Integrated Planning for Depot Location and Line Planning Problems in the Intercity Railway Network with Passenger Demand Uncertainty

Abstract

In this study, we present a mathematical model and solution approach for addressing the robust integrated intercity railway depot location and line planning problem (RIDLLPP), which encompasses the initial two stages of the railway planning process. Our primary objective is to identify depot locations that exhibit robustness across a range of likely future scenarios, and this is achieved by incorporating line planning decisions. The model focuses on five critical strategic determinations, namely the depot location, depot storage capacity, line operation, passenger assignment, and fleet allocation. To tackle this complex problem, we propose an iterative solution framework that combines the Differential Evolution (DE) algorithm with improved rounding heuristics (DE-IRH). To evaluate the effectiveness of our framework, we conduct a comparative analysis with the Gurobi solver using multiple medium-sized artificial instances. The results demonstrate that our proposed framework achieves an optimality gap of 4.87% while requiring less computational time. Furthermore, we validate the robustness of the model’s location choices across various input scenarios, thereby providing valuable insights for transportation planning agencies and railway companies that can inform their decision-making processes.

1. Introduction

In recent years, there has been a significant focus from transportation planning agencies and railway companies on the development and expansion of intercity railway networks to meet the growing travel demand. Consequently, the workload for rolling stock has increased, leading to unscheduled or periodic demands [1,2] Transportation planning agencies often consider various strategies to address these escalating demands, such as establishing new facilities or modifying existing ones. However, the expansion of high-speed rail (HSR) networks in countries like China has highlighted the challenge of aligning depot locations and capacities with transportation needs [3]. Given the substantial costs associated with facility locations and the limited flexibility to make frequent changes, depot location planning becomes a strategic decision with long-term implications spanning several decades for the financial performance of railway companies. Hence, careful attention must be given to the facility location problem to ensure optimal outcomes and avoid costly modifications within a constrained time frame.

This paper proposes a mixed integer programming model for the depot location problem with passenger demand uncertainty in an intercity railway network, which integrates the first two stages in the railway planning process. A similar problem was treated by Tönissen and Arts et al. [4], Tönissen and Arts [5], and Tönissen and Arts et al. [6], where the authors proposed robust optimization formulations for the stochastic maintenance location routing problem (SMLRP). The model presented in this paper differs significantly from that of theirs with respect to the input data and decisions. In the former, the services are given in advance and the problem consists of determining the locations of depots and the maintenance routing of rolling stocks. In contrast, as it deals with long-term decisions, services, especially train operation plans, timetables, and rolling stock workload, cannot be predetermined in advance in this paper. Our primary objective is to identify depot locations that exhibit robustness across a range of likely future scenarios, which is achieved by incorporating line planning decisions, and the problem focuses on optimizing decisions about (i) the locations and capacities of depots, (ii) the frequencies of train lines and home location of the rolling stock of train lines, and (iii) passenger assignment. Without the loss of generality, the depot capacity means the maximum number of rolling stocks that the facility can hold. We do not consider other depot capabilities, such as service and repair capability, because there is no timetable or rolling stock circulation plan to determine the work demands of the rolling stocks.

The main contributions of this paper are as follows:

• We consider the integrated depot location and line planning problem with uncertain passenger demand through a strategic modeling framework;
• The depot location, line plan, passenger path, and fleet allocation are optimized simultaneously, which yields a certain equilibrium between passengers and operational decisions;
• Multi-type train lines are considered, which makes the calculation of the required rolling stocks more accurate;
• The DE-IRH solution framework is used to deal efficiently with the NP-hard robust optimization model, where the depot location problem is solved by the DE algorithm, then the frequencies of train lines, flows on the path, and home location of the rolling stocks are determined by improved rounding heuristics.

The remainder of this paper is structured as follows. A brief review of the relevant literature is presented in Section 2. Mathematical models for the robust integrated intercity railway depot location and line planning problem (RIDLLPP) are presented in Section 3. Section 4 describes the solution framework. In Section 5, numerical examples for real-world intercity railway networks are provided to demonstrate the effectiveness of the proposed models and algorithms. The Section 6 concludes the paper and discusses future research topics.

2. Literature Review

The literature is rich in optimization models to assist various types of facility location problems, and numerous studies and reviews have been conducted on facility location, such as Aikens [7], Daskin [8], Owen and Daskin [9], and Drezner and Hamacher [10]. Recently, Laporte and Nickel et al. [11] reviewed the related models and algorithms using discrete optimization, and the development of continuous approximation models for facility location problems were reviewed by Ansari and Başdere et al. [12]. Furthermore, uncertainty, supply chain decisions, and vehicle routing often affect facility decisions. We refer to the reviews of Snyder [13] and Dönmez and Kara et al. [14] for facility location problems under uncertainty, then Melo and Nickel et al. [15] for facility location in combination with supply chain decisions, and Cabrales-Navarro and Arias-Osorio et al. [16] for the location and routing problem (LRP).

However, the depot location problem for rolling stock has essential features that make it substantially different from others in terms of logistics: (i) the customers of depots are various types of rolling stock work demand that have to travel to a depot over a fixed railway network [4], and it is necessary to plan for an empty train to reach a depot when not passing a depot for a long period; (ii) the depot location decisions should support the daily adjustment of the train line plan in different passenger demand scenarios; (iii) as long-term decisions, the line plan and railway network change annually to accommodate changing travel demands. What is more, as stated by some researchers, it does not seem appropriate to consider modifications to railway infrastructure at the network planning level without at least taking into consideration the level of traffic (both passengers and trains) that the proposed infrastructure would support in the future; thus, integrating different stages in the planning process becomes a vital research trend [17,18,19,20].

To the best of our knowledge, most papers on the depot location of rolling stocks primarily focus on integrating the depot location and maintenance routing (MLR). De Jonge [1] studied the facility location problem in the train maintenance environment and formulated a mixed-integer programming model to determine the optimal number, size, and location of train maintenance facilities. Kho and Martagan et al. [21] constructed a mixed integer linear program to solve the maintenance location routing (MLR) problem for NedTrain. Xie and Ouyang et al. [22] proposed a mixed-integer program model to optimize the location and capacity of locomotive maintenance shops. The integrated planning problem for multiple types of locomotive work facilities under location, routing, and inventory considerations was studied by Xie and Chen et al. [2]. Tönissen and Arts et al. [4] modeled the uncertainties of and changes in line planning and fleet planning using a discrete set of scenarios and provided a two-stage stochastic programming and a two-stage robust programming formulation for the maintenance location routing problem. After that, the economies of scale in recoverable robust maintenance location routing were studied by Tönissen and Arts [5]. An efficient mixed-integer programming for maintenance location routing problems under uncertainty conditions of line and fleet planning was proposed by Tönissen and Arts et al. [6]. In a recent study, Tönissen and Arts [23] studied the stochastic maintenance location routing allocation problem, where they had to locate maintenance locations and allocate fleets to these locations. The common feature of these models is their ability to determine the depot’s location and capacity, and the rolling stock’s maintenance routing, with a given rail network, candidate facilities, train service, and maintenance demand.

Recently, the combination of the depot location and rolling stock circulation has been introduced by Canca and Barrena [18]. Considering that the rolling stock departs and returns to depots every day, and the empty runs at the beginning (from depots to initial stations) and at the end of every daily route (from terminal stations to depots) seriously affect the operation efficiency of the train lines, Canca and Barrena [18] proposed a general mixed integer programming model to design rolling stock circulation plans and simultaneously consider the problem of determining the number and location of rest facilities. Lusby and Zhong et al. [3] analyzed the impact of depot location on rolling stock scheduling. They proposed a two-stage mixed-integer programming model to determine a maintenance feasible, operational rolling stock schedule while simultaneously identifying potential new depots to open. An essential assumption of the above work is the assumption of a priori train schedules.

In contrast, we consider the problem of locating depots within a railway network that involves the planning or constructing of new lines. In this context, the line plan, timetable, and rolling stock work demands are not predetermined, except for the passenger demand. This is a problem that railway companies naturally encounter during network construction. Comparatively, this problem has received much less attention from an optimization perspective. The above work is challenging to apply to such a railway network because, being that it entails a long-term decision, it is difficult to obtain more information, especially regarding train service, although some of them have taken account of uncertain or changing line planning, fleet planning, and other factors.

As the first stage of the tactical level, line planning is naturally coupled with the selection of the depot location. A line plan consists of a set of train lines, and each train line is characterized by its origin and destination station, its frequency per hour or day, the route between these two stations, and the intermediate stops at passing railway stations [24]. In the railway network, not all stations can be used as train terminals except for the stations near the depot. Thus, the location of the depot will affect the selection of train lines in the process of line planning. On the other hand, the capacity of the depot should be sufficient for the given line plan. Many papers have been published on various types of line planning problems [25,26]. Most recently, some efforts have been devoted to integrating line planning with network design (station location). Canca and De-Los-Santos et al. [27,28,29] studied the profit-oriented integration problem of station location and line planning, where they simultaneously determined and defined the infrastructure network, line planning, train capacity of each train line, fleet investment, and personnel planning. However, the influence of the depot on the train terminal is usually not considered in the line planning process, which can significantly affect the utilization of the available rolling stock and revenues, and lead to inefficient solutions.

To summarize, the main objective of this paper is to propose a new strategic planning model that is applicable to intercity railway networks with new lines. The model integrates the depot location with line planning, which is driven by an uncertain or changing passenger demand. We modeled the uncertainty of the passenger demand using a discrete set of scenarios, and the line plan, passenger assignment, fleet allocation, and location and capacity of depots are synchronously determined. We present an iterative framework to solve our robust depot location and line planning problem.

3. Mathematical Model

3.1. Problem Description and Assumption

We consider the optimization problem of where to open new depots or expand existing depots and how many storage lines should be allocated in the intercity railway network. As shown in Figure 1, the stations are sequentially numbered as $1,2,\dots$, and there are three intercity railway lines in operation: Line 1, Line 2, and Line 3. The depot d1 provides rolling stocks for Lines 2 and 3 and undertakes daily maintenance of rolling stocks. The rolling stocks running on Line 1 are provided by depot d2. Assuming that each intercity railway line operates independently, that is, the train’s initial station and terminal station belong to the same intercity line, due to the rapid growth of passenger demand, the railway planning department plans to open several new lines (the red dotted lines) to improve the convenience of passengers and increase network profit.

The new lines mean that more rolling stocks are needed, and that line plans may also be adjusted. Therefore, the railway planning department needs to decide to allocate more storage lines to the existing depots, d1, d2, and d3, or construct new depots among d3, d4 and d5, or even combine both to reduce the construction, operation costs, and travel time cost of passengers.

The model was formulated based on five important (and realistic) assumptions. First, the intercity railway network is owned and operated by a public entity whose objective is to minimize net public generalized costs. And each railway line operates independently, that is, the initial station and terminal station of each of the trains are on the same intercity line. Second, the information on the stations, existing depots, candidate depots, existing lines, and planning lines of the intercity railway network are given, and a single type of depot is considered. Third, a line pool has been defined [30,31], but the frequency and home location of the rolling stock of train lines are yet to be chosen among a given set of alternatives. Fourth, passenger demand is given by a discrete set of scenarios, which can be obtained through passenger demand surveys and passenger demand forecasts. What is more, we consider that the passenger demand of each line is symmetrical in both directions. Lastly, there is only one type of rolling stock operated on the intercity rail network.

3.2. Mathematical Notation

We consider an intercity railway network $G=\left(V,A\right)$, consisting of nodes $V$ and edges $A$. There are two types of nodes: stations, $V_1\subset V$, which are the starting and ending places of passenger trips; depots, $V_2\subset V$, which store and maintain rolling stocks. The number of storage lines of depot $\xi \in V_2$ is denoted by $n_{\xi }$, in particular, $n_{\xi }=0$ if $\xi$ is a candidate depot, and the maximum number of storage lines allowed to be constructed at $\xi$ is denoted by ${\mathcal{N}}_{\xi }$. The cost of constructing a storage line at location $\xi$ is denoted by $I_{\xi }$, which is related to the local economic level and will be measured in monetary units per day. An edge, $a\in A$, is the connection between nodes. Each edge $a$ has a length $v_a$ and travel time $t_a$. We are also given the capacity ${\mu }_a$ on the edge $a\in A$.

Let $\mathcal{L}$ be the set of all train lines. $\mathcal{L}$ can be given explicitly by a so-called line pool. Given a set of terminals, a train line $l\in \mathcal{L}$ is a route with stops in the intercity railway network that starts and ends at a terminal [30,32]. The set of edges of a train line $l$ is denoted by ${\varphi }_l$, and the stop sequence of train line $l\in \mathcal{L}$ is denoted by $r_l$. We denote by ${\mathcal{L}}_a$ the set of train lines operating on edge $a\in A$. The connection between $i\in r_l$ and $\left(i+1\right)\in R_l$ is called a riding arc and forms the train service network, and passengers can choose a directed path from origin to destination in this train service network. The set of riding arcs is denoted by $\mathcal{A}$. Note that each riding arc $\widehat{a}\in \mathcal{A}$ is associated with precisely one train line $l$, denoted by $l\left(\widehat{a}\right)$. Denoted by $F=\left\{1,2,\dots ,f_{max}\right\}$ is a set of possible frequencies at which these train lines can be operated. The number of rolling stocks required of train line $l$ with $f\in F$ is denoted by ${\epsilon }_{l,f}=[2\left({{\displaystyle \sum }}_{a\in {\varphi }_l}{t}_a+t_{turn}+\left|r_l\right|\sigma \right)/18]$, with $t_{turn}$ being the turn-back operation time at the terminal, $\sigma$ being the dwell time at the intermediate station, and the daily operation time of the intercity railway line being 18 h in general. The capacity of a train of train line $l$ is ${\eta }_l$. The fixed operation cost of a train is denoted by $C_1$, the dwell cost at an intermediate station is denoted by $C_2$, and the variable operating cost per kilometer is denoted by $C_3$.

In general, the railway operation department may specify in advance the set of possible railway lines each depot is allowed to be responsible for. This may be arranged according to the distance between the depots and the intercity railway lines. For this reason, we introduce the binary parameter ${\delta }_{l,\xi }$, ${\delta }_{l,\xi }=1$ if the train lines of an intercity line are within depot $\xi$’s service scope, and it is ${\delta }_{l,\xi }=0$ otherwise. The empty run distance between depots and the initial station of the train of a train line is denoted by $E_{\xi ,l}$.

Let $D=\left\{\left(i,j\right)|i\in V_1,j\in V_1\right\}$ be the set of all OD pairs. The number of passengers that want to travel from $i\in V_1$ to $j\in V_1$ is denoted by $q_{i,j}>0$. The possible directed passenger paths from $i\in V_1$ to $j\in V_1$ are denoted by $P_{i,j}$, and denoted by $P={\cup }_{\left(i,j\right)\in D}{P}_{i,j}$ is the set of all such paths. The travel time of path $p\in P$ is ${\tau }_p={{\displaystyle \sum }}_{a\in p}{t}_a+n\sigma$ with $n$ is the number of intermediate stations on path $p$. The unit time value of passengers is denoted by $Vot$. Since depot location is a strategic decision problem, many input parameters, especially the passenger demand, will change significantly when the lines are put into operation. We introduce a simple modeling trick [33] to deal with the uncertainty of passenger demand. The passenger demand uncertainty is described through a set of alternative scenarios: each scenario is associated with a passenger demand. Each of the input data scenarios has a positive probability to be realized. Let $\omega \in \Omega$ be the input data scenario index and $\Omega$ be the set of all possible scenarios. Each input data scenario is associated with a positive probability $p_{\omega }$. For scenario $\omega$, the number of passengers that want to travel from $i\in V_1$ to $j\in V_1$ is denoted by $q_{i,j}^{\omega }>0$.

3.3. Model Formulation

For specific input data scenario $\omega$, the relationship between multiple sets of decision variables is shown in Figure 2. The variables are classified into three groups: those related to the depot location and rolling stocks work demand assignment of the problem, those in charge of modeling transit assignment in the network, and those related to line operation decisions. The first group includes integer variable ${\gamma }_{\xi },$ describing the number of storage lines allocated to depot $\xi$, and ${\psi }_{\xi ,l}^{\omega },$ describing the value of the rolling stocks work demand provided by depot $\xi$ for train line $l$ in scenario $\omega$. The second group includes integer variables $y_p^{\mathsf{\omega }},$ counting the number of passengers on path $p$ in scenario $\omega$. And the third group includes the binary variables $x_{l,f}^{\omega }$, $x_{l,f}^{\omega }=1$ if train line $l$ is operated with frequency $f$, which equal 0 otherwise in scenario $\omega$.

The integrated maintenance location and line planning problem (Model 1) is formulated as follows:

$$\mathit{min}{Z}_{\omega }={{\displaystyle \sum }}_{\xi \in V_2}{\gamma }_{\xi }{I}_{\xi }+{{\displaystyle \sum }}_{l\in \mathcal{L}}{{\displaystyle \sum }}_{f\in F}\left(C_1+C_2\left|r_l\right|+C_3{{\displaystyle \sum }}_{a\in {\varphi }_l}{v}_a\right)x_{l,f}^{\omega }f+{{\displaystyle \sum }}_{l\in \mathcal{L}}{{\displaystyle \sum }}_{\xi \in V_2}{\psi }_{\xi ,l}^{\omega }{E}_{\xi ,l}{C}_3+Vot\cdot {{\displaystyle \sum }}_{p\in P}{y}_p^{\omega }{\tau }_p$$

(1)

subject to.

$${{\displaystyle \sum }}_{p\in P_{i,j}}{y}_p^{\omega }=q_{i,j}^{\omega },\forall \left(i,j\right)\in D$$

(2)

$${{\displaystyle \sum }}_{p:\widehat{a}\in p}{y}_p^{\omega }\le {{\displaystyle \sum }}_{f\in F}{x}_{l\left(\widehat{a}\right),f}^{\omega }f{\eta }_{l\left(\widehat{a}\right)},\forall \widehat{a}\in \mathcal{A}$$

(3)

$${{\displaystyle \sum }}_{f\in F}{x}_{l,f}^{\omega }\le 1,\forall l\in \mathcal{L}$$

(4)

$${{\displaystyle \sum }}_{l\in {\mathcal{L}}_a}{{\displaystyle \sum }}_{f\in F}{x}_{l,f}^{\omega }f\le {\mu }_a,\forall a\in A$$

(5)

$${{\displaystyle \sum }}_{\xi \in V_2}{\psi }_{\xi ,l}^{\omega }{\delta }_{l,\xi }={{\displaystyle \sum }}_{f\in F}{x}_{l,f}^{\omega }{\epsilon }_{l,f},\forall l\in \mathcal{L}$$

(6)

$${{\displaystyle \sum }}_{l\in \mathcal{L}}{\psi }_{\xi ,l}^{\omega }\le n_{\xi }+{\gamma }_{\xi },\forall \xi \in V_2$$

(7)

$${\gamma }_{\xi }\le {\mathcal{N}}_{\xi },\forall \xi \in V_2$$

(8)

$$x_{l,f}^{\omega }\in \left\{0,1\right\},\forall l\in \mathcal{L},f\in F$$

(9)

$${\psi }_{\xi ,l}^{\omega }\in Z^{+},integer,\forall l\in \mathcal{L},\xi \in V_2$$

(10)

$${\gamma }_{\xi }\in Z^{+},\forall \xi \in V_2$$

(11)

$$y_p^{\omega }\in Z^{+},\forall p\in P$$

(12)

The objective Function (1) minimizes the sum of the construction cost, line operation cost including the fixed, variable, and empty train drive costs, and passenger travel time cost, and the first item is the construction cost of the depot, the second item is the fixed and variable costs, the third item is the cost of an empty train drive between the depots and initial stations and terminal stations of train lines, and the last item is the cost of passenger travel time. In particular, the cost of empty trains can be used as a measure of the rationality of the depot location. Constraints (2) stipulate a passenger flow of $q_{i,j}^{\omega }$ for each OD pair $\left(i,j\right)\in D$. Constraints (3) enforce a sufficient transportation capacity on each riding arc. Constraints (4) ensure that a train line is operated with at most one frequency. Constraints (5) bound the sum of the line operation frequencies for each edge. Constraints (6) ensure that all rolling stock demands of the operated train line are fully satisfied. Constraints (7) ensure that the total assigned rolling stocks do not exceed the capacity of the depot. Constraints (8) limit the capacity of the depot. Constraints (9)–(12) define the related decision variables.

The robust depot location aims to find a solution that is within $\lambda \%$ (may be prespecified input of the designer) of the optimal solution for any realizable input data scenario, and the objective is to minimize a weighted sum of the total cost of all input data scenarios. For a given set of input data scenarios $\Omega$, a depot location scheme is robust if and only if:

$$Z_{\omega }-Z_{\omega }^{*}\le \lambda Z_{\omega }^{*},\forall \omega \in \Omega$$

(13)

where $Z_{\omega }^{*}$ is the optimal solution of input data scenario $\omega$.

Our robust optimization model of therobust integrated depot location and line planning problem (Model 2) is given.

$$\mathit{min} p_{\omega }{Z}_{\omega }$$

(14)

subject to Equations (2)–(13)

$$Z_{\omega }={{\displaystyle \sum }}_{\xi \in V_2}{z}_{\xi }{I}_{\xi }+{{\displaystyle \sum }}_{l\in \mathcal{L}}{{\displaystyle \sum }}_{f\in F}\left(C_1+C_2\left|r_l^{’}\right|+C_3{{\displaystyle \sum }}_{a\in {\varphi }_l}{v}_a\right)x_{l,f}^{\omega }f+{{\displaystyle \sum }}_{l\in \mathcal{L}}{{\displaystyle \sum }}_{\xi \in V_2}{\psi }_{\xi ,l}^{\omega }{E}_{l,\mathcal{l}}{C}_3+Vot\cdot {{\displaystyle \sum }}_{p\in P}{y}_p^{\omega }{\tau }_p,\forall \omega \in \Omega$$

(15)

$$Z_{\omega }\ge 0,\forall \omega \in \Omega$$

(16)

4. Solution Framework

The proposed formulation is an integer linear programming model integrating depot location and line planning, and it can be directly solved by commercial optimization software (e.g., GUROBI v9.5.1, etc.) for small-scale problems. However, in most real-world cases, it is usually difficult to find a satisfactory solution in an acceptable timeframe, e.g., the Chinese Urban Agglomeration with many intercity railway lines, and sometimes a feasible solution cannot even be found using commercial software. Most scholars hold that the heuristic or meta-heuristic algorithm effectively solves such complex engineering problems [34,35]. As illustrated by Gutiérrez and Kouvelis et al. [33] and Canca and De-Los-Santos et al. [29], decomposing the problem into several sequential steps is an effective solution strategy. Here, we introduce a solution framework that is capable of jointly solving maintenance location and line planning problems.

The proposed iterative framework consisting of the DE algorithm [36] and improved rounding heuristics [32,37] (DE-IRH) is described in Algorithm 1. The solution framework consists of two layers: the upper layer solves the depot location problem using the DE algorithm, and the transit assignment and line operation problems are solved by the improved rounding heuristics in the lower layer. Note that the robustness constraints (23) are relaxed, and the objective function will be punished when the constraints are violated. Therefore, at the lower layer, the optimization problem will be decomposed into $\left|\Omega \right|$ independent subproblems, which can be solved in parallel. Since there is no difference in the solving of subproblems of different scenarios at the lower layer, we only give a general algorithm framework here. Figure 3 summarizes the overall algorithm.

Algorithm 1: Pseudocode of DE-IRH
Data: Input data for RIDLLPP problem
While the stopping criterion is not satisfied do
	Upper layer
	Generate new depot location solutions
	Compute the associated construction cost of each solution
	Lower layer
	Solve the transit assignment and operation problems on each current solution
	Compute the total cost
	Compare and keep the best solution
end
Result: the minimum total cost and the best solution

4.1. The Upper Layer Subproblem: Depot Location

At the upper layer, the DE algorithm is used to search for the optimal depot location scheme and control the full procedure. The DE algorithm is a promising optimization algorithm that converges to the real optimum without using significant amounts of resources and has been applied in a wide range of domains and fields of technology [38]. Starting with an initial solution population, at each iteration $k$, three vectors, $v_1,v_2, \mathrm{and} v_3$, are randomly chosen from the current population to construct a new mutation vector $v_y$ by mutation operation. There are two common mutation strategies, ‘DE/rand/1/bin’ and ‘DE/best/1/bin’. The former focuses on improving the global search ability, while the latter focuses on improving global convergence. We use the latter in this paper because it can produce better solutions (see Section 5.2). The associated equations are:

DE/rand/1/bin:

$$v_y=v_1+F\left[v_2-v_3\right]$$

(17)

DE/best/1bin/:

$$v_y=v_{best}+F\left[v_2-v_3\right]$$

(18)

where $F$ is the mutation rate. The mutation process is the main distinctive component of DE and controls the evolution direction of DE. Then, the uniform crossover is used to construct trial vectors from vector $v_y$ and vector $v_x$ which is different from $v_1,v_2,$ and $v_3$. The associated equation is:

$$v_{trial}\left(i\right)=\left\{\begin{array}{c}{v}_y\left(i\right), if rand\left(0,1\right)<Cr\\ v_x\left(i\right), otherwise\end{array}\right.$$

(19)

where $Cr$ is the crossover rate. The next operation is selection, in which the trial vector $v_{trail}$ competes with the target vector $v_x$ as follows:

$$v_x=\left\{\begin{array}{c}{v}_{trial}, if f\left(v_{trail}\right)<f\left(v_x\right)\\ v_x, otherwise\end{array}\right.$$

(20)

where $f\left(·\right)$ is the objective function. In order to obtain the objective function value of the vector, we need to input the vector at the lower layer algorithm, which will be described later. If the new vector $f\left(v_{trial}\right)$ is less than or equal to the target vector $f\left(v_x\right)$, it replaces the target vector. Otherwise, the population maintains the target vector value.

4.2. The Upper Layer Subproblem: Transit Assignment and Line Operation

At the lower layer, for a given depot location scheme, we face the problem of determining the frequency, passenger assignment, and rolling stocks work demand assignment for each line. The associated model is obtained by setting the variable $\gamma$ as a constant according to the vector from the upper layer. The improved rounding heuristic is developed to solve the lower-layer model. There are four main steps of the improved rounding heuristic: First, the linear relaxation solution of the original problem is obtained by a commercial solver; second, counter the number of passengers on each riding arc, ${\pi }_{\widehat{a}}$, filter out the integer variable $x$, and update the number of passengers on each riding arc; third, recalculate the frequency of the remaining variables $x$ according to the passengers on the riding arc; fourth, recalculate the rolling stock assignment according to the variables $\psi$ of the linear relaxation solution. The detailed improved rounding heuristic is shown in Algorithm 2.

Algorithm 2: Pseudocode of improved rounding heuristic
Data: Input depot location solution (vector $v_x$)
1	Obtain LP solution ($x^{*},y^{*},{\psi }^{*}$) of the lower layer model
2	Set riding arc weights ${\pi }_{\widehat{a}}≔{{\displaystyle \sum }}_{p:\widehat{a}\in p}{y}_p^{*},\overline{\mathcal{L}}:=\varnothing ,x^{†}:=\varnothing ,{\psi }^{†}≔\varnothing$
3	for $l\in \mathcal{L}$ do
4		for $f\in F$ do
5			if $x_{l,f}^{*}=1$ then
6				$x_{l,f}^{†}:=1$
7				${\pi }_{\widehat{a}}≔\max \left\{0,{\pi }_{\widehat{a}}-f\cdot {\eta }_l\right\}$ for all $\widehat{a}\in l$
8			end
9			if $x_{l,f}^{*}$ is fractional then
10				$\overline{\mathcal{L}}≔\overline{\mathcal{L}}\cup \left\{l\right\}$
11			end
12		end
13	end
14	for $l\in \overline{\mathcal{L}}$ do
15		$\tilde{f}=\underset{f\in F\cup \left\{0\right\}}{\min }\left\{f{\eta }_l\ge {{\displaystyle \sum }}_{f\in F}{x}_{l,f}^{*}\cdot f\cdot {\eta }_l or f\cdot {\eta }_l\ge \max \left\{{\pi }_{\widehat{a}}|\widehat{a}\in l\right\}\right\}$
16		$x_{l,\tilde{f}}^{†}≔1$
17		${\pi }_{\widehat{a}}≔\max \left\{0,{\pi }_{\widehat{a}}-\tilde{f}\cdot {\eta }_l\right\}$ for all $\widehat{a}\in l$
18	end
19	Calculate the number of rolling stocks of train lines ${\kappa }_l:={{\displaystyle \sum }}_{f\in F}{x}_{l,\tilde{f}}^{†}\cdot {\epsilon }_{l,f}$
20	for $\xi \in V_2$ do
21		set depot capacity ${\vartheta }_{\xi }≔n_{\xi }+{\gamma }_{\xi }$ (according to the vector $v_x$)
22		for $l\in \mathcal{L}$ do
23			if $x_{l,\tilde{f}}^{†}=1$ and ${\psi }_{\xi ,l}^{*}>0$ then
24				${\psi }_{\xi ,l}^{†}≔\min \left\{{\vartheta }_{\xi },{\kappa }_l\right\}$
25				${\vartheta }_{\xi }≔\max \left\{0,{\vartheta }_{\xi }-{\psi }_{\xi ,l}^{†}\right\}$
26				${\kappa }_l≔\max \left\{0,{\kappa }_l-{\psi }_{\xi ,l}^{†}\right\}$
27			end
28		end
29	end
Result: IP solution ($x^{†},y^{*},{\psi }^{†}$) of the RIDLLPP model

At each iteration, the improved rounding heuristic provides the solution to the passenger assignment, line operation, and rolling stock assignment problems of the input data scenarios. It should be noted that the depot capacity constraints (20) should be checked after rolling stock assignment because they may not have sufficient capacity to store the rolling stocks of the train lines. If this happens, we will update the capacity of facilities according to the current rolling stock assignment plans.

5. Numerical Experiments

To illustrate the effectiveness and performance of our algorithm, we constructed a case based on the Pearl River Delta Intercity Network and conducted several numerical experiments. The first group of experiments compared the DE-IRH with Gurobi in different instances to illustrate the effectiveness of the proposed algorithm. The second group of experiments analyzed the influence of the control parameters on the performance of DE-IRH. Finally, the characteristics of one solution were analyzed in detail. All computations were performed in Python/GUROBI on an Intel i7-9700F 3.0 Ghz CPU with eight cores and 16 GB of RAM memory.

5.1. Scenario Setting

The topology of the intercity railway network is shown in Figure 4. The intercity network consists of 41 stations, 8 maintenance facilities, and 8 lines, where the stations are sequentially numbered as s1, s2, … s41, the lines in operation are sequentially numbered as Line 1, Line 2, …, Line 8, and the depots are sequentially numbered as d1, d2, … d8. There are some tracks (double solid lines) that are under construction. The number on the link indicates the length of the track (km).

To reasonably determine the location and capacity of the depots to sufficiently support the operation requirements after the completion of the new lines, we have designed five scenarios with the same realization probability to capture the characteristics of the passenger flow in the planning year. Based on historical operational data, we designed 150 OD pairs, and the total numbers of passengers per day of different input data scenarios are shown in Figure 5. According to the OD pairs and existing train schedule, a line pool with 153 train lines was constructed, and 1664 possible passenger paths were generated. Some train lines are shown in

Table 1. Set of possible train lines.

ID	Route	Stop-Pattern
1	s1-s2-s3-s4-s5-s6	s1-s2-s3-s4-s5-s6
2	s1-s2-s3-s4-s5-s6	s1-s3-s5-s6
3	s7-s8-s9-s10-s6-s14	s7-s8-s9-s10-s6-s14
4	s7-s8-s9-s10	s7-s8-s9-s10
5	s5-s16-s17-s10	s5-s16-s17-s10
6	s10-s11-s12-s13-s14	s10-s11-s12-s13-s14
7	s10-s11-s12-s18-s19-s20-s21-s22-s23	s10-s11-s12-s18-s19-s20-s21-s22-s23
8	s10-s11-s12-s18-s19-s20	s10-s11-s12-s18-s19-s20
9	s18-s19-s20-s21-s22-s23	s18-s20-s23
10	s26-s27-s28-s29-s30	s26-s27-s28-s29-s30
11	s26-s27-s28-s29-s30	s26-s28-s30
12	s31-s15-s32-s33-s34	s31-s15-s32-s33-s34
13	s14-s38-s37-s36-s35-s34	s14-s38-s37-s36-s34
14	s14-s38-s37-s36-s35-s34	s14-s37-s36-s35-s34
15	s14-s38-s37-s36-s35-s34	s14-s36-s35-s34
16	s20-s39-s40-s41	s20-s39-s40-s41
17	s14-s25-s24-s20-s39-s40-s41	s14-s20-s41
18	s14-s25-s24-s20-s39-s40-s41	s14-s25-s24-s20-s39-s40-s41
…	…	…

.

The basic parameters of the model are set as follows: (1) the capacity of the train is 557 passengers/train, the average speed on the links is 200 km/h, the fixed operation cost of a train is 20,000 RMB, the unit dwell cost is 2000 RMB, the variable operation cost is 36 RMB/km, and the turn-back operation time at a terminal station is 15 min; (2) the unit passenger time value is 36 RMB/h; (3) the parameters for the facilities are listed in

Table 2. Parameters about depots.

Depots	Parameters
	${\mathit{n}}_{\mathit{\xi }}$	${\mathit{I}}_{\mathit{\xi }}$(RMB)	${\mathcal{N}}_{\mathit{\xi }}$
d1	0	23,632.52	40
d2	14	77,648.79	16
d3	12	12,821.92	35
d4	0	180,118.1	30
d5	12	180,118.1	16
d6	9	180,118.1	16
d7	8	69,334.93	12
d8	8	31,791.23	20

. In particular, the depot construction cost includes the land cost, engineering cost, equipment cost, and discount rate, and it is also related to the economic level of various regions. For simplicity, only the engineering cost was considered, and the cost was converted into daily costs by considering a lifespan of 40 years with a discount rate of 4%. (4) In all computational experiments, we used $\lambda =5\%$.

5.2. Testing the DE-IRH Algorithm Performance

There are four parameters governing the behavior of the DE-IRH. The number of iterations, $K,$ and population size, $NP$, are two basic parameters of the DE. Large values of $K$ yield a better solution but mean more computing time. Small values consume little computing resources, but are more likely to drop in local best. In this paper, we set $K$ to 50. $NP$ represents the population diversity, which should take a value between $5\chi$ and $10\chi$, where $\chi$ is the dimension of the solution space. We set $NP$ to 80 because there are eight depots at the upper layer. Furthermore, the mutation rate, $F$, controlling the amplification of the differential variation, and the crossover rate, $Cr$, guaranteeing some diversities of the perturbed parameter vectors, are the main focus of our attention in this work. To this end, for the key parameters $F$ and $Cr,$ we listed the possible values [39] in

Table 3. Possible values of $F$ and $Cr$.

Parameter	Value
$F$	[0.1, 0.2, 0.3, 0.4, 0.5]
$Cr$	[0.5, 0.6, 0.7, 0.8, 0.9]

and performed a full-factorial experimental design by analyzing two response variables: the value of the objective function and the computing time.

We randomly generated six instances by inputting the parameters defined in Section 5.1. The characteristics of the instances are summarized in

Table 4. Characteristics of instances.

Model	Instances
	A	B	C	D	E	F
number of OD pairs	100	110	120	130	140	150
continuous variables	25	25	25	25	25	25
Number of binary variables	45,500	51,000	53,500	57,000	72,500	91,800
Number of integer variables	9508	10,668	11,513	11,793	13,758	14,453
Number of constraints	3793	4118	4383	4588	5488	5688

. We ran three tests for all possible combinations of $F$ and $Cr$. As mentioned in Section 4, we also tested two different mutation strategies. In summary, we performed $2\times 6\times 5\times 5$ tests in total. In addition, we recorded the sampled point during the algorithm iteration to avoid repeated calculations on the same sampling point.

For each combination of factors for a given instance, we compute the mean of three tests, and we represent the response surface corresponding to the objective function (Figure 6) and the computing time (Figure 7). We also represent the convergence curve of the optimal objective function (Figure 8). It should be noted that we only represent and analyze the tuning process of ‘DE/best/1/bin’ because it yields optimality with a smaller gap.

• We can draw some conclusions from this analysis:

There is an interesting phenomenon: for fixed values of $F$, large values of $Cr$ can yield smaller objective function values; for fixed values of $Cr$, smaller objective function values will be found near a small value of $F$(see Figure 6);

• In all cases, the best objective function values are obtained for a mutation rate $F$ near 0.2 and crossover rate $Cr$ near 0.9, and the worse objective function values are found at $F=0.5$, $Cr=0.5$ (see Figure 6).
• In all cases, for fixed values of $F$, large values of $Cr$ imply low computation times. And in most cases, the lowest computation times are found for factor $F$ near 0.3 (shown in Figure 7);
• Low values of $Cr$ imply high computation times (shown in Figure 7).
• In all cases, the DE-IRH converges within a given number of iterations (shown in Figure 9);
• With the increase in instances scale, the DE-IRH needs more computation times to reach convergence (shown in Figure 8).

We also compare our results with the optimal results obtained from Gurobi (set TimeLimit to 3600 s), and the output of Gurobi (set TimeLimit to 3600 s) is summarized in

Table 5. Computational performance of Gurobi.

Instances	Obj	Time (s)	MIPGAP
A	13,188,000.00	3600	0.17%
B	16,002,000.00	3600	0.47%
C	15,899,000.00	3600	0.34%
D	21,438,000.00	3600	0.62%
E	27,586,000.00	3600	0.74%
F	-	-	-

‘MIPGAP’: The GAP between the upper and lower bounds of Gurobi; ‘-’: no feasible solution within the given time.

, the output of DE-IRH is summarized in

Table 6. The best results of DE-IRH with different mutation strategies.

Instances	DE-IRH with ‘DE/rand/1/bin’		DE-IRH with ‘DE/best/1/bin’
	Obj	Time (s)	Obj	Time (s)
A	14,501,824.65	938.79	13,829,959.85	718.09
B	18,338,516.82	926.57	17,353,798.75	645.39
C	18,066,611.33	1041.21	17,205,817.85	831.86
D	24,045,703.38	722.45	23,355,074.36	521.48
E	32,093,562.05	823.33	30,428,477.18	510.31
F	34,690,824.31	1525.29	34,033,401.15	1130.33

, and the best combination of $F$ and $Cr$ of DE-IRH is summarized in

Table 7. The best combination of $F$ and $Cr$ of DE-IRH.

Instances	A	B	C	D	E	F
‘DE/rand/1/bin’	(0.3,0.9)	(0.4,0.8)	(0.5,0.7)	(0.4,0.6)	(0.4,0.7)	(0.5,0.9)
‘DE/best/1/bin’	(0.1,0.9)	(0.1,0.9)	(0.2,0.9)	(0.3,0.9)	(0.3,0.9)	(0.1,0.9)

. The results show that: (1) Gurobi could not obtain the optimal solution within one hour, and with the increase in the instance scale, the MIPGAP of the upper and lower bounds has a growth trend. In particular, no feasible solution can be found in instance F within 1 h. Thus, its resolution through exact procedures becomes inadequate as soon as one deals with large real-world instances; (2) the ‘DE/rand/1/bin’ mutation strategy is more likely to find a better solution with a large value of $F$ and it reduces the computing time by 3/4. The relative error between the best solution and optimal solution (obtained from Gurobi) is 9.96% (instance A); (3) it is more likely to find a better solution with the combination of a small value of $F$ with a large value of $Cr$ when taking the ‘DE/best/1/bin’ mutation strategy, and the computing time is less than 1/3 of Gurobi’s. The proposed algorithm yielded an optimality gap from 4.87% (instance A) to 10.30% (instance E) in our experiment. (4) For the optimal problem considered in this work, compared with the ‘DE/rand/1/bin’ mutation strategy, the ‘DE/best/1/bin’ mutation strategy reduces the objective value by 5.18% and computing time by 245.04 s on average on the whole.

Comparing the best results of commercial solvers against the average results of the DE-IRH, we can conclude that commercial solvers are not adequate to solve this complex problem, and the DE-IRH algorithm can obtain fairly good solutions, especially for large-scale instances.

5.3. Analyzing the Characteristics of the Solution

After analyzing the performance of the proposed DE-IRH in comparison with some state-of-the-art commercial solvers, we focus on the characteristics of solutions. Taking instance F, which is close to the real-life case, as an example, we represent some indicators of the best solution (obtained in Section 4.2) in

Table 8. The result of the robust model in different scenarios.

Scenarios	Construction Cost (RMB)	Operation Cost (RMB)	Travel Time Cost (RMB)	${\mathit{Z}}_{\mathit{\omega }}$	${\mathit{Z}}_{\mathit{\omega }}^{*}$	$\mathit{\epsilon }$
$\omega =1$	3,932,372.428	20,823,928	8,336,192.58	30,189,464.55	29,212,482.74	3.34%
$\omega =2$		22,020,040	9,255,625.78	32,305,009.75	31,164,074.52	3.66%
$\omega =3$		19,424,392	8,292,099.24	28,745,835.21	27,517,926.71	4.46%
$\omega =4$		22,588,880	9,223,253.82	32,841,477.79	31,915,826.52	2.90%
$\omega =5$		21,947,076	8,593,656.18	31,570,076.15	30,590,628.12	3.20%

and

Table 9. Depot location, capacity, and assignment of the rolling stocks of trains of train lines.

Scenarios		$\mathit{\omega }=1$	$\mathit{\omega }=2$	$\mathit{\omega }=3$	$\mathit{\omega }=4$	$\mathit{\omega }=5$
Number of train lines		697	743	660	763	728
Number of rolling stocks		127	131	118	134	134
Empty runs (km)		2859	3217	2787	3112	2959
Depot capacity and rolling stock work demand assignment	d1($\gamma =21$)	15	21	20	17	16
	d2($\gamma =11$)	23	22	24	25	24
	d3($\gamma =34$)	44	45	36	46	43
	d4($\gamma =0$)	0	0	0	0	0
	d5($\gamma =9$)	21	18	17	19	21
	d6($\gamma =2$)	11	8	7	9	10
	d7($\gamma =1$)	5	8	5	8	9
	d8($\gamma =3$)	8	9	9	10	11

. Table 8 shows the objective function values for different input data scenarios, where the deviation between the objective values of the robust optimization model (Model 2) and the ones of the non-robust model (Model 1) are all less than the given threshold (5%), in particular, a deviation of 2.9% in scenario $\omega =4 \mathrm{was} \mathrm{observed}$.

For specific input data scenario $\omega$, the number of train lines and the number of rolling stocks required of the train lines are listed in Table 9. In order to obtain a robust depot location scheme and support the line planning scheme in different input data scenarios, it is recommended to construct 21 storage tracks at depot d1, 11 storage tracks at depot d2, 34 storage tracks at depot d3, 9 storage tracks at depot d5, 2 storage tracks at depot d6, 1 storage track at depot d7, and 3 storage tracks at depot d8, and no storage tracks are constructed at depot d4. Among them, depot d3 has the most rolling stock management tasks, while few rolling stocks are assigned to depot d7, because of its different locations. The average empty train length is 23.2 km over different scenarios, which is an acceptable value for China’s large-scale intercity railway network.

Table 10. Train routes and assignment of rolling stocks of trains of train lines in different scenarios.

Routes	$\mathit{\omega }=1$		$\mathit{\omega }=2$		$\mathit{\omega }=3$		$\mathit{\omega }=4$		$\mathit{\omega }=5$
	d1	d2	d1	d2	d1	d2	d1	d2	d1	d2
s31 → s34	8		9		8		9		9
s14 → s34	7		12		12		8		7
s1 → s6		9		11		10		12		10
s5 → s5		13		11		12		12		13
s14 → s5		1
s5 → s10						2		1		1
	d3	d5	d3	d5	d3	d5	d3	d5	d3	d5
s7 → s6	8		7		7		8		5
s8 → s14	2		3		3		3		2
s8 → s6	1		1		1		1		2
s10 → s10	20		18		12		19		12
s10 → s23	13		13		13		13		19
s5 → s10			3				2		1
s7 → s10									2
s26 → s23		6		5		5		4		6
s6 → s30		9		7		6		7		9
	d6	d7	d6	d7	d6	d7	d6	d7	d6	d7
s14 → s34	11		8		7		9		10
s10 → s18 → s20		3		3		2		4		3
s10 → s25 → s20		1		2		1		1		1
s18 → s23		1		2		1		2		4
s10 → s18				1		1		1		1
	d8		d8		d8		d8		d8
s20 → s41	5		7		7		8		6
s14 → s41	3		2		2		7		5

and Figure 9 show the train routes and the number of rolling stocks provided by the depot for each train route in different input data scenarios. The first column lists all the train routes that appear, and the remaining columns show the number of rolling stocks provided by the depot in different input data scenarios in Table 10. We can conclude that: (1) The depot d3 is the busiest because it needs to meet the rolling stock work demand for seven train routes, while depot 6 just needs to meet the rolling stock work demand of one train route; (2) several intercity railway lines require more than one depot to cooperate with each other to undertake the task of rolling stock work demand. For example, Line 3 requires depots d2 and d3 to provide rolling stocks, and depots d3 and d5 together provide rolling stocks for the train of Line 4 (Figure 9); (3) the uncertainty of the input data has an impact on the train route, for example, train route ‘s14-s5′ only appears in the scenario $\omega =1$, and train route ‘s5-s10′ only appears in scenarios 3, 4, 5, etc. The detailed train routes are shown in Figure 10.

6. Conclusions

This work addresses the robust integrated depot location and line planning problem characterized by uncertainty in the input data in an intercity railway network with planning lines. The key factor in our problem is considering the dynamic relationship between depot location decisions and line planning decisions. We have proposed an integer linear programming model integrating depot location, line planning, and rolling stock work demand assignment to minimize the cost related to depot construction, the cost due to line operation (including fixed, variable, and empty train drive cost), and passenger travel time cost. The proposed model reflects the service provider’s point of view but considers the user’s travel time cost as well.

We have described the DE-IRH solution framework consisting of the DE algorithm and improved rounding heuristics for integrated depot location and line planning faced by a transit planning agency capable of solving instances of realistic sizes. We evaluated the impact of two different mutation strategies and the control parameters on the performance of the DE-RH through a series of numerical experiments. We compared the results with Gurobi to verify the performance of the DE-IRH. As reported in Section 4.2, our algorithm can yield an optimality gap of 4.87% within less computing time. In the real-size scenario, the solver cannot find a feasible solution in an acceptable time, while our algorithm can quickly provide the decision-maker with a better solution. The numerical results reveal the influence of the uncertainty of the input data on strategic decision-making, and the outputs obtained through the model can provide useful information for strategic and tactical decisions of railway departments.

The proposed method is more applicable for the single type of maintenance depot location problem, that is, there is only one type of rolling stock in operation. The maintenance capacity of the depot is not discussed in this paper. As further work, it would also be interesting to consider multi-type facilities and maintenance capacity. This also includes the development of a new efficient heuristic algorithm for solving very large instances of the model.