key: cord-0077822-7lygmabx authors: Cao, Zhichao; Xu, Ting; Zhang, Silin; Ceder, Avishai Avi; Sun, Yuyao title: Comparative Evaluation: Passenger Satisfaction and Operation Efficiency of Different Even-Headway and Even-Load Public Transport Timetables date: 2022-05-07 journal: KSCE J Civ Eng DOI: 10.1007/s12205-022-1231-x sha: 154fe0865c92fdafc4eb0cfc6803b497f66d5fd3 doc_id: 77822 cord_uid: 7lygmabx Even-headway timetables with easy to memorize departure times are commonly in use in public transport (PT) service. In addition, some PT services are using timetables where their departure times are based on even-passenger-load at the max-load stop along the route. In practice, these two types of timetables are implemented in daily operation plans, allowing for their merits, with the former maintaining the initial, even-interval departure rules and conventions, especially when a new PT line is planned without available demand information, while the latter is demand-oriented, exhibiting fluctuation in time and space. The principles of creating the aforementioned timetables have been presented in the literature, but very few evaluations feasibly attest to performance by means of mathematical analysis. Concentrating on building a modelling framework for PT timetabling problems during the planning stage, this work is taking into account 1) passengers’ service-frequency satisfaction, in terms of waiting times, 2) passengers’ perception of riding comfort, in terms of seat availability, and 3) planned passengers’ load ratio (concerning vehicle’s capacity) linked with operation efficiency. Yielding to the models developed in a theoretical manner, the comparative evaluation is conducted through multiple timetable schemes that include four different dimensions: (A) even-headway departures with same headway throughout the time horizon, (B) even-load departures with same load throughout the time horizon, (C) even-headway departures for each demand-dependent period, and (D) even-load departures for each demand-dependent period. Finally, a case study in Nantong, China examines the comparison performances produced and the best plans for operational timetables. The results indicate that travelers attributed great importance to higher satisfaction preferences on frequency or headway-related experience linked with their waiting time. waste some PT vehicles loading only a few passengers in offpeak hours. Contrarily, from the user standpoint, passengers become somewhat confused by the irregular arrival times of PT vehicles, given that timetables are announced a priori, but not easy to remember as is the case with even-interval departures. The intention to improve PT service level of even-interval departures tends to result in amply delivering user satisfaction in relation to travel habits, while demand-based departures have the advantage of reducing the likelihood of passengers left-behind due to capacity constraints. Subsequently, we are confronted by the question of the difference between even-load, even-headway, and some of the relevant integrated timetables in addressing serviceability improvement. This poses the question of which is preferable and whether a best scheme exists. That is, the scheme of a timetable with specific setting of departure times. The absence of a valid comparative evaluation of the main PT timetabling plans tends to implicate the inability to reasonably improve service quality, due to the nature of the space-time distribution for peak and off-peak hours. Clearly, PT resources are scarce (Ceder et al., 2013b) during peak hours, whereas they seem to be adequate during off-peak hours. Some of the literature provides evidence that the level of service as perceived by passengers heavily depends on their subjective evaluation of the sectional travel experience, such as out-of-vehicle wait, in-vehicle congestion (Ceder and Marguier, 1985; Liu et al., 2013; Muñoz et al., 2014) . Alternatively, from the viewpoint of systematic PT planning optimization, either even-headway timetables (Shafahi and Khani, 2010; Ibarra-Rojas and Rios-Solis, 2012; Liu et al., 2013; Kang et al., 2015) or even-load timetables (Niu and Zhou, 2013; Sun et al., 2014; Sun et al., 2018 , Zhang et al., 2020 , tend to be offered as one option of input to produce sub-optimal tactics, such as, 1) route design, 2) timetable development, 3) vehicle scheduling, 4) crew scheduling, and 5) real-time monitoring and control (Ceder, 2016) . However, few studies have focused on their essential characteristics, particularly ignoring the efficiency of PT resources, i.e., the average load factor. It should be noted that although even-headway timetables and even-load timetables were always employed independently, neither the quantitative comparison nor if and how they achieved consistency have been taken into account in the literature. Moreover, considering that the main stakeholders are essentially associated with passengers and PT companies, the trade-off between them inspired us to build a multi-measure, criteria-based modelling framework in this study. This study is comprised of five sections including this introductory section. Section 2 provides a literature review of PT timetable design and optimization. Section 3 elaborates a mathematical framework including four specified models for the comprehensive, comparative evaluation, concluding with a feasible algorithm proposed to resolve the issues. Section 4 validates the proposed solution through a case study analysis. Finally, Section 5 concludes the study and proposes future extensions of our work. Recent literature in timetable design and optimization has been reviewed, and its outlines are presented here. Ceder et al. (2013a) investigated the uncertainty influence of out-of-vehicle times on PT passengers' desire to choose transfer routes. Nesheli et al. (2016) addressed a qualitative and quantitative study of estimating PT users' perceptions and decisions using tactic-based guidelines. They focused on two dimensions of decision-making problems: (a) delay on PT users' perceptions and decisions to change route or mode; and (b) evaluation of users' decisions based on various real-time operational tactics. The case was examined and verified based on data from Auckland, New Zealand, and Lyon, France. Yang et al. (2017) was concerned with bus-bridging service with the aid of passenger flow control, to resolve oversaturated passenger demand, proposing a two-stage mathematical programing model. Jin et al. (2014) advanced integrating a localized bus service and enhancing metro network resilience to resolve some of the potential disruptions. A two-stage stochastic programming model is proposed and affirmed by actual travel demand data from the Singapore public transit system. Jin et al. (2016) introduced smartly designed bus bridging services considering commuter travel demand at the time of the disruption. Three basic procedures were proposed to address metro network disruption problems: 1) demand-response bus route generation, 2) path-based multicommodity network flow model implementation, and 3) optimal allocation of available fleet size, respectively. Nesheli and Ceder (2015) built a robust, tactic-based, real-time framework for the PT transfer network to maximize the number of synchronizations. Ibarra-Rojas and Rios-Solis (2012) formulated the synchronization problem of timetabling to facilitate passenger transfers and avoid bus bunching along the PT network. The problem proposed was proved to be an NP-hard problem and the feasible solution space of the model was analyzed to improve the solution efficiency. Ibarra-Rojas et al. (2014) proposed two integer linear programming models to find the trade-off between the timetabling and vehicle scheduling problems. A ∈-constraint method is used to solve this integrated bi-objective problem jointly. Wang et al. (2017) formulated an integrated model to optimize the train schedule and circulation plan simultaneously based on the demand analysis, which was transformed into mixed integer linear programming problems and efficiently solved by the CPLEX solver. Zhang et al. (2020) presented a comprehensive optimization model for feeder bus timetabling and a procurement scheme considering environmental impact. Multiple vehicle type schemes were compared on the basis of two standards that were 1) passengers' waiting time, and 2) the operator's costs, including the environmental criteria of PT systems. Zhang et al. (2018) proposed two nonlinear non-convex programming models to optimize metro timetables with the objective of minimizing overall passenger travel times. Liu et al. (2013) proposed a model PT timetable design considering vehicles' random travel time with three classical objectives: 1) to minimize overall passenger waiting time, 2) overall passenger riding time and 3) total operating cost. A Genetic Algorithm Incorporating Monte Carlo Simulation is used to solve the timetabling model. Aiming to minimize the waiting time at PT transfer stops, Shafahi and Khani (2010) formulated a timetable optimization model throughout the adjustment of the first vehicle's departure time based on the rule of even-headway intervals. Kang et al. (2019) developed an integrated last train timetable optimization and bus bridging coordination model with bi-objectives. One is to maximize last train connections and the other is to minimize waiting times for rail-to-bus passengers. Liu and Ceder (2020) focused on the battery-electric PT vehicle scheduling problem considering stationary battery chargers with the objective of minimizing both fleet size and total number of battery chargers. Unlike the existing literature that addresses the aim to reduce passenger travel time or to optimize fleet size, without considering onboard passengers' load and load ratio (concerning vehicle's capacity), this study analyzes the integration of multifaceted interests to compare, and thereby, develop an integration platform in order to make a comparison on a fair basis. The model developed in our study is formulated with three measures: 1) passengers' service-frequency satisfaction, in terms of waiting times, 2) passengers' perception of riding comfort, in terms of seat availability, and 3) planned passengers' load ratio (concerning vehicle's capacity) linked with operation efficiency, allowing us to better depict the multi-scheme comparison to identify good candidates of timetables for various passenger demands. This section addresses the notations and mathematical formulations of the model. Fundamental assumptions are proposed in Section 3.1. Parameter definition and estimation criteria are introduced in Section 3.2 and Section 3.3, respectively. Then, modelling constraints are provided in Section 3.4. Finally, a solution algorithm is presented in Section 3.5. The framework of facilitating the mathematical model to allow for the comparison between the operational timetables is shown Fig. 1 . First, the necessary hypothesis of modelling is shown with regard to the time-dependent passenger demand and enables testing the timetables and schedules developed during the planning horizon. The fundamental passenger demand information is handled stably based on the historical data, irrespective of its fluctuation. The number of passengers who are about to board the vehicles of the given PT corridor during the time-interval (t-1, t] is represented as Q(t). Moreover, the sum of Q(t), t ∈ T indicates the integral demand of a fixed PT line. Second, the inputs of the models are involved with dual components to a significant degree, i.e., both demand distribution (that can be divided into different time-windows, and thereby treated separately) and various timetables need to be examined. Third, the service ability to provide during the planning level (the number of trips, used as a decision variable m) is also especially noteworthy besides the departure rules determined. Taking the fleet size into consideration, we check the possibility of valuing m by the means of enumerations to conduct the comparison analysis. Finally, the outputs of the model developed are evaluated and analyzed quantitatively based on the threefold measures: (i) passengers' perception of waiting time (survey-based), (ii) seat availability (represents user's comfort; model's result), and (iii) passengers' onboard load (represents operator's control variable related to the fleet size). Moreover, the best suited timetabling schemes are captured, subject to related constraints in order to seek conclusions on the trade-off between them. To facilitate the model formulation, reasonable assumptions proposed in the study are presented below. Assumption I: The practical PT operations precisely carry out the designed schedule. In other words, neither the major distributions nor operation interruptions are taken into account in the study. Assumption II: The model developed is based on an independent route, with random passengers' arrival. Assumption III: Fluctuations of passengers' average waiting The maximum capacity of one vehicle, i.e., the sum of seats and standing space. C 0 Average demand on the basis of even-load timetable. times affect their satisfaction evaluation, and thereby the satisfaction criteria specified through the logical questionnaire for the public are supposed to be feasible. Assumption IV: Overall PT vehicles are homogeneous. This means that their operation average speeds are identical. Assumption V: The strict capacity threshold of allowing the number of passengers to board is considered so that overloading is prohibited. The following symbols and the corresponding definitions are introduced in Table 1 . With the accumulated passenger arrivals process, how to reasonably determine the perception of classified service is the concern of our first measure in the study. Indeed, distinctions of different service assessments from individual users exist (Wu et al. 2013 ). The perceived satisfaction degree can be represented by the means of a random decision variable ξ kf . Here, the term 'ξ kf ' refers to a perceived (headway/frequency-dependent) satisfaction degree of the f-th proportion of users on the k-th headway, ξ kf ∈ [0, 1]. In order to clarify the relationship between headways and satisfaction of waiting, we first determine quantitative performance criteria throughout the questionnaire survey taken from about 100 volunteers, shown in Table 2 . Herein, the attribute of volunteers is presented in Table 1 of Supplementary Material File. Considering complex scenarios of both off-peak hours and consecutive, fully loaded vehicles, as well as traffic jams, we investigate the entire span of headway intervals from 5 to 25 minutes. Herein, a majority of interviewees indicate a significant satisfaction on headways of less than 5 minutes. It means the average waiting time is about 2.5 minutes as shown by Liu et al. (2013) . Hence, we do not penalize them. Moreover, we divide the full span of headways into five components by 5-minute intervals in Table 2 . Subjective perception ξ kf of different waiting times can be categorized into five items, 1) highly satisfied, 2) satisfied, 3) normal, 4) dissatisfied, and 5) highly dissatisfied, which correspond to different scores that was normalized within [0, 1]. For example, in the 2 nd column of Table 2 , k = 1 is for the 1 st headway whose range is from 5 to 10 minutes. Corresponding to the two red modules in Fig. 2 , there are 40 percent of users-f = [0, 40%]-express one hundred percent satisfaction. In the similar way, (40%, 60%] of them have satisfaction degree ξ kf = 0.8, which is according to the orange module depicted by Fig. 2 . Alternatively, (60%, 100%] of users vote for the satisfaction degree ξ kf = 0.6, highlighted by two green modules of Fig. 2 . It should be noted that the distinction of multiple headways is essentially derived from subjective perception of the headway/ frequency that is implying on the perception for the waiting times. Figure 2 has its appeal in its graphical nature and visual simplicity, which is transformed from Table 2 . For example, we focus on the first column (k = 1) of Fig. 2 . Given the headway of one certain planned timetable equal to 6 min, Eq. (1) derives the calculation of the number of satisfied passengers more precise: , where we deduce the number of satisfied passengers w cs on the first vehicle with headway h=6 by the use of the arrival rate Q(t), t ∈ T. Typically, it is desirable for a PT company to implement an attractive service for the users and at the same time to minimize the required resources to attain this advantageous service. With this in mind, we can pose a question on what 'an attractive service' from users' perspective is. Based on the literature review we can focus on three measures that affect users' satisfaction from a PT service: average waiting time (service headway or frequency dependent), seat availability (for riding comfort), and average passengers' load ratio (operator's control variable relating to the fleet size). The function of the combined measure is , where F obj is the sum of three weighted measures; that is, f ps is the measure of passengers' average waiting time, f pc is the measure related to seat availability, and f le is the measure of average load ratio. The weights of Eq. (2) w 1 , w 2 and w 3 are specifying the importance of each of the three measures, respectively. Subject to the lower bound of fleet size, the number of PT service trips m should satisfy Eq. (3). where m min is the minimal number of required trips based on the stipulated regulation and m max is the maximal number of trips specified by the PT company operator and it is fleet-size dependent. The first measure is the percentage of passengers who are satisfied from service's headway (directly) or who found the waiting time acceptable (indirectly). This is strictly from the perspective of the PT users. The following is the formulation of Measure 1: , where w cs and w td are, respectively, the number of satisfied passengers and overall passengers, which result from Eqs. (5) and (6): . Practically, it is desirable for PT vehicles to accommodate passengers' traveling experience, such as available seats, personal spaces, and acceptable congestion, even with the close-to-favoriteheadways. Thus, we further specify the passengers' load situation f pci of vehicle i, in Eq. (7), wherein, we need to consider Z i as the average number of passengers onboard at the route's maxload stop/point. This is related to seating availability. Yielding to the rigorous constraint of prohibiting over-loading in Assumption V, Eq. (8) specifies the number of waiting passengers allowed to board for time interval [t i , t i+1 ], . Finally, Eq. (9) indicates Measure 2 as the average measure of seating availability where for f pci equals or greater than 1.0 there are available seats; thus, Eq. (9) can serve as a measure of passengers' travelling comfort. The formulation is as follows: . In the study, the operation horizon T during the planning level is written as [a, b] can be represented by the use of Eq. (8). In addition, Eq. (7) copes with the situations of having a seat (Z i ≤ C g ) and some standing (Z i ≥ C g ). Realistically, it is desirable for a PT company to achieve the balanced load of overall vehicle services subject to the fleet size. Eq. (10) represents Measure 3 as the average load ratio between passengers' load at the max-load stop Z i and vehicle's capacity, where this measure is related to the fleet size, thus a measure of operator's perspective. (10) We model the PT timetabling problem in consideration of timespace demand distribution. Thereby, we expand on fourfold models to design demand-dependent timetables: 1. Model A: even-headway timetable for the entire daily time horizon (same headways). The use of an even-headway timetable has the advantage of convenience in facilitating vehicle scheduling and crew scheduling operations. In addition, its performance criteria can be necessary measures for evaluating other timetables. 2. Model B: even-load timetable at the passengers' max-load stop for the entire daily time horizon (same max load), which exhibits the changes of departure times based on the fluctuations of the demand. 3. Model C: even-headway timetable, but with same headway only for each given time window usually associated with peak and off-peak hours. 4. Model D: even-load timetable at the max-load stop determined at each given time window usually associated with peak and off-peak hours; thus, the even load is different between time windows. Particularly, Models C and D need to be primarily assigned to the reasonable number of vehicles on the basis of different levels of passenger demand. For the sake of simplification, the four aforementioned timetable schemes are represented by different colors, meanwhile corresponding to dynamic demand peaks in time and space, shown in Fig. 3 . Most importantly, the key issue of timetable design is how to deduce the series of departure times t i , throughout the regulation of different timetables. Headway throughout the Time Horizon Individually, Figs. 4 − 7 demonstrate the four aforementioned timetable scenarios so as to further explain how the principle of determining departure intervals works. Briefly, the planned timespace trajectory departures of i-th vehicle ( ) from origin stop j = 1 are investigated as the underlying four examples. The black line is the time-space trajectory while the red line is the headway interval. Figure 4 depicts the operation of an even-headway timetable throughout the whole daily time horizon, whose planned departure/ arrival time performs identical time intervals, namely the even headway. Observation 1: If, for a set of departures t i A in timetable A and a fixed set of required trips m, all trips start and end within the schedule horizon [t 0 , t D ] and no deadheading trip insertions are allowed, then a series of departure times in T is equal to the sum of t 0 , plus the corresponding sequence of headway . The beginning of the study horizon T is t 0 = a. Considering the identical departure intervals when devising the even-headway timetable, we manage the departure times of subsequent vehicles t i , , in Eq. (11): , . ( 1 1 ) As illustrated in Fig. 5 , the headways for the overall trips are usually diverse (depicted by red lines), but their load factor is nearly identical according to historical demand data. Typically, the number of trips accounts for the overwhelming proportion of the peak-hours demand (marked by the red zone in Fig. 5 ). On the contrary, fewer vehicles are assigned to feed the demand due to few passengers in the green zone in Fig. 5 . It should be noted that the observed average even-load of the overall demand is defined as C 0 , traversing the number of PT services m during the horizon T in Eq. (12): . With regard to vehicle i ∈ I, the number of accumulated passengers during departure interval [t i , t i + 1], can be rewritten as Furthermore, knowing t 0 = a, if a ≤ t ≤ t i+1 , for each fixed value t i , the definite integral is supposed to form a corresponding value, hereby the indicated fixed load C 0 . Thus, we build an auxiliary function Φ(x), a ≤ x ≤ t i . Then, Eq. (14) can be replaced by . (14) Next, we have Φ(x)' as the derivative of Φ(x) and say that , where . (17) Next, we propose two models featuring the merits of evenheadway and even-load timetables. Replicating the classical bimodal pattern demand distribution, in our study the daily operation horizon is classified into five periods: where the five specified demand-periods can be rewritten as , i ∈ I, p n ∈ P, n ∈ N. Therefore, considering Eqs. (18) − (19), the measure of Model C can be reiterated in Eq. (20): Generally speaking, the demand during the peak hours is able to reach the upper boundary of capacity of the PT system, C d . This enables identifying the number of services yielding to the fleet size constraint at a pre-assigned level. Under the preprocessing, optimized timetables in cases with and without ample PT resources, results are analytically obtained, which permit the further study of the influence of the fleet size (or the frequency M) involved in Model D. As stated in Fig. 7 , Model (D) captures the sound, a priori number of trips in period d ∈ D, where the departure intervals are re-determined according to the time slices [t i , t i+1 ]. Considering the even-load features of multiple demand-periods, in Eq. (21) C d indicates the average demand accumulated in the period d, d = n + 1, d ∈ D. , n ∈ N. Thus, via adding constraints C d and t i D , the measures can be formulated, i.e., , i ∈ I. An enumeration algorithm solution is developed to solve Models A, B, C, and D. The brief procedure of the algorithm, along with its input and output are systematically shown in the flowchart in Fig. 8 . The algorithm solution is coded in Matlab R2018a. All results for the four models are acquired in the range from 5s to 38s by implementation on two personal computers with Intel ®core TM i7-9750H CPUs 2.60GHZ and 16GRAM. Below is the general outline of Fig. 8 . Step 1: Initialization: specifying the input parameters of Models A, B, C, D. Step 2: Demand file: determining the minimum and maximum number of departures. Step 3: Attributes of timetables: directly obtaining the departure times of Models A via Eq. (11) and C in the light of Eq. (19), or alternatively capturing the average load of overall trips in Models B and D, which refers to Eqs. (12) − (14), (21). Furthermore, by use of the differentiating method, deducing the departure times of Models B and D by the means of Eqs. (15) and (22), respectively. Step 4: Estimation criteria: deriving perceived satisfaction degree of the j-th proportion of users on the k-th headway based on criteria index ξ kf as per Section 3.1.4. Step 5: Measures: Acquiring the values of the measures as per Eqs. (6), (9) − (10), respectively. Step 6: Iteration: While (T > T D and i < m max ) do // Search ergodicity of updating the overall reasonable decision variables. Realistically, the effectiveness of the models and the proposed algorithm solution are examined by creating multiple timetables for the No. 7 PT in Nantong, China. The No. 7 PT in Nantong which includes 31 stops is specified for the case study. The first and last departure times of PT are 6:00 am and 9:00 pm, respectively. Moreover, the morning and evening peak hours are defined as 7:00 am -9:00 am and 5:00 pm -7:00 pm, respectively. The detailed calculation parameters are presented in Table 3 . Data acquisition is based on a typical workday, based on the collection of passenger arrivals over time at each stop along the route. Generally, we primarily acquire the data of the passenger arrival rate at each stop within the multiple time slices, and finally fitting the passenger arrival rate Q(t) over times along the PT line, shown in Fig. 9 . The original data is presented in Table 8 . Each vehicle has 50-seat and capacity of 100 passengers. Based on the data collected we selected initially between 95 to 180 trips for the case study. This section is to verify the calculation of the specific comprehensive criteria using the combination of different modes of operation. Overall models yield solutions within the fleet size constraint and satisfaction degree criteria. From the standpoints of bestmeasure comparison, Model A archives the maximum result, the trend of which holds increasingly with decision variable m. Furthermore, users' satisfaction on equal and short departures systematically account for the best result of Model A. In the meanwhile, the performance of Model A exhibits better than the overall situation of Model C. Subject to the fleet size constraint during one certain period, users' satisfaction decreases especially in off-peak hours because of the longer waiting times. By comparing the measures of Models B and D, we can draw conclusions with regard to the conditions under which the load ratio in balance can produce better results than those of Model A with only a relative lack of resources (fleet size of PT), and vice versa. Certainly, it is possible for more waiting passengers to board the vehicles in even-load timetables (Models B and D) than by use of an even-headway timetable (Model A) which make more sense. Furthermore, in Fig. 10 we also can conclude that the trends of Models B and D perform similar fluctuations, as marked by the green ellipses in Fig. 10(a) including the option that both of them obtain simultaneously when m = 144. Based on the summary of Fig. 10(a) , the option of the four timetable candidates with reference to different demand features is decided mathematically, subject to the fleet size constraint to a major degree. Theoretically, the four models were to attain similar measures, depicted by the blue ellipse of Fig. 10(a) . Additionally, the basic data of the measures including the data of Fig. 9 are provided in the tables of Tables 5 − 7. Satisfaction Perceived on Waiting (headway/ frequency dependent) In terms of passengers' perceived experience about the frequency of service that impact their waiting times at stops, Fig. 10(b) demonstrates a more attractive service mode with even-headway departures. In practical terms, most users are accustomed to an even-headway timetable, accepting it more easily. Hence, Model A yielded the best results, as expected. In the set of decision variable m, we observe an interesting finding for f ps results of Model A fluctuating to a rather tiny extent although they seem to form a straight line. In fact, it ranged with 4.89 × 10 -5 % in each m, but was not linear, which is proved by data in Table 6 . Compared with Model A, Model C performs at a lesser degree of satisfaction during some demand periods, expect for m ∈ [117, 137] where the values of f ps are still more than 0.78. The reason is that the limited number of vehicles are assigned in advance, such that the longer headway meanwhile leads more average waiting times, which evidenced a reduction of satisfaction in the multiperiods separately. The variation of Models B and D exhibits a similar tendency. This is surveyed graphically in Fig. 10(b) , which indicates that even-load timetables can acquire a higher satisfaction degree when the number of trips is lower. Indeed, it has an advantage for reducing the likelihood of left-behinds due to overload. However, unbalanced departure intervals may create some inconveniences that interfere with frequent travelers' habits once service resources are adequate (m ≥ 117). Moreover, Models B and D produce the optimal results when m is equal to117, which is much closer to those of Models A and C. As summarized in Fig. 10(b) , real life scenarios are employed to validate that users are inclined to choose the regular PT service styles within their natural desire. With regard to evaluation of if and how in-vehicle congestion occurs, Fig. 10 (c) demonstrates that the two closely related evenload timetables represented by Models B and D perform greater service satisfaction than others in a general horizon. This suggests that if allowing for the conformable accommodation to seats and ample standing space for all trips, even-load timetables enable reducing numerous complaints of being left behind, and thus, drastically increase the degree of satisfaction. As long as the service is sufficient (m ≥ 149 in Model B and m ≥ 158 in Model D), this high satisfaction degree coincides with less number of left-behind passengers owing to vehicle overload. Because of even-headway departures, the lack of close attention to demand resulted in the inferior performances of Models A and C. Besides, the satisfaction degree for Models A and C indeed increases gradually with the shorter headway that means higher operation efficiency/capacity. From the Fig. 10(c) summary, it is reasonable to draw an immediate conclusion that applying the even-load timetables did improve the satisfaction quality, comfort-based, of overall in-vehicle travel experience on the average by 11.21% (Models B to A) and by 50.37% (Models D to C), respectively. As summarized in Table 4 , eleven scenarios of different weights, w i , i = 1,2,3, were tested to evaluate and compare the proposed model based on different operational preferences; we set two weights as 1, and set the other weight as 0.1, 0.5, 1, 10, 50, and 100, respectively. The sensitivity analysis results of all these weights for the four models are listed in Tables 2 − 5 of Supplementary Material, where the measure of m, based on different weights, is executed with the corresponding measures. The term scheme refers to a timetable with specific setting of departure times. Table 4 summarizes the main results of the performance indicators, an analysis of which follows: 1. Multiple performance criteria: Overall measure (F obj ), the satisfaction degree of waiting frequency-related (f ps ) and measure related to seat availability (f pc ) as well as average load ration of vehicles (f le ) were computed to evaluate feasibilities for the four proposed models. 2. Trip order: The algorithm solution developed was carried out to find the best trip sequence. Available fleet size yielded the best performance, as expected. As data stated in Tables 2 − 5 of Supplementary Material, we distinguish the ranking of the four models developed throughout different weights/items in Table 4 . It may also be observed from Table 4 that the classical even-headway timetables and even-load timetables contribute to a higher percentage of better schemes. Except the weights w 1 : w 2 : w 3 =1:1:0.5, the greatest value is achieved in Model D in which vehicles experience a reasonable load ratio. For sixteen items, the result of number of trips m with reference to different weights are acquired, and shown graphically in Fig. 11 . Generally speaking, good schemes achieved 8 times by Model A and 7 times by Model B. It is evidence of the obvious potential of the even-headway timetabling and even-load timetabling models for deployment in real life. The longer waiting times and heavier fluctuant load ratio may account for the low-ranking performances of Model C. Particularly, once adding the value of weight w 3 , the gap between Model A and Model C quickly becomes wider (as may be observed in items 14, 15, and 16 in Table 4 ), the percentage of which is more than 99%. Beyond that, the average improvement percentage by comparing Model A to Model C is 24.61%. To provide users with a well-satisfied, demand-oriented alternative to public transport (PT) agencies, our work develops four models to provide the adaptive candidates to determine a feasible timetable scheme during the operation level. These timetabling schemes have been quantitatively compared and widely discussed in a mathematical manner. Up to now, to the best of the authors' knowledge, there has been no specific investigation of the performance criteria of applying various timetables particularly taking passengers' load ratio (with reference to vehicle's capacity) and users' perceptions (of service frequency and seat availability) into consideration. Naturally, the best-selected plan a priori represents major advantages for PT operators and passengers. We propose four models to contribute to the evaluation comparison subject to the respective individual constraints: (A) even-headway departures with same headway throughout the time horizon, (B) even-load departures with same load throughout the time horizon, (C) evenheadway departures for each demand-dependent period, and (D) even-load departures for each demand-dependent period. Furthermore, these models are compared in ternary-dimensions: 1) passengers' service-frequency satisfaction, in terms of waiting times, 2) passengers' perception of riding comfort, in terms of seat availability, and 3) planned passengers' load ratio linked with operation efficiency. To procure the reasonable comparison and to propose the performance standard, a users' perceived satisfaction degree criteria was built to investigate their evaluation regarding various departure intervals. To further analyze the difference between them and the ensuing results, data from the case study in Nantong, China provided an example. In the meanwhile, the enumeration results proved the effectiveness of the models and the algorithm solution developed. A large majority of passengers serviced by the Nantong PT system in reality attributed strong importance to service satisfaction situations in terms of waiting time. This was evident from the highest percentage of best schemes where each scheme represents the result of a timetable (departure times) with reference to one of the four models. Moreover, the typical even-headway timetable has better service potential and market prospects for PT makers. We set the available number of trips as the decision variables to conduct a quantitative analysis. On the basis of the graphical tendencies of the four models, we find different measures and make a general comparison. Finally, the sensitivity analysis and discussion were undertaken to evaluate the diverse demandpreferences throughout a wide comparison. The consequences addressed in the study will undoubtedly support PT operators in choosing and/or devising a more reasonable, attractive PT timetable. Thus, operators are advised to make a feasible decision based on the available service sources. The limitation of the present work is that criteria for the perceived satisfaction degree is not verified extensively using other cities. Indeed, the questionnaire with reference to about 100 volunteers was collected online due to the circumstances of Covid-19 in China. Implementing the classical method of comparative analysis facilitated enlightening a more flexible timetable accepted by and highly attractive to the public. The inferences made in the work are only based on PT data from Nantong, China. Currently, a majority of routes of Nantong PT system are employing the scheme of Model C that included in the comparison and shown to offer nearly the worst performance. In fact, Nantong PT system has been criticized for not being very satisfying. Therefore, our study lends significance to helpful prospects in relation to improving the current service level by devising a better timetable for the planning horizon. Future research may focus on the different aspects of users' perceptions of public transport timetables. That is, to investigate a strategy-based timetable optimization approach considering multiobjective formulation also reflecting users' satisfaction besides minimizing the cost of the users and operator. Methods for creating bus timetables Public transit planning and operation: Modeling, practice and behavior, second edition Modelling public transport users' behaviour at connection point Approaching even-load and evenheadway transit timetables using different bus sizes Passenger waiting time at transit stops Transit timetables resulting in even max-load on individual vehicles Integrating short turning and deadheading in the optimization of transit services An integrated approach for timetabling and vehicle scheduling problems to analyze the tradeoff between level of service and operating costs of transit networks Synchronization of bus timetabling Enhancing metro network resilience via localized integration with bus services Optimizing bus bridging services in response to disruptions of urban transit rail networks A practical model for last train rescheduling with train delay in urban railway transit networks Last train timetabling optimization and bus bridging service management in urban railway transit networks Battery-electric transit vehicle scheduling with optimal number of stationary chargers Bus stop-skipping scheme with random travel time Comparison of dynamic control strategies for transit operations A robust, tactic-based, real-time framework for public-transport transfer synchronization Public transport user's perception and decision assessment using tactic-based guidelines Optimizing urban rail timetable under timedependent demand and oversaturated conditions A practical model for transfer optimization in a transit network: Model formulations and solutions Demand-driven timetable design for metro services A bi-objective timetable optimization model for urban rail transit based on the time-dependent passenger volume Integrated optimization of regular train schedule and train circulation plan for urban rail transit lines Boundedrationality based day-to-day evolution model for travel behavior analysis of urban railway network Optimizing passenger flow control and bus-bridging service for commuting metro lines This study was supported by National Natural Science Foundation of China (72101127) and (72101126) 100 110 110 111 121 97 114 100 109 111 110 120 96 113 99 108 112 109 118 95 112 98 107 113 108 117 94 111 97 106 114 107 116 93 110 96 105 115 106 115 93 109 95 104 116 105 114 92 108 94 103 117 104 113 91 107 94 102 118 103 112 90 106 93 101 119 103 111 90 105 92 100 120 102 111 89 104 91 100 121 101 110 88 94 76 89 78 85 142 86 93 75 88 77 84 143 85 93 75 88 77 84 144 85 92 74 87 76 83 145 84 91 73 86 76 82 146 84 91 73 86 75 82 147 83 90 72 85 75 81 148 82 90 72 85 74 81 149 82 89 72 84 74 80 150 81 88 71 84 73 80 151 81 88 71 83 73 79 152 80 87 70 82 72 79 153 80 87 70 82 72 78 154 79 86 69 81 71 78 155 79 86 69 81 71 77 156 78 85 68 80 70 77 157 78 84 68 63 74 65 71 170 72 78 63 74 64 70 171 71 78 62 73 64 70 172 71 77 62 73 64 69 173 71 77 62 72 63 69 174 70 76 61 72 63 69 175 70 76 61 72 63 68 176 69 75 61 71 62 68