.

Box 2 GDGN for tree expansion.

Input: Graphs ${\mathcal{G}}_{1,1}$, ${\mathcal{G}}_{1,2}$, ${\mathcal{G}}_{2,1}$, ${\mathcal{G}}_{2,2}$, number of grids in the first sorting segment X₀, and previous action a_x − 1.

1: Set the sorting segment index e as 1 if 1 ≤ x ≤ X₀, and as 2 otherwise;

2: If the vertex corresponding to a_x − 1 has incident edges in ${\mathcal{G}}_{e,1}$

3: If self-loop exists at vertex a_x − 1

4: Set a_x − 1 as a_x;

5: Else

6: Randomly select one edge incident to a_x − 1 in ${\mathcal{G}}_{e,1}$ and set its connected vertex as a_x;

7: End if

8: Else

9: Randomly select one edge incident to a_x − 1 in ${\mathcal{G}}_{e,2}$ and set its connected vertex as a_x;

10: End if

11: Decrease the grid quantity attribute of vertex a_x by 1 in both ${\mathcal{G}}_{e,1}$ and ${\mathcal{G}}_{e,2}$;

12: If the grid quantity attribute of a_x = 0

13: Remove all edges incident to a_x from ${\mathcal{G}}_{e,1}$ and ${\mathcal{G}}_{e,2}$;

14: End if

Output: Selected action a_x.

Overall DT-SMCTS framework

Now, we summarize the complete proposed DT-SMCTS framework. We first provide a detailed timeline of how and how often simulations are carried out using the PSDTS within the integrated DT-SMCTS optimization framework. We then describe the DT-SMCTS procedure in detail, clarifying how data flow is handled in each update and how the sorting plans are updated. Finally, we discuss the deployment challenges and limitations of the proposed framework.

Timeline in CBSS: the CBSS timeline can be divided into two parts: the online phase and the offline phase. The parcel-sorting shift corresponds to the online phase, during which the IIoT platform acquires and uploads parcel-sorting data W and bin-packing data G. At the end of the sorting shift, the online phase terminates, and W and G have been fully uploaded to the cloud. Since a sorting shift occurs daily, the online phase recurs once per day. The interval between two successive sorting shifts constitutes the offline phase, during which the sorting plan is updated and then deployed to the CBSS. Specifically, plan updates are performed once every three offline phases (i.e., every 3 days). In each updating offline phase, DT-SMCTS aggregates the uploaded data from the most recent three shifts and invokes the PSDTS to perform simulation-based sorting optimization, thereby producing the updated sorting plan. Notably, the online and offline phases of CBSS correspond, respectively, to the data-acquisition and plan-update phases of the DT-SMCTS framework. Accordingly, DT-SMCTS operates as a macroscopic online framework, in which data from the most recent 3 days are continually collected and leveraged to update the sorting plan.

Data flow and plan update in DT-SMCTS: after each parcel-sorting shift, parcel-sorting data W and bin-packing data G are uploaded to the cloud via the IIoT platform. Once these data have been collected over three consecutive sorting shifts (i.e., 3 days), DT-SMCTS triggers a sorting plan update as detailed in Box 3 and in Fig. 7.

During each update cycle, the PSDTS first uses parcel-sorting data W to reset and calibrate the virtual parcel-generation module and uses bin-packing data G to update the maximum-capacity and bin-packing time parameters of the virtual grid module, so that the PSDTS reflects the most recent parcel-arrival and bin-packing characteristics. The DT-INLP module in DT-SMCTS also takes the new parcel-sorting data W as input and uses it to compute the empirical distribution of parcel volumes over parcel destinations. Based on this distribution, DT-INLP determines an appropriate number of grids to allocate to each destination. Next, leveraging this grid-quantity information, the GDGN module in DT-SMCTS produces a grid-selection rule, which serves as the expansion policy for MCTS and helps ensure both high search efficiency and high-quality solutions. Finally, the MCTS component in DT-SMCTS executes its standard four-step iterations (i.e., selection, expansion, simulation, and backpropagation). In each iteration, a candidate sorting plan is evaluated by the PSDTS, and the evaluation result is used to update the node values in MCTS. Once the iterations terminate, we obtain an updated sorting plan.

Notably, in each update cycle, the four-step MCTS procedure is not initialized from a fully stochastic sorting plan generated from scratch. Instead, we keep approximately 80% of the grid-to-destination mappings from the previous sorting plan (obtained 3 days earlier) and allow DT-SMCTS to rearrange only about 20% of the mappings. This design is motivated by an empirical trade-off observed in real-world parcel-sorting data from SF Express, as detailed in Supplementary Note 1. Selecting approximately the top 20% destinations by parcel volume already covers a substantial fraction of both the total parcel volume and demand drift, where the latter is quantified by the variance of destination-share fluctuations within the window. In contrast, increasing the fraction of destinations updated in each cycle further improves coverage but incurs higher computational cost. Thus, there is no need to substantially modify their associated grid-to-destination mappings, and the resulting partial update is computationally more efficient.

Deployment challenges: the deployment of our proposed method involves several practical considerations, including packer training, hardware integration with legacy systems, and coordination with downstream systems.

First, packers in PDCs are responsible for retrieving and affixing printed labels to packed bins. In this study, we assume that all packers are well-trained, so the total bin-packing time is calculated as the sum of the base packing time for replacing the bin and the label-fetching time, which depends on the distance between the grid and the nearest label printer. However, if the sorting plan is updated more frequently (e.g., every 2 days or less), packers may not receive sufficient training, which could reduce efficiency and slightly increase label-fetching times. To address this challenge, it is essential to adjust the label-fetching time simulated in our model according to the sorting plan’s update frequency, rather than treating it as a constant function of the grid-to-printer distance.

Second, DT-SMCTS relies on the IIoT platform for sensing and storing specific parcel data, which introduces system-level requirements. As detailed in the Results and Discussion section, the IIoT platform must collect the following data: (i) delivery timestamp with millisecond resolution, (ii) index of the induction station, (iii) scanning error code, (iv) index of the parcel scanner, and (v) parcel destination. For PDCs with legacy systems, adopting our proposed method requires updating the system to enable the sensing and storage of these data. Specifically, this may involve upgrading hardware components, such as scanners and sensors, as well as modifying communication protocols to ensure seamless data flow between the IIoT devices and the central system. These updates are essential for the successful adoption of our method in PDCs.

Finally, the deployment of DT-SMCTS will also impact downstream systems, such as truck scheduling. Specifically, DT-SMCTS assigns destinations to grids and updates these assignments every 3 days. Consequently, downstream systems must be informed of these updates on a three-day cycle to make necessary schedule adjustments, ensuring the overall efficiency of the logistics network.

Limitations: the proposed DT-SMCTS is subject to several limitations, including data dependency, computational complexity, and limited adaptability.

First, the establishment of the PSDTS requires key operational data from the PDCs, such as the positions of key sorting devices, conveyor travel speeds, bin capacity, and bin packing times. Additionally, the DT-SMCTS framework relies on historical parcel data collected by IIoT systems. Any inaccessibility of these data or discrepancies in their quality may adversely affect the performance of the proposed method.

Second, numerical results show that our method performs well for PDCs with up to 600 grids, which is currently considered a large PSS by SF Express. However, as the number of grids continues to increase over time, the computational demands and processing times of DT-SMCTS will rise significantly due to its tree-search nature. Should the sorting plan generation time exceed the update period, the effectiveness of our method would be compromised.

Finally, the proposed DT-SMCTS is developed for the CBSS in PDCs, and its adaptability depends on the underlying sorting mechanism of the PSS. For other conveyor-based sorting systems, the overall PSDTS pipeline can be largely transferred with system-specific re-parameterization and limited mechanism-level customization. In particular, the IIoT platform can be reused with minor adjustments to sensor placement and data interfaces, and the virtual parcel generation module remains directly applicable as long as parcel-level event logs (e.g., induction timestamps) are available. Besides, the sorting simulation module and visualization platform are also reusable, while the CBSS mechanism module primarily requires customization to match the target system’s conveyor layout, tray and chute configurations, scanning positions, and grid locking and packing rules.

Meanwhile, extending PSDTS to AGV-based sorting systems requires a more substantial redesign of the mechanism module to capture AGV motion, routing, dispatching, and interaction-induced congestion, together with corresponding adjustments of the IIoT instrumentation to record AGV trajectories and key operational events for DT calibration. Nevertheless, the virtual parcel generation module remains applicable, and the sorting simulation module and the visualization platform can be retained after being adapted to interface with the redesigned mechanism module. Importantly, for both the conveyor-based and AGV-based sorting systems, the underlying grid-to-destination assignment sorting optimization problem remains well defined, and DT-SMCTS can be applied as the DT-based optimization framework once the system-specific DT simulator is available for evaluating candidate plans.

Box 3 Overall DT-SMCTS framework.

Input: Portion of parcels targeted to each destination ${\left\{P_d\right\}}_{d=1}^D$, number of grids X, importance factors ω₁, ω₂, and terminating iterations threshold H_th.

DT-INLP module construction:

1: Determine the grid quantity assignment α^* based on ${\left\{P_d\right\}}_{d=1}^D$ according to Box 1;

GDGN module construction:

2: For each sorting segment e = 1, 2, construct graphs ${\mathcal{G}}_{e,1},{\mathcal{G}}_{e,2}$ based on the geographic adjacency and α^*;

Sorting plan derivation via DT-SMCTS:

3: Denote Π^* as the currently best sorting plan and initialize its throughput performance as R^* = − ∞;

4: Let current_episode = 0;

5: While current_episode < H_th

6: Begin from the root node s₁;

Selection:

7: While the node at the current depth is not the leaf node

8: Select the node at the next depth according to Eqs. 10 and 11;

9: End while

10: Denote the depth of the reached leaf node as L;

Expansion:

11: Input a_L to ${\mathcal{G}}_{e,1},{\mathcal{G}}_{e,2}$, and derive a_L + 1 according to Box 2;

Simulation:

12: Randomly select one action for each depth from L + 2 to X that satisfies the constraints in Eqs. 6, 7, and 8,

thereby obtaining the sorting plan Π;

13: Send Π to the PSDTS and acquire C_avg(Π) and C_peak(Π);

Backpropagation:

14: Update the attributes of the nodes from s_L + 2 to s₁ using Eq. 12;

15: If ω₁C_avg(Π) + ω₂ C_peak(Π) > R^*

16: Π^* = Π, R^* = ω₁ C_avg(Π) + ω₂ C_peak(Π);

17: End if

18: current_episode = current_episode + 1;

19: End while

Output: Sorting plan Π^*.

Author contributions

R. Li, W. Zhang, and C. Huang conceived the project. R. Li, W. Zhang, Y. Tian, and C. Huang wrote the manuscript. R. Li, W. Zhang, and C. Huang performed the modeling, algorithm design, and DT-based numerical experiments. W. Zhang, R. Liu, L. Liao, and X. Song supported the industrial implementation, data acquisition, and field tests. Y. Tian, R. Liu, B. Huang, L. Liao, and C. Huang contributed to data analysis and visualization. X. Qin, H. Zhang, S. Cui, and C. Huang provided critical revisions and supervised the study. All authors reviewed and approved the final manuscript. R. Li and W. Zhang contributed equally to this work and are listed alphabetically by last name. Both authors reserve the right to list their name first in their resumes.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

The authors disclose support for the research of this work from the Shenzhen High-Tech Zone Development Special Program Innovation Platform, under the auspices of the Nanshan District Science, Technology and Innovation Bureau of Shenzhen, in Shenzhen High-Tech Zone Project with Grant No. KC2022KCCX0041; the Basic Research Project No. HZQB-KCZYZ-2021067 of Hetao Shenzhen-HK S & T Cooperation Zone; the Guangdong Provincial Key Laboratory of Future Networks of Intelligence (Grant No. 2022B1212010001); and the Shenzhen Key Laboratory of Big Data and Artificial Intelligence (Grant No. SYSPG20241211173853027).

Data availability

The anonymized data associated with this study have been deposited in the Zenodo repository⁶⁸ under https://doi.org/10.5281/zenodo.21319488. These data include the anonymized proportions of parcels destined for each destination during the sorting shifts at the four parcel distribution centers reported in the paper, and the anonymized proportions of parcels destined for each destination over 8 consecutive days. The parcel-related raw operational data are available under restricted access because they contain proprietary and commercially sensitive information owned by SF Express and are subject to strict confidentiality restrictions under the data use agreement with the data provider. Access can be obtained by contacting songxiang1@sf-express.com. Requests will be reviewed by the corresponding author and SF Express within 60 days. Access may be granted for non-commercial research purposes and for verification of the results reported in this study, subject to approval by SF Express and completion of a data use agreement. Once access has been granted, the data will be available for an initial period of 1 month, renewable upon approval. Source data are provided with this paper.

Code availability

The source code for the proposed DT-SMCTS framework has been deposited in a GitHub repository and archived in Zenodo⁶⁸ under https://doi.org/10.5281/zenodo.21319488. The source code for the PSDTS is proprietary to SF Express and is available under restricted access because it contains proprietary and commercially sensitive implementation details of the company’s operational system and is subject to confidentiality restrictions under the data use agreement with the data provider. Access to the PSDTS source code may be requested by contacting songxiang1@sf-express.com and is subject to approval by the corresponding author and SF Express and completion of a confidentiality agreement.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41467-026-76915-4.