A Double-Level Calculation Model for the Construction Schedule Planning of Urban Rail Transit Network

Abstract

The construction of urban rail transit (URT) guides and promotes urban development. Different URT line construction schedule, including construction sequence (priority order of line construction) and construction timing (when to build), will have different effects on urban traffic and development. Therefore, the planning of construction schedule is an important part of URT network planning. At present, the determination of construction schedule is mainly based on qualitative analysis methods (i.e., experience, comparisons with other cities, and expert opinion) in engineering practice. In this study, based on an analysis of the main factors affecting the construction sequence and the construction timing data of existing URT lines, a quantitative double-level model of a construction schedule is proposed. The model consists of construction sequence and construction timing sub-models. The construction sequence sub-model employs an improved Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) with Rough Set method; the construction timing sub-model takes the results of the construction sequence model and the factors associated with urban development characteristics into account and presents an improved Logistic-β method. The model is verified using the Chengdu rail transit network as the case study. The results of the study show that the double-level calculation model could provide quantitative theoretical support for the construction schedule planning of URT network.

1. Introduction

The construction of urban rail transit (URT) guides and promotes the development of the city. Different construction schedule of URT lines, including construction sequence (priority order of line construction) and construction timing (when to build), will affect urban traffic and development differently. If the construction sequence is able to meet the present needs and the future development of the city, then it can be regarded as scientific and reasonable, and will promote urban development. If the construction timing is too early, it will lead to insufficient passenger flow and result in a heavy financial burden to the city. Meanwhile, if it is started too late, it will not be able to alleviate urban traffic congestion. Therefore, the planning of construction schedule is an important part of URT network planning.

At present, deciding on the construction schedule in engineering practice is performed mainly based on qualitative analysis methods, such as experience, comparisons with other cities, and expert opinion. Based on a study of Wuhan urban planning and Wuhan URT network planning reports, Song and Yang [1] determined the construction schedule of Wuhan rail transit network using the methods of experience and analogy with other cities. In recent years, some scholars have begun to introduce quantitative methods into the URT network planning field. However, most of this research is focused on network structure, network scale, passenger flow forecasting, or station location [2,3,4,5,6,7,8], and there are few studies focusing on construction schedule. Luo [9] designed a grey relational analysis (GRA) model with fixed attributes to determine the construction sequence. However, the model did not include construction timing, and the model with fixed index is also not universal. Traffic demand is a core factor in the construction schedule planning of a line. Peng et al. [10] proposed a construction schedule optimization model with a travel demand constraint to minimize the total discounted cost of URT with the aim of ensuring the maximum return on investment. Cheng and Schonfeld [11] proposed a construction scheduling method for rail transit line extension projects that maximized the net present value. Based on Cheng’s research, Sun [12] proposed a bilevel model for optimizing the construction schedule of URT extension lines.

Some researchers have devoted themselves to studying the construction schedule of road network lines or high-speed railway lines. Some scholars have established construction schedule models by analyzing the factors that affect them. Saphores and Boarnet [13] analyzed the impact of population growth on the optimum construction timing to maximize the return on highway project investment. Hosseininasab and Shetab-Boushehri [14] prioritized the travel time when solving the construction schedule of urban roads. Xu et al. [15] analyzed the construction schedule of high-speed railway lines for a city from the perspective of traffic accessibility. Others have focused on obtaining greater investment returns through optimizing the construction schedule. Kuby et al. [16] proposed a heuristic reverse model for the construction schedule of railway network lines to increase the return. Tao and Schonfeld [17] proposed a Lagrangian Relaxation Heuristic Algorithm approach to deciding the construction sequence of transportation projects. Tao and Schonfeld [18] developed an island model to schedule the construction of highway network lines when the project construction cost is uncertain. Kim et al. [19] presented a construction schedule model for the highway network, which sought to obtain higher returns under constrained investment. Lo and Szeto [20] used cost recovery principles to develop a model for transportation projects in order to determine the construction timing for transportation projects. Weng and Qu [21] presented a model for a road construction schedule with a limited budget. Shayanfar et al. [22] designed a construction schedule model with the aim of minimizing the total cost of road projects. He et al. [23] proposed a model to analyze the construction schedule of road-widening projects, in which the goal was to minimize the total social cost.

In summary, the models mentioned above seek mainly to obtain the maximum return on investment by optimizing the construction schedule. However, for URT construction, the cost is not the only issue, it is closely related to the development of a city, so some factors associated with urban development should be considered in the models.

To that end, a quantitative double-level model for deciding the construction schedule of URT lines is proposed in this study. The model is divided into two sub-models, one is the construction sequence sub-model (CSM), and the other is the construction timing sub-model (CTM). The CSM employs the improved TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) with Rough Set method, while the CTM takes the results of the construction sequence model and the factors associated with urban development characteristics into account and presents an improved Logistic-β function.

This paper is organized as follows. In Section 2, the double-level calculation model of the construction schedule is established. In Section 3, network planning for Chengdu rail transit of China is analyzed as a case study. In Section 4, the major conclusions and further research directions are presented.

2. Model Formulation

This paper is devoted to establishing a complete quantitative calculation model framework for determining the construction schedule of URT lines. The double-level calculation model consists of CSM and CTM sub-models. The goal of CSM is to determine the construction priority of each line. Once the construction sequence of the lines has been determined, the CTM is responsible for determining a construction time that is coordinated with urban development.

2.1. Rough Set and Improved TOPSIS Construction Sequence Sub-Model

The problem of deciding the construction sequence of URT lines can be described in mathematical terms as follows: there are m lines in a URT network, and there are n attributes affecting the sequence of line construction; by comprehensively considering all n attributes, the construction sequence of all m lines is determined. Thus, the problem of deciding the construction sequence of URT lines belongs to ranking alternatives of MCDM (Multiple Criteria Decision Making) problems.

At present, there are many methods available for solving the problems of this type, for example TOPSIS (The Technique for Order Preference by Similarity to an Ideal Solution), AHP (Analytic Hierarchy Process), SAW (Simple Additive Weighting), COPRAS (Complex Proportional Assessment), ELECTRE (ELimination Et Choix Traduisant la REalité), and so on [24]. TOPSIS offers several advantages over other approaches, such as strong adaptability and easy implementation and decision making [25]. It has been demonstrated to be one of the best methods for solving ranking issues and is used in many research fields [26,27,28,29,30]. Recently, TOPSIS has also begun to be applied in the field of transportation planning, in areas such as transportation system evaluation [31,32], the ranking of urban traffic system performance [33], and ranking of the importance of URT network nodes [34].

However, the TOPSIS method has some limitations, for example, qualitative decision making attribute weight problems, and it sometimes fails to rank alternatives. The following improvements have been made in this study.

(1)

The weightings used in the TOPSIS method are determined using qualitative methods, such as expert scoring (ES), and weighted arithmetic average (WAA). This will affect the accuracy of the ranking results. Since Rough Set theory was first proposed by Pawlak in 1982 [35,36], it has been used to improve the weight problem in many studies [37,38,39]. In this study, the Rough Set classification mechanism was used to intelligently calculate the attribute weight to solve the problem of the subjectivity of attribute weighting in TOPSIS.

(2)

The TOPSIS method just sets up two ideal solutions, namely the positive ideal solution (PIS) and the negative ideal solution (NIS). Its core idea is to find an optimal solution close to the positive ideal solution and far away from the negative ideal solution. However, when the distances between the alternatives and the ideal solution are equal, the TOPSIS method cannot rank the alternatives. In this study, an ideal solution set is designed to solve this limitation of TOPSIS.

The improved TOPSIS process flow is as follows:

Step 1: Establish a multi-attribute decision indicator system.

The lines are set as $U=\{u_1,u_2,\cdots ,u_m\}$. The decision indicators are set as $C=\{c_1,c_2,\cdots c_n\}$; then, the attribution value set of the lines is set as $V=\left\{v_{i,j}\left|i\in [1,m],j\in [1,n]\right.\right\}$; the initial decision indicator table is shown in

Table 1. Ordered initial decision indicator table of the construction sequence.

Line	Indicator
	c1	c2	⋯	cj	⋯	cn
$u_1$	$v_{1,1}$	$v_{1,2}$	⋯	$v_{1,j}$	⋯	$v_{1,n}$
$u_2$	$v_{2,1}$	$v_{2,2}$	⋯	$v_{2,j}$	⋯	$v_{2,n}$
⋯	⋯	⋯	⋯	⋯	⋯	⋯
$u_i$	$v_{i,1}$	$v_{i,2}$	⋯	$v_{i,j}$	⋯	$v_{i,n}$
⋯	⋯	⋯	⋯	⋯	⋯	⋯
$u_m$	$v_{m,1}$	$v_{m,2}$	⋯	$v_{m,j}$	⋯	$v_{m,n}$

. It should be noted that the decision indicators are selected in line with factors influencing the characteristics of urban development.

Step 2: Normalize the data.

To avoid the problem of reverse order, the indicators are standardized. Let $X=\left\{x_{i,j}\left|i\in [1,m],j\in [1,n]\right.\right\}$ be the normalized attribution value set. Then, the benefit indicators and cost indicators are treated as in Equations (1) and (2), respectively, and the normalized decision matrix DM is obtained as shown in Equation (3):

$$x_{i,j}=\left\{\begin{array}{cc}\frac{v_{i,j}–\min \left\{v_{k,j}\left|k\in [1,m]\right.\right\}}{\max \left\{v_{k,j}\left|k\in [1,m]\right.\right\}-\min \left\{v_{k,j}\left|k\in [1,m]\right.\right\}},\hfill & \max \left\{v_{k,j}\left|k\in [1,m]\right.\right\}\ne \min \left\{v_{k,j}\left|k\in [1,m]\right.\right\},\\ \hfill 1\hfill & \max \left\{v_{k,j}\left|k\in [1,m]\right.\right\}=\min \left\{v_{k,j}\left|k\in [1,m]\right.\right\}.\end{array}\right.$$

(1)

$$x_{i,j}=\left\{\begin{array}{cc}\frac{\max \left\{v_{k,j}\left|k\in [1,m]\right.\right\}-v_{i,j}}{\max \left\{v_{k,j}\left|k\in [1,m]\right.\right\}-\min \left\{v_{k,j}\left|k\in [1,m]\right.\right\}},\hfill & \max \left\{v_{k,j}\left|k\in [1,m]\right.\right\}\ne \min \left\{v_{k,j}\left|k\in [1,m]\right.\right\},\\ \hfill 1\hfill & \max \left\{v_{k,j}\left|k\in [1,m]\right.\right\}=\min \left\{v_{k,j}\left|k\in [1,m]\right.\right\}.\end{array}\right.$$

(2)

$$DM=\left[\begin{array}{cccccc}{x}_{1,1}& x_{1,2}& & \dots & x_{1,j}& \dots \\ x_{2,1}& x_{2,2}& & \dots & x_{2,j}& \dots \\ ⋮& ⋮& & \dots & ⋮& \dots \\ x_{i,1}& x_{i,2}& & \dots & x_{i,j}& \dots \\ ⋮& ⋮& & \dots & ⋮& \dots \\ x_{m,1}& x_{m,2}& & \dots & x_{m,j}& \dots \end{array}\begin{array}{c}{x}_{1,n}\\ x_{2,n}\\ ⋮\\ x_{i,n}\\ ⋮\\ x_{m,n}\end{array}\right]$$

(3)

Step 3: Construct the weighted decision matrix.

Let $sig(c_j)$ be the information entropy importance of the indicator j; the function is shown as Equation (4).

$$\begin{array}{ll}sig(c_j)=& -{\displaystyle \sum _{k=1}^{l_j}p(X_k^j)\ln [p(X_k^j)]}-{\displaystyle \sum _{k=1}^{l_C}p(X_k^C)\ln [p(X_k^C)]}\hfill \\ & +{\displaystyle \sum _{k=1}^{l_{Cj}}p(X_k^{Cj})\ln [p(X_k^{Cj})]},\forall j\in [1,n]\hfill \end{array}$$

(4)

where $l_j$ is the number of categories for lines according to indicator $c_j$, $X_k^j$ denotes the set of lines which are divided into class k according to the indicator $c_j$, $p(X_k^j)$ is the probability of dividing the line into class k according to the indicator $c_j$, $l_C$ is the number of categories of lines classified according to the whole indicator set, $p(X_k^C)$ is the probability of dividing the line into class k according to the whole indicator set, $l_{Cj}$ is the number of categories for lines after removing the indicator $c_j$ of the whole indicator set. $p(X_k^{Cj})$ is the probability of dividing the line into class k after removing the indicator $c_j$ of the whole indicator set.

The weighting of each indicator is shown in Equation (5):

$${\omega }_j=\mathrm{sig}(c_j)/{\displaystyle \sum _{f=1}^n\mathrm{sig}(c_f)}$$

(5)

where ${\omega }_j$ is the weighting of indicator $c_j$.

Let $Y=\{y_{i,j}\left|i\in [1,m],j\in [1,n]\}\right.$ be the weighting attribution value set. Then the weighting decision matrix $D{M}_{\omega }$ is obtained as shown in Equation (6):

$$\begin{array}{cc}\hfill D{M}_{\omega }& =\left[\begin{array}{cccccc}{x}_{1,1}& x_{1,2}& & \dots & x_{1,j}& \dots \\ x_{2,1}& x_{2,2}& & \dots & x_{2,j}& \dots \\ ⋮& ⋮& & \dots & ⋮& \dots \\ x_{i,1}& x_{i,2}& & \dots & x_{i,j}& \dots \\ ⋮& ⋮& & \dots & ⋮& \dots \\ x_{m,1}& x_{m,2}& & \dots & x_{m,j}& \dots \end{array}\begin{array}{c}{x}_{1,n}\\ x_{2,n}\\ ⋮\\ x_{i,n}\\ ⋮\\ x_{m,n}\end{array}\right]\left[\begin{array}{cccccc}{\omega }_1& 0& 0& \dots & 0& \dots \\ 0& {\omega }_2& 0& \dots & 0& \dots \\ ⋮& ⋮& & \dots & ⋮& \dots \\ 0& 0& & \dots & {\omega }_i& \dots \\ ⋮& ⋮& & \dots & ⋮& \dots \\ 0& 0& & \dots & 0& \dots \end{array}\begin{array}{c}0\\ 0\\ ⋮\\ 0\\ ⋮\\ {\omega }_m\end{array}\right]\hfill \\ & =\left[\begin{array}{cccccc}{y}_{1,1}& y_{1,2}& & \dots & y_{1,j}& \dots \\ y_{2,1}& y_{2,2}& & \dots & y_{2,j}& \dots \\ ⋮& ⋮& & \dots & ⋮& \dots \\ y_{i,1}& y_{i,2}& & \dots & y_{i,j}& \dots \\ ⋮& ⋮& & \dots & ⋮& \dots \\ y_{m,1}& y_{m,2}& & \dots & y_{m,j}& \dots \end{array}\begin{array}{c}{y}_{1,n}\\ y_{2,n}\\ ⋮\\ y_{i,n}\\ ⋮\\ y_{m,n}\end{array}\right]\hfill \end{array}$$

(6)

Step 4: Determine the PIS and NIS.

Let $U^{\prime }=\{u_1^{\prime },u_2^{\prime },\cdots u_m^{\prime }\}$ be the ideal solution set, let $s_{i,j}$ be the sort value of $c_j$, where $s_{i,j}=sorted(y_{1,j},y_{2,j},\cdots ,y_{m,j})[i]$ ($sorted()$ represents a function that sorts the values from large to small), $u_i^{\prime }=(s_{i,1},s_{i,2},\cdots s_{i,n})$.

Then, the PIS ($u_1^{\prime }$) is as shown in Equation (7), and the NIS ($u_m^{\prime }$) is as shown in Equation (8):

$$u_1^{\prime }=(1,1,1,\dots ,1)$$

(7)

$$u_m^{\prime }=(0,0,0,\dots ,0)$$

(8)

Step 5: Calculate the Euclidean distances.

Let $d_{i,h}$ be the Euclidean distance from $u_i=(y_{i,1},y_{i,2},\cdots ,y_{i,n})$ to $u_h^{\prime }=(s_{h,1},s_{h,2},\cdots s_{h,n})$. It is calculated by Equation (9).

$$d_{i,h}=\sqrt{{\displaystyle \sum _{j=1}^n{\left(s_{h,j}-y_{i,j}\right)}^2}},\forall i\in [1,m],h\in [1,m]$$

(9)

Given that $q_a=\min \left\{d_{a,h}\right\},(h=1,2,\cdots ,m)$, $q_b=\min \left\{d_{b,h}\right\},(h=1,2,\cdots ,m)$. If $q_a>q_b$, line a is after line b in the construction sequence; if $q_a<q_b$, line a is before line b in the construction sequence; if $q_a=q_b$, the closeness of ${\gamma }_a$ and ${\gamma }_b$ are defined according to Equations (10) and (11).

$${\gamma }_a=d_{a,m}/(d_{a,1}+d_{a,m})$$

(10)

$${\gamma }_b=d_{b,m}/(d_{b,1}+d_{b,m})$$

(11)

If ${\gamma }_a>{\gamma }_b$, line a is after line b in the construction sequence; if ${\gamma }_a<{\gamma }_b$, line a is before line b in the construction sequence.

Step 6: Determine the construction sequence.

The optimized construction sequence is obtained as shown in Equation (12):

$$U_o=\{u_{o1},u_{o2},\cdots ,u_{om}\}$$

(12)

2.2. Logistic-β Construction Timing Sub-Model

After determining the construction sequence of the line, it is necessary to determine the optimal construction timing coordinated with urban development. Based on the construction timing data of existing URT lines, this paper studies the growth curve of URT network scale. In this study, construction timing includes the start and the end time of line construction, determined by the URT network growth curve.

2.2.1. The URT Network Growth Curve

The reasonable growth curve of URT network scale can be plotted by finding a fit to the scale of growth of URT developed cities in the world. Mileage growth data for URT networks in seven representative cities around the world were collected (Figure 1), namely Beijing, Shanghai, Guangzhou, Hong Kong, Tokyo, Seoul, and Singapore. These data were fitted with S-type curves using the Gompertz, Logistic [40], and von Bertalanffy [41] equations. Both the Gompertz and the Logistic equations have fixed inflection points, while the von Bertalanffy equation has variable inflection points, as shown in

Table 2. Common S-type curves.

Name	Function	Inflection Point y	Inflection Point x	Extremum y
Gompertz	$y=A{e}^{-b{e}^{-Kx}}$	$A/e$	$\ln B/K$	$A$
Logistic	$y=\frac{A}{1+B{e}^{-Kx}}$	$A/2$	$\ln B/K$	$A$
Von Bertalanffy	$y=A{\left(1-B{e}^{-Kx}\right)}^3$	$8A/27$	$\ln 3B/K$	$A$

. The fits of results of the network scale growth curves for seven cities with three S-type curves are shown in

Table 3. Fitting results of network scale growth curves with the three kinds of S-type curves.

Name	Cities
	Beijing	Shanghai	Guangzhou	Hongkong	Tokyo	Seoul	Singapore
Gompertz	–	–	–	–	0.993	–	–
Logistic	0.978	0.995	0.966	0.979	0.992	0.974	0.943
Von Bertalanffy	–	0.989	–	0.981	0.987	0.967	0.953

.

It can be seen from Table 3 that the best fit is found with the Logistic curve. Therefore, the Logistic curve was used to describe the reasonable growth curve of network scale. The function is shown as Equation (13):

$$s=f(t)=A/(1+B{e}^{-Kt})$$

(13)

where s is the cumulative operation scale of URT, t is the cumulative construction time of URT, and A, B, and K are calibration parameters.

2.2.2. Parameter Analysis

The variations in growth curves of different cities are mainly controlled by parameters A, B, and K, as shown in Figure 2. In actual projects, the final scale of the URT network should match the demand for urban development, and the overall mileage scale of the network is affected by the value of parameter A (Figure 2a).

The timing and speed of URT construction are affected by parameter B. There are two construction stages, the start-up period and the improvement period, in parameter B. The larger parameter B is, the longer the start-up period and the slower the construction speed, which results in a longer period of centralized construction and optimization, and vice versa. Parameter B should take a lower value from the perspective of operation (see Figure 2b).

The construction period is reflected by parameter K. The larger the parameter K is, the shorter the construction period and the faster the stage will completed, and vice versa. In engineering practice, the value of parameter K should be adapted to city development such that it is neither so large that it puts pressure on urban finance nor so small that URT construction lags behind the demand for urban development (Figure 2c).

2.2.3. Construction Timing

Let S_T denote the network scale at time T, then the function is as Equation (14):

$$S_T=S_0+{\displaystyle \sum _{i=1}^n{l}_i},i\in N$$

(14)

where S₀ is the network scale at T = 0, T is the construction planning period of the URT, and l_i is the construction mileage.

As shown in Figure 2, the line mileage is open range. To solve this problem, a proximity substitution point is set: $P_0^{\prime }=\left(0,S_0^{\prime }\right)$, $P_T^{\prime }=\left(T,S_T^{\prime }\right)$. Let β denote the closeness between the logistic curve calibrated by the proximity substitution point and the ideal logistic curve, as shown in Equation (15).

$$\beta =(S_T-S_0^{\prime })/S_T=S_T^{\prime }/S_T,\beta \in \left(0,1\right)$$

(15)

The larger the value of β is, the closer the network growth curve will be to the ideal logistic growth curve. The URT network growth curve scale is defined as shown in Equation (16):

$$s=f\left(t\right)=\left\{\begin{array}{cc}{S}_T/(1+\beta /(1-\beta )e^{\frac{2\ln (1-\beta )/\beta }{T}\cdot t})& S_0=0,\\ S_T/(1+(S_T-S_0)/S_0{e}^{\frac{\ln S_0\left(1-\beta \right)/\left(S_T-S_0\right)\beta }{T}\cdot t})& S_0>0.\end{array}\right.\begin{array}{cc}& t\in \left[0,T\right]\end{array},\beta \in \left(0,1\right)$$

(16)

where s is the cumulative operation scale of the URT, t is the cumulative construction time of the URT, S_T is the total network scale (km), S₀ is the length of constructed lines, T is the construction planning period of the URT, and β is the resolution of the growth curve.

In the light of the construction sequence of the URT $U_o=\{u_{o1},u_{o2},\cdots ,u_{om}\}$ (see Equation (12)), the reasonable growth curve of URT network scale is S = f(t), and the data of line length $L_o=\{l_{o1},l_{o2},\cdots ,l_{om}\}$ and the construction end time of each line $t_{oi}^{end}$ can be obtained as shown in Equation (17):

$$t_{oi}^{end}=f^{-1}(S_i)=f^{-1}(S_0+{\displaystyle \sum _{j=1}^i{l}_{oj}}),i\in N$$

(17)

The reasonable construction period is then:

$$T_o=\left\{T_{o1},T_{o2},\cdots ,T_{on}\right\},\forall T_{oi}\in T_o,t_{oi}^{end}-T_{oi}\ge 0,i\in N$$

(18)

The construction starting time of each line $t_{oi}^{begin}$ is then as shown as Equation (19):

$$t_{oi}^{begin}=f^{-1}(S_0+{\displaystyle \sum _{j=1}^i{l}_{oj}})-T_{oi},i\in N$$

(19)

3. Case Study

The Chengdu rail transit network in China is taken as the case study. According to the Chengdu rail transit network planning report (2016–2050) (http://www.chengdurail.com (accessed on 6 May 2020)), the network consists of 46 lines, including 23 metro lines, 16 express lines, and 7 regional railways. 15 of the 23 metro lines are the focus of the study. Seven of the 15 lines have already been put into operation (solid lines, Figure 3), while eight lines are under construction (dotted lines, Figure 3). The actual construction schedule of the seven lines that have been put into operation is shown in

Table 4. Construction schedule of the seven metro lines used in this case study.

Line	Line Length (km)	Actual Construction Sequence	Actual Construction Timing
Line 1	51.88	1	28 December 2005
Line 2	46.42	2	29 December 2007
Line 4	60.89	3	23 February 2012
Line 3	52.49	4	1 January 2012
Line 10	46.25	5	1 January 2014
Line 7	38.62	6	1 January 2013
Line 5	55.85	7	1 September 2015

.

3.1. Input Data and Results

Using the data for the lines of the Chengdu rail transit network, the CSM and CTM sub-models were used to calculate the construction sequence and timing.

3.1.1. Input and calculate Data

The data and decision indicators are taken from the Chengdu rail transit network planning report (2016–2050). Seven decision indicators (i.e., C₁–C₇) were selected to form initial decision indicators matrix (

Table 5. Decision indicators for Chengdu (extracted from the Chengdu rail transit network planning report (2016–2050)).

Indicator	Specification
C1	Passenger flow intensity
C2	Passenger flow turnover
C3	Peak hour maximum sectional flow
C4	Traffic location coefficient
C5	Coverage of passenger flow distribution center
C6	Network accessibility contribution degree
C7	Urban location coefficient

and

Table 6. Initial decision matrix for the construction sequence of Line 1~15.

Line	Indicator
	C1	C2	C3	C4	C5	C6	C7
Line 1	2.75	937.93	3.69	2.4873	16.46	0.280591291	0.085790
Line 2	2.17	754.35	3.80	2.5275	17.62	0.125724324	0.090079
Line 3	2.84	1544.21	4.02	2.5877	6.33	0.079717956	0.079061
Line 4	2.19	1015.59	3.63	2.3241	7.59	−0.06363601	0.067603
Line 5	3.65	1668.81	4.91	2.5195	1.27	0.274976983	0.060436
Line 6	3.38	2262.90	4.72	2.5783	2.53	−0.292315108	0.072636
Line 7	4.97	1083.80	3.30	2.2613	30.38	1.006768672	0.132170
Line 8	2.72	1204.10	3.65	2.2780	1.27	0.168429513	0.076736
Line 9	2.43	1627.48	4.19	2.2217	7.59	1.043929557	0.065091
Line 10	1.83	1163.53	3.29	2.2461	15.19	−0.042388455	0.016284
Line 11	1.43	326.16	2.58	2.4593	1.27	0.210285834	0.095867
Line 12	1.37	731.40	2.30	2.4556	1.27	−0.521358532	0.071406
Line 13	1.49	1693.21	3.73	2.2085	8.86	0.101369899	0.009523
Line 14	1.40	1331.85	3.57	2.2417	3.80	−0.056742955	0.000000
Line 15	2.08	749.41	2.89	2.3135	3.80	−0.564448979	0.033854

). The number of categories was divided into five: extremely urgent, more urgent, urgent, not urgent, and least urgent. The weighted decision matrix based on the Rough Set of the decision table is presented in

Table 7. Weighted decision matrix for the construction sequence of Line 1~15.

Line	Indicator
	C1	C2	C3	C4	C5	C6	C7
Line 1	0.056518	0.014200	0.047416	0.162573	0.130466	0.060530	0.085790
Line 2	0.032764	0.009939	0.051168	0.186051	0.032616	0.049437	0.090079
Line 3	0.060204	0.028272	0.058673	0.221139	0.043489	0.046141	0.079061
Line 4	0.033583	0.016002	0.045369	0.067455	0.054361	0.035873	0.067603
Line 5	0.093377	0.031164	0.089033	0.181399	0.000000	0.060128	0.060436
Line 6	0.082320	0.044954	0.082551	0.215641	0.010872	0.019493	0.072636
Line 7	0.147438	0.017586	0.034112	0.030814	0.250060	0.112545	0.132170
Line 8	0.055289	0.020378	0.046051	0.040522	0.000000	0.052496	0.076736
Line 9	0.043412	0.030205	0.064472	0.007731	0.054361	0.115207	0.065091
Line 10	0.018839	0.019436	0.033771	0.021951	0.119594	0.037395	0.016284
Line 11	0.002457	0.000000	0.009551	0.146247	0.000000	0.055494	0.095867
Line 12	0.000000	0.009406	0.000000	0.144128	0.000000	0.003087	0.071406
Line 13	0.004915	0.031731	0.048780	0.000000	0.065233	0.047692	0.009523
Line 14	0.001229	0.023343	0.043322	0.019371	0.021744	0.036367	0.000000
Line 15	0.029078	0.009824	0.020126	0.061276	0.021744	0.000000	0.033854

.

The start time is set as 28 December 2005; the end time is set as 2050. The total scale of the network will be 2450.4 km, and the total construction period will be 44 years, so S₀ is set to 0, β is set to 99.6%. According to the construction period of the lines in operation, T_oi is set to 5.8.

3.1.2. Output Results

Python was used to construct the model and calculate the sequence and timing of construction. The results are shown in

Table 8. Calculation results of construction schedule of lines 1~15.

Line	Construction Sequence	Construction Timing
Line 7	1	November 2005
Line 1	2	July 2009
Line 2	3	December 2010
Line 3	4	April 2012
Line 5	5	June 2013
Line 6	6	August 2014
Line 4	7	June 2015
Line 11	8	June 2015
Line 12	9	June 2016
Line 9	10	February 2017
Line 10	11	August 2017
Line 8	12	March 2017
Line 15	13	June 2018
Line 13	14	February 2018
Line 14	15	September 2019

.

3.2. Discussion

The best way to verify the validity of the model is to compare its accuracy and precision with that of previous studies. However, engineering practice usually uses qualitative analysis methods; there are few quantitative research models at present. Of those available, the grey correlation analysis (GRA) construction sequence model [9] with fixed attributes may not be universally applicable, and the construction schedule optimization model [10] focuses on the total discount cost factor, which is inconsistent with the scope of this study. Thus, they are not suitable for use in a comparative analysis with our proposed model. Instead, we will verify the validity of the model by comparing the model calculation results with the actual construction schedule.

Since the operating lines have been tested in practice and have an objective evaluation on their construction sequence, the lines that have been put into operation (Line 1, Line 2, Line 3, Line 4, Line 5, Line 7, and Line 10) are selected for comparative analysis in the study.

3.2.1. Analysis of the Construction Sequence

The calculated and actual construction sequences are shown in

Table 9. Comparison of calculated and actual construction sequence.

Construction Sequence	I	II	III	IV	V	VI	VII	VIII	IX	X	XI	XII	i	ii	iii
Model results	7	1	2	3	5	6	4	11	12	9	10	8	15	13	14
Actual results	1	2	4	3	10	7	5	Non-operating lines

. The model calculation results are basically consistent with the actual construction sequence. However, there are significant differences in the case of three lines (i.e., Line 7, Line 4, and Line 10) which are shown in red bold in Table 9.

The model placed Line 7 at the start of the construction sequence. Line 4 and Line 10 should be given lower priority. The actual construction results are the opposite. The specific analysis is shown below.

(1)

Decision indicators

C₁ is the passenger flow intensity, which is the passenger flow per unit length (Equation (20)):

$$Q_i=P_i/l_i$$

(20)

where P_i is the daily average passenger volume of the line, and $l_i$ is the length of the line.

C₂ is the passenger flow turnover, which is the passenger volume per average transport distance (Equation (21)):

$$Z_i=P_i\cdot s_i$$

(21)

where P_i is the daily average passenger volume of the line, and $s_i$ is the average haul distance.

C₃ is the peak hour maximum sectional flow which means transportation capacity of the line during peak hours and its role in urban traffic.

C₄ is the traffic location coefficient, which is explicated the importance of the area served by the line. The more important the area served is, the higher in the construction sequence would be. The function is shown as Equation (22).

$$T{L}_i={\displaystyle \sum _{j=1}^{m_i}{R}_{ij}}/l_i$$

(22)

where $m_i$ is the number of urban roads within the service scope of the line, $R_{ij}$ is the length of the urban road $j$ within the service scope of line $i$, and $l_i$ is the length of line i.

C₅ is the coverage of passenger flow distribution centers, which is calculated by the proportion of the total number of urban transport junctions served by the line. In the study, airports, railway stations and bus terminals were set as the transport junctions. According to the planning of Chengdu, 36 points which are divided into four levels, were chosen. C₅ is obtained based on the importance of each point. The function is shown as Equation (23).

$$P{L}_i={\displaystyle \sum _{j=1}^{m_i}{D}_{ij}{h}_j}/{\displaystyle \sum _{k=1}^m{D}_k{h}_k}$$

(23)

where $m_i$ is the number of passenger flow distribution points within the attraction range of the line i, $D_{ij}$ is the number of passenger flow distribution points of type j within the attraction range of the line i, $h_j$ is the coefficient, m is the number of passenger flow distribution points within the attraction range of the network, $D_k$ is the number of passenger flow distribution points of type k, $h_k$ is the coefficient.

C₆ is network accessibility contribution degree which is defined in terms of the reduction in the efficiency of the URT network after removing a certain line. The URT network was described by topological network constructed by using a 0–1 adjacency matrix between 450 stations in 15 lines (Figure 4), and then URT network efficiency could be expressed as Equation (24).

$$A{C}_i=\frac{E_{net}-E_{n-i}}{l_i\cdot E_{net}}=\frac{\frac{1}{M\left(M-1\right)}{\displaystyle \sum _{p\ne q\in S}\frac{1}{d_{pq}}}-\frac{1}{M_i^{\prime }\left(M_i^{\prime }-1\right)}{\displaystyle \sum _{p\ne q\in S_i^{\prime }}\frac{1}{d_{pq}}}}{l_i\cdot \frac{1}{M\left(M-1\right)}{\displaystyle \sum _{p\ne q\in S}\frac{1}{d_{pq}}}}$$

(24)

where $E_{net}$ is the URT network efficiency, $E_{n-i}$ is the network efficiency after removing line i, M is the number of nodes, $M_i^{\prime }$ is the number of nodes after removing line i, $S$ is the collection of nodes, $S^{\prime }$ is the collection of nodes after removing line i, $d_{pq}$ is shortest connected distance of node p to node q, and $l_i$ is the length of line i.

C₇ is the urban location coefficient, which is calculated according to data in the overall urban planning of Chengdu (2016–2030) and research on the future development of Chengdu (2016 version). It represents the importance of the area served by the line. The more important the area served by the line, the greater the priority that should be given to the construction of the line. The function for this is shown as Equation (25).

$$R_i={\displaystyle \sum _{\mathrm{j}=1}^M{l}_{ij}/li}\cdot r_j$$

(25)

where $M$ is the number of grades classified by the importance of urban regional development, $l_{ij}$ is the mileage of the line i in the developmental grades of j, $l_i$ is the mileage of line j, and $r_j$ is the development coefficient of the region with the second development level.

(2)

The construction sequence

The decision indicator values of the seven lines are shown in radar charts (Figure 5). Specifically, the value of each indicator in the radar chart is relative to its origin. The higher the value, the better the performance will be.

The discussion of the lines with striking differences between the calculated and actual sequence and timing (Lines 7, 4, and 10) is as follows.

For Line 7, which is ranked first in the model calculation construction sequence (Table 8), the values of four indicators (C₁, C₅, C₆, and C₇) are the highest of any of the lines (Figure 5f). This can be attributed to the fact that Line 7 is in the most densely populated area (between the second and third ring roads of Chengdu). Meanwhile, it connects three large railway stations (Chengdu Station and of Chengdu East Station and Chengdu South Station). In addition, it can be seen from Figure 3 that Line 7 is a loop line which can connect with all other lines. Therefore, the model indicator values C₁, C₅, C₆, and C₇ are excellent, and the calculation result of Line 7 is the highest.

The values of indicators C₂ and C₃ are calculated from forecast data from the Chengdu Metro Network Planning Report (2016). The report predicted that indicators C₂ and C₃ (average passenger volume and peak hour passenger numbers) of Line 7 would be relatively low (see the Appendix A 

Table A1. Forecast data from the Chengdu Metro Network Planning Report (2030).

Lines	Line Length (km)	Daily Passenger Volume (Ten Thousand People)	Average Transportation Distance (km)	Peak Section Flow (Ten Thousand People/h)
Line 1	51.88	141.42	6.81	3.58
Line 2	46.42	98.63	7.94	3.74
Line 3	52.49	145.08	10.79	3.77
Line 4	60.89	110.47	8.09	3.60
Line 5	55.85	198.51	8.10	4.54
Line 6	73.51	242.58	9.21	4.13
Line 7	38.62	183.43	6.26	3.08
Line 8	44.05	115.30	10.40	3.38
Line 9	68.92	158.68	10.75	3.71
Line 10	46.25	81.92	12.36	2.79
Line 11	30.05	41.92	7.74	2.26
Line 12	58.31	76.94	9.24	2.01
Line 13	91.69	124.73	12.18	3.27
Line 14	75.13	94.05	12.89	3.16
Line 15	54.07	109.07	5.68	2.63

and

Table A2. Forecast data from the Chengdu Metro Network Planning Report (2050).

Lines	Line Length (km)	Daily Passenger Volume (Ten Thousand People)	Average Transportation Distance (km)	Peak Section Flow (Ten Thousand People/h)
Line 1	51.88	144.20	6.33	3.80
Line 2	46.42	102.92	7.05	3.86
Line 3	52.49	153.22	9.94	4.26
Line 4	60.89	155.82	7.30	3.66
Line 5	55.85	209.66	8.25	5.27
Line 6	73.51	253.78	9.03	5.30
Line 7	38.62	200.26	5.09	3.51
Line 8	44.05	124.39	9.72	3.92
Line 9	68.92	175.64	8.82	4.67
Line 10	46.25	87.17	15.08	3.78
Line 11	30.05	43.89	7.47	2.89
Line 12	58.31	83.08	9.05	2.59
Line 13	91.69	148.90	12.54	4.18
Line 14	75.13	116.86	12.42	3.98
Line 15	54.07	116.31	7.56	3.15

). However, according to the real-time data from the Chengdu Rail Transit Company (Chengdu, China) (November and December 2020) (https://www.chengdurail.com/index.html (accessed on 3 February 2021)), the passenger flow and passenger transport intensity of Line 7 were actually higher than those of other lines (Figure 6). This shows that the construction sequence should be advanced. The calculation result of the model is more reasonable than the original network construction schedule.

For Line 4, the model calculation ranks it 7th (see Table 8), and the values of its indicators are mediocre, as can be seen in Figure 5d. Passenger flow, traffic location, and urban development location are all at a disadvantage. The reason for giving priority to the construction of Line 4 in that it is a main line running east to west through Chengdu.

For Line 10, the model calculation ranks it 11th (see Table 8), the value of five indicators (C₁, C₃, C₄, C₆, and C₇) are the lowest of any line, as shown in Figure 5g. Meanwhile, the real-time passenger flow data (Figure 6) also proved that the construction of Line 10 was premature, and the passenger flow did not meet the expected levels.

The network accessibility contribution degree (C₆) of Line 10 is also the lowest of any line. Not all lines have a positive contribution to the network accessibility of the whole network, the contribution degree of Lines 4 and 10 are negative. In fact, their existence decreases the network efficiency. The main reasons for this are that the radiation distance to the periphery of the network is too large, and that the topological distance between the stations on these two lines and other stations in the network is large. The connection and transfer stations between these lines and other lines are insufficient, and their use has little influence on the shortest path between other lines. This finding indicates that the route or transfer station setting of these lines is not based on the network, but more on urban development requirements and other factors. In contrast, Line 7 increases network efficiency.

In summary, it is calculated by the double-level model that the construction sequence is basically consistent with the actual sequence. Where there are differences, Line 7 should have been the first line constructed, while Lines 4 and 10 should have been placed later in the schedule. The analysis shows that the results calculated by the model are better than the original construction sequence plan.

3.2.2. Analysis of Construction Timing

The construction timing is determined by the URT network growth curve, so the growth curve of Chengdu rail transit is analyzed first in this section, and then the construction timing is discussed.

(1)

The growth curve of Chengdu rail transit network

Analyzing the construction process of Chengdu rail transit in the past 15 years (from 2005 to 2020), the reasonable growth curve of Chengdu rail transit was calculated by Equation (16). The growth curve of Chengdu rail transit is shown in Figure 7. It can be seen that the construction timing lags behind at the beginning, but it coincides with the reasonable growth curve in the twelfth year of construction.

The actual construction time of several segments in the initial lagged slightly, so that the scale of the actual network could not reach the reasonable scale level. To a certain extent, this reduced the service capacity of the URT in the initial stages and resulted in additional resources being made available in later construction phases to improve the construction speed.

(2)

The discussion of construction timing

On the basis of the actual construction stage of the Chengdu rail transit, the lines were divided into different construction segments. The construction timing of each segment calculated by the double-level model is shown in Figure 8.

The red line in Figure 8 shows the calculated construction timing calculated by the model, and the blue line shows the actual construction timing. The calculated construction timing of several lines (Line 1 segment 3, Line 3 segment 2, Line 5, Line 7, and Line 10 segment 2) are consistent with the actual construction timing.

At the same time, the actual construction timing of several lines lagged that of the model. The construction timing of Line 1 segment 1 lagged by 2.25 years; and that of Line 1 segment 2 lagged by 21 years. Meanwhile, Line 2, Line 4 segment 1, Line 3 segment 1, Line 4 segment 2, and Line 10 segment 1 lagged about 1 year. This is because, in practice, the construction time of Line 1 segment 1 was delayed, resulting in the delay of the construction time of subsequent lines. The poor geological conditions in Chengdu are also another reason for the delay in follow-up line construction.

Fortunately, the construction speed of follow-up lines has gradually increased, marking up for the delays in the early stages. Chengdu rail transit will soon enter the key period of centralized construction. The network scale that needs to be completed in the initial stage has basically reached its goal. The construction timing is becoming more reasonable, which meets the needs of urban development.

Overall, the effectiveness of the double model is verified by the case study, and thus, the model can provide quantitative theoretical support for the construction schedule planning of future URT networks. It should be noted that the CSM still has some limitations in terms of the selection of decision attributes, and more URT construction data, especially for small and medium-sized cities, should be included in the CTM.

4. Conclusions

This paper proposed a novel double-level quantitative calculation model framework for determining the construction schedule of URT lines. The factors affecting the construction sequence were analyzed, and a construction sequence sub-model (CSM) was developed based on a Rough Set and improved TOPSIS approach. Based on the theory of network scale growth and city-specific characteristics, a Logistic-β construction timing sub-model (CTM) was built. The model was verified using a case study based on network planning for the Chengdu rail transit network. The study illustrates that the double-level calculation model can provide quantitative theoretical support for the construction schedule planning of future URT networks.

At the same time, although the CSM uses an improved TOPSIS approach, the selection method of decision attributes needs to be further studied before ranking. In addition, because many small and medium-sized cities are constructing URT networks, more URT construction data should be included in future versions of the model to make the CTM results more accurate. Finally, traffic demand is the core factor in the construction schedule planning of lines; how to integrate traffic volume into the models in a more appropriate way, and how to use newly developed software, such as BIM, in the models still requires further study.