Mathematical Modeling and Optimization of a Two-Layer Metro-Based Underground Logistics System Network: A Case Study of Nanjing

Abstract

With the surge in urban logistics demand, traditional surface transportation faces challenges, such as traffic congestion and environmental pollution. Leveraging metro systems in metropolitan areas for both passenger commuting and underground logistics presents a promising solution. The metro-based underground logistics system (M-ULS), characterized by extensive coverage and independent right-of-way, has emerged as a potential approach for optimizing urban freight transport. However, existing studies primarily focus on single-line scenarios, lacking in-depth analyses of multi-tier network coordination and dynamic demand responsiveness. This study proposes an optimization framework based on mixed-integer programming and an improved ICSA to address three key challenges in metro freight network planning: balancing passenger and freight demand, optimizing multi-tier node layout, and enhancing computational efficiency for large-scale problem solving. By integrating E-TOPSIS for demand assessment and an adaptive mutation mechanism based on a normal distribution, the solution space is reduced from five to three dimensions, significantly improving algorithm convergence and global search capability. Using the Nanjing metro network as a case study, this research compares the optimization performance of independent line and transshipment-enabled network scenarios. The results indicate that the networked scenario (daily cost: CNY 1.743 million) outperforms the independent line scenario (daily cost: CNY 1.960 million) in terms of freight volume (3.214 million parcels/day) and road traffic alleviation rate (89.19%). However, it also requires a more complex node configuration. This study provides both theoretical and empirical support for planning high-density urban underground logistics systems, demonstrating the potential of multimodal transport networks and intelligent optimization algorithms.

1. Introduction

Urban logistics, primarily reliant on city road networks, has evolved into a high-pollution, high-consumption, and low-value-added industry. Against the backdrop of surging logistics demand, its adverse effects on urban traffic, the environment, and public safety have become critical societal concerns [1,2,3], manifesting as traffic congestion, pollutant emissions, energy consumption, accidents, acoustic and visual disturbances, inefficient land use, and infrastructure deterioration [4,5,6]. As an emerging and sustainable transport infrastructure, the Underground Logistics System (ULS) offers an optimal solution for transforming urban logistics, particularly given the constraints of limited urban land resources [7,8,9].

The ULS is a dedicated subterranean transportation network consisting of specialized tunnels and node facilities that connect suburban logistics hubs or distribution centers with urban customers [10,11]. By operating independently of surface roads, ULS alleviates traffic congestion, employs clean energy sources to mitigate urban pollution, and remains unaffected by external disruptions, ensuring greater reliability and efficiency in freight transportation. In 2019, the Outline for Building a Transport Powerhouse issued by the Central Committee of the Communist Party of China and the State Council explicitly emphasized the development of “urban underground logistics distribution systems” to establish a safe, convenient, efficient, green, and cost-effective modern transportation network [7,12,13]. Moreover, during major public emergencies, such as the COVID-19 pandemic, urban logistics costs surged, driver shortages became severe in heavily affected areas, and additional manual inspection and quarantine procedures for inbound and outbound goods increased human contact risks. ULS presents a viable solution for addressing such emergency logistics challenges effectively.

Among various underground logistics solutions, the metro-based underground logistics system (M-ULS), which leverages urban metro networks for the integrated transportation of passengers and freight, is considered a priority development model for future urban logistics due to its relatively low construction costs and minimal retrofitting requirements [14,15,16]. The advantages of an M-ULS are evident, as demonstrated by a series of point-to-point, single-line urban rail freight projects implemented in Dresden, Amsterdam, Zurich, Bremen, and Paris, among other European cities [17,18,19]. In China, a pilot project has been launched in Tongzhou, Beijing [20,21]. When fully operational, the M-ULS network can achieve comparable socio-environmental benefits to independently constructed ULS networks. Additionally, co-developing an M-ULS alongside metro systems significantly reduces construction costs and optimizes underground space utilization. By capitalizing on existing metro design, construction, and operational practices, an M-ULS can be rapidly deployed and scaled, thereby minimizing investment risks and accelerating commercial implementation.

However, research on the M-ULS remains in its early stages, and existing network planning knowledge fails to address practical questions, such as “What are the fundamental structural forms of an M-ULS network?” or “How should an M-ULS network be hierarchically structured in a real urban environment?” Current studies primarily focus on assessing the feasibility of an M-ULS and developing simplified theoretical network models [22,23,24]. For instance, Motraghi et al. [25] conducted a discrete-event simulation analysis of mixed passenger–freight metro operations on Newcastle’s metro network in the UK. Behiri et al. [26] formulated a train scheduling model for metro locomotives under the passenger–freight shared track mode. Zhao et al. [27] developed a freight station location evaluation model based on Shanghai Metro. Sun et al.’s [28] study proposed effective modeling and optimization techniques for planning a city-wide M-ULS network. A mixed integer programming model, with a well-matched solution framework combining a multi-objective PSO algorithm and an A* algorithm, were developed to optimize the location–allocation–routing (LAR) decisions of the M-ULS network. Hu et al. [29] proposed an equilibrium chance-constrained programming (ECP) approach for the M-ULS network planning problem (MULNP) considering uncertain demand and costs. Chen et al. [30] explored the decision making mechanisms of M-ULS network expansion based on multi-source data collection and mathematical programming. A model with three interrelated modules was proposed and applied in the Beijing city case. To address these research gaps, this study develops an optimization framework for an M-ULS network layout in real urban environments with complex spatial constraints, leveraging mathematical modeling and algorithmic simulation techniques. The proposed framework aims to provide theoretical support for the planning and implementation of metro freight systems and other forms of ULS in China.

The planning and implementation of an M-ULS represent a transformative approach to mitigating urban freight-related congestion, pollution, and inefficiencies. Despite growing interest, existing studies often overlook the structural complexity and hierarchical coordination required for large-scale metro freight networks. This study aims to bridge this gap by addressing the following research questions:

(1)

What is an effective mathematical framework for optimizing the location allocation of multi-tier nodes (including surface terminals, metro freight stations, and underground logistics hubs) within an M-ULS network?

(2)

How can algorithmic efficiency be improved to solve large-scale M-ULS network planning problems under real-world spatial and capacity constraints?

(3)

What are the comparative advantages of networked (with transshipment) versus independent line operational modes in terms of cost and freight volume?

By answering these questions, this study contributes to both the theoretical understanding and practical planning of sustainable urban freight systems.

The structure of this paper is as follows: Section 2 systematically analyzes the characteristics of metro-based freight networks and the key challenges in their planning. Section 3 develops a coverage model and a mixed-integer programming model for node location allocation, along with a corresponding solution algorithm. Section 4 applies the proposed models to the Nanjing metro network, comparing the optimization outcomes between independent line operations and networked transfer scenarios. Finally, Section 5 summarizes the key findings and discusses future research directions.

2. Literature Review

For a long time, the academic community both domestically and internationally has made significant progress in research on metro-based freight systems, particularly in the areas of feasibility, network layout, operational scheduling, and economic benefit evaluation. Regarding feasibility, Hu et al. [31] proposed an innovative concept for a metro-based freight system prototype, describing the demand flow, hierarchical structure, facility characteristics, and technical features of the M-ULS network, demonstrating that the M-ULS is technically feasible. In terms of operational scheduling, Behiri et al. [26] developed a train scheduling model for metro locomotives operating in a “passenger-freight shared line” mode. With regard to network layout, Dong et al. [16] constructed a bi-line cross-type node location model for an M-ULS. For economic benefit evaluation, Xu et al. [8] employed a system dynamics approach to quantify the impact of metro-based freight implementation on urban logistics performance under public health emergency scenarios.

In recent years, resilience and sustainability have been incorporated into research on metro-based underground logistics systems (M-ULS). Scholars have focused not only on node location optimization during the planning phase but also on resilience assessment during the operational phase. Xue et al. [32] developed a two-stage stochastic programming model to maximize the disaster resilience of underground logistics networks under disaster scenarios, solving large-scale instances via a hybrid heuristic algorithm. Liu et al. [13] designed a multi-level Underground Waste Collection System (UWCS) and formulated a Mixed-Integer Linear Programming (MILP) model to minimize total cost; this model was effectively solved using a combination of a genetic greedy algorithm and variable neighborhood search (GGA-GVNS) to address the optimization challenges of complex networks. Meanwhile, Zheng et al. [33] applied complex network theory to construct a metro station importance evaluation system and a bi-level programming location model. They emphasized the necessity of comprehensively considering multiple factors in the planning phase—such as nodal topological attributes, regional accessibility, total cost, and customer satisfaction—thus providing a scientific decision making framework for an M-ULS node layout. Li et al. [34] proposed a resilience assessment method for an M-ULS from the perspective of multi-layer interdependent networks. Through simulation analysis, they revealed the impact of factors like logistics facility failures and train service interruptions on system performance, offering a quantitative tool for stability during the operational phase. These two studies have enriched the theoretical system of an M-ULS from the dimensions of “planning optimization” and “operational resilience,” respectively, collectively advancing the development of metro-based logistics systems toward higher efficiency, reliability, and sustainability.

Despite these advancements, several research gaps remain to be addressed. Most existing studies focus on single-line or simplified network architectures, with limited attention paid to complex structures that support multi-level transfers. Furthermore, there is still a lack of integrated models capable of simultaneously addressing multidimensional challenges, such as node location, flow allocation, capacity constraints, hierarchical coordination, and resilience recovery, in large-scale urban environments. From a methodological perspective, although heuristic and meta-heuristic algorithms like Genetic Algorithms (GA), Immune Clone Selection Algorithms (ICSA), and variable neighborhood search (VNS) have been widely applied, there is a continued need to develop more adaptive and efficient solution frameworks to tackle the NP-hard nature of large-scale network design problems.

This study aims to address these gaps by proposing a comprehensive optimization framework for a two-layer metro-based underground logistics network, incorporating mixed-integer programming and an improved algorithm for enhanced convergence and solution quality. The case study of Nanjing metro network serves to validate the model’s applicability and performance in a real-world setting.

3. Model Development

3.1. Problem Statement and Notations

The earliest planning and operational optimization of urban freight transportation based on metro systems must fully account for the following characteristics [35,36,37,38]:

(1) Limited system service capacity: This constraint is primarily reflected in three aspects: the cargo handling capacity of different network nodes, the service coverage of each node, and the transport capacity of metro tunnels.

(2) Complex hierarchical structure of multimodal transport networks: While metro freight networks serve as the final segment of the logistics supply chain, the underground infrastructure itself follows a layered structure. Additionally, various external facilities (such as urban logistics hubs, newly constructed surface terminals, and customer demand points) interact with the metro freight network.

(3) Complexity of passenger–freight mixed operations: Metro networks are planned and developed based on urban and regional passenger flows. The implementation of mixed passenger–freight transportation must not compromise the functionality and efficiency of passenger services. Therefore, the differences in flow patterns between passengers and freight must be carefully considered to ensure the scientific selection and planning of metro freight network locations.

A valid operation model of an M-ULS has been proposed by Dong et al. [16]. In the model, the freight vehicles (FVs) were designed to share the collinear rail track with passenger vehicles. They depart from the urban distribution center (designated as UDC) and then drive into the metro network through a connected tunnel. FVs discharge and upload the pallets at a bypass platform in the metro stations with the freight function (designated as metro freight stations, MFSs). The automated internal transport system at each MFS will dispatch the pallets through the dedicated shaft to the upper level. The ground electric vehicle will complete the delivery to each customer within the service scope. The process also includes packaging and sorting. Reverse operation is the whole process of goods entering the M-ULS from the ground.

The service scope of the single line is obviously limited due to the dispersion of customers. Thus, this study constructs an M-ULS network consisting of four metro lines. Transfer stations are set up to support the network’s operation. The special station is named the underground logistics hub (ULH). The system characteristics of M-ULS network planning are as follows.

First, the system service capacity of the M-ULS is limited, which is mainly reflected in three aspects:

• The handling capacity of the depot. An MFS is reconstructed based on the traditional metro station. The capacity of the MFS is restricted by the efficiency and availability of the transfer equipment, the limitation of storage space, and the delivery schedule. An MFS should support rapid deliveries and ground transshipment. When goods arrive at the underground destination, they are transferred above quickly for further packaging and sorting. A reconstructed station with a double-decked structure is available for underground temporary storage, which leaves a whole floor underground for freight order processing.
• The impact of the “last mile delivery.” The last mile of the M-ULS refers to the ground distribution of goods from the MFS to customers. It should be emphasized that the travel distance of end supply would be optimized to ensure minimal burden on traffic mobility.
• The transport capacity of the metro tunnel. The departure frequency of vehicles is limited. Accordingly, sometimes, the freight schedule must be compressed or rearranged to assure passenger service priority.

Second, an M-ULS network involves not only the physical space above or underground but also supply chain activity subjects, such as upstream suppliers, customers, and carriers. Network nodes should be configured with different functions to optimize network efficiency. Figure 1 illustrates the framework of system behavior, in which the nodes are classified into four levels according to their position in the network. A multi-level node layout under different functional combinations is important for M-ULS network planning.

Third, the operational organization of the hybrid underground transport mode is complicated. The metro stations are planned to be located to meet passengers’ travel needs. However, the directions of passenger flow and freight flow are not always consistent. The freight Origin–Destination (OD) matrix needs to be evaluated to measure whether the underground logistics line is worth building. A reasonable network layout should improve these scenarios with minimal cost.

To simplify the modeling process, the following assumptions are made:

(1)

The freight transportation schedule cannot alter the original metro timetable.

(2)

Transfer times at MFSs are not considered, but transfer costs are accounted for.

(3)

The metro–freight system does not accommodate customer-to-customer (C2C) logistics within the city.

(4)

The OD matrix represents the daily freight volume exchanged between each demand point and various logistics hubs.

(5)

The impact of the urban road network on last-mile ground distribution is disregarded, simplifying this process to a point-to-point direct connection, which can be solved using the P-median model.

(6)

Transfer MFSs are not permitted to provide distribution services to surface terminals.

(7)

Each candidate surface terminal is exclusively assigned to a single metro freight node, meaning that the freight source is unique.

(8)

The construction and storage conditions of metro freight nodes are not considered.

The parameters and corresponding definitions used in the proposed model are summarized in

Table 1. Notation of sets, parameters, and variables.

Symbol	Definition
Set
$U\left(j\right)$	Set of surface terminals covering demand points ${\beta }_i$
$V\left(i\right)$	Set of all demand points covered by surface terminals ${\alpha }_j$
$Y_{ij}$	Freight allocation coefficient from ${\alpha }_j$ to ${\beta }_i$
Constant parameter
$C_j^D$,$C_u^H$	Construction and maintenance cost of candidate MFS $j\in R_D$ and ULH $u\in R_D$, respectively
$c_{ux}^{k_{xj}}$,$c_{dx}^{k_{ij}}$	Shipping price of unit OD pair $Q_i^x$ at arc $\left(x,j\right)$ and arc $\left(i,j\right)$, respectively
${\delta }^{k_{xj}}$	Transshipment price of underground OD pair $\left(x,j\right)$
$\left[Q_{\max }^j\right]$,$\left[D_{\max }^j\right]$	Freight handling capacity and maximum ground distribution distance at candidate MFS $j\in R_D$
$\left[{\omega }_{\max }^x\right]$,$\left[T_{\max }^u\right]$	Transport capacity of metro line $x\in N$ and underground transfer capacity at ULH $u\in R_D$
$Q_i^x$	Freight OD pair between customer $i\in {\mathsf{\Gamma }}_C$ and UDC $x\in N$
Decision variables
$X_j$	Surface terminal is established at location ${\alpha }_j$
$m_j$	1, if candidate metro station $j\in R_D$ is selected as MFS, and 0 otherwise
$f_i^x$	1, if customer $i\in {\mathsf{\Gamma }}_C$ is selected to accept M-ULS service from UDC $x\in N$, and 0 otherwise
$z_{ij}$	1, if customer $i\in {\mathsf{\Gamma }}_C$ is visited by the EV from MFS $j\in R_D$, and 0 otherwise
$v_j^x$	1, if MFD $j\in R_D$ exists on the metro line directed by UDC $x\in N$, and 0 otherwise
$v_u^x$	1, if ULH $u\in R_D$ successfully transfer OD pairs from other UDCs to a certain line X, and 0 otherwise
$y_j^x$	1, if sub arc $\left(x,j\right)$ is routed by the freight OD from UDC $x\in N$, and 0 otherwise

.

3.2. E-TOPSIS-Based Freight Flow Screening Model

For Decision Object 1, key information was extracted from the urban freight demand O–D characteristics to construct three evaluation indicators: freight flow singularity, regional accessibility, and order priority. These indicators are used to screen O–D pairs suitable for metro transportation and determine the service scope of the M-ULS.

(1) Freight Flow Singularity

Metro freight should leverage its advantages of long-distance and underground direct delivery to concentrate supply toward areas with high demand, thereby achieving economies of scale. Priority is given to transporting goods to areas that are farther from the source and have higher demand, as expressed in Equation (1).

$$F_s={\displaystyle \sum _{i\in {\mathsf{\Gamma }}_C} f_{ix}\cdot d_{ix}}\cdot {\left(Q_{ix}\right)}^{{\rho }_i^x} \forall x\in N$$

(1)

A larger $F_s$ indicates a higher requirement for centralized transportation of goods from park x; ${\rho }_i^x$ is the freight volume correction coefficient, taken in this study as [0.1, 0.45].

(2) Regional Accessibility

The efficiency of ground distribution is often affected by road conditions. Prioritizing the underground transportation of goods in congested areas helps improve traffic flow. The regional accessibility indicator is characterized by a time–road response function, represented as the ratio of cumulative delivery time to average delivery path length, as proposed by Masson et al. [39], shown in Equation (2).

$$A_s={\displaystyle \sum _{i\in {\mathsf{\Gamma }}_C}\left[1-\left(\frac{t_{ix}\cdot f_{ix}}{T_{ix}^h}\cdot {‖\frac{S_{ix}^h}{d_{ix}}‖}^{{\phi }_h}\right)\right]}\forall x\in N$$

(2)

where ${\phi }_h$ is the congestion coefficient, $S_{ix}^h$ is determined based on the road network layout in the case study, and $t_{ix}$ and $T_{ix}^h$ are obtained by dividing the corresponding path length by the transport speed of the mode (M-ULS or road vehicle).

(3) Order Priority

Similarly to through-type small distribution centers, the MFS does not provide storage for large quantities of goods. When there are many orders, priority management for M-ULS transportation becomes crucial. This is modeled as a function between unit transportation profit and response time satisfaction.

$$P_s={{\displaystyle \sum _{i\in {\mathsf{\Gamma }}_C}{Q}_{ix}\cdot b_{ix}\cdot \left(\frac{L_{\max }-{\mu }_{ix}}{L_{\max }-L_{\min }}\right)}}^{\gamma }\forall x\in N$$

(3)

where $b_{ix}$ is the order profit in the traditional mode; ${\mu }_{ix}$ is the delivery time window, satisfying $L_{\min }<{\mu }_{ix}<L_{\max }$ within the maximum–minimum response time range; and $\gamma$ is the time sensitivity factor, with a value in [0.5, 1].

The entropy method calculates the information entropy from observed values of each indicator to determine the indicator weights. The TOPSIS method ranks multi-objective decisions by calculating the distance between evaluation alternatives and the optimal or worst system service. The E-TOPSIS method combines both, using the indicator weights obtained from entropy calculation as input for multi-objective evaluation. It is an effective and commonly used evaluation method that avoids subjective decision making. The construction process of the E-TOPSIS model for metro freight O–D screening is as follows.

Step 1: Construct a dimensionless discriminant matrix $R=\left\{F_s,A_s,P_s\right\}=\left(w_{ijk}\right)$ for the indicators, where $w_{ijk}$ is the observed value of the j-th subsystem (i.e., network flow selection alternative) relative to the k-th logistics park for evaluating the i-th indicator. The best value for the i-th indicator is taken as $w_i^{best}$ and the worst value as $w_i^{worst}$. Introduce the efficacy coefficient $\eta$ for transformation to obtain the normalized data matrix E.

$$E={\left[w_{ijk}\right]}^{\prime }={\displaystyle \frac{w_{ijk}-w_i^{worst}}{w_i^{best}-w_i^{worst}}}\cdot \eta +\left(1-\eta \right)$$

(4)

Step 2: Define the characteristic proportion $P_{ij}={\displaystyle \sum _{k=1}^l\left(w_{ijk}/{\displaystyle \sum _{j=1}^n{w}_{ijk}}\right)}$ of the j-th subsystem under indicator i, and calculate the entropy value ${\epsilon }_i=\left(-{\displaystyle \frac{1}{\ln \left(n\right)}}\right){\displaystyle \sum _{j=1}^n{P}_{ij}\ln \left(P_{ij}\right)}$ of the i-th indicator. Define the differentiation coefficient $a_i=1-{\epsilon }_i$ to reflect the comparative role of the indicator in the system, and obtain the normalized indicator weight coefficient $Z={\left(z_1,z_2,\dots ,z_m\right)}^T$, where $z_i=a_i/{\displaystyle \sum _{i=1}^m{a}_i}$.

Step 3: Calculate the relative closeness ${\lambda }_j={\upsilon }_j^{-}/\left({\upsilon }_j^{+}+{\upsilon }_j^{-}\right)$ of the flow selection alternatives in this stage to the positive and negative ideal solutions under the weight coefficient X, where ${\upsilon }_j^{+\left(-\right)}$ is the Euclidean distance between the current alternative and the optimal or worst system service. The relative closeness of an alternative ranges between 0 and 1. Finally, based on the critical closeness value, rank ${\lambda }_j$ in descending order and sequentially select the corresponding customers and O–D pairs to be included in the M-ULS service scope until the maximum network capacity is reached. Denote the O–D screening model as Model P(1), expressed as follows.

$$\overline{\mathsf{\Omega }}={\mathsf{\Omega }}_i^x\cdot H_j ;\forall H_j=\left\{\begin{array}{l}{\left(1,1,\dots ,1\right)}^T, 1\ge {\lambda }_j\ge {\overline{\lambda }}_j\\ {\left(0,0,\dots ,0\right)}^T, 0\le {\lambda }_j\le {\overline{\lambda }}_j\end{array}\right.$$

(5)

3.3. Coverage Model for Surface Terminal Location

Based on the parameter definitions provided in Table 1 above, the following mixed-integer programming model P(2) is formulated.

$$\mathrm{Min} {\displaystyle \sum _{j,{\alpha }_j\in {\xi }_A}{X}_j}$$

(6)

$${\displaystyle \sum _{j,{\alpha }_j\in V\left(i\right)}{Y}_{ij}}=1,\hspace{1em}\forall {\beta }_i\in {\xi }_A$$

(7)

$${\displaystyle \sum _{x=1}^n{\displaystyle \sum _{i,{\beta }_i\in U\left(j\right)}{Q}_i^x{Y}_{ij}}}\le \left[Q_{\max }^r\right]X_j\hspace{1em}\forall r\in {\mathsf{\Gamma }}_K$$

(8)

$$L\left(i,j\right)\le R,\forall {\beta }_i\in U\left(j\right);{\alpha }_j\in V\left(i\right)$$

(9)

$$X_j\in \left\{0,1\right\},\forall {\alpha }_j\in V\left(i\right)$$

(10)

The objective function (6) aims to minimize the number of surface terminals required for complete coverage. Constraint (7) ensures that the total demand allocated to all terminal nodes matches the underground freight OD volume. Constraints (8) and (9) define the maximum cargo handling capacity and service radius of each surface terminal, respectively. Equation (10) specifies the feasible range of the decision variables.

To standardize the formulation, the above model is expressed in matrix form as $W\cdot K\le b$, where W is a matrix and K and b are vectors.

The matrix W is a 2n times n(n + 1) matrix. Specifically, in the first n rows, the first n columns contain only zeros; the diagonal elements from column (n + 1) to 2n are denoted as $D_{11},\cdots ,D_{n1}$, while all other elements are zero, following this pattern. In rows (n + 1) to 2n, the diagonal elements in the first n columns are $-\left[Q_{\max }^r\right]$, while all other elements in these columns are zero. From column (n + 1) to 2n, the elements in the (n + 1) th row are $D_{11}^r{Q}_1^x,\cdots ,D_{n1}^r{Q}_n^x$, while all other rows contain only zeros, following the same pattern. The matrix D is defined as follows:

$$D_{ij}=\left\{\begin{array}{l}1,\hspace{1em}L\left(i,j\right)\le R\\ 0,\hspace{1em}L\left(i,j\right)>R\end{array}\right.$$

(11)

$$W={\displaystyle \sum _{x\in N}\stackrel{\mathrm{n}\cdot (\mathrm{n}+1) \mathrm{column}}{\left[\stackrel{\cdots \cdots \cdots \cdots \cdots \cdots \mathrm{n}\cdots \cdots \cdots \cdots \cdots \cdots }{\begin{array}{cccc}0& 0& \cdots & 0\\ 0& 0& \cdots & 0\\ ⋮& \ddots & \cdots & ⋮\\ 0& 0& \cdots & 0\\ -[Q_{\max }^r]& 0& \cdots & 0\\ 0& -[Q_{\max }^r]& \cdots & 0\\ ⋮& \ddots & \cdots & ⋮\\ 0& 0& \cdots & -[Q_{\max }^r]\end{array}}\stackrel{\cdots \cdots \cdots \cdots \cdots \cdots \mathrm{n}\cdots \cdots \cdots \cdots \cdots \cdots }{\begin{array}{cccc}{D}_{11}& 0& \cdots & 0\\ 0& D_{21}& \cdots & 0\\ ⋮& \ddots & \cdots & ⋮\\ 0& 0& \cdots & D_{n1}\\ D_{11}{Q}_{1x}& D_{21}{Q}_{2x}& \cdots & D_{n1}{Q}_{nx}\\ 0& 0& \cdots & 0\\ ⋮& \ddots & \cdots & ⋮\\ 0& 0& \cdots & 0\end{array}}\stackrel{\cdots \cdots \cdots \cdots \cdots \cdots \mathrm{n}\cdots \cdots \cdots \cdots \cdots \cdots }{\begin{array}{cccc}0& 0& \cdots & 0\\ 0& D_{22}& \cdots & 0\\ ⋮& \ddots & \cdots & ⋮\\ 0& 0& \cdots & D_{n2}\\ 0& 0& \cdots & 0\\ D_{12}{Q}_{1x}& D_{22}{Q}_{2x}& \cdots & D_{n2}{Q}_{nx}\\ ⋮& \ddots & \cdots & ⋮\\ 0& 0& \cdots & 0\end{array}}\begin{array}{c}\cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \\ \cdots \end{array}\stackrel{\cdots \cdots \cdots \cdots \cdots \cdots \mathrm{n}\cdots \cdots \cdots \cdots \cdots \cdots }{\begin{array}{cccc}{D}_{1n}& 0& \cdots & 0\\ 0& D_{2n}& \cdots & 0\\ ⋮& \ddots & \cdots & ⋮\\ 0& 0& \cdots & D_{nn}\\ 0& 0& \cdots & 0\\ 0& 0& \cdots & 0\\ ⋮& \ddots & \cdots & ⋮\\ D_{1n}{Q}_{1x}& D_{n2}{Q}_{2x}& \cdots & D_{nn}{Q}_{nx}\end{array}}\right]}}$$

$$\begin{array}{l}K={\left\{X_1,\cdots ,X_n,Y_{11},\cdots ,Y_{n1},Y_{12},\cdots ,Y_{n2},\cdots ,Y_{1n},\cdots ,Y_{nn}\right\}}^T;\\ b={\left\{-1,\cdots ,-1,0\cdots ,0\right\}}^T\end{array}$$

(12)

The vector K is a $n+n^2$-dimensional vector. The vector b consists of −1 for the first n rows and 0 for the remaining n rows.

All solutions K that simultaneously satisfy conditions $W\cdot K\le b$, $0\le Y_{ij}\le 1$, and $i,j\in V_R^{\ast }$ constitute the feasible solution space for surface terminal location selection, ensuring compliance with both the maximum service capacity constraint and the full coverage requirement of the system service area.

3.4. Metro Freight Node Location Allocation Model

The MIP model of an M-ULS network can be formulated as follows:

$$\begin{array}{l}\mathrm{minimize}{\displaystyle \sum _{x\in N}{\displaystyle \sum _{u,j\in R_D}\left(m_j{C}_j^D+{\upsilon }_j^x{C}_u^H\right)}}\\ +{\displaystyle \sum _{i\in {\mathsf{\Gamma }}_C}{\displaystyle \sum _{u,j\in R_D}{\displaystyle \sum _{x\in N}\left(c_{dx}^{k_{ij}}{m}_j{z}_{ij}+c_{ux}^{k_{xj}}{m}_j{f}_i^x+{\delta }^{k_{xj}}\left(1-{\upsilon }_j^x\right)\right)}}}\end{array}$$

(13)

subject to

$${\displaystyle \sum _{i\in {\mathsf{\Gamma }}_C}{Q}_i^x{f}_i^x}+{\displaystyle \sum _{i\in {\mathsf{\Gamma }}_C}{\displaystyle \sum _{j\in R_D}{Q}_i^t{z}_{ij}{\displaystyle \sum _{u\in R_D}{\vartheta }_u^x}}}\le \left[{\omega }_{\max }^x\right],\hspace{1em}\forall t,x\in N,t\ne x$$

(14)

$${\displaystyle \sum _{i\in {\mathsf{\Gamma }}_C}{\displaystyle \sum _{x\in N}{Q}_i^x{f}_i^x{z}_{ij}{m}_j}}\le \left[Q_{\max }^j\right],\hspace{1em}\forall j\in R_D$$

(15)

$${\displaystyle \sum _{i\in {\mathsf{\Gamma }}_C}{\displaystyle \sum _{j\in R_D}{\displaystyle \sum _{x,t\in N}{Q}_i^t{z}_{ij}{\vartheta }_u^x}}}\le \left[T_{\max }^u\right],\hspace{1em}\forall u\in R_D, \forall t\ne x\underset{x\to \infty }{\lim }$$

(16)

$$\Vert Z_i-Z_j\Vert f_i^x{z}_{ij}\le \left[D_{\max }^j\right]m_j,\hspace{1em}\forall i\in R_D.j\in R_D$$

(17)

$${\displaystyle \sum _{j\in R_D}{z}_{ij}}=1, {\displaystyle \sum _{i\in {\mathsf{\Gamma }}_C}{z}_{ij}}\ge 1,\hspace{1em}\forall f_i^x=1, \forall i\in {\mathsf{\Gamma }}_C, \forall j\in R_D$$

(18)

$$z_{ij}\le m_j, \forall f_i^x=1, \forall i\in {\mathsf{\Gamma }}_C, \forall j\in R_D$$

(19)

$${\displaystyle \sum _{(x,j)\in K}{\displaystyle \sum _{x\in N}{y}_j^x}}={\displaystyle \sum _{j\in R_D}{m}_j},\hspace{1em}\forall j\in R_D$$

(20)

$${\displaystyle \sum _{u\in R_D}{\vartheta }_u^x}\le 2,\hspace{1em}\forall x\in N,\forall \left(x,j\right)\in K$$

(21)

$$m_j,f_i^x,z_{ij},{\upsilon }_j^x,{\vartheta }_u^x,y_j^x\in \left\{0,1\right\},\hspace{1em}\forall i\in {\mathsf{\Gamma }}_C,\forall j\in R_D,\forall u\in R_D,\forall x\in N$$

(22)

The objective function (13) represents the minimum cost of M-ULS network setting and operation, which consists of the fixed cost of depot infrastructures and the transportation cost of freight OD. Constraints (14), (15), and (16) ensure that the capacity of the metro tunnel, the terminal facility, and underground transshipment are not violated, respectively. Constraint (17) represents the distance restriction of last-mile delivery from MFD to customers. Constraint (18) guarantees that each involved customer is connected to at least one MFD, for which ownership is unique. Constraint (19) guarantees that the OD flow can only be allocated to the metro station with cargo function. Constraint (20) ensures that each selected MFD is visited by both ground and underground vehicles. Constraint (21) specifies for any OD pair that the frequency of its underground transshipment is no more than twice; Constrain (22) consists of the range of relevant decision variables.

4. Algorithm Solution

4.1. Model Decomposition and Simplification

The above mathematical model is an NP-Hard problem with complex network characteristics integrating multiple layers: a customer layer, an intermediate layer (surface terminals), an underground connection layer (metro freight nodes and MFS hubs), and a supply layer (urban logistics hubs). The key challenge in solving this problem lies in the fact that no single layer’s node information can be fully retrieved, and the choice of nonlinear optimization approach significantly impacts algorithmic complexity. Once the supply relationships and surface terminal generation strategy are determined, the MC-LAP problem simplifies into a capacity-constrained multi-stage joint transportation network design problem. To enhance computational efficiency, the following solution strategy is adopted:

(1) Reducing the supply layer: The opening decisions of urban logistics hubs and activation decisions of transfer MFS hubs are treated as control variables.

(2) Reducing the customer layer: The E-TOPSIS method is employed to evaluate the characteristics of demand points within a randomly generated set of surface terminals. Instead of assigning individual demand points, a set of candidate surface terminals is used, thereby simplifying the customer layer.

(3) Optimizing metro freight node selection: Under the current opening strategy, an optimal subset of surface terminals is selected and assigned to metro freight nodes. A shortest-path-based routing mechanism is integrated for network navigation.

(4) Dimensionality reduction: The solution space is reduced from five-dimensional ($X\times R_D\times R_H\times {\mathsf{\Gamma }}_K\times {\mathsf{\Gamma }}_C$) to three-dimensional ($X\times R_D\times {\mathsf{\Gamma }}_K$). Within this lower-dimensional space, the Immune Clone Selection Algorithm (ICSA) is applied to determine the priority opening locations of metro freight nodes and the optimal placement of surface terminals.

(5) Iterative updating: Served demand points are removed, and real-time flow and allocation results are fed back into the upper-level model for further evaluation and iteration. This process continues until no new surface terminals can be generated or the system’s service capacity limit is reached.

The solution procedure comprises three stages. First, we perform dimensionality reduction of decision variables across the supply, customer, and connection layers to structurally decompose the complex location–allocation–routing model and employ the E-TOPSIS method to screen and evaluate candidate surface terminals, thereby simplifying demand-side modeling complexity. Second, an exact set-covering algorithm is applied to optimize the siting of surface terminals. Finally, an improved ICSA is used to configure metro freight nodes; by integrating an adaptive Gaussian mutation mechanism and real-time freight flow feedback, the algorithm enhances global search capability and mitigates premature convergence [40,41]. The algorithm workflow and model decomposition are illustrated in Figure 2.

4.2. Exact Algorithm for Set Covering

Based on the results of Model P(1), the exact algorithm for solving the set covering model P(2) for surface terminal location is implemented as follows, and the algorithm pseudocode is shown in

Table 2. Parameters of Nanjing M-ULS.

Parameter	Value	Unit	Parameter	Value	Unit
${\delta }^{k_{xj}}$	$U\left(0.2,0.5\right)$	CNY/each parcel	$\left[T_{\max }^x\right]$	3 × 105	Parcels/per day
$C_j^D$	$U$(10,000, 50,000)	CNY/per day	$\left[Q_{\max }^j\right]$	$U$(80,000, 120,000)	Parcels/per day
$C_u^H$	50,000	CNY/per day	$\left[D_{\max }^j\right]$	2	Kilometer
$\left[{\omega }_{\max }^x\right]$	7.5 × 105	Parcels/per day	$c_{ux}^{k_{xj}}$/$c_{dx}^{k_{ij}}$	40/100	CNY/per k parcel·km

.

Step 1: Initialization. Set all $X_j=0,Y_{ij}=0$, and clear sets $U\left(j\right)$ and $V\left(i\right)$.

Step 2: Selecting the next surface terminal. Select $X_j=0$ from ${\mathsf{\Gamma }}_C$, and identify $j^{\prime }$ such that $\left|\left.U\left(j^{\prime }\right)\right|\right.=\max \left\{\left|\left.U\left(j\right)\right|\right.\right\}$ holds. Set $X_{j^{\prime }}=1$, and remove $j^{\prime }$ from set ${\mathsf{\Gamma }}_C$.

Step 3: Determining the coverage of candidate surface terminal $j^{\prime }$. Assign elements in $U\left(j^{\prime }\right)$ to $j^{\prime }$ in ascending order based on $V\left(i\right)$, until the capacity reaches either $Q_{\max }^{j^{\prime }}=0$ or $U\left(j^{\prime }\right)=\varnothing$. If $i\in U\left(j^{\prime }\right)$ and $Y_{ij}\le 1$ hold, then if $\overline{Q_i}\left(1-Y_{ij}\right)\le Q_{\max }^{j^{\prime }}$, set $Y_{i{j}^{\prime }}=1-Y_{ij}$, $Q_{\max }^{j^{\prime }}=Q_{\max }^{j^{\prime }}-\overline{Q_i}\left(1-Y_{ij}\right)$, $Y_{ij}=1$, and remove demand point i from $U\left(j^{\prime }\right)$ and ${\mathsf{\Gamma }}_C$. If $\overline{Q_i}\left(1-Y_{ij}\right)>Q_{\max }^{j^{\prime }}$, set $Y_{i{j}^{\prime }}=Q_{\max }^{j^{\prime }}/\overline{Q_i}$, $Y_{ij}=Y_{ij}+Y_{i{j}^{\prime }}$, $Q_{\max }^{j^{\prime }}=0$ accordingly.

Step 4: Termination check. If ${\mathsf{\Gamma }}_C$ is empty, terminate. Otherwise, update sets $U\left(j\right)$ and $V\left(i\right)$, and return to Step 2.

Algorithm pseudo-code: Coverage model for surface terminal location

1:  Initialization $X_j=0,Y_{ij}=0$,${\mathsf{\Gamma }}_C$ 2:  clear $U\left(j\right)$,$V\left(i\right)$ 3:  while $X_j=0$ do 4:  for $j^{\prime }\in {\mathsf{\Gamma }}_C$, find $\left|U\left(j^{\prime }\right)\right|=\max \left\{\left|U\left(j\right)\right|\right\}$ 5:    if true then 6:     let $X_{j^{\prime }}=1$, remove $j^{\prime }$ from ${\mathsf{\Gamma }}_C$ 7:    end if 8:  end for 9:  for  $j\in U\left(j\right)$ do 10:    assign to $j^{\prime }$ subsequently according to length $V\left(i\right)$ 11:    untile $Q_{\max }^{j^{\prime }}=0$ or $U\left(j^{\prime }\right)=\varnothing$ end 12:   for $i\in U\left(j^{\prime }\right)$ and $Y_{ij}\le 1$ do 13:    if $\overline{Q_i}\left(1-Y_{ij}\right)\le Q_{\max }^{j^{\prime }}$ then 14:     let $Y_{i{j}^{\prime }}=1-Y_{ij}$, $Q_{\max }^{j^{\prime }}=Q_{\max }^{j^{\prime }}-\overline{Q_i}\left(1-Y_{ij}\right)$, $Y_{ij}=1$ 15:     remove $i$ from ${\mathsf{\Gamma }}_C$ 16:    end if 17:    if $\overline{Q_i}\left(1-Y_{ij}\right)>Q_{\max }^{j^{\prime }}$ then 18:     let $Y_{i{j}^{\prime }}=Q_{\max }^{j^{\prime }}/\overline{Q_i}$, $Y_{ij}=Y_{ij}+Y_{i{j}^{\prime }}$, $Q_{\max }^{j^{\prime }}=0$ 19:    end if 20:  check ${\mathsf{\Gamma }}_C\ne \varnothing$ 21:  if true then 22:    return to line 4 23:  end if 24:  end while

4.3. Immune Clonal Selection Algorithm

The ICSA is inspired by the clonal selection principle in biological immune systems, which explains the fundamental characteristics of adaptive immune responses under antigenic stimulation. Due to its strong search capability, ICSA has been widely applied to combinatorial optimization, intelligent optimization, and production scheduling problems [40,42,43]. However, ICSA exhibits slow convergence, with relatively fixed immune and cloning probabilities, resulting in limited adaptability when solving complex problems [41,44,45]. To address the characteristics of a large feasible solution space and frequent information feedback at different stages in UD, this study introduces a normal distribution-based adaptive mutation mechanism. This approach ensures uniform and dynamic high-probability mutations within the local neighborhood of each antibody that meets the mutation rate, enhancing search randomness and stability tendency. Consequently, it effectively mitigates the risk of getting trapped in local optima. The heuristic algorithm framework is illustrated in Figure 3.

The specific steps of the ICSA for solving the metro freight node location allocation model are as follows.

Step 1: Generate the initial population

The initial antibodies are derived from the memory unit population, which consists of antibodies randomly generated from the feasible solution space. Each antibody encodes the metro freight node opening scheme and the surface terminal location information obtained from E-TOPSIS feedback. The initial antibody population is denoted as $K_0\left(RC\right)$, comprising RC randomly generated antibodies.

Step 2: Evaluate solution diversity

From the parent population $K_n\left(RC\right)$, select the individual with the highest fitness ($a=RC\times 20\%$) and the individual with the lowest affinity ($b=RC\times 20\%$) to form the solution vector. The Lehmer mean is used to construct an affinity expression, which quantifies the matching degree between antibodies and antigens. The affinity of the u-th solution vector is then computed.

$$I_u={\displaystyle \frac{1}{1+Q\left(u,v\right)}};\hspace{1em}\forall Q\left(u,v\right)={\displaystyle \frac{{\displaystyle \sum _{u\ne v}{‖F_b\left(u\right)-F_b\left(v\right)‖}^2}}{{\displaystyle \sum _{u\ne v}‖F_b\left(u\right)-F_b\left(v\right)‖}}}$$

(23)

Here, $I_u$ represents the affinity of the u-th solution vector, $F_b$ corresponds to the objective function, and $Q\left(u,v\right)$ denotes the Lehmer mean, which quantifies the average Euclidean distance between antibody u and all other antibodies  $u\ne v$.

Step 3: Cloning operation

The cloning ratio is determined based on antibody–antigen affinity and similarity among antibodies. A total of $a+b\pm t$ selected antibodies is replicated to generate the cloned population $Z_n\left(N_c\right)$. Specifically, the total number of clones produced by all selected antibodies is given by

$$N_{RC}={\displaystyle \sum _{u=1}^{a+b\pm t}round\left(RC\left(1-I_u\right)\right)}$$

(24)

Step 4: Genetic mutation operation

A subset of clones is selected from $Z_n\left(N_{RC}\right)$ and subjected to Gaussian mutation, generating a new population $S_n$. The mutation rate follows an adaptive strategy and is correlated with the fitness value $f_k$ of each antibody. This mutation process is expressed as $c_j=normrnd\left(c_j,\sigma ,1,1\right)$, where $c_j$ represents the j-th attribute of the clone and $normrnd$ is a random variable following a normal distribution with a mean of $c_j$ and standard deviation of $\sigma$. The local search radius $\sigma$ of each antibody is dynamically adjusted based on its fitness and affinity, adapting to $\sigma =\omega \cdot I_u/f_k$ accordingly.

Step 5: Immune selection operation

A subset of antibodies with the highest $f_k$ values is selected from $S_n\cup Z_n$ to form the memory population Y. Then, the top 30% (k) antibodies with the highest fitness from Y are chosen to replace an equal number of the lowest-fitness antibodies in $K_n$, generating the offspring population $K_{n+1}$.

Step 6: Node search reset

Upon reaching the maximum iteration count (N), the algorithm checks $I_y^{\left(n+1\right)}\left(i\right)$. If there exists an i such that $I_y^{\left(n+1\right)}\left(i\right)>\max I_t^{\left(n+1\right)}$, the objective cost $F_b\left(i\right)$ is updated, and information on unmet demand points is fed back into the E-TOPSIS evaluation model for a new iteration. If no such i exists, the algorithm terminates.

5. Case Study: Nanjing M-ULS Network

5.1. Background and Parameter Setting

Nanjing city is an important transportation hub in east China. Express delivery volume in Nanjing reached 630 million parcels in 2017, and the total amount of social logistics is nearly CNY three trillion. Currently, the Nanjing Metro network is composed of 10 metro lines and 174 stations. The total length of the network is 378 km, which solves about 3.5 million trips per day.

Based on real geographic information, five major logistics centers around Nanjing were selected. Nanjing Metro Line one (whole line), Line three (whole line), Line two (section), and Line four (section) were also selected as the empirical context to analyze the optimal M-ULS network with a hub-and-spoke layout and a multi-line transfer situation. The proposed scenario contains a total of 80 candidate MFDs, six ULHs, which are activated by default, as well as 312 customer clusters in the range of 388 km². The total freight OD volume between five UDCs and all customers is set to be 1.9 million parcels per day. The other parameters were obtained through the Monte Carlo simulation, as listed in Table 2.

The ICSA algorithm was programmed using the MATLAB R2017b platform. All experiments were run on a desktop computer with Windows 10, an Intel Core I7-7700K 4.2GHz processor, and 32 GB of RAM. The initial population size $p=100$, with a maximum iteration number $G_{\max }=100$.

To analyze the impact of network connectivity on M-ULS location planning, this study designs two metro freight transportation scenarios based on the operational status of transfer MFS hubs.

(1)

Network scenario with transfer functionality

In this scenario, six existing metro transfer stations in Nanjing—Nanjing Railway Station (Lines 1 and 3), Gulou (Lines 1 and 4), Xinjiekou (Lines 1 and 2), Nanjing South Railway Station (Lines 1 and 3), Jiming Temple (Lines 3 and 4), and Daxinggong (Lines 2 and 3)—are designated for underground freight transshipment. Freight locomotives determine the optimal routing and transfer strategy within the metro tunnel network based on the distance to the target node. Freight flows originating from different logistics hubs can be transferred across metro lines, converging at a designated metro freight node, from which they are further allocated to associated ground terminals (GTs) for processing and final last-mile delivery.

Given the high freight volume transported between ground terminals and metro freight nodes, the travel distance between a metro freight node and the next-level freight node is constrained to within 3 km. If the distance between a ground terminal and its corresponding metro freight node exceeds 3 km, a cost penalty coefficient is applied to mitigate the impact on urban road traffic at the final distribution stage.

(2)

Independent Line Scenario

1. In this scenario, all transfer MFS nodes remain non-operational for freight transshipment, meaning that each metro freight line exclusively serves a specific logistics hub. Consequently, metro freight nodes on different lines can directly distribute shipments to demand points within their respective service areas. However, if a demand point is located far from a particular metro line, it cannot receive goods from the corresponding logistics hub. As a result, most customers in this scenario may only receive partial or no underground logistics services, depending on their proximity to available metro freight routes.

2. The 3 km last-mile delivery constraint remains applicable, ensuring that demand points selected through the E-TOPSIS method are concentrated near metro lines. This constraint aims to reduce travel distances in last-mile distribution and minimize urban road congestion.

3. Under the independent line scenario, metro freight nodes replace the function of ground terminals, thereby simplifying the network’s structure. Introducing an additional ground terminal layer between metro freight nodes and customer demand points would be redundant. Instead, directly supplying customers from metro freight nodes improves overall efficiency. Because each metro freight node has a single inbound freight direction, its sorting, distribution, and dispatching operations become more streamlined and efficient.

However, this scenario does not guarantee optimal performance, as it presents several critical planning challenges. These include limited service coverage, incomplete logistics support, a higher number of metro freight nodes, underutilized nodes, and low network robustness. Specifically, if the freight volume from a designated logistics hub drops significantly while other hubs experience surges in demand, nodes along a specific metro line cannot assist in redistributing freight flow, leading to inefficiencies. To facilitate a comparative analysis, the model in Section 4 has been modified accordingly to optimize the multimodal transport network layout under the independent line scenario.

During the planning process, logistics hubs are assigned to nearby metro lines based on their proximity, establishing the following correspondences:

• Dingjiazhuang Logistics Hub (UDC1) → Metro Line 1, terminal node: Maigaoqiao.
• Cangbomen Logistics Hub (UDC2) → Metro Line 2, terminal node: Maqun.
• Yongning Logistics Center (UDC3) → Metro Line 3, terminal node: Linchang.
• Wangjiawan Logistics Center (UDC4) → Metro Line 4, terminal node: Wangjiawan.
• JD Logistics Center (UDC5) → Metro Line 3, terminal node: Dongda Jiulonghu.

5.2. Computation Results for the Network Scenario with Transfer Functionality

In the first stage of demand point selection, only the freight input data for each metro line are known, while inter-line transfer volumes remain uncertain. Therefore, before applying the E-TOPSIS method, it is necessary to set an upper limit on freight input from logistics hubs to their corresponding metro lines. Additionally, an estimated maximum transfer volume is used to ensure that the sum of direct input and transferred freight does not exceed the maximum daily transport capacity of the metro line.

This study assumes that approximately 50% of the metro line’s transport capacity is reserved for transferred freight. Taking a single metro line’s freight capacity (million parcels/day) as a reference, the direct freight input for each line is maintained at a million parcels/day. A 90 million parcels/day limit is applied to metro lines serving multiple logistics hubs. The E-TOPSIS method is then employed to evaluate freight OD pairs suitable for underground logistics, with the results presented in

Table 3. E-TOPSIS method selection results under the underground network scenario.

Logistics Park	$\overline{{\mathit{\lambda }}_{\mathit{j}}}$	$\mathbf{max}{\mathit{\lambda }}_{\mathit{j}}$	Service Demand Points Quantity	Total Freight Input (in 10,000 Units)	Notes Coverage Rate
UDC1 UDC2 UDC3 UDC4 UDC5	0.2209 0.0914 0.1173 0.2409 0.1091	0.9268 0.9408 0.9061 0.9602 0.8622	195 159 190 167 171	89.6 89.1 63.4 55.1 24.2	62.58% 50.96% 60.89% 100% 54.81%

.

Among the 312 demand points in the city, 195 points can receive metro freight services from UDC1, achieving a coverage rate of 62.58%. The total freight input from UDC1 to its corresponding metro line reaches 896,000 parcels/day, which is the maximum allowable input capacity for that line. A similar pattern is observed for UDC2, but due to differences in the freight volume per OD pair, the number of demand points served by UDC2 is lower than that of UDC1, with a coverage rate of only about 50%. UDC3 and UDC5 share Metro Line 3, and their combined freight input reaches the capacity limit of the line. However, compared to UDC3, UDC5 generates OD pairs with higher regional accessibility and order priority index values. As a result, 72.4% of the freight entering Line 3 originates from UDC3, while UDC5 contributes only 242,000 parcels/day, accounting for 27.6%. Despite differences in OD pair freight volumes, the number of demand points served by UDC3 and UDC5 is relatively similar, reaching 190 and 171 points, respectively. Metro Line 4 operates as an independent supply line, with UDC4 generating OD pairs with only 167 out of the 312 demand points. Consequently, 100% of its freight is allocated to underground transport, reaching a total input volume of 551,000 parcels/day.

Aggregating the results from all five logistics hubs, the network scenario with transfer functionality achieves the following: 278 demand points across the city receive metro freight services. The daily freight volume entering the metro network reaches 3.214 million parcels, accounting for 55.2% of the total urban demand (5.82 million parcels/day). Most of these 278 demand points can fully receive freight from all five logistics hubs, while a minority can only receive partial underground logistics services. The latter depends on the degree of service area overlap among the five logistics hubs. Detailed results are presented in

Table 4. Service range of the M-ULS under the network scenario with transfer functions.

	Service Demand Points Quantity	Total Freight Input (in 10,000 Units)	Freight Coverage Rate
Service area	278	321.4	55.2%

.

The ground terminals are subject to dual constraints: service radius and freight volume. Using the ground terminal set cover algorithm, an optimal solution for the selection of 44 ground terminals was obtained, and their locations and the allocation of demand points are shown in Figure 4. In terms of the number of serviced demand points, the following five ground terminals provide service to the highest number of points, reaching 12 points each: GT-1, GT-2, GT-9, GT-11, and GT-27. Regarding freight handling capacity, GT-23 has the highest freight volume, while GT-40 has the lowest, handling 99,702.9 million parcels/day and 11,247.5 million parcels/day, respectively, with nearly a nine-fold difference in freight volume. In terms of service radius, GT-9 has the largest coverage radius of 1989.84 m, while GT-17 has the smallest radius at only 257.12 m, representing a 66,110-fold difference in covered area.

The ICSA heuristic algorithm was used to select several subway stations from a total of 70 candidate subway stations in the network as subway freight nodes and to determine the optimal allocation of the 44 ground terminals. The results of the algorithm’s operation are shown in Figure 5. It can be observed that with each iteration, the ICSA algorithm converges around the 60th iteration, which demonstrates the effectiveness of the optimization program developed in this study. The objective cost value in Equation (8) decreased from the initial solution of CNY 2.159 million/day to the optimal cost of CNY 1.743 million/day, indicating a significant optimization effect.

Under the optimal site selection and allocation scheme, a total of 16 regular MFS nodes are opened across the network, located at the following subway stations: Line 1: Xinmofan Road Station, Zhangfuyuan Station, Sanshan Street Station, Software Avenue Station, Heding Bridge Station, Xiaolong Bay Station, and Tianyin Avenue Station; Line 2: Muxuyuan Station, Mochouhu Station, Xinglong Street Station, Yurun Street Station, and Youfangqiao Station; Line 3: Yuhuamen Station and Dongda Jiulonghu Station; Line 4: Jiuhuashan Station and Longjiang Station. The optimal allocation of the 44 ground terminals is shown in Figure 5. The ground transportation distance between the terminals is mainly controlled within 3 km, and, in terms of the number of associated ground terminals, Tianyin Avenue Station on Line 1 has the most with five terminals, while the lowest number is two terminals. In terms of cargo handling capacity, Zhangfuyuan Station handles the highest daily freight volume at 325,200 units, almost reaching saturation, while Xinmofan Road Station on Line 1 has the lowest at 96,200 units per day. The rest of the stations handle between 150,000 and 250,000 units per day on average.

Figure 6 shows the configuration results of the optimized subway freight network. It is evident that the distribution of cargo flow between subway freight nodes and logistics parks in different directions changes depending on the distance of the node from the target logistics park. For example, Dongda Jiulonghu Station, located near the JD Logistics Park, receives no freight flow, which can be understood as the station is closer and more suitable for traditional ground-based distribution, directly supplying demand points within its associated ground terminal’s service area. This freight flow does not need to enter the subway freight system. The benefits of the subway freight system are more evident when long-distance underground transportation is involved.

In terms of underground transport distance, the total underground mileage from the five logistics parks to most of the subway freight nodes exceeds 100 km, including both direct routes on the same line and transshipment across different lines. The total underground transport distance to Tianyin Avenue Station on Line 1 is the longest, reaching 167.3 km, which far exceeds the combined length of the four subway freight lines (114.57 km). Consequently, certain segments of the subway lines are reused multiple times, indicating that the volume of cargo transshipment between the lines is both substantial and frequent.

The transshipment situation between the various lines is shown in Figure 7. Among the 16 subway freight nodes, Jiuhua Mountain Station has the highest transshipment rate of 94.05%, indicating that nearly all of the cargo supplied to this node requires transshipment via two transfer MFSs on Line 4, and the reverse is also true. The transshipment rate for the other nodes remains above 60%. The high frequency of underground cargo transshipment results in the actual freight volume carried by the subway lines being much greater than the direct input volume of each line. For example, Line 1 actually handles 1.436 million parcels daily, which is 536,000 parcels more than the total freight supplied directly from UDC1 to Line 1, reaching 81.8% of the line’s full capacity. Therefore, it is essential to reserve a portion of the line’s freight capacity for transshipment in the earlier planning stages. In the given case, a 60% reserve for transshipment is a reasonable value, which will vary depending on the urban demand distribution and the scope of the planning.

Line 3 is the line with the most subway stations and the longest tunnels among all of the lines, but its actual load is the lowest of the four lines, with only two open subway freight nodes handling 329,000 parcels/day, which accounts for 37.5% of the freight supply for this line. The main reason for the low load on Line 3 is its proximity to Line 1, resulting in a high overlap in service areas between the two lines. Due to cost considerations, the algorithm chose to transship most of the cargo from UDC3 and UDC5 to the subway freight nodes on Line 1 rather than establishing more subway freight nodes on Line 3. Therefore, when two freight subway lines are in close proximity or the subway network’s density is high, the actual freight flow is concentrated on several lines, leading to fewer freight nodes and lower loads on other lines.

Figure 8 illustrates the traffic volume distribution of the entire subway–freight system in the network scenario with transshipment functionality compared to the traditional point-to-point surface delivery mode. In the table, GTV and UTV represent the traffic volumes of the freight flow from the logistics park to the ground terminals in the ground and underground portions (measured in ten thousand parcels·km), obtained by multiplying the OD amount of the freight flow direction by the corresponding transport distance in either the ground or underground portion. It can be observed that the integration of the subway–freight system results in a significant reduction in road freight traffic. The path with the highest reduction occurs at GT-16. In the traditional model, the ground traffic volume from each logistics park to GT-16 is 1.1845 million parcels·km. After it receives subway freight service from Yuhuamen Station on Line 3, the original GTV is replaced by 1.7137 million parcels·km of UTV and 15.4 thousand parcels·km of GTV, representing a multimodal transportation mode of both ground and underground. The reduction in ground freight traffic reaches 98.7%. For other ground terminals, the reduction in ground freight traffic is generally over 80%, showing that the subway–freight system significantly reduces ground freight traffic, thereby releasing urban road service capacity.

When examining the traffic volumes of the underground and ground sections of the subway–freight system itself, we can observe that the ratio of UTV to total traffic volume for each path is generally above 90%, meaning that in each delivery activity, about 90% of the traffic volume occurs underground. This indicates that the subway–freight system will eliminate a significant portion of the freight transport that takes place on the urban ring and other urban roads. The subway freight network can provide nearly 100% underground logistics service, significantly improving the flow of urban road traffic and alleviating traffic pressure on congested road segments.

5.3. Calculation Results for the Independent Line Scenario

In the independent scenario, no transshipment is possible between the underground freight lines. Due to the limited extension of each subway line, its service area is smaller compared to the underground network scenario. Additionally, in this scenario, no ground terminals are set up, meaning the goods exiting the subway freight nodes will directly flow to the customers. Consequently, the subway freight nodes will bear the heavy burden of last-mile delivery and will also function as end storage facilities for the goods. In this context, a cargo handling capacity constraint of 100,000 parcels/day and a delivery radius limit of 3 km are applied to the service capacity of the subway freight nodes in the independent line scenario. Similarly, the E-TOPSIS method is first used to determine the optimal service range for each subway freight line, as shown in

Table 5. E-TOPSIS method selection results under the independent line scenario.

Metro Line	$\overline{{\mathit{\lambda }}_{\mathit{j}}}$	$\mathbf{max}{\mathit{\lambda }}_{\mathit{j}}$	Service Demand Points Quantity	Total Freight Input (in 10,000 Units)	Notes Coverage Rate
Line 1 Line 2 Line 3 Line 4	0.1507 0.073 0.0828 0.2672	0.5534 0.9478 0.8624 0.5171	153 110 202 43	73.45 65.96 76.71 27.69	49.04% 35.26% 64.75% 25.75%

.

Due to the radius limitations, none of the four lines reach the designated cargo input capacity of 900,000 parcels/day. Among them, UDC3 and UDC5 supply 767,100 parcels/day to Line 3, making it the line with the highest daily underground freight flow. At the same time, 64.7% of the parcel delivery tasks from the two logistics parks can be handled by Line 3. In contrast, Line 4 has the smallest volume of goods, with only 276,900 parcels/day, and this line can only provide underground logistics services to 43 out of 167 demand points that require supply from UDC4, resulting in a node coverage rate of 25.75%.

Table 6. Service scope of the metro–freight system under the independent line scenario.

	Service Demand Points Quantity	Total Freight Supply	Freight Coverage Rate
Service area	283	243.82 × 104 units	41.89%

summarizes the overall service coverage of the metro–freight system in the independent line scenario. The total number of demand points is 283, and the total supply capacity of the four metro freight lines amounts to 2.4382 million parcels/day, accounting for 41.89% of the total regional demand.

The optimization results of the metro freight network in the independent line scenario obtained using the ICSA algorithm are shown in Figure 9. A total of 33 metro freight nodes were established across the four lines, with 11 nodes on Line 1, 9 nodes on Line 2, 10 nodes on Line 3, and 3 nodes on Line 4. As it can be observed, the distribution of the nodes across the lines is relatively even, with a freight node placed approximately every few metro stations until reaching the terminal points of the lines.

Figure 9 displays the optimized freight information for the nodes. Among the 33 nodes on all four lines, 13 nodes have a total freight volume exceeding 90,000 parcels/day, approaching saturation. The highest freight volume is at Yunnan Road Station on Line 4, with 99,429 parcels/day, while the lowest is at Xiamafang Station on Line 2, with 16,657.98 parcels/day. The node serving the most demand points is Chengxin Avenue Station on Line 3, supplying 23 demand points, while the fewest demand points are served by Xuanwu Gate Station on Line 1 and Xiamafang Station on Line 2, each serving only 4 demand points.

As shown in Figure 10, the metro–freight system in the independent line scenario can effectively alleviate road freight traffic pressure while maintaining a high proportion of underground logistics. In the freight OD on all four lines, the final metro freight node on Line 1, Nanjing Jiaoyuan Station, has the highest road freight traffic relief rate. After the metro–freight system is introduced, 96.08% of the traffic volume originally reaching its nine attached demand points was transferred underground, leaving only 22.6 thousand parcel·km of ground transportation. Most nodes achieved around 80% traffic relief in terms of road traffic volume reduction. Regarding underground logistics, Nanjing Jiaoyuan Station, located at the terminal of Line 1, achieved the highest underground transportation share, with 97.67% of the traffic volume serving its demand points occurring underground. Throughout the network, the underground transportation volume at other nodes accounts for more than 90% of the total transportation volume, indicating that the metro freight network’s accessibility is fully leveraged.

5.4. Comparison of Two Metro Freight Network Planning Schemes

The results of the metro freight network planning for Nanjing Metro Lines 1 to 4 are presented in

Table 7. The comparison between the independent line scenario and the underground network scenario.

	Underground Network Scenario	Stand-Along Scenario
Metro freight line	Lines 1–4	Lines 1–4
Optimal objective cost	CNY 1,743,157.91/d	CNY 1,960,351.02/d
Service area	325.72 km2	345.94 km2
Freight node configuration	16 regular MFSs, 6 transfer MFSs, and 44 ground terminals	33 regular MFSs
Freight delivery mode	Direct/transfer	Direct
Total freight volume in underground network	321.4 million parcels/d	243.82 million parcels/d
Overall transfer rate in underground network	67.34%	0
Total metro freight GTV	412.89 million parcels·km	467.65 million parcels·km
Total freight volume	7478.38 million parcels·km	4914.87 million parcels·km
Average freight traffic mitigation rate	89.19%	87.37%
Average underground logistics proportion	93.17%	90.89%
Average underground transport distance of freight flow	23.18 km	17.29 km

.

In terms of the freight distribution model, Scheme 1’s underground network with transfer functions adopts a combined multi-level distribution model that integrates direct underground delivery or underground transfer with a two-tier ground distribution network. The advantage of this model is that it can enhance system robustness and risk resistance by efficiently managing the distribution of freight flow across different network layers. When there is a surge in freight volume on a particular line, other nodes can quickly respond and share the load. However, the downside is the complexity of the multimodal network, which presents significant challenges in coordinating and linking the various layers of nodes. Scheme 2, which involves the independent line scenario, utilizes direct metro delivery combined with ground-level direct distribution. This network structure is relatively simpler and does not face issues related to underground freight transfer or secondary distribution of ground freight. However, the downside is that metro freight nodes bear a heavy workload, as they not only handle the delivery tasks but also serve as warehouses. Additionally, the single-line network can only provide fixed, single-source logistics services to demand points, resulting in lower service completeness.

In terms of freight node configuration, Scheme 1’s underground network scenario includes 16 general MFS nodes assigned for freight to leave or enter the metro system, 6 transfer MFS nodes for inter-line transfer, and 44 ground terminals for customer service, making the total number of related freight nodes 66. In contrast, Scheme 2’s independent line scenario only sets up 33 metro freight nodes, without the need for transfer MFS or ground terminals.

Cost optimization: After algorithm optimization, the optimal costs for Scheme 1 and Scheme 2 are CNY 1.74 million per day and CNY 1.96 million per day, respectively. These costs include the construction costs of the nodes as well as the operational costs for both underground and ground freight transportation.

System service coverage: In terms of system service coverage, Scheme 1 serves 278 demand points within a 325.72 square kilometer area, while Scheme 2 serves 283 demand points within a 345.94 square kilometer area.

Underground freight transportation: In Scheme 1, the combined metro freight network handles a total of 3.214 million parcels per day, with 67.34% of the freight requiring transfer between metro lines to reach the respective metro freight nodes. The average underground transport distance for a single freight flow is 23.18 km. In Scheme 2, the four independent metro freight lines handle a total of 2.438 million parcels per day, with the freight flow traveling an average distance of 17.29 km along a single metro line to reach its destination, without the need for transfer.

Traffic volume distribution: In Scheme 1, the total road traffic generated by the metro–freight system amounts to 4.1289 million parcels·km, which can alleviate 89.19% of the road freight traffic in the traditional ground distribution mode. All freight transportation activities in the underground system result in 74.7838 million parcels·km of traffic, accounting for 93.17% of the total traffic volume. In Scheme 2, the metro–freight system generates traffic volumes of 4.6765 million parcels·km on the surface and 49.1487 million parcels·km underground, with the average freight traffic mitigation rate and underground logistics ratio reaching 87.37% and 90.89%, respectively.

Comparison and analysis: Overall, Scheme 1 offers advantages in terms of operational costs, handling larger freight volumes, and relieving road freight traffic pressure while making full use of the accessibility of the metro network to achieve a higher proportion of underground logistics. However, this comes with higher node configuration requirements, more frequent freight transfers between metro lines, and a more complex freight delivery mode. Goods from logistics hubs in different directions need to go through multiple rounds of transfer between ground and underground networks before reaching the customer, and the distances traveled on metro lines are longer.

The difference in the system’s service coverage and the ground freight traffic generated in the final stage of delivery to customers between Scheme 1 and Scheme 2 is not significant. However, the total freight volume in Scheme 1 is much larger than in Scheme 2. Therefore, overall, the underground network scenario with transfer functionality provides more comprehensive underground logistics services and performs better in addressing last-mile delivery challenges and system scalability.

6. Conclusions

This study addresses the network planning problem of an M-ULS by proposing a mixed-integer programming model that integrates a multi-tier node layout, capacity constraints, and transshipment function optimization. To solve the model, an improved ICSA is employed. An empirical analysis based on the Nanjing metro network is conducted to evaluate the planning performance under two typical scenarios: a transshipment-enabled network and an independent line network. The key findings reveal that the transshipment-enabled network outperforms the independent line network in terms of total cost (CNY 1.743 million/day vs. CNY 1.960 million/day), underground freight volume (3.214 million parcels/day vs. 2.438 million parcels/day), and road freight alleviation rate (89.19% vs. 87.37%). Furthermore, the multi-tier node configuration—comprising 16 conventional MFS nodes, 6 transshipment nodes, and 44 ground terminals—enhances network coordination and operational efficiency. This study systematically addresses the core research problem of “M-ULS network structural forms and hierarchical construction mechanisms,” providing theoretical support for network planning in complex urban environments.

The innovations of this study are three-fold. First, a novel M-ULS network planning framework integrating coverage models and multi-objective optimization is developed, incorporating underground transshipment nodes and ground terminals into the decision making process for the first time. Second, an adaptive mutation mechanism based on a normal distribution is proposed, significantly improving the search efficiency and stability of the ICSA algorithm in complex solution spaces. Third, the empirical analysis unveils the dynamic allocation patterns of freight flows under a metro network-sharing scheme, providing data-driven insights for multi-line coordinated operations.

The findings of this study offer significant implications for both policymaking and practical implementation in the field of sustainable urban logistics. At the policy level, it is advised that governmental bodies proactively formulate transportation policies with an emphasis on supporting metro-based underground freight systems, thereby enhancing the integration of surface and underground transport modes. In the short term, increasing investment in underground logistics infrastructure is essential to secure adequate funding for advanced technological equipment, which will improve urban logistics efficiency, reduce pollution and greenhouse gas emissions, and enhance social and environmental benefits. At the practical level, this study provides novel insights into multimodal transport strategies combining surface and underground networks for urban goods distribution and offers policymakers and planners valuable references for decision making in city logistics and underground system development. Moreover, the implementation of the proposed two-layer metro-based logistics system facilitates the establishment of a more environmentally friendly and efficient urban logistics network, helping to address urban transportation challenges and promote sustainable urban development.

Due to the unavailability of precise data, this study adopts several stringent assumptions, which may limit the applicability of the model to M-ULS network optimization. For example, the impacts of the road network on M-ULS distribution and the transfer time of MFS operations were not considered. Meanwhile, practical barriers remain insufficiently addressed, such as regulatory uncertainties, multi-stakeholder coordination, and safety standards for mixed passenger–freight operations. Future research could incorporate these assumptions as model constraints while further exploring dynamic multi-period demand and dynamic route response mechanisms, developing dynamic scheduling algorithms based on digital twin technology. In practice, integrating smart logistics technologies to enhance node automation levels is essential. Additionally, fostering the convergence of an M-ULS with emerging technologies, such as unmanned aerial vehicles and autonomous driving, could facilitate a more efficient and sustainable urban logistics transition.