Abstract
A proper public transportation system should be economically efficient to improve its sustainability and competitiveness. The superfluous provision of conventional bus services in areas with a weak or scattered travel demand distribution might result in a waste of transport capacity. This study proposes an operation plan optimization model based on the characteristics and cost composition of conventional and customized bus services, which systematically analyzes the feasibility of fully or partially replacing conventional bus services with customized bus services. A mixed-integer planning model and a hybrid genetic algorithm are used to optimize the operation plan for specific travel demands and supply capacities. Then, a sensitivity analysis of key factors is carried out to identify the optimal operation mode under different circumstances. The results show that with an optimized fleet size, service frequency, and service area, integrated operation modes can reduce operating costs without sacrificing the passenger time.
1. Introduction
In recent years, the urbanization process of major economies worldwide has continued to develop rapidly, as has the scale of cities. Furthermore, the urban traffic structure has also undergone significant changes; motorized travel has escalated rapidly, and urban traffic congestion has become increasingly serious. These factors have put tremendous pressure on the urban transportation system. Public transportation (PT) has been proven as an effective way to alleviate urban traffic pressure and respond to the development of urbanization. Therefore, many countries have put forward the clear strategy of giving priority to the development of public transport as an important path to the alleviation of traffic congestion. For example, China has proposed a binding national development goal of 100% coverage of public transport stations in built-up areas in cities with a population of more than 1 million [1].
A proper PT system should not only increase the attractiveness of PT and promote the coordinated development of urban and rural regional transportation but should also be economically efficient to improve its competitiveness. Due to the basic public service attributes of PT, the PT management department usually opens new bus lines or extends the original bus lines in remote areas of the city, even in sparsely inhabited areas, to increase PT coverage. However, the provision of fixed PT services in areas with low travel demand and scattered distribution would inevitably lead to a waste of supply capacity to ensure that the fixed departure interval is within a proper range. Thus, customized public transport is one of the potential ways to solve this issue.
Customized bus services are one form of demand-responsive transport (DRT), which also exists as flexible transport services (FTS), dial-a-ride services, or paratransit services; these services represent flexible bus services for which the travel needs of passengers are investigated in advance, and the bus route is then optimized according to a set of time windows and capacity restrictions [2]. Some studies in the field of customized PT have explored the cost difference between conventional and variable PT methods. These studies agree that, in some cases, the replacement of conventional PT with variable PT can achieve the goals of not affecting the use and reducing costs [3, 4]. However, few studies have systematically analyzed the feasibility of the partial replacement of conventional bus routes with customized bus services in different operating environments.
This study aims to fill this gap via the use of a mixed-integer planning model with an improved genetic algorithm, and the following is achieved. (1) A bus system operating cost model based on the operating characteristics and cost element composition of conventional and customized bus services is established. (2) An operation plan selection and optimization model, as well as a solution algorithm, is proposed for the identification of an appropriate operation plan to use customized buses to fully or partially replace conventional bus lines for specific travel demands and supply capacities. (3) A sensitivity analysis is carried out, the best operation plan for corresponding travel demand is determined, and the potentially applicable environment in which a particular operation mode has advantages is identified.
The remainder of this paper is organized as follows. A literature review is presented in Section 2, and the proposed model is presented in Section 3. Then, the solution algorithm of the problem is detailed in Section 4. The details of numerical experiments and the corresponding results are presented and discussed in Section 5. Finally, Section 6 concludes this article.
2. Literature Review
With the continuous deepening of PT research, innovative flexible bus transit services, such as customized buses, community buses, circular routes, and feeder services, have emerged. Kirby and Bhatt [5] analyzed and identified the characteristics of subscription bus services, an original type of customized bus service, such as the increased focus on long-distance trips, high service reliability, and satisfactory personalized features. Daganzo [6] conducted a preliminary study on dial-a-ride services, and the results showed that as the demand level decreases, the demand-responsive system becomes relatively more attractive than the fixed-route system, and the checkpoint system may become more cost-effective. Khattak and Yim [7] explored the demand for a personalized DRT service via a survey conducted in the San Francisco Bay Area, which revealed the potentially high demand of people willing to pay for the service. Davison et al. [8] explored factors that affect the competitiveness of the demand-responsive transit systems in Great Britain as well as their underlying mechanisms. Wang et al. [9] conducted a comprehensive analysis of a successful customized bus system based on practical subscription data. Later, Wang et al. [10, 11] analyzed the mechanism of customized bus user loyalty modeling as well as the spatial dependence and spillover effects in customized bus demand; the results demonstrated that areas with poor accessibility have a larger market for customized bus service.
Some studies have compared flexible bus services with regular bus services with fixed routes and have explored the feasibility of replacing the fixed-route policy with the flexible-route policy of a branch line system without disturbing the existing coordination between the main transit and branch line services [2, 12, 13]. With the goal of minimizing the average cost (operating and passenger costs), Chien et al. [14] concluded that operators can reduce daily operating costs by providing conventional services during periods of high demand and operating customized services during off-peak periods. Li and Quadrifoglio [15] explored the operation model of switching between a demand-responsive and a fixed-route operating policy and concluded that the demand-responsive policy is more preferred when the demand falls between 10 and 50 passengers/mile2/h. Velaga et al. [16] explored public transport provisions in rural and remote areas illustrated with experience from Scotland and demonstrated that flexible transportation services provide considerable potential for PT systems in rural areas. Nourbakhsh and Ouyang [17] proposed an alternative flexible-route bus system within a predetermined area by seeking the best network layout, the service area of each bus, and the headway of the bus to minimize the total cost of the system; they found that the proposed flexible-path system tends to have the lowest system cost under low- to medium-demand levels. Qiu et al. [18] proposed a model with which to choose between fixed-route and flexible-route policies for transit services based on the assessment of the service quality function under varying demand. Ronald et al. [19] studied the performance changes of a fixed-time flexible-route scheme and a fully self-organizing scheme using MATSIM and demonstrated that the two programs have different effects on operators and passengers.
Although high-quality research has been conducted in recent decades to optimize the timetabling and vehicle scheduling of conventional and customized bus services, it is difficult to formulate a cohesive solution for the planning problem of the entire bus network via sequential approaches. Kim and Schonfeld [20, 21] employed optimization models to analyze conventional-only, flexible-only, and variable-type (can switch between conventional and flexible services as demand changes over time) services by jointly considering the bus size and service type; the results confirmed that variable-type operation can reduce the system cost by changing the operation type as the demand density changes. Later, Kim and Schonfeld proposed probabilistic optimization models to integrate and coordinate original and flexible bus services to connect one terminal and multiple local regions [22]. Liu et al. [23] compared the constant-frequency conventional bus dispatching mode with the dynamic-frequency demand-responsive bus dispatching mode; they then proposed a hybrid operating mode that combines fixed- and dynamic-frequency services to minimize the average waiting time of passengers and maximize the operator’s profit. In summary, most previous studies have separately focused on the optimization of the schedule or fleets of only conventional bus services, customized bus services, or variable-type bus services; research on the joint optimization of conventional buses, customized feeder buses, and customized point-to-point buses is comparatively lacking. Therefore, the methods for the optimization of integrated customized and conventional bus services require further exploration.
3. Method
Assume that a bus line provides travel services between the city center and the suburbs. This line that has been built or is about to be constructed has bus stations with known departure intervals and operating service times. Moreover, the travel demands are known and stable.
Table 1 lists a portion of the sets, indices, and parameters used in this study.
3.1. Operation Mode
3.1.1. Conventional Operation Mode
The conventional operation mode (Figure 1) is the most widely used operation mode of conventional bus systems, in which buses always run on urban roads according to fixed routes and fixed schedules. Passengers walk to the appropriate station from various origins, take the bus, get off at the appropriate station, and then walk to their destinations.
3.1.2. Integrated Operation Mode
In the integrated operation mode, PT services are jointly completed by two types of buses, namely, conventional and customized buses. Specifically, in low-demand districts, customized bus services are used to replace portions of the original conventional bus services, and the remaining stations are still served by conventional buses. In this study, the integrated operation mode is divided into two types, namely, (1) integrated operation mode I: customized buses (feeder transit) and conventional buses and (2) integrated operation mode II: point-to-point (p2p) customized buses and conventional buses.
(1) Integrated Operation Mode I: Customized Buses (Feeder Transit) and Conventional Buses. In this mode (Figure 2), a customized bus provides a connection service in its service area, picks up passengers according to the needs in the area, and transfers them to the transfer station; a conventional bus completes the rest of the transportation service. Specifically, a customized bus starts from the outer terminal station of the existing conventional bus route, picks up passengers according to their specific location within the stations, and transfers them back to the transfer station to complete connection services. Then, the entire service is completed by the conventional bus covering the remaining stations.
(2) Integrated Operation Mode II: Customized Buses (p2p) and Conventional Buses. In this mode (Figure 3), customized buses transfer some passengers from their origin to their destination, which is a type of “direct” public transport service. Passengers are no longer required to go to the departure bus station or get off at the arrival bus station. In other words, the p2p customized bus departs from the depot, picks up passengers according to their specific location within the stations, transfers them to their destination, and finally returns to the depot. Other passengers continue to be served by a conventional bus.
3.2. Cost Element
The total operating cost of a bus service includes the operating costs of both the conventional and customized buses, i.e.,where is the conventional bus operating cost, is the customized bus operating cost, and is the total operating cost. The conventional bus operating cost includes the fixed input cost and variable operating cost , and the same is true for the customized bus operating cost.
3.2.1. Fixed Input Cost
The fixed input cost is the sum of the fixed cost of each vehicle in the fleet. The fixed cost is generally related to the size of the fleet, the operating time, the drivers’ salaries, vehicle purchase fees, vehicle depreciation, insurance fees, etc. Formally, the fixed input costs of conventional buses and customized buses can, respectively, be expressed aswhere is the fixed input cost per conventional bus hour, is the size of the conventional bus fleet, is the fixed input cost per customized bus hour, is the size of the customized bus fleet, and is the length of the bus service time. can be calculated based on the departure interval by equation (4), where is the average driving speed of the bus during the bus service time, and is the total mileage of one cycle. Moreover, is related to the number of customized bus departures and each departure time.
3.2.2. Variable Operating Cost
The variable operating cost is generally related to the fuel cost, which is associated with the total mileage of the fleet. The variable operating costs of conventional buses and customized buses can, respectively, be expressed aswhere is the fuel cost of conventional buses per mile, is the total mileage of conventional bus , is the set of conventional buses, is the fuel cost of customized buses per mile, is the set of customized buses, is the mileage of customized bus driving from point to point , and is the set of route sections of customized bus . Thus, is the total mileage of customized bus .
3.3. Assumption
As it would be difficult to determine the specific departure and destination locations of passengers during the operation of the conventional mode, certain assumptions are made regarding the characteristics of passenger travel demand based on actual conditions:(1)The operation mode has no effect on bus travel demand, which means that conventional bus travel demand would be fully transferred by customized bus services in areas where the conventional operation mode has been changed.(2)The location and travel time of passengers are randomly generated based on an assumed distribution.(3)Customized bus provides pickup and drop-off service; therefore, the passenger’s walking time is ignored.
3.4. Objective Formulation
The research goal is to determine the optimal bus operation plan and the related operating parameters after clarifying the corresponding travel needs. Therefore, 0-1 integer programming was used to construct the objective function of the operation mode selection and optimization model. The objective function is divided into three parts, and its expression iswhere represents the bus operation mode. When , is the total operating cost of the conventional operation mode; when , is the total operating cost of integrated operation mode I; and when , is the total operating cost of integrated operation mode II. Moreover, is the operating cost of the optimal bus operation plan, and is a binary variable.
As the conventional operation mode does not include customized buses, the operating cost of customized buses is considered as 0, and the expression of iswhere is the size of the conventional bus fleet in the conventional operation mode and is the set of conventional buses in the conventional operation mode.
The operating costs of the integrated operation modes, namely, and , include two parts, namely, the operating cost of conventional bus operation and the operating cost of customized bus operation. The corresponding expressions are as follows:
3.5. Constraints
Because the goal of the model is to reduce the operating costs while ensuring that the needs of passengers are met, the main constraints of this model are given as follows:
(8) is the mode selection constraint; there is only one variable, which corresponds to the operation mode of the optimal bus operation plan. (9) is the vehicle load constraint, which stipulates that the number of served passengers in any route section of customized bus cannot exceed its capacity. (10) is the average passenger time constraint, which guarantees that the average passenger time of plan in the integrated operation mode does not exceed the average passenger time in the conventional operation mode. (11) restricts the number of replacement stations to not exceed the total number of original conventional bus stations . (12) and (13) indicate that for each passenger who uses a customized bus, there is only one customized bus to provide service, where is the set of origins of passengers using a customized bus. Moreover, represents that customized bus travels from point to point ; otherwise, . Finally, (14) indicates that the customized bus that arrives at point and departs from point must be the same bus.
3.5.1. Total Passenger Time
(1) Total Passenger Time. in the Conventional Operation Mode. The total passenger time on a fixed route is , which includes the passenger walking time , the passenger waiting time , and the passenger travel time .where is the passenger walking time for the conventional bus mode, which is equal to the total passenger walking distance divided by the walking speed, i.e., , and is the summation of the walking distance of all passengers from the origin to the departure station and from the arrival station to the destination, namely, .
is the passenger waiting time for conventional buses. Assuming that the time for passengers to arrive at the departure stations is uniformly distributed, can be calculated from the departure interval , namely, , where is the number of passengers waiting for conventional buses.
is the passenger travel time on conventional buses. The total travel time of passengers is the sum of the in-vehicle time of all passengers, which is equal to the multiplication of the number of passengers in each section by the section travel time. Hence, the calculation of is converted into the calculation of the sum of the travel time of passengers in each interval between adjacent stations, where represents the set of intervals between adjacent stations on the fixed route, is the number of people within conventional bus in interval , and represents the travel time of conventional bus through interval .
In summary, the expression of the total passenger time in the conventional operation mode is as follows:
(2) Total Passenger Time. in the Integrated Operation Mode. The total passenger time in the integrated operation mode includes the total passenger time on a fixed route and the total passenger time on a customized route .
is related to the type of customized bus in the integrated operation mode. The total passenger time for integrated operation mode II includes the passenger waiting time and passenger travel time . In addition, the passenger transfer time must also be calculated for integrated operation mode I because passengers may need to wait for a conventional or customized bus at a transfer station at this time.
is the summation of the customized bus arrival times exceeding the origin time window, namely, . When a customized bus arrives at a passenger’s origin within a reasonable time window, the passenger does not generate a waiting time. Moreover, a passenger waiting time will only occur when a customized bus arrives at an origin outside its time window.where is the waiting time at point , is the arrival time of customized bus , the reasonable time window is , and the length of the time window is , namely, .
Moreover, is related to the routes of customized buses. Assuming that customized buses will depart times, represents the number of passengers on the bus when customized bus passes through , and represents the travel time of customized bus passes through . Then, the expression of is as follows:
is the sum of the waiting times when passengers wait for a customized/conventional bus at a transfer station. Therefore, it is composed of the total waiting time of passengers transferring from a customized bus to a conventional bus and the total waiting time of passengers transferring from a conventional bus to a customized bus , which is denoted as .
As discussed previously, if the operation mode of integrated operation mode is and a customized bus replaces the service for stations, the total passenger time of mode in integrated operation mode can be expressed as follows:
4. Solution Algorithm
The optimization of the proposed model is fundamentally an NP-hard problem, for which the use of intelligent algorithms is currently an efficient method.
In this research, a hybrid genetic algorithm with a local search is used to solve the proposed model. In addition to the known constants, the fleet size and mileage of conventional and customized buses are variables that directly affect the results of the model and are the key variables to be optimized under the premise of satisfying the model constraints. The algorithm is designed as follows.
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5. Case Study
5.1. Numerical Information
As shown in Figure 4, in this experiment, the total length of the original bus line was 8.4 km, the line passed 15 stations, the distance between adjacent stations was 600 m, and the departure interval of conventional buses was 15 min. Furthermore, to be consistent with actual circumstances, the passengers’ origin and destination, departure and arrival stations, and ideal travel time met the following experimental assumptions:(1)The number of boarding passengers decreases linearly from the urban district to the low-demand district, and the demand of the farthest station is 20% of the first station in the urban district(2)The average travel distance of passengers is 3 km(3)The passengers’ origin and destination locations are randomly located within 500 m of their departure and arrival stations(4)The ideal travel time of a passenger is randomly generated within the operating service time
5.2. Results Analysis
The experiment was carried out in the MATLAB 2018b environment. The total travel demands were set to 200 passengers per hour, and the values of the known constants are reported in Table 2. An experiment dealt with 200 demands, as the service time was 1 hour.
Because the passenger demand was randomly generated, the experiment was repeated five times, and the average value was taken as the final result. The operation plans under each operation mode are exhibited in Table 3. The optimal operation plan was integrated operation mode I (feeder transit and conventional buses) that 4 of 15 stations were replaced and the passengers were transfered with 4 customized buses. Compared with the conventional operation mode, the operating cost of this plan was found to be reduced by 124.08 yuan per hour.
Figure 5 exhibits the change in the operating cost per capita with the increase in the number of replaced stations. The following can be observed: (1) the operating costs per capita of the two integrated operation modes were both found to decrease and then increase with the increase of the number of replaced stations and (2) the numerical range of the reduction of the operating cost per capita of integrated operation mode I was found to be more affected than that of integrated operation mode II.
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Figure 6 presents the comparison of the operating costs between the conventional and customized buses in the two integrated operation modes. It was found that the steady increase in operating costs with the increase in the number of replaced stations influenced by the cost of operating customized buses. The cost proportion of conventional bus operating costs in integrated operation mode I was found to increase at a slower rate than that in integrated operation mode II, i.e., at the same number of replaced stations, the customized bus operating cost in integrated operation mode I was lower than that in integrated operation mode II.
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5.3. Sensitivity Analysis
Governments and bus companies are facing diverse operating environments. Therefore, it is necessary to explore the impact of changes in some key parameters on the results.
5.3.1. Travel Demand
Travel demand is a key parameter that affects the change of the optimal bus operation plan. To study the impact of travel demand, the optimal solutions for different values of passenger demand were compared.
Figure 7 exhibits the changes in the operating cost per capita of the two integrated operation modes under changes in both the travel demand and the number of replaced stations. It should be noted that although the degrees of change in the two integrated operation modes were different, the trends were the same. Regarding the travel demand, under low-demand conditions (less than 150 passengers per hour), the reduction in the operating cost was found to increase with the increase of the number of replaced stations, especially for integrated operation mode I; this indicates that integrated operation mode I exhibited an obvious advantage in reducing operating costs in low-demand districts. Regarding the medium-demand conditions (greater than or equal to 150 and less than 300 passengers per hour), integrated operation mode I also exhibited stable economic benefits when few stations were replaced.
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Table 4 summarizes the optimal bus operation plans for different travel demands. With the increase in the number of passengers, the change of the optimal bus operation plan exhibited three obvious characteristics: (1) the mode changed from integrated operation mode I to integrated operation mode II and then to the conventional operation mode, (2) the number of replaced stations was gradually reduced, and (3) the reduction in the cost per capita was gradually reduced. The analysis results support the use of the integrated operation mode to replace the conventional operation mode in some cases and reveal the economic benefits of both integrated operation modes.
Furthermore, Tables 5 and 6 report the optimal plan indicators of the two integrated operation modes under different travel demands. It is evident that the two integrated operation modes exhibited similar trends in terms of the reduction in the cost per capita, but the operating cost of integrated operation mode I tended to remain unchanged after 500 passengers per hour whereas that of integrated operation mode II continued to increase.
5.3.2. Service Coverage Radius of a Station
The maximum walking distance of passengers for a new conventional bus line is limited in most large cities with developed PT [24]. However, this criterion is clearly excessively stringent and does not reflect reality in low-demand areas. Therefore, the influence of the service coverage radius is discussed in this section. The service range radius r was respectively set to 250 m, 500 m, 750 m, and 1 km. Each experiment was carried out three times, and the average value was used as the outcome.
Table 7 presents the optimal operation plans for different radii. It was found that more stations can be replaced as the service coverage radius limit of a station increased. This means that the integrated operation mode has a distinct advantage in areas with a limited number of bus stops. Furthermore, in all scenarios, integrated operation mode I was found to be the best option, thereby demonstrating its versatility in a variety of situations.
5.3.3. Departure Interval of Conventional Bus
The departure interval of a conventional bus is the key factor affecting the operating cost. Figure 8 reveals that the operating cost of the conventional operation mode was found to gradually decrease with the increase of the departure interval, whereas the operating costs of the other two integrated operation modes began to increase after failing to a certain degree due to the overload demand of customized buses.
6. Conclusion
This paper proposed an optimization model for bus operation planning based on cost analysis for low-demand districts. In addition to the conventional operation mode, two integrated operation modes were proposed to simultaneously optimize key operation variables such as the headway, fleet size, and service area under certain demand. The results of experiments revealed that the proposed model and solution algorithm can provide reliable results under different circumstances. Nevertheless, this study faced some limitations as follows: (1) the impacts of passenger loyalty and continuous usage intention were not considered, which could have resulted in differences in the experimental environment as a result of the reduced customized bus demand, and (2) in this study, the Euclidean distance between passengers and stations was used as the passenger walking distance to simplify the calculation. More accurate computation results could be obtained by applying the real walking distance of passengers according to the road network geography. Future studies should also consider the use of real data to validate the proposed model.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Fund of Chinese Central Government for Basic Scientific Research Operations in Commonweal Research Institutes (No. 20224808).