Multi-Modal Urban Traffic Transfer Schedule Timetable Bi-Objective Optimization: Model,Algorithm,Comparison,and Case Study

Abstract

The optimization of the connection between urban rail transit and the bus is an essential issue that benefits passenger travel and the urban structure and has social benefits, which can be realized by reasonably adjusting the bus departure schedule. This study is necessary because the development status quo of China’s urban transportation network planning is unreasonable, travel efficiency is not high, and operating costs are high. This paper sets up the decision variables of bus departure time and departure interval at each station, establishes a dual-objective optimization model with the minimum schedule change and the minimum transfer time, and studies the application of the augmented Chebyshev algorithm in the dual-objective optimization model. Secondly, based on the Shenzhen metro and public transportation integrated circuit card data, the case analysis uses the generalized Chebyshev algorithm and the non-dominated sorting genetic algorithm, respectively. The optimization results show that using the improved augmented and generalized Chebyshev algorithm in the bus schedule alteration time within a reasonable range can maximize the total transfer time, which compared with the original scheme is shortened by 68.06%. In contrast, genetic algorithms will make the complete bus schedule alteration prominent, and the whole transfer time is substantially increased. The results show that the improved augmented generalized Chebyshev algorithm is more suitable for solving the dual-objective rail transit connection problem.

Keywords

transfer connections bus schedules multi-objective optimization augmented weighted Chebyshev algorithm Genetic algorithm

In recent years, with the acceleration of urbanization, establishing a sound urban public transportation system is conducive to alleviating the contradiction between urban transportation supply and demand. Conventional buses are still the main body of public transportation, bearing the majority of the traffic. However, with the further development of urban rail transit and conventional buses and the gradual increase of rail transit lines, more attention must be paid to the integrity and coordination of the public transportation system. In the urban public transportation system, passengers often need to transfer to buses through metro transportation to reach their destinations. Still, the transfer between urban rail and buses generally has problems such as long transfer distances, inappropriate transfer time, and unreasonable designed transfer routes, resulting in low transfer efficiency. At the same time, most of the current public transportation preparation schedule does not consider the coordination with rail transit. Most interchange passengers do not want to face the bus situation, especially passengers with travel time requirements, so thoroughly weighing the coordination between urban rail transit and bus interchange can, to a certain extent, reduce the possibility of the above situation.

Transfer time is the time consumed by passengers to walk from the urban rail to the platform of the connecting bus and finally get on the connecting bus. Xu et al. ( 1 ) proposed the concept of passenger waiting time, based on which this paper creatively divides all transfer passengers into two categories: non-stay passengers and stay passengers, that is, their transfer time is divided into non-stay passenger transfer time and dwell passenger transfer time, which is closer to real life. Therefore, our paper first analyzes the actual situation of interchange passengers and establishes a total interchange time model. Moreover, based on the advantages of higher flexibility of bus departures and relatively lower operating costs, this paper considers optimizing the connection between buses and urban rail transit by debugging a suitable bus schedule. However, significant changes in bus schedules often cause some passengers to be uncomfortable. For this situation, a minimum change in bus schedules is used as one of the objective functions in this paper. Combined with the objective of minimizing the overall transfer time of passengers, a dual-objective rail transit connection optimization model is developed in this paper. In the algorithm part, a Chebyshev algorithm based on the Pareto idea is proposed to set the incremental model to approximate the problem’s non-dominated frontier and find the non-dominated solution set with good convergence ( 2 ). Finally, it is used to search for the optimal solution. The non-dominated sorting genetic algorithm (NSGA-II) is the most commonly used method to study the interchange between rail transit and public transportation. Through experimental comparison, it is found that compared with the NSGA-II, the augmented Chebyshev algorithm can solve the non-dominated solution with better adaptability and significantly optimize the passenger transfer time. In this paper, we comprehensively consider the non-stagnant and stagnant transfer phenomena from rail transit to bus vehicles, establish a dual-objective optimization model of an urban rail transit connection scheme with bus schedule change as the decision variable, and solve it by the improved Chebyshev algorithm, which makes the final optimization scheme more adaptable. It can provide the foundation for a sound decision solution for public transportation operators.

Based on the above description, The main contributions of this paper can be summarized as follows.

Interchanges between multiple modes of transportation are considered, rather than just studying a particular method of transportation. Our study is more realistic than other studies evaluating urban rail transit or bus travel.

A dual-objective optimization model is established according to the metro and bus operation characteristics. Meanwhile, unlike other multi-objective studies, multiple objectives tend to be parallel or congruent. Our goals reflect a direct contradiction in time, and the two purposes are mutually constrained and must be considered together.

A new algorithm is used: the augmented Chebyshev algorithm. In previous studies, the NSGA-II is a more common algorithm for computing the bi-objective optimization model. This study uses the augmented Chebyshev algorithm to find practical non-dominated solutions. By building an augmented model, we obtain a better non-dominated frontier so that the original two conflicting objectives can be more optimized to a certain degree. This algorithm has better solution efficiency than other algorithms, and the quality of the solution results is high.

Applying the non-dominated Pareto solution to bus schedule changes, different decision-makers with different needs because of other geographical and economic conditions, and so forth, can achieve their purpose by adjusting the weights so that the adjustment results have real benefits. The rest of this paper is organized as follows. The second section summarizes the relevant research results of previous scholars, the third section specifies the model, and the fourth section introduces the principle of the augmented Chebyshev algorithm in detail and analyzes the process of establishing and solving the dual-objective optimization model. In the fifth section, we translate and compare two different algorithms based on actual data to verify the superiority of the augmented Chebyshev algorithm. Finally, the conclusions are summarized in the sixth section.

Literature Review

Since the last century, researchers have been studying the optimization of public transportation interchanges one after another. With respect to research subjects, bus interchange research predates urban rail transit because of bus systems’ more general and widespread role in urban transportation. In contrast, urban rail transit systems are more specialized and specific to certain cities. Research on the bus interchange has focused on bus stop planning, optimizing interchange strategies, and improving operational efficiency. Some early papers focused on the patterns of passenger transfer behavior, such as Zahir et al. ( 3 ) who, through field surveys, studied passenger arrival patterns at stops, bus departure times at terminals and stops, and stopping times. Attention is also paid to the problem of interchange station location and planning, for example, Chien et al. ( 4 ) combined geographic and demographic information to obtain information such as the optimal number of bus routes and their locations using genetic algorithms. More papers focus on the study of interchange time and efficiency, such as Bookbinder and Désilets ( 5 ), who focus on interchange optimization considering the effect of interchange time randomness, while Belletti et al. ( 6 ) and Ismail and Ang ( 7 ) both aim to weigh the interchange between buses through the rational scheduling of bus vehicles and crew. Ismail and Ang ( 7 ) even set the objective model of maximizing revenue and minimizing operating costs and fleet size by scientific methods. Research on rail transit interchange started relatively late, focusing on passenger mobility and urban rail transit line design. For example, Zheng et al. ( 8 ) explored the walking time pattern of interchange passengers in stations from a statistical perspective, using Beijing rail transit as an example. Blanco et al. ( 9 ) aimed to reduce operating costs and achieve better passenger guidance by optimizing rail lines and schedules, which will undoubtedly entail a huge engineering effort. Optimizing schedules is certainly a more realistic research direction when urban rail lines are built, so Wong et al. ( 10 ), Xu et al. ( 1 ), Yin et al. ( 11 ), and Chen et al. ( 12 ) conducted research in this way. Still, they have built different objective models or solution methods. Wong et al. ( 10 ) considered the minimum transfer waiting time for all passengers and develop an optimization-based heuristic algorithm for model calculations. Xu et al. ( 1 ) wanted to reduce passenger wait times and rail vehicles’ energy consumption during peak and off-peak hours. Yin et al. ( 11 ) not only considered the problem of minimizing the waiting time for urban rail transit transfers but also wanted to satisfy the need for minimum operating costs and developed a Lagrangian relaxation-based heuristic algorithm to optimize rail vehicle schedules. Chen et al. ( 12 ) solved the single-objective model problem of more successful passenger interchanges by the genetic algorithm and Dijkstra algorithm. Over time, the study of bus–rail interchanges has evolved to a comprehensive perspective, and researchers have begun to focus on the following areas. (a) Integration of bus and urban rail transit, such as considering the overall operating cost, for example, Chien and Schonfeld ( 13 ) used a continuous substitution method to find the optimal solution of the objective function to minimize the overall cost of the bus and urban rail transit network. As well as considering the accessibility of the interchange process, a related study using regression models was conducted by Wang et al. ( 14 ). (b) Optimizing the interchange strategy. Zhao and Tan ( 15 ) conducted a study on interchange efficiency from a microscopic perspective but did not propose a specific approach to solve the model. Wang et al. ( 16 ) considered macroscopic spatial topological relationships to study adjustments related to interchange bus routes along rail transit completion, although this will undoubtedly face cumbersome planning problems. (c) Enhancing the passenger experience. In today’s human-centered world, this is undoubtedly the focus of current research, most of which aims to reduce passenger travel time or transfer waiting time by conveniently adjusting schedules, thus increasing passenger satisfaction. Most studies have targeted this, for example, Shu-xia et al. ( 17 ) studied the average waiting time for urban rail transit transfers to conventional buses by considering factors such as platform distance and pedestrian walking speed. Zhang and Liu ( 18 ) used a hybrid genetic algorithm and a hill-climbing algorithm to solve the single-objective model with minimum total passenger travel time. Ren and Ouyang ( 19 ) also considered the rail inter-district transport capacity and solved it using the Cplex solver, although this approach tends to be time-consuming. Tian-wei et al. ( 20 ) considered the issue more deeply, built a multi-objective optimization model with minimum total passenger waiting time, total number of moment adjustments, and total moment adjustment time, and used the NSGA-II combined with simulation methods to solve the model. Wang et al. ( 21 ) considered a special case: the interchange problem in the case of sudden disruption of urban rail transit operation, aiming to reduce the interchange waiting time and avoid exchange failure, and used an improved simulated annealing algorithm for the case study.

With respect to research methods, it is easy to see that in the beginning, the research on interchange optimization of public vehicle schedules was mainly focused on building single-objective models, which aim to optimize a specific objective of the interchange process, such as minimizing the interchange time of passengers, minimizing the total cost of interchange, and so forth. The trend in research is now to consider multiple objectives to optimize interchange tasks in a more comprehensive and integrated manner. Model computation is attempted using, for example, linear programming, integer programming, heuristic algorithms, metaheuristic algorithms, and so forth. For example, Tang et al. ( 22 ) combined the genetic algorithm and simulated annealing algorithm to solve the problem of minimizing passenger waiting time and bus operation costs. Li et al. ( 23 ) used a genetic algorithm to calculate a bi-objective optimization model with minimum travel time along the bus route and full passenger waiting time at the stop in the bus transfer problem. Bie et al. ( 24 ) developed a multi-objective optimization model for minimizing the sum of departure time delay, energy consumption, and transit procurement cost in a transit transfer study and solved it using the NSGA-II. Han et al. ( 25 ) proposed a bus schedule dynamic optimization model with the objective function of minimizing bus operation cost and passenger travel cost, which was solved using the advanced adaptive NSGA-II (AANSGA-II). The NSGA-II is a meta-heuristic algorithm based on a genetic algorithm, and it is not difficult to find that it is widely used in the above multi-objective optimization permutation problem. Still, it may inevitably have issues such as convergence to locally optimal solutions and slow convergence, so more computational methods must be explored.

With the rapid development of information technology, research on bus and urban rail transit interchange has begun to focus on the application of technologies such as intelligent transportation systems (ITSs), big data, and smartphone applications, and researchers have started to explore how these technologies can be used to improve the efficiency and convenience of interchange. Transfer recognition is the basis of passenger trip recognition. It has essential research significance for multi-modal transportation connections, which generally require obtaining travel origin–destination (OD) data through integrated circuit (IC) card data and judging the transfer behavior in space and time. Trépanier et al. ( 26 ) derived the IC card data based on trip chains and interchange strategies for the off-board stations, and analyzed the line network passenger flow based on the OD data. Jang ( 27 ) used IC card data to evaluate the characteristics of travel time, interchange, and travel time distribution of passengers in multiple travel modes and finally analyzed the interchange behavior of passengers between particular areas. Munizaga and Palma ( 28 ) developed a multi-modal public transportation network travel OD derivation model based on IC card and Global Positioning System (GPS) data and identified interchange stations by extrapolating distances with interchange times. The study of interchange identification can be used to obtain accurate information on passenger travel OD, which can be used to analyze and optimize the operation of buses and urban railways. Therefore, this study conducted a transfer study using IC card data.

Model Formulation

In this section, we introduce a new dual-objective optimization model with minimal transfer time–bus moment changes to balance the depreciation of the overall transfer time for urban rail transit-to-bus transfer passengers while ensuring that the bus moment changes are not too significant. Before we formulated the model, for simplicity of modeling, the model used the following assumptions to ensure the reliability of the model formulation:

the upstream and downstream lines of conventional buses and rail transit are independent research objects, and only the interchange stations in the sequences are considered;

the exact route runs the same type of bus vehicles, running at the same speed and specific fleet size, ignoring the waiting time after the bus arrives and the time required for passengers to get on and off the bus;

the number of bus departures during the study period is known, and the transfer passengers randomly select the next bus from the bus routes that stop at the station as the transfer object.

Notations and Parameters

This section lists all the relevant notations and parameters to build and describe the model we need and the problem under study. Table 1 shows this section’s complete list of symbols and parameters.

Table 1.

Symbols and Parameters Used in This Article

Symbol	Definition
$s$	Urban rail transit interchange stations, $s = 1, 2, . . ., S$
$Rl$	Rail lines involved in interchange, $l = 1, 2, . . ., L$
$t_{Rl}^{s}$	Rail train arrival time at the interchange station $s$
$w$	Number of train departures on rail lines, $w = 1, 2, . . ., W$
$n$	Bus lines, $n = 1, 2, . . ., N$
$q$	Interchange bus stop, $q = 1, 2, . . ., Q$
$k$	Number of bus line departures, $k = 1, 2, . . ., K$
$d$	The direction of bus operation, $d = 1, 2, . . ., D$
$t_{RB}$	Transfer time of passengers from urban rail transit to the bus
$t_{n, k}^{q}$	Bus arrival time at stop $q$
$H_{n, min}$	The lower limit of bus departure interval for bus line $n$
$H_{n, max}$	The upper limit of bus departure interval for bus line $n$
$t_{max}$	Maximum interchange time for passengers
$H_{n, k - 1}$	Departure interval between bus number $k$ and bus number $k - 1$ on line $n$
$t_{n, k}^{q}$	Departure time of bus number $k$ of line $n$ at station $q$

Decision Variables

In the model we developed, the bus departure interval and the bus departure time at the interchange station are used as decision variables, which are as follows.

$H_{n, k - 1}$ : as the departure interval between bus trip $k - 1$ and bus trip $k$ , it is restricted to a specific range, which we will use as an example in the fifth section, and its range must match the actual situation.

$t_{n, k}^{q}$ : as the departure moment of the bus vehicle at the interchange station, this decision variable can assist in representing the interchange time of passengers and has a crucial role in both objective functions.

Objective Function

Minimum Transfer Time

Coordination between conventional and rail transit can only be achieved by providing a reliable bus operation service. Minimal bus schedule changes have the most negligible impact on existing bus operations. To avoid large fluctuations in the departure interval during the schedule adjustment process, which would affect passengers riding at other stops on the route, the minimum bus schedule change is used as an objective function:

min Z_{1} = \sum_{n = 1}^{N} \sum_{k = 1}^{K} \sum_{q = 1}^{Q} | t_{n, k}^{q} - \bar{t_{n, k}^{q}} |

(1a)

In Equation 1a, to facilitate subsequent algorithmic operations, let $y_{n, k}^{q} = | t_{n, k}^{q} - \bar{t_{n, k}^{q}} |$ , and the linearization of Equation 1a is expressed as Equation 1b:

{\begin{matrix} min \sum_{n = 1}^{N} \sum_{k = 1}^{K} \sum_{q = 1}^{Q} y_{n, k}^{q} \\ y_{n, k}^{q} > (t_{n, k}^{q} - \bar{t_{n, k}^{q}}) sgn \\ y_{n, k}^{q} \leq (t_{n, k}^{q} - \bar{t_{n, k}^{q}}) sgn \\ sgn = {\begin{matrix} 1, y_{n, k}^{q} \geq 0 \\ - 1, y_{n, k}^{q} < 0 \end{matrix} \end{matrix}

(1b)

Minimum Transfer Time

The time consumed by passengers to get off the rail train, walk to the platform of the connecting bus, and finally get on the connecting bus is called the transfer time of passengers, which can be expressed as $t_{RB}$ . Then, the transfer time is divided into two parts: the first part is the time passengers spend getting off the rail train and walking to the platform of the connecting bus, and the second part is the waiting time of passengers at the bus transfer station. Suppose the rail train arrives at a specific moment, and the transfer passengers arrive at the transfer station after a period of walking time. Suppose most passengers wait for a short time or do not need to wait for the time to catch the nearest transfer bus. In that case, this situation is an ideal state of the transfer as the waiting time of the passengers in the transfer process is short, and the transfer efficiency is high. However, most passengers who get off the rail train to transfer arrive at the transfer station under normal circumstances. The last connecting bus has already left or just left the station. Passengers still need to wait some time to get on the next bus, then the passenger transfer time is the time for the next bus to the station minus the arrival moment of the rail transit. When the interchange passengers exceed the capacity of the connecting bus, the excess passengers also have to wait for another interval between the departure of the connecting bus. When the passenger transfer time is more significant than $t_{max}$ , passengers end the wait to choose other modes of transportation to travel. Figure 1 shows a schematic of the interchange time definition.

Figure 1.

Schematic diagram of interchange time definition.

With advanced ITS technology, it is easy to obtain information on the number of passengers participating in rail-to-bus transfers and their respective transfer stations during the same period from the IC card data. The number of interchanges from the urban rail transit station $s$ to the bus stop $q$ of the $Rl$ rail line is defined as $P_{Rl, s}^{q}$ . All interchange passengers are divided into two categories: non-stay passengers and stay passengers, and the interchange time is divided into non-stay passenger interchange time and stay passenger interchange time. Also, the symbol $A_{n, k}^{s}$ is used to distinguish whether the transfer passengers exit at the $s th$ metro station and transfer to the $k th$ trip of the $n th$ bus, as shown in Equation 2:

A_{n, k}^{s} = {\begin{matrix} 1, transfer to the k th trip of the n th bus \\ 0, others \end{matrix}

(2)

i. The transfer time of a single non-stay passenger starts from getting off the urban rail transit and ends when the bus arrives at the transfer station, and the transfer time is expressed by Equation 3:

t_{Rl, s}^{q 1} (i) = t_{n, k}^{q} - t_{Rl}^{s}

(3)

The number of all non-stay passengers is denoted as $P_{Rl, s}^{q^{1}}$ . It is ensured that it is the number of non-stagnant people at all interchange stations, as we are looking at the whole system and not individual lines or stations:

P_{Rl, s}^{q^{1}} = \sum_{n = 1}^{N} \sum_{k = 1}^{K} \sum_{l = 1}^{L} \sum_{q = 1}^{Q} \sum_{s = 1}^{S} \sum_{d = 1}^{D} P_{Rl, s}^{q} A_{n, k}^{s}

(4)

So, the non-stay passenger transfer times are found by Equation 5:

\begin{matrix} t_{Rl, s}^{q^{1}} = P_{Rl, s}^{q^{1}} \times t_{Rl, s}^{q^{1}} (i) \\ = \sum_{n = 1}^{N} \sum_{k = 1}^{K} \sum_{l = 1}^{L} \sum_{q = 1}^{Q} \sum_{s = 1}^{S} \sum_{d = 1}^{D} P_{Rl, s}^{q} A_{n, k}^{s} (t_{n, k}^{q} - t_{Rl}^{s}) \end{matrix}

(5)

ii. The transfer time of a single stranded passenger starts from getting off the urban rail transit. It ends when the $(k + r) th$ bus arrives at the transfer station after the $k$ th trip, or when the passenger chooses another way to leave after reaching the maximum transfer time as expressed in Equation 6, where $t_{min}^{s, q}$ represents the shortest transfer time for transferring passengers, which can generally be regarded as the normal walking time for passengers from the drop-off point to the transferring bus stop. If the value is 0, the passenger gives up the transfer and no longer expects the bus, which is the last thing we aim for and try to avoid:

t_{Rl, s}^{q^{2}} (i) = {\begin{matrix} (t_{n, k + r}^{q} - t_{Rl}^{s}), t_{min}^{s, q} \leq t_{Rl, s}^{q^{2}} (i) \leq t_{max} \\ 0, t_{Rl, s}^{q^{2}} (i) > t_{max} \end{matrix}

(6)

The symbol $B_{n, k}^{s}$ is used to distinguish whether the transfer passenger exits at the $s th$ metro station and transfers to the $(k + r) th$ bus of the $n th$ bus, as shown in Equation 7:

\begin{matrix} B_{n, k}^{s} = {\begin{matrix} 1, transfer to the (k + r) th trip of the n th bus, \\ (r = 1, 2 . . ., m, m \leq K - k) \\ 0, others \end{matrix} \end{matrix}

(7)

The number of passengers in this segment is expressed as $P_{Rl, s}^{q^{2}}$ in Equation 8:

P_{Rl, s}^{q^{2}} = \sum_{n = 1}^{N} \sum_{k = 2}^{K} \sum_{l = 1}^{L} \sum_{q = 1}^{Q} \sum_{s = 1}^{S} \sum_{d = 1}^{D} P_{Rl, s}^{q} B_{n, k}^{s}

(8)

So, the stranded passenger transfer times are obtained from Equation 9:

\begin{matrix} t_{Rl, s}^{q 2} = P_{Rl, s}^{q^{2}} \times t_{Rl, s}^{q^{2}} (i) \\ = \sum_{n = 1}^{N} \sum_{k = 2}^{K} \sum_{l = 1}^{L} \sum_{q = 1}^{Q} \sum_{s = 1}^{S} \sum_{d = 1}^{D} P_{Rl, s}^{q} B_{n, k}^{s} (t_{n, k + r}^{q} - t_{Rl}^{s}) \end{matrix}

(9)

The total transfer time for all transfer passengers must add up the transfer time for non-stay and stay passengers and is expressed by Equation 10:

t_{RB} = \sum_{n = 1}^{N} \sum_{k = 1}^{K} \sum_{l = 1}^{L} \sum_{q = 1}^{Q} \sum_{s = 1}^{S} \sum_{d = 1}^{D} (t_{Rl, s}^{q^{1}} + t_{Rl, s}^{q^{2}})

(10)

The constraint function that minimizes the total permutation time in Equation 11 is our second objective function:

\min Z_{2} = \sum_{n = 1}^{N} \sum_{k = 1}^{K} \sum_{l = 1}^{L} \sum_{q = 1}^{Q} \sum_{s = 1}^{S} \sum_{d = 1}^{D} [P_{R l, s}^{q} A_{n, k}^{s} (t_{n, k}^{q} - t_{R l}^{s}) + P_{R l, s}^{q} B_{n, k}^{s} (t_{n, k + r}^{q} - t_{R l}^{s})]

(11)

Constraints

In this section, some conditions related to bus schedule development are described to adjust bus schedules. There are several constraints described below.

(a) Maximum and minimum constraints on the departure interval

The departure interval between any two adjacent bus vehicles on the $n th$ route is subject to maximum and minimum departure interval constraints. Here, $H_{n, min}$ is the lower limit of the departure interval of the bus line $n$ , while $H_{n, max}$ is the upper limit of the departure interval of the bus line $n$ :

H_{n, min} \leq H_{n, k - 1} \leq H_{n, max}

(12)

(b) Number of departures constraint

During the study period, the number of departures of the $n th$ bus vehicle should be limited and combined with the actual number of departures to control the maximum number of bus line departures:

1 \leq k \leq K_{n, max}

(13)

Let the maximum transfer time of passengers be $t_{max}$ . When passengers’ maximum transfer time is greater, they will end their waiting and choose other transportation modes with which to travel:

0 \leq t_{Rl, s}^{q} \leq t_{max}

(14)

(d) Passenger interchange constraint

The maximum number of interchange passengers per rail trip is limited to a certain number of regular bus trips:

\sum_{k = 1}^{K} f_{nlwk} \leq 1

(15)

where $f_{nlwk}$ is equal to 1, which means that passengers of the $w$ th train of the rail line Rl successfully transfer to the $k th$ train of the bus line $n$ . This is equivalent to 0, which means no transfer.

Overall Statements

Our dual-objective model with minimal changes in transfer times–bus moments is realistic. On the one hand, we strive for a more reasonable connection between urban rail transit and bus, to improve the level of public transportation services, and for passengers to have a better interchange experience with minimal overall interchange time, which is achieved by adjusting the bus departure schedules at each station, but this change is not arbitrary. Otherwise, it will increase the discomfort of bus drivers and passengers who often participate in interchanges, which is unrealistic. The next issue we will discuss is reconciling these two conflicting objectives while satisfying certain constraints, that is, finding reasonable algorithms for solving the problem and getting solutions that benefit both goals.

Algorithms

In studying bi-objective optimization problems, the final optimal solution is often an optimal set of keys, forming a Pareto front. To find the answer with better adaptability, weighing the two objectives in different situations is a crucial problem for research in multi-objective optimization problems. To this end, in this paper, we propose an augmented Chebyshev algorithm to find a better convergent non-dominated solution set and obtain the Pareto front of the problem for the above dual-objective model with a minimum bus schedule change and minimum transfer time.

Overview of the Augmented Chebyshev Algorithm

The traditional Chebyshev algorithm is insensitive to $d_{1}$ ( 29 ). It has poor convergence, while the augmented Chebyshev algorithm adds the $l_{\infty}$ -parameter of infinite parity (i.e., $P = \infty$ ) with the penalty term $σ$ to the traditional Chebyshev algorithm, which makes the derived non-dominated solution set converge better and obtains an approximate Pareto front by continuously adjusting the weight vector.

Extreme Value Point

Based on the minimum value obtained from the previous dual-objective optimization model, the minimum value is used as the minimal value of the non-dominated frontier, that is, ( $Z_{1}^{\min}$ , $Z_{2}^{\min}$ ), then $Z_{1}^{\min} = Z_{1} (t)$ , $Z_{2}^{min} = Z_{2} (t)$ . Since the great value cannot be obtained by a direct method, the dictionary order method is used in this paper to receive it. The specific calculation formula is as follows:

\begin{array}{l} Z_{1}^{max} = min Z_{1} (t) \\ s . t . Z_{2} (t) = Z_{2}^{min} \cup Equation (12) - Equation (15) \end{array}

\begin{array}{l} Z_{2}^{max} = min Z_{2} (t) \end{array}

(16)

\begin{array}{l} s . t . Z_{1} (t) = Z_{1}^{min} \cup Equation (12) - Equation (15) \end{array}

(17)

Quantification

Since the two objective functions of the bus schedule change and transfer time are measured in different units, the linear mid-threshold method is used to avoid the influence of dimensionality of the objective function values before applying the extended Chebyshev algorithm. It transformed into a unified index evaluation value using the tremendous and minimal values calculated earlier. Meanwhile, the objective function value is the specific formula given in Equation 18:

\bar{Z_{i}} (t) = \frac{Z_{i} (t) - Z_{i}^{min}}{Z_{i}^{max} - Z_{i}^{min}} (i = 1, 2)

(18)

Augmentation Model

Using Chebyshev’s algorithm, the bi-objective model is transformed into a single-objective incremental model by adding a $σ$ penalty factor to find the non-dominated solution set. The effective weight vector $w = (w_{1}, w_{2})$ is set and satisfied $| | w | | = 1$ . For the objective function of the original problem, let $w_{1}$ and $w_{2}$ ( $w_{1}, w_{2} \geq 0$ ) be the weight vector of the total bus schedule change objective and the weight vector of the total interchange time objective, respectively. Here, $σ$ is introduced into the augmented model as a penalty factor to control the illegality of the constraints so that the sequence of single-objective optimal solutions approximates the optimal solution infinitely in the feasible domain, thus finding the set of non-dominated solutions ( 30 ); $ρ$ is the decision variable of the objective, which represents the weighted Chebyshev distance between all feasible points in the objective space and the minimal value ( 31 ). The specific expression of the augmented model is as follows:

\begin{array}{l} min Z^{-} = ρ + σ [\bar{Z_{1}} (t) + \bar{Z_{2}} (t)] \\ s . t . {\begin{cases} w_{i} \bar{z_{i}} (t) \leq ρ \forall i = 1, 2 \\ t \in T \\ Equation (12) ~ Equation (15) \end{cases} \end{array}

(19)

Specific Implementation Steps of the Algorithm

The exact steps for solving the problem using the augmented Chebyshev algorithm are as follows.

Step 1: Calculate the minimum value. Optimize only objective (1) to solve the original model to find the optimal urban rail transit connection scheme $x_{1}$ and the minimum total bus schedule change $Z_{1}^{min} = Z_{1} (x_{1})$ ; optimize only objective (2) to solve the original model to find the optimal urban rail transit connection scheme $x_{2}$ and the minimum total transfer time $Z_{2}^{min} = Z_{2} (x_{2})$ .

Step 2: Calculate the maximum value. Based on the dictionary sequence method of Equation 16, add the constraint of $Z_{2} (x) = Z_{2}^{min}$ , which optimizes objective (1) only. Then, it solves the original model and obtains the optimal rail transit connection scheme $x_{1}^{'}$ and the maximum total bus schedule change $Z_{1}^{max} = Z_{1} (x_{1}^{'})$ ; based on the dictionary sequence method of Equation 17, add the constraint of $Z_{1}^{min} = Z_{1} (x_{1})$ , optimize objective (2) only, solve the original model, and obtain the optimal rail transit scheme $x_{2}^{'}$ and the maximum total transfer time $Z_{2}^{max} = Z_{2} (x_{2}^{'})$ .

Step 3: Dimensionless transformation of the objective function value. Through the method of thresholding in a straight line, the objective function value is transformed into an objective function value between 0 and 1 by taking the extreme and minimal values of the previous two objective function values to eliminate the interference of the dimension.

Step 4: Find the non-dominated solution. Firstly, the value of the penalty factor is determined. The penalty factor $σ$ of the augmented single-objective model is to make the solved answer satisfy the constraints set by the model, that is, to control the constraint illegality. Since the augmented model is to find the minimum value, it is obtained by subtracting the sum of the values of all feasible solutions from the great value of the two objective function values. Then, uniform weights are applied to the two objectives ( 32 ) to generate $h$ sets of weight vectors ${w^{k} = (w_{1}^{k}, w_{2}^{k}), k = 1, 2, . . ., h}$ in the interval $[0, 1]$ . Then, the weights of the total bus schedule change objective are set to $w_{1} = 1 - (k - 1) / (h - 1)$ , and the weights of the total interchange time objective are set to $w_{2} = 1 - w_{1}$ . Next, the weighted Chebyshev distance $ρ$ is obtained from the weights and all feasible points with the maximal values. Finally, the optimal solution ${x^{*}, ν^{*}}$ is obtained by solving the augmented single-objective model to obtain the non-dominated solution set ${Z_{1} (x^{*}), Z_{2} (x^{*})}$ . A flow chart of the algorithm is shown in Figure 2.

Figure 2.

Flow chart of the augmented Chebyshev algorithm.

Numerical Example

Illustration of the Calculation

We obtained the IC swipe card data about Shenzhen bus and rail transportation in December 2016. Meanwhile, after strict data screening and pre-processing, we select specific day data, for instance, validation. Shenzhen 66 Bus Road has five bus stops near access to four different metro stations, including the first bus stop, three terminal stations, and the last station, as shown in Figure 3. The bus stops numbered from 1 to 5 are the Window of the World, Baishizhou 3, Xili Fating 2, the zoo station, and Wangjingkeng Country. The average time for each bus stop to run to the next stop, the total number of interchanges per day at each stop, and the number of interchanges in directions 1 and 2 are shown in Tables 2 and 3, with each stop marked virtually. The operation period is from 6:00 a.m. to 11:00 p.m. In that period, 125 buses, including the direction from site 1 to site 5, had 1764 interchange passengers. This paper mainly studies the interchange of sites 1–5, where site 1 is the first station and site 5 is the terminal.

Figure 3.

Shenzhen 66 Bus Road and urban rail transit connection diagram.

Table 2.

Time Period of Bus Operation for Each Station

Bus stop	1–2	2–3	3–4	4–5
Arrival time at the following interchange, min	3	10	10	12
Maximum arrival time at the following interchange, min	2	8	7	20
Minimum arrival time at the following interchange, min	6	16	14	9

Table 3.

Interchange Data of the Four Stations

Site	1 (First station)	2	3	4	5 (Terminal)
Total number of interchanges, person	796	131	720	117	NA
Direction 1 interchange number, person	796	68	364	62	NA
Direction 2 interchange number, person	NA	63	356	55	NA

Note: NA = not available.

We mark whether a transfer passenger is a stranded passenger (0,1) based on the data of getting off the urban rail transit and getting on the transfer bus, and mark the order of departure of the previous bus from 1 to 125, where we assume the longest and shortest transfer time from the urban rail transit to the bus is shown in Table 2, and we believe the bus departure interval is 3–20 min. For stranded passengers, the default is to take the next bus. Table 4 shows the shortest transfer times from urban rail transit to bus after our changes.

Table 4.

Minimum Transfer Time From Urban Rail Transit to the Bus

Bus stop	1	2	3	4
Minimum interchange time, min	2	2	5	7

Because of the incompleteness of each station’s departure data and the difficulty in determining them, this paper fits the departure time of a station based on the original departure, and the results show that it belongs to a normal distribution. Then, this normal distribution is used to randomly generate 125 departure times for four stations in direction 1 and 125 departure times for four stations in direction 2. Because the fitting is based on historical data and then the departure times of the four stations are estimated, the random variation of the departure times of the stations is more reasonable for solving the optimal solution of the dual-objective model of the original problem.

Calculation Results

In this paper, the bi-objective model with minimum bus schedule changes and minimum transfer time is programmed in Cplex software, and the data of each arithmetic case is substituted to solve its optimal solution. In the algorithm arithmetic example, the same model is optimized in directions 1 and 2, respectively. This paper shows that the exact algorithm has a good test effect for the rail–bus connection model. In the parameter setting, the value of $h$ was set to 10 based on the issue of the operation efficiency of the software. We generate 10 sets of weight vectors ${w^{k} = (w_{1}^{k}, w_{2}^{k}), k = 1, 2, . . ., h}$ in the $[0, 1]$ interval range. Using $w_{1} = 1 - (k - 1) / (h - 1)$ , the weight of the total bus schedule change target can be found, and after that, the weight of the total transfer time target can be found according to $w_{2} = 1 - w_{1}$ .

From the results of the Cplex software run, it can be seen that in direction 1, the values of penalty factors $σ = 6246.04$ , minimal $z_{1}^{min} = 184.00$ , $z_{2}^{min} = 654.09$ , and maximal, $z_{1}^{max} = 4430.40$ , $z_{2}^{\max} = 1, 002, 200$ are calculated, and in direction 2, the values of penalty factors $σ = 6.60$ , minimal $z_{1}^{min} = 1061.00$ , and maximal, $z_{1}^{max} = 4430.40$ , $z_{2}^{\max} = 4437.00$ are calculated. Eight representative non-dominated solutions are obtained in direction 1, and seven non-dominated solutions are obtained in direction 2. The non-dominated solutions correspond to the objective function values of the augmented model, and the specific results of the function values and weights of the two single objectives are shown in Tables 5 and 6. The tables show that the augmented Chebyshev algorithm finds eight different non-dominated solutions in direction 1 and seven different non-dominated solutions in direction 2. Each solution shows a large difference in the rail–bus connection solution, which reflects the importance of changing the bus schedule as the decision variable. It reflects the significant impact of using bus schedule change as the decision variable to solve the problem of low time efficiency of the rail–bus feeder solution, thus illustrating the algorithm’s effectiveness.

Table 5.

Calculation Results of Direction 1 of the Augmented Chebyshev Algorithm

Non-dominated solutions	Bus schedule change, min	Target weights	Total interchange time, min	Target weights	Broadening model target values
1	243.08	0.89	12115.84	0.11	5217.95
2	409.63	0.78	9858.66	0.22	4002.63
3	620.57	0.67	8418.79	0.33	3409.42
4	928.54	0.56	7156.19	0.44	3103.09
5	1247.78	0.44	6286.01	0.56	3078.03
6	1867.26	0.33	5185.85	0.67	3421.73
7	2478.73	0.22	4578.95	0.78	4080.35
8	2713.93	0.11	4464.59	0.89	4413.02

Table 6.

Calculation Results of Direction 2 of the Augmented Chebyshev Algorithm

Non-dominated solutions	Bus schedule change, min	Target weights	Total interchange time, min	Target weights	Broadening model target values
1	1091.00	0.89	7737	0.11	412.64
2	1319.80	0.78	6324.04	0.22	506.76
3	1666.20	0.67	5181.49	0.33	379.92
4	1956.76	0.56	4613.51	0.44	504.29
5	2377.92	0.44	4146.67	0.56	582.19
6	2422.42	0.33	4115.67	0.67	452.06
7	2559.23	0.22	4047.27	0.78	332.56

From Tables 5 and 6, it can be seen that in direction 1, when the weight of target 1 is $w_{1} = 0.44$ and the weight of target 2 is $w_{2} = 0.56$ , the rail–bus connection scheme has the best effect, and the total passenger transfer time is 4464.59 (min), which is 68.06% shorter than the original scheme. In direction 2, when the weight $w_{1} = 0.22$ for objective 1 and the weight $w_{2} = 0.78$ for objective 2, the best results are obtained for the rail–bus connection scheme, and the total passenger transfer time is 4047.27 (min), which is 63.18% shorter than the original scheme. The specific bus schedule changes for each stop in each direction are shown in Tables 7 and 8. The color shades in the tables represent the magnitude of the changes, with darker colors indicating more remarkable changes.

Table 7.

Selected Bus Schedule Changes for Four Stops in Direction 1

Table 8.

Selected Bus Schedule Changes for Four Stops in Direction 2

In addition, because of the high concentration of passengers during the morning and evening peak hours, government departments respond by increasing the number of operating vehicles. To further understand the situation of each station at a specific moment, we conducted statistics on the changes in bus schedules at each of the four stations in direction 1, as shown in Figure 4. As shown in Figure 4, the bus schedules of the four stations in direction 1 are changed more in the morning peak and the last bus. It is easy to understand that in recent years, the urban population has been growing, and more rural people are moving to the city to work, which leads to an increase in the number of passengers going to work in the morning and an increase in the number of people transferring to the bus. Because of the high concentration of passengers during the morning and evening peak hours, government departments will respond by increasing the number of vehicles in operation. So, the previous bus schedule has not been able to adapt to the situation today, and there is a need for appropriate adjustments to the bus schedule to alleviate the stranded bus platform transfer passengers.

Figure 4.

Distribution of bus schedule changes at the four stops in direction 1.

In practice, the weighting of these two objectives is chosen as the best solution based on the decision-maker’s preference and the location of the bus stop. When the decision-maker prefers the most minor bus schedule change, the weighting factor $w_{1} > 0.5$ for objective I and the weighting factor $w_{2} > 0.5$ for objective II when the decision-maker chooses the shortest total transfer time for passengers. For example, there are five stops on bus line 66 in Shenzhen: Window of the World, Baishizhou 3, Xili Fating 2, zoo station, and Wangjingkeng Country. The Window of the World bus station is a bus line hub connecting many bus lines and urban rail transit lines. The Window of the World is located in the city center, and the number of interchanges is high, so in the rail–bus connection problem, decision-makers tend to pay more attention to reducing the waiting time of interchange passengers to avoid the phenomenon of passengers being stranded at the bus station. So, the weight of objective (2) takes up more weight, and the seventh or eighth non-dominated solution corresponds to the solution. Baishizhou 3 bus station has the most concentrated and large-scale farmers’ houses in Shenzhen, so the number of farmers is primary, and the living standard is low.

The passengers at this station tend to pay more attention to the cost of travel and have a greater demand for taking the bus. Therefore, decision-makers will pay more attention to the goal of minimal changes in the bus schedule to accommodate the needs of this type of passenger, so the weight factor of objective (1) is significant, and the solution corresponds to the first or second non-dominated solution chosen. The Xili Fating 2 and zoo sites have many parks, attractions, squares, and other entertainment places in Shenzhen, located in the city center. At the same time, they are adjacent to universities’ business and technology parks. Most of those who take the bus are tourists, office workers, and working people. They pay more attention to time efficiency and hope that the transfer time is shorter, so the weight of goal two is a relatively large. At this time, the decision-maker may choose the seventh or eighth non-dominated solution corresponding to the solution. Finally, the Wangjingkeng Country site is located in residential areas, as the ride is mostly for older adults. It is influenced by the habits of users, who prefer to follow the previous way and time and, if the bus schedule changes are significant, they often cause discomfort to elderly passengers; this should make goal one better so that the bus departure time changes as little as possible. At this point, decision-makers often choose the first or second non-dominated solution corresponding to the solution.

Comparison of Algorithms

To verify the effectiveness of the augmented Chebyshev algorithm, the same model is tested in this paper using the multi-objective NSGA-II, which we invoked Tian et al. ( 33 ) to run the multi-objective optimization model of this paper and compare it with the augmented Chebyshev algorithm. Among them, the relevant parameters of the NSGA-II are set as follows: the population size is 200, the number of iterations is 100 million, the cross-variance probability is 0.85, and the variance probability is 0.15. Finally, four representative non-dominated solutions are obtained in direction 1, and seven in direction 2 represent non-dominated solutions. The results of two objective function values are shown in Tables 9 and 10.

Table 9.

Calculation Results of the Non-Dominated Sorting Genetic Algorithm Direction 1

Non-dominated solutions	Bus schedule change, min	Total interchange time, min	Total interchange time optimization ratio, %
1	1297.6	7002.61	49.90
2	1532.96	6536.42	53.24
3	2138.08	5350.15	61.73
4	3468.8	4581.28	67.23

Table 10.

Calculation Results of the Non-Dominated Sorting Genetic Algorithm Direction 2

Non-dominated solutions	Bus schedule change, min	Total interchange time, min	Total interchange time optimization ratio, %
1	659.42	9811.33	10.73
2	1056.64	8168.48	25.68
3	1476.16	6880.96	37.39
4	2037.14	5573.29	49.29
5	2880.32	4632.32	57.85
6	3358.29	4253.92	61.30
7	3764.1	4191.2	61.87

In the case test, the comparison results of the two algorithms to explore the set of representative non-dominated solutions are shown in Figures 5 and 6. AWTA represents the augmented Chebyshev algorithm and NSGA-II describes the genetic algorithm. It can be seen from the figures that both algorithms can find non-dominated solutions. However, the augmented Chebyshev algorithm finds more representative non-dominated solutions than the genetic algorithm, and it is significantly better than the genetic algorithm with respect to the quality of solutions. From here, it can be seen that the NSGA-II has too much change in the departure time of the target bus schedule in the problem model, which also leads to the negative effect of the long total passenger transfer time; therefore, although the NSGA-II can obtain four representative non-dominated solutions, the optimization effect is not apparent even with significant changes in the bus schedule. This is reflected in the NSGA-II not being robust to the problem of optimizing models for rail–bus connections. It is not the optimal method with respect to optimization effectiveness, and does not provide the decision-maker with a better optimization solution.

Figure 5.

Pareto front-facing ratio for the direction 1 algorithm.

Figure 6.

Pareto front-facing ratio for the direction 2 algorithm.

It can be concluded that using the augmented Chebyshev algorithm compared with the NSGA-II not only ensures the quality of the solution but also converges better and yields more solutions. It has a larger optimization ratio for the total transfer time, which can significantly reduce the passengers’ transfer waiting time, which has an essential impact on improving passenger satisfaction and provides a good optimization solution for the decision-maker. In addition, the solution time of the non-dominated solution of the model is 37,164.59 s for the NSGA-II and 184.80 s for the augmented Chebyshev algorithm, indicating that the traditional genetic algorithm is more complex than the augmented Chebyshev algorithm in solving the original problem model, which leads to a slower solution speed.

In summary, from the analysis of the examples in this section, it can be concluded that the improved augmented generalized Chebyshev algorithm is more suitable for solving the dual-objective rail transit connection problem than the multi-objective NSGA-II.

Analysis of Bus Schedule Changes

With the established studies on rail transit connections, there are more studies on optimizing rail metro schedules and fewer studies on bus schedule changes. This paper proposes adjustments to conventional bus schedules to reduce passenger transfer times and improve the efficiency of transfer management. Although many factors influence bus schedule changes, if there are large changes, they will not achieve the purpose of optimization and will cause dissatisfaction among bus riders, which will lead to a decrease in the number of bus riders and further increase the operating costs of bus operators. However, if the bus schedule is changed appropriately, the rail–bus connection problem will also be optimized. In this paper, the data in Tables 9 and 10 show that the total passenger transfer time decreases as the magnitude of bus schedule changes increases, and introducing a penalty factor in the incremental model improves the robustness of the model solution. It can be seen that changing the bus schedule has a significant impact on optimizing the total passenger transfer time. However, decision-makers should consider the actual situation of different stops on different routes, reasonably weigh the weight of the two objectives, and assess the operating cost of public transportation and the quality of interchange service to appropriately adjust the bus schedule to reduce the total passenger transfer time.

Conclusions

This study proposes a schedule bi-objective optimization model from two perspectives: overall passenger transfer time and bus schedule alteration, with bus departure interval and departure, which stops at each transfer station, as decision variables. Objective (1) minimizes the overall passenger transfer time, and objective (2) minimizes the bus schedule changes. This dual-objective consideration allows the study to integrate the passenger demand and operational efficiency of the transit system, which is of practical application. Then, we design an augmented Chebyshev algorithm to solve the model, which obtains the Pareto front of the problem by getting a good set of non-dominated solutions with good convergence. A group of actual IC swipe card data is used for validation, which increases the credibility and practicality of the study results. The calculated results are also compared with the conclusions derived from the NSGA-II. The results show that the augmented generalized Chebyshev algorithm is better than the NSGA-II in optimization, and the total passenger transfer time is reduced by 68.06%, which indicates that the use of the augmented generalized Chebyshev algorithm to solve this kind of bi-objective optimization model has greater advantages and application potential.

In essence, urban rail transit transfer time is regulated collaboratively between the metro and bus, so we must consider both urban rail transit and bus factors. Based on the limitations of personal competence, this study has some shortcomings, such as some possible assumptions and simplifications in model building, which may limit the accuracy and applicability of the study. Also, the quality and reliability of the IC swipe data can affect the accuracy of the results of the study. As well as solving the model, other algorithms may be more suitable for the problem, or parameter tuning of the algorithm can improve the results. Therefore, a more comprehensive selection and comparison of algorithms must be considered to ensure a more comprehensive and reliable conclusion. These limitations and shortcomings do not negate the value and contribution of the study; rather, they help us to improve further and expand the direction of the study. In future studies, we will first consider the urban rail transit’s running and departure times rather than just the bus’s departure and running times. Secondly, our analysis considers bus route interchanges only under ideal conditions, and we can continue to discuss the impact of the optimization on other non-interchange stations. Moreover, we would like to use some optimization algorithms to perform a more accurate hyperparameter search for the NSGA-II, aiming to obtain a more scientific comparison of the algorithm results. In addition, we would like to apply the model with appropriate adjustments to other optimization contexts to verify further the superiority and adaptability of the augmented Chebyshev algorithm.

Footnotes

Acknowledgements

All three authors share the full results.

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: F. Tian, J. Liang, R. Chen; data collection: F. Tian, J. Liang, R. Chen; analysis and interpretation of results: F. Tian, J. Liang, R. Chen; draft manuscript preparation: F. Tian, J. Liang, R. Chen. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

A Multi-Objective Subway Timetable Optimization Approach with Minimum Passenger Time and Energy Consumption. Journal of Advanced Transportation, Vol. 50, No. 1, 2016, pp. 69–95.

Özpeynirci

Ö.

Köksalan

An Exact Algorithm for Finding Extreme Supported Nondominated Points of Multiobjective Mixed Integer Programs. Management Science, Vol. 56, No. 12, 2010, pp. 2302–2315.

Zahir

U. M.

Matsui

Fujita

Investigate the Effects of Bus and Passenger Arrival Patterns and Service Frequency on Passenger Waiting Time and Transit Performance of Dhaka Metropolitan Area. WIT Transactions on the Built Environment, Vol. 49, 2000, p. 10.

Chien

S. I. J.

Tsai

F. M.

Hou

. Optimization of Multiple-Route Feeder Bus Service: An Application of GIS. Presented at 82th Annual Meeting of the Transportation Research Board, Washington, D.C., 2003.

Bookbinder

J. H.

Désilets

Transfer Optimization in a Transit Network. Transportation Science, Vol. 26, No. 2, 1992, pp. 106–118.

Belletti

Davini

Carraresi

Gallo

BDROP: A Package for the Bus Driver’s Rostering Problem. Computer Scheduling of Public Transport, Vol. 2, 1985, pp. 319–324.

Ismail

Ang

Integer Programming Approach in Bus Scheduling and Collection Optimization. Jurnal Teknologi, Vol. 43, 2005, pp. 1–14.

Zheng

J. C.

C. X.

Gao

Research on Walking Time of Transfer Pedestrians in Urban Rail Transit Station. Applied Mechanics and Materials, Vol. 610, 2014, pp. 1053–1056.

Blanco

Conde

Hinojosa

Puerto

An Optimization Model for Line Planning and Timetabling in Automated Urban Metro Subway Networks. A Case Study. Omega, Vol. 92, 2020, p. 102165.

10.

Wong

R. C.

Yuen

T. W.

Fung

K. W.

Leung

J. M.

Optimizing Timetable Synchronization for Rail Mass Transit. Transportation Science, Vol. 42, No. 1, 2008, pp. 57–69.

11.

Yin

Yang

Tang

Gao

Ran

Dynamic Passenger Demand Oriented Metro Train Scheduling with Energy-Efficiency and Waiting Time Minimization: Mixed-Integer Linear Programming Approaches. Transportation Research Part B: Methodological, Vol. 97, 2017, pp. 182–213.

12.

Chen

Mao

Bai

T. K.

Timetable Synchronization of Last Trains for Urban Rail Networks with Maximum Accessibility. Transportation Research Part C: Emerging Technologies, Vol. 99, No. 2019, pp. 110–129.

13.

Chien

Schonfeld

Joint Optimization of a Rail Transit Line and Its Feeder Bus System. Journal of Advanced Transportation, Vol. 32, No. 3, 1998, pp. 253–284.

14.

Wang

Correia

G. H. d. A.

Chen

A Network-Based Model of Passenger Transfer Flow Between Bus and Metro: An Application to the Public Transport System of Beijing. Journal of Advanced Transportation, Vol. 2020, 2020, pp. 1–12.

15.

Zhao

Tan

Research on Coordination and Optimization of Conventional Bus Routes in Environment of Rail Transit. Proc., 2017 3rd International Forum on Energy, Environment Science and Materials (IFEESM 2017), Atlantis Press, Atlantis, 2018, pp. 83–89.

16.

Wang

Liu

Lai

Analysis on Optimization of Normal Bus Lines Along Urban Rail Transit. Journal of Physics: Conference Series, Vol. 1972, No. 1, 2021, p. 012094.

17.

Shu-xia

Xu-mei

Lei

Dong-fang

Average Waiting Time Model for Transfer from Rail Transit to Buses. Journal of Transportation Systems Engineering and Information Technology, Vol. 10, No. 2, 2010, p. 143.

18.

Zhang

Liu

W. S.

The Optimization Method to Transfer Bus Routes for Rail Transit Terminal Station. Applied Mechanics and Materials, Vol. 253, 2013, pp. 1869–1875.

19.

Ren

Ouyang

Coordinated Passenger Flow Control and Bus Connection Setting During Peak Hour of Urban Rail Transit. Proc., 6th International Conference on Transportation Engineering, American Society of Civil Engineers, Reston, VA, 2019, pp. 86–94.

20.

Tian-wei

En-jian

Yang

Ning

Lin

Multimodal Timetable Optimization Between Urban Transport Hubs Considering Elastic Demand. Journal of Transportation Systems Engineering and Information Technology, Vol. 21, No. 1, 2021, p. 16.

21.

Wang

Yuan

Cao

Optimal Bus Bridging Schedule with Transfer Passenger Demand During Disruptions of Urban Rail Transit. Journal of Transportation Engineering, Part A: Systems, Vol. 147, No. 10, 2021, p. 04021071.

22.

Tang

Yang

A Hybrid Algorithm for Urban Transit Schedule Optimization. Physica A: Statistical Mechanics and its Applications, Vol. 512, 2018, pp. 745–755.

23.

Qiu

Liu

Yang

Dynamic Optimization of Bus Line Schedule in Commuter Corridor Based on Bus IC Card Data. Journal of Advanced Transportation, Vol. 2022, 2022, pp. 1–10.

24.

Bie

Wang

Optimization of Electric Bus Scheduling Considering Stochastic Volatilities in Trip Travel Time and Energy Consumption. Computer-Aided Civil and Infrastructure Engineering, Vol. 36, No. 12, 2021, pp. 1530–1548.

25.

Han

Ning

Yang

Yin

Train Timetabling in Rail Transit Network Under Uncertain and Dynamic Demand Using Advanced and Adaptive NSGA-II. Transportation Research Part B: Methodological, Vol. 154, 2021, pp. 65–99.

26.

Trépanier

Tranchant

Chapleau

Individual Trip Destination Estimation in a Transit Smart Card Automated Fare Collection System. Journal of Intelligent Transportation Systems, Vol. 11, No. 1, 2007, pp. 1–14.

27.

Jang

Travel Time and Transfer Analysis Using Transit Smart Card Data. Transportation Research Record: Journal of the Transportation Research Board, 2010. 2144: 142–149.

28.

Munizaga

M. A.

Palma

Estimation of a Disaggregate Multimodal Public Transport Origin–Destination Matrix from Passive Smartcard Data from Santiago, Chile. Transportation Research Part C: Emerging Technologies, Vol. 24, 2012, pp. 9–18.

29.

Boyd

J. P.

Rangan

Bucksbaum

Pseudospectral Methods on a Semi-Infinite Interval with Application to the Hydrogen Atom: A Comparison of the Mapped Fourier-Sine Method with Laguerre Series and Rational Chebyshev Expansions. Journal of Computational Physics, Vol. 188, No. 1, 2003, pp. 56–74.

30.

Calafiore

G. C.

El Ghaoui

Optimization Models. Cambridge: Cambridge University Press, 2014.

31.

Coghetto

Chebyshev Distance. Formalized Mathematics, Vol. 24, No. 2, 2016, pp. 121–141.

32.

Liu

Jiao

Sun

MOEA/D with Adaptive Weight Adjustment. Evolutionary Computation, Vol. 22, No. 2, 2014, pp. 231–264.

33.

Tian

Cheng

Zhang

Jin

PlatEMO: A MATLAB Platform for Evolutionary Multi-Objective Optimization [Educational Forum]. IEEE Computational Intelligence Magazine, Vol. 12, No. 4, 2017, pp. 73–87.