Optimization of Heterogeneous Passenger Subway Transfer Timetable Considering Social Equity

Abstract

With the accelerated operation of subway networks, the increasing number of subway transfer stations results in inefficient passenger travel. The target of this paper is to solve the research question of how to reduce transfer waiting time (TWTT) for heterogeneous passengers. The key problem is to determine the optimal concerted train timetable considering the transfer walking time (TWKT) of the passengers. On the basis of field survey data, the regression method was used to establish a TWKT prediction model for general passengers (G) and vulnerable passengers (V), including the elderly, passengers traveling with children, and those carrying large luggage. Afterward, a two-objective integer programming model was formulated to minimize the subway operating costs and TWTT for each group, in which V is given the priority weight to ensure social equity. The headway, loading capacity, and TWKT of heterogeneous passengers were set as optimization model constraints. A genetic algorithm (GA) was designed to find the optimal solution. A case study in which the Beijing Jianguomen Station was selected as the key transfer station was conducted to verify the performance of the proposed model. Key results show that the total TWTT for V and G can be reduced by 18.6% and 27.2%, respectively, with one train saved. Results of the parameter sensitivity analysis reveal the interconnection between the operating cost, heterogeneous passenger proportion, and transfer time. The proposed model can be used for improving transfer efficiency for passengers while considering the enterprise operating costs.

1 Introduction

Subways, which can help alleviate road traffic congestion, play a vital role in society’s sustainable development. Travelers in the city are increasingly becoming concerned with the quality of their travel. Thus, additional attention is paid to whether safe and effective evacuation is available to passengers inside subway stations [1]. Low transfer efficiency affects passenger satisfaction. Transfer stations are nodes that connect subway networks, and they directly influence the benefit and service levels of the entire subway network. As shown in Fig. 1, developed and developing countries have numerous subway transfer stations. Data on subway transfer stations were obtained from the official websites of cities in different countries in February 2022. Clearly, optimizing the transfer station timetable to benefit passengers and enterprise is an important scientific problem worth studying.

Fig. 1

Number of subway transfer stations in cities in developed and developing countries

Passengers traveling in subways are heterogeneous in terms of age, walking behavior, and load [2]. From the perspective of social equity, passengers can be divided into two groups by considering the convenience of transferring. The first group is the vulnerable passenger (V) group, which includes the elderly, passengers with children, and passengers with luggage. This group has reduced mobility and needs additional help. The other group is the general passenger (G) group. Heterogeneous passengers at transfer stations are shown in Fig. 2, in which V and G members are waiting at a transfer platform. Accordingly, focus should be given to service equity interest in urban public transport.

Fig. 2

Heterogeneous passengers at a transfer station

Total transfer time (TTT) is an important factor in evaluating transfer efficiency, and includes transfer walking time (TWKT) and transfer waiting time (TWTT). The two lines that transfer passengers successively are called feeder line $p$ and receiving line $q$ [3]. That is, TTT for passengers is the time lag between the arrival of the feeder train and the departure of the receiving train. If the time lag is equal to TWKT, TWTT is zero. If the time lag is more than TWKT, then passengers have to spend more time to wait for the next train. TWKT consists of the time passengers spend walking at different transfer facilities, and it is determined by the passengers’ walking speed, transfer facilities, and passenger flow intensity, among others. Greater train departure frequency related to the train timetable is considered a straightforward and effective solution to improve operational service levels [4]. Train timetables are closely related to the service level and costs of subways. However, service level and costs are two objectives that are in conflict with each other. The longer the train headway, the longer the waiting time of passengers. Hence, the interests of operating enterprises and the transfer efficiency of passengers should be coordinated and optimized. In consideration of the TWKT of heterogeneous passengers, coordinating the arrival and departure of trains between two lines in the transfer station is one of the effective methods for reducing waiting time and improving transfer efficiency. The headway during off-peak periods is larger than that during rush hour, thereby easily influencing passengers’ transfer efficiency.

Suppose one transfer station has two lines. The train arrives at feeder line $p$ according to the timetable. Heterogeneous passengers get off the train and transfer to wait for the next train on the receiving line $q$. The headway of the trains on line $p$ is 4 minutes, which is the reference line. Assuming that two passengers—one G and one V—get off each feeder train, TWKTs for G and V are 2 and 3 minutes, respectively, as shown in Fig. 3a. The transfer process of heterogeneous passengers in a subway in three train timetables is shown in Fig. 3b–d. The headways of trains on line $q$ vary, making the time lag between the trains on lines $p$ and $q$ different, thereby influencing the transfer efficiencies of V and G.

Fig. 3

Illustration of the TWTT of heterogeneous passengers in different train timetables. a TWKT distribution, b train timetable 1, c train timetable 2, d train timetable 3

The detailed train timetables and TWTTs for heterogeneous passenger transfers in three train timetables are shown in Table 1. The TWTTs of the total passengers shown in Fig. 3b–d are 15, 15, and 11 minutes, respectively. The average TWTTs of V and G shown in Fig. 3b–d are 3, 3, 1.3 minutes and 2, 2, 2.3 minutes, respectively. As shown in Fig. 3b, c, the total TWTT and average TWTT of heterogeneous passengers is the same under different numbers of trains on receiving line $q$. Although the number of trains on line $q$ in train timetables 2 and 3 are the same, the average TWTT of passenger V is reduced by 55.56%. Compared with train timetable 2, the total waiting time for heterogeneous passengers on train timetable 3 is reduced by 26.67%. An efficient train timetable can be easily inferred to reduce the number of trains and also optimize the total TWTT of heterogeneous passengers from the perspective of fairness.

Table 1 Train timetable and waiting time of heterogeneous passengers

To the best of our knowledge, previous methods have focused on TWTT optimization for total passengers, without considering the different characteristics among passengers. From the social fairness aspect, TWTT for V should be prioritized. In this study, the service level related to heterogeneous passengers and costs of subways are considered simultaneously. The main contributions of this study are twofold:

• 1. Passengers are classified into G and V based on transferring convenience. To obtain the TWTT of V and G, the average TWKT is calculated by developing the TWKT prediction model.

• 2. A two-objective integer programming model is established to minimize the operational costs and total TWTT of heterogeneous passengers in a vital transfer station. In the optimization model, V is given a certain priority by setting the weight parameter. The relationship between the interests of the subway operating enterprise and TWTT for passengers is analyzed.

Given that the average walking speed of heterogeneous passengers is related to the type of transfer facilities and section passenger flow, the advantage of the proposed prediction model is that it can predict the average TWKT of V and G under different passenger flows.

The remainder of this paper is organized as follows: Section 2 gives a brief literature review. Section 3 presents the TWKT prediction model and a timetable optimization model. The genetic algorithm (GA) is used to solve the optimization model. Section 4 provides the case study of a key transfer station. Section 5 presents a discussion of the sensitivity analysis of parameters. Lastly, Sect. 6 presents the conclusions and future research topics. A flow diagram of this study is shown in Fig. 4.

Fig. 4

Flow diagram of the study

2 Literature Review

2.1 Heterogeneity of Passengers

Passengers in subways are not homogeneous and are regarded as different individuals with heterogeneous characteristics. Heterogeneous passengers have different requirements for a successful transfer probability in trips. Characteristics such as age, luggage, gender, and walking direction result in remarkable differences in walking speed. Patra et al. [5] indicated that the walking speed of younger people is faster than that of middle-aged people and the elderly. Passengers carrying luggage walk more slowly than those without luggage. To provide some differences of individuals with disabilities within a crowd, Stuart et al. [6] conducted a crowd experiment with heterogeneous crowds, including individuals with disabilities. Chen et al. [7] proposed a cellular automaton model to describe the movement of children with and without group behavior. Xie et al. [8] proposed a schedule-based itinerary-choice model to address the independently and identically distributed assumptions used in random utility models and the heterogeneity of passengers’ perceptions. Li et al. [9] discovered that heterogeneous passengers in the passenger flow would increase total boarding time and decrease boarding efficiency. Each group differs in velocity, and interaction within a crowd varies as well. From the aforementioned studies, passengers are regarded as a heterogeneous group because of differences in age, space occupied, and disability compared with general passengers.

2.2 Fairness of Service

Social sustainability, particularly that associated with social equity, should be given considerable attention [10]. A subway timetable synchronization optimization model was formulated to optimize passengers’ waiting time while relatively limiting the waiting time of all transfer stations in a subway network. Departure time, running time, dwelling time, and headways for all directions were adjusted to improve the worst transfer in a subway network [11]. Qi et al. [12] proposed a novel method to deal with the train timetabling problem in consideration of women-only passenger cars on a subway line. A simulation-based model was established by dividing train cars into two types: general passenger cars and women-only passenger cars. Karakoc et al. [13] proposed a novel method to incorporate the concepts of social equity and social vulnerability with the measures of infrastructure network restoration scheduling.

2.3 Transfer Time

Many passengers may have no choice but to make at least one transfer when they travel from origin to destination in urban subways. Transfer time is a vital index in evaluating transfer efficiency, and TWKT and TWTT are its two components. The TWKT of passengers is affected by numerous factors during the transfer process. Mohring et al. [14] found that passengers’ perception of their waiting time was nearly twice the actual time. Transfer distance is an important factor affecting the walking time of passengers, and transfer mode determines transfer distance. Among all transfer modes, passengers must walk a long distance in the two modes station hall and channel transfers. Zhou et al. [15] established a prediction model for subway passengers’ transfer and walking time according to transfer passenger flow and the types of transfer facilities. Average waiting times of passengers on subway platforms were lower than the headway but more than half the headway [16]. Arrival patterns of passengers can be grouped into two categories: (i) one group arriving randomly (e.g., when the timetable is not known) and (ii) another group arriving according to a beta distribution. Ingvardson et al. [17] proposed a general framework to estimate passenger waiting times. The results showed that even with a 5-minute headway, 43% of passengers arrived at the station on time when timetables were available.

2.4 Schedule Optimization

2.4.1 Single-Objective Schedule Optimization

Wong et al. [18] presented a mixed-integer-programming optimization model to minimize the total TWTT of passengers. Niu et al. [19] established an integer programming model that considers linear constraints to minimize the waiting time of total passengers with given time-varying origin-to-destination passenger demand matrices. Hassannayebi et al. [20] presented a path-indexed nonlinear formulation of the train timetabling problem with the objective of minimizing the average waiting time per passenger. Yuan et al. [21] developed a mixed-integer linear programming model to minimize the total waiting time of passengers by considering station passing, platform load, and train transport capacities. Shi et al. [22] proposed an integrated integer linear programming model to minimize total passenger waiting time. Veelenturf et al. [23] formulated an integer linear programming model to reduce the number of canceled and delayed train services. Zhang et al. [24] formulated two nonlinear nonconvex programming models to optimize timetables, aiming to minimize total passenger travel time. To minimize systematic energy, Wang et al. [25] proposed an integrated energy-efficient train operation method. Two dynamic programming algorithms were used to obtain the global optimal solution.

2.4.2 Multiobjective Schedule Optimization

An increasing number of studies have focused on the timetable problem by formulating a multiobjective optimization model. Yin et al. [26] developed a stochastic programming model for the subway train rescheduling problem to jointly optimize passengers’ delay time, traveling time, and energy consumption of the train. Hassannayebi et al. [27] established a multiobjective stochastic programming model to minimize passenger time and overload. Yin et al. [28] proposed a mixed-integer linear programming approach for the train scheduling problem to minimize energy consumption and passenger waiting time. Mo et al. [29] considered various system constraints and proposed a flexible subway train scheduling approach to minimize energy cost and passenger waiting time. Guo et al. [30] formulated a mixed-integer programming approach to improve the last train schedule planning to optimize transfer efficiency for homogeneous passengers. The result showed that the average waiting time for key stations improved by 15.07%. Nguyen et al. [31] built a multiobjective optimizer to minimize passengers’ journey times, transfer rates, and operator cost.

For convenience of comparison, recent publications on train scheduling problems are listed in Table 2. The optimization objectives address two aspects: (i) passenger level and (ii) enterprise level. The factors in each kind are listed at the bottom of Table 2. Although the TWKT was considered for the transfer time calculation in a few studies, it is treated as a constant. The heterogeneity of the transfer passengers is rarely discussed. The walking speed and TWKT vary among different passengers, especially with different station structures and numbers of facilities. In this study, the objective function is set up with precise calculation for heterogeneous passengers. At present, demands exist for the disabled, the elderly, passengers carrying luggage, and those traveling with children and often require some specific services (e.g., particular seats, auxiliary facilities). Once the transfer behavior of passengers is involved, the complexity of the problem increases substantially. When they have to transfer in the subway, the train timetable can effectively increase their transfer convenience and improve travel satisfaction.

Table 2 Characteristics of closely related studies compared in this work

3 Model Formulation and Solution

3.1 Notations

For modeling convenience, Table 3 lists all the relevant sets, parameters, and decision variables used in the formulation.

Table 3 Indices, parameters, and variables

3.2 Basic Assumptions

Without loss of generality, this research is based on the following assumptions:

• (1) All trains run according to the planned schedule. The time of the train doors opening is the arrival time of the train. The time of the train doors closing is the departure time of the train.

• (2) The provided total train capacity is sufficient, and no passenger is detained on the platform.

• (3) Passengers choose to take the train closest to them and will not wait for the next train.

• (4) No delay occurs for passengers in the transfer process.

• (5) In a certain period, the number of transfer passengers in a transfer direction of the subway station remains unchanged.

3.3 TWKT Prediction Model

The TWKT prediction model is established in accordance with factors affecting the walking speed of passengers [see Formula (1)]. The factors are passenger type, transfer facility, and passenger flow. The model is composed of four parts, and each part is the time that passengers walk on platforms, horizon passages, stairs, and escalators. $t_{\rm spqia}^{\rm Walk}$ is calculated as follows:

$$t_{\rm spqia}^{\rm Walk} = \sum\limits_{b = 1}^{4} {t_{\rm sba}^{\rm Walk} } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S,{\kern 1pt} {\kern 1pt} {\kern 1pt} i \in K_{p} ,{\kern 1pt} {\kern 1pt} a \in A,{\kern 1pt} {\kern 1pt} {\kern 1pt} b \in B,$$

(1)

where $t_{\rm spqia}^{\rm Walk}$ is the average walking time for type $a$ ($a$ = 1, 2; 1 stands for the V, 2 stands for the G) passengers on train $i$ from line $p$ to line $q$ at transfer station $s$, and $b$ is the transfer facility ($b$ = 1, 2, 3, 4; 1 stands for the platform, 2 stands for the horizon passage, 3 stands for the stairs, 4 stands for the escalator). For the type $a$ passengers at transfer station $s$, $t_{\rm sba}^{\rm Walk}$ represents the average walking time on transfer facility $b$. $t_{\rm sba}^{\rm Walk}$ is calculated as follows:

$$t_{\rm sba}^{\rm Walk} = \frac{{L_{\rm sba} }}{{v_{\rm sba} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S,{\kern 1pt} {\kern 1pt} {\kern 1pt} a \in A,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} b{\kern 1pt} \in B,$$

(2)

where $L_{\rm sba}$ is the walking distance on transfer facility $b$ of type $a$ passengers at transfer station $s$, and $v_{\rm sba}$ is the average walking speed on transfer facility $b$ of type $a$ passengers at transfer station $s$.

Du et al. [3] and Zhou et al. [15] indicated that the distribution of passengers on the platform is uniform. $L_{sp1a} (x)$ is the distance from a passenger’s position $x$ to the position of passage entrance on the platform. When passengers transfer from feeder line $p$ to the platform of the receiving line $q$, they often gather at the door near the exit and wait for the train. $L_{s1a}$ represents the average walking distance on the platform at transfer station $s$ for the type $a$ passengers, which is calculated as follows:

$$L_{s1a} = \int_{0}^{{l_{\rm sp1} }} {\frac{{L_{sp1a} (x)}}{l}} dx + \frac{{l_{sq1} }}{2(R + 1)}\quad \forall s \in S,{\kern 1pt} {\kern 1pt} {\kern 1pt} a \in A,$$

(3)

where $l_{\rm sp1}$ is the length of the platform of feeder line $p$ at transfer station $s$. $l_{sq1}$ represents the length of the platform of receiving line $q$ at transfer station $s$, and $R$ is the number of exits on the platform of the receiving line $q$.

$F_{\rm sb}$ is the section passenger flow of transfer facility $b$ at transfer station $s$; when $b$= 1, 2, 3, $F_{\rm sb}$ is calculated as follows:

$$F_{\rm sb} = \frac{{Q_{\rm sb} }}{{B_{\rm sb} \times t}} \times \frac{1}{R}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} b{\kern 1pt} \in B,$$

(4)

where $Q_{\rm sb}$ is the number of passengers passing through the section of facility $b$ at transfer station $s$, $B_{\rm sb}$ is the width of transfer facility $b$, and $t$ is the time of passengers passing through transfer facility $b$. $v_{\rm sba}$ ($b$= 1, 2, 3) represents the average walking speed of type $a$ passengers walking on transfer facility $b$ at transfer station $s$, which is calculated as follows:

$$v_{\rm sba} = \varphi_{\rm sba} F_{\rm sb}^{2} + \chi_{\rm sba} F_{\rm sb} + \varepsilon_{\rm sba} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} a \in A,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} b \in B,$$

(5)

where $\varphi_{\rm sba}$, $\chi_{\rm sba}$, and $\varepsilon_{\rm sba}$ are the estimable parameters.

When passengers walk on the stairs, the two forms are upward and downward stairs. For the type $a$ passengers at transfer station $s$, the average walking time on stairs is marked by $t_{\rm s3a}^{\rm Walk}$ , which is calculated as follows:

$$t_{\rm s3a}^{\rm Walk} = \frac{{L_{s3}^{\rm Up} }}{{v_{\rm s3a}^{\rm Up} }} + \frac{{L_{s3}^{\rm Down} }}{{v_{\rm s3a}^{\rm Down} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} a \in A,$$

(6)

where $L_{s3}^{\rm Up}$ is the slope length of the upward stairs, and $L_{s3}^{{D{\text{own}}}}$ is the slope length of the downward stairs. For the type $a$ passengers at transfer station $s$, $v_{\rm s3a}^{\rm Up}$ and $v_{\rm s3a}^{\rm Down}$ represent the average walking speed of passengers on the upward and downward stairs, respectively.

The length and speed of the escalator are fixed. When passengers are on the escalator, the time they spend on it is certain. $t_{s4a}^{\rm Walk}$ is the time that passengers spend on the escalator, which is calculated as follows on the basis of queuing theory:

$$t_{s4a}^{\rm Walk} = \frac{{L_{\rm s4} }}{{v_{\rm s4} }}{ + }W_{\rm se}^{\rm Del} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S,{\kern 1pt} {\kern 1pt} {\kern 1pt} a \in A,$$

(7)

where $L_{\rm s4}$ is the slope length of the escalator and $v_{\rm s4}$ is the stable escalator speed. $W_{\rm se}^{\rm Del}$ is the delay time for passengers waiting before the escalator, which is calculated by Formula (8):

$$W_{\rm se}^{\rm Del} = \left\{ \begin{gathered} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \lambda \le I_{4} \hfill \\ \frac{{L_{\rm se}^{\rm Que} }}{\lambda }{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \lambda {\kern 1pt} > I_{4} \hfill \\ \end{gathered} \right.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S,$$

(8)

$$\lambda = \frac{{Q_{\rm s4} }}{{t_{\rm se}^{\rm Rea} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S,$$

(9)

where $L_{\rm se}^{\rm Que}$ is the average queue length, $\lambda$ is the passenger average arrival rate, and $I_{4}$ is the border service intensity. $\lambda$ is calculated by Formula (9). $Q_{\rm s4}$ represents the number of passengers passing through the section of the escalator at transfer station $s$, and $t_{\rm se}^{\rm Rea}$ is the total time for passengers reaching the escalator.

3.4 Two-Objective Optimization Model

3.4.1 Objective

(1) Fleet size

The total number of trains $K$ is the sum of trains from receiving line $q$ during the research period: the larger the size of $K$, the larger the fleet size, and the larger the operational cost expenditure by the enterprise. $K$ is expressed by $n_{q}$, as shown in Formula (10):

$$K = n_{q}$$

(10)

(2) TWKT

In Formula (11), $t_{\rm spqia}^{\rm Wait}$ is the TWTT for each passenger of type $a$ transferring from train $i \in K_{p}$ at station $s \in S$, $t_{spi}^{Arr}$ is the arrival time of train $i$ on feeder line $p$ at transfer station $s$, $t_{sqj}^{\rm Dep}$ is the departure time of train $j$ on receiving line $q$ at transfer station $s$, and $t_{\rm spqia}^{\rm Walk}$ is the average TWKT of type $a$ passengers transferring from train $i$ at transfer station $s$. $t_{\rm spqia}^{\rm Walk}$ can be calculated by Formula (1). $\psi$ is a binary variable which determines whether passengers can make a successful transfer in Formula (12). If the transfer is successful, $\psi$ is equal to 1; otherwise, it is set as 0.

$$t_{\rm spqia}^{\rm Wait} { = (}t_{sqj}^{\rm Dep} - t_{spi}^{Arr} - t_{\rm spqia}^{\rm Walk} ){\kern 1pt} {\kern 1pt} \psi {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} a \in A,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i \in K_{p} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} j \in K_{q}$$

(11)

$$\psi \left\{ \begin{gathered} 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} t_{sqj}^{\rm Dep} \le t_{spi}^{Arr} { + }t_{\rm spqia}^{\rm Walk} \le t_{{sqj{ + }1}}^{\rm Dep} \hfill \\ 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{otherwise}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hfill \\ \end{gathered} \right.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} a \in A,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i \in K_{p} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} \forall j \in K_{q}$$

(12)

$t_{\rm spqv}^{\rm Wait}$ is the total TWTT of V transferring from line $p$ to line $q$ at transfer station $s$, which is calculated by Formula (13). $t_{\rm spqg}^{\rm Wait}$ is the total TWTT of G, which is calculated by Formula (14). $Q_{\rm spqi}$ is the total number of passengers transferring from train $i$ on feeder line $p$ to train $j$ on the receiving line $q$. $t_{\rm spqiv}^{\rm Wait}$ and $t_{\rm spqig}^{\rm Wait}$ are the TWTT of each passenger of V and G transferring from train $i$ at station $s$. $\theta_{v}$ and $\theta_{g}$ are the percentages of V and G, respectively, and the values of $\theta_{v}$ and $\theta_{g}$ are between 0 and 1, with the sum of 1. $\theta_{v}$ and $\theta_{g}$ are used to calculate the total TWTT for V and G, respectively. They are given by the proportion of the two types of passengers from a field survey in the subway station. $T^{\rm Wait}$ represents the total TWTT of the heterogeneous passengers, where $\delta_{v} \in N +$ and $\delta_{g} \in N +$ are the weight coefficients for V and G, respectively. If there is no distinction between V and G, which means that the two types of passengers are considered homogeneous, the values of $\delta_{v}$ and $\delta_{g}$ are both set as 1. From the perspective of social equity, V should be given more priority on the index TWTT than G, and $\delta_{v}$ is set greater than $\delta_{g}$. If G is given more priority on the index TWTT, $\delta_{g}$ is set larger than $\delta_{v}$.

$$t_{\rm spqv}^{\rm Wait} = \theta_{v} \sum\limits_{i = 1}^{{n^{p} }} {Q_{\rm spqi} t_{\rm spqiv}^{\rm Wait} } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i \in K_{p} {\kern 1pt} {\kern 1pt}$$

(13)

$$t_{\rm spqg}^{\rm Wait} = \theta_{g} \sum\limits_{i = 1}^{{n^{p} }} {Q_{\rm spqi} t_{\rm spqig}^{\rm Wait} } \quad {\kern 1pt} \forall s \in S,{\kern 1pt} {\kern 1pt} {\kern 1pt} i \in K_{p} {\kern 1pt}$$

(14)

$$T^{\rm Wait} = \delta_{v} t_{\rm spqv}^{\rm Wait} + \delta_{g} t_{\rm spqg}^{\rm Wait} \quad \forall s \in S$$

(15)

(3) Objective function

The proposed model aims to minimize operating costs and the total TWTT of heterogeneous passengers. The objective function of the optimization model is shown in Formula (16). Considering the different dimensions of $K$ and $T^{\rm Wait}$, in this paper, the linear normalization method is applied to normalizing $K$ and $T^{\rm Wait}$. $\overline{{T^{\rm Wait} }}$ and $\overline{K}$ are the standardization of $T^{\rm Wait}$ and $K$ in Formulas (17) and (18), respectively. To achieve a trade-off between operating cost and TWTT, weight coefficients $\gamma_{1}$ and $\gamma_{2}$ are set. Values of $\gamma_{1}$ and $\gamma_{2}$ range from 0 to 1, with the sum of 1. Generally, the total TWTT of passengers and the operating cost of the enterprise can be treated as equally important, and the values of $\gamma_{1}$ and $\gamma_{2}$ are both set as 0.5. If more importance is attached to passenger satisfaction with travel, $\gamma_{1}$ will be set greater than 0.5. If more consideration is given to conserving operational resources and reducing operational costs, $\gamma_{2}$ should be set greater than 0.5. The value of $\gamma_{1}$ and $\gamma_{2}$ can be determined according to the actual situation of the subway station operation by the decision-maker. $T_{\min }^{\rm Wait}$ and $T_{\max }^{\rm Wait}$ are the minimum and maximum TWTT for each $K$. $K_{\min }$ and $K_{\max }$ are the minimum and maximum fleet size, respectively.

$$\min Z = \gamma_{1} \overline{{T^{\rm Wait} }} + \gamma_{2} \overline{K}$$

(16)

$$\overline{{T^{\rm Wait} }} = {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{T^{\rm Wait} - T_{\min }^{\rm Wait} }}{{T_{\max }^{\rm Wait} - T_{\min }^{\rm Wait} }}$$

(17)

$$\overline{K} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{K - K_{\min } }}{{K_{\max } - K_{\min } }}$$

(18)

3.4.2 Constraints

(1) Headway constraints

The safe operation of trains on the feeder and receiving lines is guaranteed by Constraint (19). $h_{q\max }$ is the maximum headway on receiving line $q$, and $h_{q\min }$ is the minimum headway on receiving line $q$. The values of $h_{q\max }$ and $h_{q\min }$ are determined by survey.

$$h_{q\min } \le t_{{sqj{ + }1}}^{Dep} - t_{sqj}^{Dep} \le h_{q\max } \quad {\kern 1pt} {\kern 1pt} \forall s \in S,\quad j \in K_{q}$$

(19)

(2) First and last train constraints

The arrival time of the first train on line $q$ should be equal to or later than the beginning time $t_{beg} \in N$ in Constraint (20) during the research period. The departure time of the first train on line $q$ during the research period should not exceed $h_{q\max }$ of the train before the first train. Constraint (21) ensures that the departure time $t_{{sqn_{q} }}^{Dep}$ of last train $n_{q}$ on receiving line $q$ is earlier than the end time $t_{end} \in N{ + }$. $t_{{\rm spqn_{p} v}}^{\rm Walk}$ is the average TWKT of V transferring from train $n_{p}$. Moreover, this train can transport the passengers of the last train on feeder line $p$.

$$t_{beg} \le t_{sq1}^{Dep} \le t_{sq0}^{Dep} + h_{q\max } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S$$

(20)

$$t_{{spn_{p} }}^{Arr} + t_{{\rm spqn_{p} v}}^{\rm Walk} \le t_{{sqn_{q} }}^{Dep} \le t_{end} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S$$

(21)

(3) Loading capacity constraints

The train is assumed to have sufficient capacity during the research period, which is expressed in Constraint (22). In particular, $n_{q}$ is the fleet size, which is the total number of trains on line $q$; $c_{q}$ is the number of grouped trains; $m_{q}$ is the load capacity of the carriage; and $\tau_{q}$ is the load factor of trains. $Q_{q}$ in Constraint (23) is the total passenger flow in line $q$ during the research period. $Q_{sqj}^{Abo}$ is the number of passengers boarding train $j$ on receiving line $q$ at transfer station $s$, and $Q_{sqj}^{Ali}$ is the number of passengers alighting on train $j$ on receiving line $q$ at transfer station $s$. The values of $c_{q}$, $m_{q}$, and $\tau_{q}$ are obtained by survey. The value of $Q_{q}$ can be calculated from automatic fare collection (AFC) data.

$$n_{q} c_{q} m_{q} \tau_{q} \ge Q_{q}$$

(22)

$$Q_{q} = \sum\limits_{o = 1}^{{o_{q} }} {\sum\limits_{j = 1}^{{n_{q} }} {(Q_{sqj}^{Abo} } } + Q_{sqj}^{Ali} ){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} o \in O,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j \in K_{q}$$

(23)

(4) Heterogeneous passenger transfer constraints

$t_{\rm spqig}^{\rm Walk}$ must be less than $t_{\rm spqiv}^{\rm Walk}$ according to Constraint (24). Meanwhile, $r$ is the minimum TWKT, which is used for the selection of the transfer station.

$$t_{\rm spqiv}^{\rm Walk} > t_{\rm spqig}^{\rm Walk} > r{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall s \in S,{\kern 1pt} {\kern 1pt} i \in K_{p} {\kern 1pt} {\kern 1pt}$$

(24)

3.5 Solution Algorithm

It has been proven that the timetable synchronization problem belongs to the NP-hard class [32]. Artificial intelligence techniques are commonly used to solve these problems. Compared with the branch-and-bound (B&B) method, the performance of the GA is more satisfactory in solution time and quality [33]. The GA is simple and convenient, with fast solution speed and low cost. It can be well adapted to engineering practice and can efficiently optimize and solve subway train timetables [34]. As the most widely used stochastic optimization procedure, the GA is used to solve the model in this study. Therefore, it is chosen to achieve the optimized timetable in the MATLAB toolbox. The algorithm steps to solve the two-objective optimization model are as follows:

Step 1. Initialize parameters $t_{beg}$, $t_{end}$, $\theta_{v}$, $\theta_{g}$, $\delta_{v}$, $\delta_{g}$, $Q_{\rm spqi}$, $c_{q}$, $m_{q}$, $\tau_{q}$.

Step 2. Calculate $t_{\rm spqiv}^{\rm Walk}$ and $t_{\rm spqig}^{\rm Walk}$ using the TWKT prediction model.

Step 3. Calculate $K_{\min }$ and $K_{\max }$ using Formulas (25) and (26), respectively.

$$K_{\min } = \left\lceil {\frac{{Q_{q} }}{{e_{q} m_{q} \tau_{q} }}} \right\rceil$$

(25)

$$K_{\max } = \left\lfloor {\frac{{t_{end} - t_{beg} }}{{h_{q\min } }}} \right\rfloor$$

(26)

Step 4. $K$ for $K_{\min }$, $K_{\min } { + }1$,…, $K_{\max }$, encoded mode is real-coded. Initialize the population $(t_{sq1}^{Dep} ,t_{sq2}^{Dep} ,t_{sq3}^{Dep} ,...,t_{{sqn_{p} }}^{Dep} )$; the number of the initial population is set as 200, and generation is 500.

Step 5. The fitness function shown in Formula (27) is used to determine the optimal timetable of different $K$.

$$fit = \delta_{v} t_{\rm spqv}^{\rm Wait} + \delta_{g} t_{\rm spqg}^{\rm Wait}$$

(27)

Step 6. Calculate $Z$ using Formula (16).

Step 7. Find the optimal solution.

4 Case Study

4.1 Key Transfer Station Selection

The selection principle is as follows:

• 1. The transfer passenger flow of the selected transfer station is larger than that of other transfer stations.

• 2. V accounts for a relatively large proportion.

• 3. The transfer structure of the selected transfer station is the channel transfer or station hall transfer.

4.2 Parameter Estimate of the TWKT Prediction Model

4.2.1 Data Acquisition

By considering heterogeneous passengers under different facilities and passenger flows, Jianguomen (G), Jintailu (J), and Puhuangyu (P) stations in Beijing, China, were chosen. These stations were selected in accordance with the preceding principles. We chose different survey times on weekdays, given the effect of different periods in a day on changes in passenger flow. Table 4 shows detailed information on the investigated transfer stations.

Table 4 Detailed information on the investigated transfer stations

The investigated passengers include V and G. The passengers surveyed were familiar with the transfer route. Survey data were divided into learning and test groups. The learning group included 907 investigated passengers. Data sizes of the transfer facilities are listed in Table 5. The section passenger flow of the facilities was recorded every 1 min.

Table 5 Information on the investigated transfer facilities

4.2.2 Parameter Estimate

As shown in Table 6, the maximum, minimum, and average speeds of V and G in the different facilities of the three transfer stations were acquired from the observation data.

Table 6 Speeds of five groups of passengers at different facilities

Based on Formula (5) the average TWKT of V and G on the platform, in the horizon passage, and on the upward and downward stairs is fitted by quadratic polynomials. The relationship between the average walking speed and passenger flow of V and G on the platform and in the horizon passage are shown in Fig. 5a, b. For passengers walking on the platform, as shown in Eqs. 28 and 29, the R² values of the significant degree of regression are 0.9297 and 0.9334, respectively. For V and G in the horizon passage, as shown in Eqs. 30 and 31, the R² values of the significant degree of regression are 0.8801 and 0.8649, respectively.

$$v_{s1v} = 0.4692F_{s1}^{2} - 1.2227F_{s1} + 1.2635$$

(28)

$$v_{s1g} = 0.7586F_{s1}^{2} - 1.7825F_{s1} + 1.5292$$

(29)

$$v_{s2v} = - 1.5814F_{s2}^{2} + 0.1318F_{s2} + 1.2155$$

(30)

$$v_{s2g} = - 1.1539F_{s2}^{2} - 0.0516F_{s2} + 1.4373$$

(31)

Fig. 5

Relationship between average walking speed and section passenger flow for two types of passengers: a on the platform, b on the horizon passage, c on the upward stairs, and d on the downward stairs.

Figure 5c, d show the relationship between the walking speed and passenger flow of V and G on the upward and downward stairs. For passengers walking on the upward stairs, as shown in Eqs. 32 and 33, the R² values of the significant degree of regression are 0.9043 and 0.9488, respectively. For passengers walking on the downward stairs, as shown in Eqs. 34 and 35, the R² values of the significant degree of regression are 0.9213 and 0.9393, respectively.

$$v_{s3v}^{\rm Up} = 0.0991F_{s3}^{2} - 0.2550F_{s3} + 0.6370$$

(32)

$$v_{s3g}^{\rm Up} = 0.6309F_{s3}^{2} - 1.2880F_{s3} + 1.1385$$

(33)

$$v_{s4v}^{\rm Down} = 0.2396F_{\rm s4}^{2} - 0.4385F_{\rm s4} + 0.7202$$

(34)

$$v_{s4g}^{\rm Down} = 0.5248F_{\rm s4}^{2} - 1.0978F_{\rm s4} + 1.0839$$

(35)

The R² values show that the passenger walking speed is strongly related to section passenger flow on the platform, in the horizon passage, and on the upward and downward stairs. As section passenger flow increases, walking speed gradually decreases. The change in the walking speed of passengers on the platform is greater than that in the horizon passage. When the section passenger flow is small, the average walking speed of V and G is different on the upward and downward stairs, which is relatively high. The average walking speed of passengers on the upward stairs is 0.55 m/s when the section passenger flow is 0.8 passengers/meter·hour. By contrast, the walking speed of passengers on the downward stairs is 0.6 m/s when the section passenger flow is 0.66 passengers/meter·hour.

4.3 Results

4.3.1 Data Preparation

Key nodes in transit networks, such as subway transfer stations, are closely related to urban functions and high-density land use [35]. The subway network of Beijing is shown in Fig. 6. This map shows that the geographical location of transfer stations varies. Some stations are near recreational facilities, such as the Beijing Zoo and the Universal Beijing Resort, several stations are near hospitals, and some stations are near hubs and airports.

Fig. 6

Subway network of Beijing

On the basis of the principle of selecting stations, the Jianguomen Station, which is the transfer station of Lines 1 and 2, was chosen. Each line was considered as two separate lines. According to the intensity of transfer passenger flow and the proportion of V, the direction of Gucheng in Line 1 was regarded as the feeder line. The direction of Beijingzhan in Line 2 was the receiving line. Details of the transfer directions of the Jianguomen Station are shown in Fig. 7.

Fig. 7

Details of the Jianguomen transfer station

During off-peak time, train resources are prone to waste. In general, passengers have to wait for long times owing to the long headway. The selected research period was from 10:00 to 11:03:20.

As shown in Table 7, the value of $Q_{q}$, $Q_{\rm spqi}$, $c_{q}$, $m_{q}$, $\tau_{q}$, $h_{q\max }$, and $h_{q\min }$ in the optimization model were obtained through AFC and survey data. The minimum transfer time at Jianguomen Station was investigated and $r$ was set as 120 s. Since the passenger flow in Jianguomen Station was different between 10:00–10:20 and 10:20–11:00, $t_{\rm spqiv}^{\rm Walk}$ and $t_{\rm spqig}^{\rm Walk}$ were calculated by the TWKT prediction model in different study periods. Through field survey, the percentage of V in Jianguomen Station was 24% and the percentage of G was 76%. Therefore, $\theta_{v}$ was set as 0.24 and $\theta_{g}$ was set as 0.76. In the optimization model, from the perspective of social fairness, $\delta_{v}$ was set as 5 and $\delta_{g}$ was set as 1 to give priority to V. The sensitivity analysis of weight $\delta_{v}$ and $\delta_{g}$ will be discussed in detail in Sect. 5.2. $\gamma_{1}$ and $\gamma_{2}$ were both set as 0.5, respectively, without any bias towards passenger travel satisfaction or enterprise operating costs. The results of sensitivity analysis of weight $\gamma_{1}$ and $\gamma_{2}$ will be presented in Sect. 5.3.

Table 7 Main parameters of the optimization model

4.3.2 Optimized Train Timetable and Initial Train Timetable

According to the arrival time of the train in the direction of Gucheng of Line 1, the departure time of the train in the direction of Beijingzhan of Line 2 was solved. The optimized train timetable of Beijingzhan is shown in Table 8.

Table 8 Comparison of the initial and optimized train timetables

Pedestrian simulation can dynamically and effectively reflect the pedestrian movement characteristics in the subway station [36]. The simulated transfer data can directly avoid the error of the calculation results using the model and verify the validity of the previous model.

The GA provides satisfactory solutions rather than exact solutions. Obtaining an exact solution is unrealistic due to problem scale and complex constraints. Therefore, using the GA can obtain an adequate solution that meets practical requirements and can be achieved within an acceptable time and cost range. Figure 8 shows the iterative process of the GA in solving the proposed optimization model. When $n_{q}$ is 14, with an obvious convergence, the optimization speed slows and reaches a stable state after 300 iterations. When the GA is iterated to 500 generations, the fitness value is 91,779 with 57 s, while when it is iterated to 1000 generations, the best fitness value is 88,920. The difference between them is 3.1%, which is feasible and satisfactory for the practice. The fitness value is calculated according to Eq. (27). The objective value $Z$ is 0.18, which is obtained at the 500th iteration.

Fig. 8

Convergence test of the GA

By inputting the calculated train timetable into AnyLogic pedestrian simulation software, the TWTT for heterogeneous passengers was obtained by simulation. Indicator values of the initial and optimized train timetables are given in Table 9. Evidently, subway enterprise and passengers have benefited when weight coefficients $\gamma_{1}$ and $\gamma_{2}$ are 0.5. During the research period, the number of trains running on the direction of Beijingzhan of Line 2 was reduced from 15 to 14. Compared with the initial train timetable, the total TWTT $T^{\rm Wait}$ of passengers in the optimized train timetable is reduced by 24.5%. The TWTTs of V and G are decreased by 18.6% and 27.2%, respectively. From 10:00 to 11:00 in Table 8, V can save about 145 minutes in the Jianguomen Station. The average TWTTs of V and G are decreased by 17.0% and 27.2%, respectively. The result proves the correctness of the parameter and effectiveness of model optimization.

Table 9 Comparative analysis of optimization indexes

5 Discussion

For an improved understanding of the mechanism and effect of the model, a sensitivity analysis of the model parameters is conducted in sequence.

5.1 Sensitivity Analysis of Number of Trains

To analyze the relationship between the objective value $Z$ and TWTT, the change in the two values with $n_{q}$ is shown in Fig. 9. As $n_{q}$ continues to increase, the curve of $Z$ and total TWTT decreases initially but continuously increases thereafter. The optimal objective of $Z$ is 0.18 when $n_{q}$ is 14, which is one fewer train on the receiving line than the initial train timetable. In this case, the TWTT of the total passengers is 37 hours. Meanwhile, the minimum TWTT is 36 hours, which obtained 18 at $n_{q}$. This difference is caused by the model, considering the benefits of passengers and the operating cost of the subway. When $n_{q}$ is 14, the TWTT of total passengers is only 2.7% longer than $n_{q}$, which is 18. With an increase in $n_{q}$, the headway constraints lead to limitations in finding the optimal solution, leading to an increase in total TWTT. Therefore, the optimized train timetable can achieve a win–win balance between passengers and the subway enterprise.

Fig. 9

Total TWTT under different numbers of trains in the receiving line

A difference in transfer time is observed between vulnerable passengers and the general group under the condition of insufficient trains (Fig. 10). Overall, when the number of trains on receiving line $q$ increases, the average TWTT of V and that of G have a certain volatility. The initial train timetable has 15 trains on the receiving line. The average TWTT of V and G are 118 and 103 s, respectively. When $n_{q}$ is over 13 and 12, the average TWTT of V and G are less than that when the train timetable is the initial timetable. The model clearly has a good optimization effect. When $n_{q}$ increases, the average TWTT difference between V and G tends to be consistent and gradually increases. Under these circumstances, the heterogeneity of passengers is less evident.

Fig. 10

Average TWTT of V and G by the number of trains on the receiving line

5.2 Sensitivity Analysis of the Weights of V and G

The influence of the heterogeneity diversity is illustrated in Fig. 11. To simplify without loss of generality, $\delta_{g}$ is set as 1. Thereafter, $\delta_{v}$ can be taken as the single variable to analyze the comparison. For social fairness, the value of $\delta_{v}$ should be no less than $\delta_{g}$. By the preceding model, when the two parameters are equal, although the TWTT of the total passengers decreased by 16% compared with the initial timetable, the optimal effect is 19% for G but only 4% for V. This result indicates that the optimization of the transfer time means increased difficulty for people who need more help than the average person. If the weight $\delta_{v}$ is increased, the model can bring more benefits to vulnerable groups. The difference in the transfer time between V and G gradually decreases.

Fig. 11

Optimization effect of average TWTT of V and G with different weights of $\delta_{v}$

5.3 Sensitivity Analysis of the Weights of Cost and Waiting Time

From the perspective of the transportation system and society, the improvement of passengers’ benefits calls for the payment of the operation agent, which has a game relationship between them. In the proposed model, $\gamma_{1}$ and $\gamma_{2}$ represent the weights of TWTT and operating cost. If $\gamma_{1}$ is larger, the optimization objective is to minimize the waiting time for passengers as much as possible, which may result in a shorter headway between train schedules but also potentially increase operating costs. By contrast, if $\gamma_{2}$ is larger, the optimization objective is to minimize the operating cost on a large scale, which may result in a longer headway between the train timetable and increased waiting time for passengers. The values of $\gamma_{1}$ and $\gamma_{2}$ are due to the management concept of the subway system, which is affected by the operating conditions of the enterprise and the intensity of passenger flow. In practical applications, the balance between the importance of operating cost and TWTT must not only be determined by the enterprise but must also be guided by the government. Push-and-pull is revealed as waves, as shown in Fig. 12. The vertical axis represents the value of the objective function $Z$, with the optimal value marked by the large triangle. A plane is generated with the two tuples ($\gamma_{1}$,$\gamma_{2}$) and the number of trains $n_{q}$ on the receiving line. When $\gamma_{1}$ and $\gamma_{2}$ are equal to 0.5, the optimal objective value $Z$ is obtained when $n_{q}$ is 14. If $\gamma_{1}$ is set from 0.1 to 0.4, which means the enterprises receive extra privileges, then operating costs can be reduced with smaller feet size than with more than 15 trains. The objective value $Z$ ranges from 0 to 0.27. However, when $\gamma_{1}$ is set to 0.6–0.9, passengers earn preferential treatment. Heterogeneous passengers can save more TWTT by $n_{q}$ of 14 and 17, which is larger than the first case. $Z$ ranges from 0.06 to 0.23, which is more optimal than the front stage.

Fig. 12

Relationship between the objective function value and the weight $\gamma_{1}$ and $\gamma_{2}$

In particular, the proposed model can be applied for the simplex objective. When $\gamma_{1}$ is 0, the model has only one objective, to minimize the cost of the enterprise, apart from the TWTT of passengers. When $\gamma_{1}$ is 1, regardless of the operating cost of the enterprise, the sole goal is to minimize the total TWTT of passengers. Compared with the previous study, which focused on the single objective schedule optimization (see Table 2), the proposed model can address either of the objectives and consider the benefits of passengers and enterprises. Hence, the proposed model has good universality.

5.4 Further Implications

Further implications of the proposed method are addressed from the following perspectives.

• 1. Travel experience improvement. By optimizing subway timetable and transfer arrangement, the proposed model can reduce the TWTT for the passengers. The travel experience can be improved with greater convenience. With regard to prioritizing V, the key station for the timetable adjustment can be chosen based on the proportion of V, which is affected by the geographical location and land use of the station. Numerous elderly passengers in transfer stations are directed to clinics and hospitals. Passengers with children take a domain near the park. In the transfer station to the railway station and airport, numerous travelers carry large luggage. For example, Fig. 6 shows that Station 1, which is near the Beijing Zoo, and Station 3 can transfer to Line 7 heading to Universal Studios. Many excited children who are fond of pandas and minions take the trip to these two transfer stations on weekends. Stations 1 and 4 are near the hospital. Numerous elderly passengers travel to these stations. Stations 2, 5, and 6 are near or accessible to the railway station and airport. These stations should focus in particular on the parameter setting for a friendly service. With the aging of society, a preference for increasing travel needs of the elderly group is appreciated.

• 2. Operating cost saving. The bi-objective optimization model can also help subway operators reduce operating costs by efficiently allocating resources. It makes sense for the growth stage of the subway network when weight $\gamma_{2}$ of the cost is increased. For cities that have only two or three lines, the enterprise should use the fewest trains to obtain the largest profit to achieve the sustainable development of enterprises. When the issue of the trade-off between improving passenger transfer efficiency and minimizing subway operating costs is solved, under these circumstances, in cities with accomplished networks such as Beijing, Tokyo, London, and New York, the operating profit of the subway company can gradually increase with increasing passenger flow. Simultaneously, a more efficient and passenger-friendly subway system can enhance the competitiveness of a city or region, thus attracting more commuters and visitors and promoting economic growth.

• 3. Application in different scale networks. The results of the sensitivity analysis reveal that the method results are feasible for various subway networks with different scales, characteristics, and passenger needs. The proposed model can also be applied to multi-transportation systems with different travel modes to save the transfer time in the micro-hubs and terminals for travelers.

6 Conclusion

On the basis of passenger and enterprise perspectives, a two-objective linear programming model was established to minimize the cost of subway enterprises and the TWTT of heterogeneous passengers. With the Jianguomen transfer station taken as an example, the actual passenger flow, surveyed proportion of vulnerable and general passengers, train timetable, and other data for Jianguomen transfer station were used. The parameter of the prediction model was estimated, and its effectiveness was verified by the GA. The main findings of this study are as follows:

• 1. The heterogeneity of transfer passengers is found when section passenger flow is large. When the section passenger flow increases, the heterogeneity difference between the two types of passengers gradually decreases.

• 2. The objective of the proposed model is to optimize the TWTT of G and V at the vital transfer station and reduce the operational costs of the subway enterprise. Constraints of the model include the headway, loading capacity, and transfer of heterogeneous passengers. One train can be saved, and the passengers’ TWTT can be reduced by 24.5%.

• 3. Results of the parameter sensitivity analysis reveal the interconnection among the factors of transfer time. As $n_{q}$ increases in a maturing network, the average TWTT difference between V and G tends to be consistent, whereas the heterogeneity of passengers becomes less evident. The proposed model can provide more benefits for the vulnerable group if $\delta_{v}$ is increased. A higher $\gamma_{1}$ can shift the priority from enterprises to passengers. If $\gamma_{1}$ is given as 0 or 1, then the proposed model is equivalent to single-objective optimization.

The limitation of this paper is that the renovation of facilities is not taken into account. Passengers’ walking time can be reduced by retrofitting transfer facilities, such as converting stairs into escalators or adding conveyor belts. It can also be affected by passengers’ familiarity with the environment. These factors can be considered in future studies.