In this paper, the bilevel programming model of the public transport network considering factors such as the per capita occupancy area and travel cost of different groups was established, to alleviate the urban transportation equity and optimize the urban public transport network under fairness constraints. The upper layer minimized the travel cost deprivation coefficient and the road area Gini coefficient as the objective function, to solve the optimization scheme of public transport network considering fairness constraints; the lower layer was a stochastic equilibrium traffic assignment model of multimode and multiuser, used to describe the complex selection behavior of different groups for different traffic modes in the bus optimization scheme given by the upper layer. The model in addition utilised the noninferior sorting genetic algorithm II to validate the model via a simple network. The results showed that (1) the travel cost deprivation coefficient of the three groups declined from 33.42 to 26.51, with a decrease of 20.68%; the Gini coefficient of the road area declined from 0.248 to 0.030, with a decrease of 87.76%; it could be seen that the transportation equity feeling of low-income groups and objective resource allocation improved significantly; (2) before the optimization of public transport network, the sharing rate of cars, buses, and bicycles was 42%, 47%, and 11%, respectively; after the optimization, the sharing rate of each mode was 7%, 82%, and 11%, respectively. Some of the high and middle income users who owned the car were transferred to the public transportation. It could be seen that the overall travel time of the optimized public transport network reduced, enhancing the attraction of the public transport network to various travel groups. The model improves the fairness of the urban public transport system effectively while ensuring the travel demand of the residents. It provides theoretical basis and model foundation for the optimization of public transit network, and it is a new attempt to improve the fairness of the traffic planning scheme.
Equity is a topic of concern to the whole society. The promotion of equity is more important than the increase of wealth to a great extent. Although the issue of equity has aroused great concern in society, the transportation equity has not received enough attention. At present, urban traffic planning theory attaches importance to traffic efficiency and neglects transportation equity, making it more difficult to be guaranteed.
In recent years, certain scholars have begun to explore the scheme of transportation equity. For instance, Litman [
The improvement of transportation equity has also attracted the attention of researchers. Farrington et al. [
It is not difficult to find that the current research focuses on equity assessment, mainly on the equity analysis of travel expenses, road resource allocation, and congestion charging policies, and proposes specific countermeasures from the aspects of public participation, traffic demand analysis, and operational design. The fairness of public transport system means that, through the public transit network optimization, more road traffic resources can be provided for bus travel groups. However, little attention is paid to the lack of fairness in urban transportation planning practice, especially in the case of relatively fixed resource elements including urban spatial layout, land use form, and road network structure, which planning pattern can be used to maximize the fairness and efficiency.
As the leading travel mode in urban transportation system, public transport and the optimization of its network guarantees basic accessibility for low-income groups effectively, which directly affects urban transportation equity. Thus, the public transport system is of great significance to improve transportation equity. Its service level is related to the transportation rights of most residents, and it is also the adjustment lever of transportation equity among different social groups. It thereby improves the overall accessibility of the public transport system and shortens the gap between the bus travel groups and other travel groups. Finally, the bus travel groups and other travel groups have the same opportunities to participate in social activities. Obviously, it is helpful to meet the travel needs of different social groups by improving the fairness of public transport system. In particular, it is of great significance for the improvement of the travel environment of vulnerable groups. However, the traditional public transport network optimization schemes are mostly optimized with the minimum passenger travel time, the highest passenger flow rate, the lowest line overlap factor, and the largest bus economic benefit [ The complex travel behavior of urban residents is analyzed systematically by constructing multimode and multiuser traffic network. The travel cost relative deprivation coefficient and the road area Gini coefficient are proposed to characterize the fairness of the public transport system, based on travel perception and resource allocation. Under the fairness constraint, the bilevel planning model of urban public transport network is constructed. Moreover, the algorithm is designed, and the validity of the model is verified by examples.
The structure of the paper is arranged as follows: (1) construct multimode and multiuser traffic network and analyze the complex travel behavior of urban residents; (2) construct the bilevel planning model of public transport network considering fairness constraints, and give algorithm for solving the model; (3) design an example to verify the actual calculation and validity of the model; (4) give the main conclusions of this paper.
In order to facilitate the modeling and analysis of this paper, three travel modes, car, bus and bicycle, are considered in the traffic network, and the travelers are classified into three travel groups, high, medium, and low, according to income levels. A simple transportation network is constructed as shown in Figure
Sample of transportation network.
After each travel group chooses the traffic modes, the mapping to urban transportation network presents different subnetwork structure, that is, the above networks are divided into high income group travel subnets, medium income group travel subnets, and low-income group travel subnets. Also, according to the traffic mode, different groups of travel subnets are subdivided into car subnets, bus subnets, and bicycle subnets. In this study, the initial public transport network among different groups is all the same. There are structural differences between the car and the bicycle subnet, and these differences do not change with the adjustment of the public transit network. Therefore, it can be considered that the multimode and multiuser traffic network mentioned above is composed of nine layers of subnets. Each subnet consists of a single group and a single travel mode. Moreover,
Assuming that the travel demand of the multimode and multiuser transportation network is known and invariant, the travel demand of different traffic modes should meet the following constraints:
The traveler's choice of transportation is affected by many factors. This paper only considers factors such as transportation cost and travel time. Combining the car ownership ratios of different groups, the Logit model is used to deal with traffic mode choice. After determining the mode, the traveler makes a route selection for the traffic mode subnet. For each type of traveler, the travel demands of the road segment and the route have the following relationship:
Considering the difference in the average passenger capacity of each traffic mode, the travel demand of different traffic modes needs to be converted into a unified road flow. The conversion formula is
In the multimode and multiuser traffic network, assuming that the three types of travel groups select routes according to the Logit mode, the probability that the
The optimization of public transport network considering transportation equity constraint is a special network optimization problem. The optimization of the public transport network leads to the redistribution of traffic resources inevitably, resulting in changes in the traffic mode sharing rate and general travel cost of various travel groups. Moreover, the distribution of bus travel demand of different groups is also affected, which leads to differences in residents' travel costs and per capita road occupancy area. The bilevel optimization model of public transport network considering transportation equity in this paper is an effective improvement on the existing bilevel optimization model of public transport network. In the bilevel optimization model of public transport network, the planners first propose the optimization scheme for the original one, and the lower layer model completes the multimode and multiuser traffic allocation according to the optimization scheme. The two objective functions of the upper model are calculated by the traffic and travel time of each traffic mode to evaluate the adaptability and fairness of the optimization scheme. The model structure is shown in Figure
Model structure.
The assessment of transportation equity is mainly based on the subjective feelings of residents and the allocation of objective resources. The generalized travel cost and the possession of road resources determine the choice of residents' travel modes and routes. Hence, the objective function of the upper model mainly includes the travel cost relative deprivation coefficient and the road area Gini coefficient. The travel cost relative deprivation coefficient is used to express the subjective feelings of residents, and the road area Gini coefficient is used to express the allocation of objective resources.
Travel cost relative deprivation coefficient refers to certain travelers who measure the cost of completing their travel behavior and other groups by comparing the generalized travel costs between different groups to reflect the coefficient of deprivation of a certain travel group.
According to the above discussion, the Logit model was used to describe the random traffic behavior of different traffic modes between the OD pairs based on the principle of the minimum travel cost. The formula is as follows:
In order to quantify the difference of road occupation area per capita among different groups, the Lorentz curve and Gini coefficient were used to measure the fairness of road resource allocation. As shown in Figure
Lorenz curve for transport resource allocation.
The radians of Lorentz curve are mainly determined by three groups of numerical values: the ratio of low-income group
Therefore, it can be determined that the area
Considering that it is impossible for travelers to master the traffic state of the road network completely, the route selection should be a stochastic process. Therefore, the stochastic user equilibrium assignment model was adopted to characterize the route choice behavior for travelers, of which the travel mode of each group was determined first. Further, the traffic assignment results which solved by the optimal target of minimum travel time in terms of the process of users’ route choice behavior were put into the upper model. The optimization objective function of the lower model is minimizing the sum of the travel time of each subnet, and the specific model is as follows:
Due to the conflict of the objective functions, the travel cost relative deprivation coefficient and the road area Gini coefficient in the upper optimization only obtain multiple sets of noninferior solutions or suboptimal solutions, and it is impossible to obtain the optimal solution simultaneously. Hence, the decision maker chooses a set of noninferior solutions based on the degree of preference. The traditional multiobjective optimization method uses weighted method and transforms multiple targets into single-objective function to solve the problem. The disadvantage of this method is that the weights of each objective are subjective and there is no alternative. The best method is noninferior sorting genetic algorithm II, which gets multiple Pareto optimal solutions after one program runs. With the advantages of speediness, diversity, and uniformity, it has been considered as the most suitable simulation algorithm for solving multiobjective problems [
Algorithm parameters such as population size PopSize and evolutionary algebra GenMax were set; the car and bicycle subnet data were initialized; the initial public transport network population was generated randomly by 0-1 coding; the length of bit strings was determined by the number of bus lines searched by depth-first algorithm; for example: 001010001 means that there were 9 bus lines between origin and destination, and the lines 3, 5, and 9 were selected to form the public transport network.
The users with cars in the three groups were divided into three travel modes, and the users without cars were divided into bus and bicycle travel modes by Logit model.
For each individual in each generation (public transport network) and other mode subnets to achieve SUE traffic allocation, the MSA algorithm was used to solve 9 mode subnets at the same time. In addition, the upper objective function was calculated by using the distribution result, and the fitness function was the objective function itself.
(1) The number of individuals was selected via the tournament strategy to determine the tournament population; (2) the cross-operation was performed by the OX-like method, and the exchange operation was performed on two variation points in the individual by the exchange method; (3) the upper objective function value of each individual in the new population was calculated after the merging of parent and child populations; (4) the fast undominated sort was made based on the value of the objective function, and the crowding distance of the individual in the population is calculated; (5) the former PopSize individuals were selected to generate a new generation population, according to the frontier order value and the crowding distance of each individual.
If the maximum number of iterations is reached, the algorithm terminates, producing a Pareto optimal solution set. Otherwise, turn to Step
Taking the simple road network shown in Figure
Relevant data of the sample of public transport network.
Segment | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| 0.196 | 0.186 | — | — | 0.165 | 0.164 | — | — | — | — | — | 0.157 |
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| 900 | 600 | — | — | 800 | 700 | — | — | — | — | — | 700 |
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Segment | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
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| 0.196 | — | — | — | — | — | 0.195 | 0.206 | — | — | 0.197 | 0.187 |
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| 900 | — | — | — | — | — | 600 | 700 | — | — | 900 | 700 |
Take
Relevant data of different travel groups.
Travel Groups | Percentage of total travel | Car ownership ratio | Time value |
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High income group | 25% | 90% | 25 |
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Medium income group | 40% | 45% | 16 |
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Low income group | 35% | 5% | 6 |
Relevant data of different traffic modes.
Traffic Modes | Car unit conversion factor | Average number of passengers per vehicle | Vehicle usage fee | Road area occupied by vehicle |
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Car | 1.00 | 2 | 12 | 30 |
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Bus | 1.50 | 30 | 2 | 45 |
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Bicycle | 0.25 | 1 | 0 | 7.5 |
Note: due to the particularity of the network structure in the example, the lengths of all travel routes between origin and destination are equal, and the transportation cost of the car is set as a fixed value.
In the NSGA-II algorithm, the population size is 100, the crossover rate is 0.8, the mutation rate is 0.1, and the evolutionary algebra is 500. The distributions of the Pareto optimal solution set and other solution set are shown in Figure
Calculation results.
Optimization | Bus lines | Target value | Optimization | Bus lines | Target value | ||
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| | | | ||||
1 | 1 | 22.35 | 0.061 | 4 | 1 | 26.13 | 0.022 |
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2 | 1 | 23.64 | 0.050 | 5 | 1 | 29.78 | 0.004 |
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3 | 1 | 24.24 | 0.043 | 6 | 1 | 32.70 | 0.003 |
Distribution of Pareto optimal solutions and other solutions.
Compared the different traffic mode ridership.
Compared the different traffic mode travel time.
For the three travel groups, the travel cost relative deprivation coefficient is 33.42, and the road area Gini coefficient is 0.248 before the optimization of the public transport network; after optimization, as shown in Table
Before the public transit network optimization, a large number of transportation resources were occupied by high income groups to meet the needs of private car travel. In contrast, low-income groups without car purchasing power only chose buses with less comfort, convenience, and accessibility. After optimization, the preferences of the three groups changed significantly. As can be seen from Figure
(1) The relative deprivation coefficient of travel cost described the influence of the generalized travel cost difference of different groups on transportation equity, while the Gini coefficient of road area reflected the equilibrium of traffic resources distribution among different groups. In this paper, the bilevel optimization model of public transport network was established under the condition of traffic fairness constraint, and the relationship between the public transit network optimization and travel opportunities of different groups was discussed. The upper level optimization aimed at minimizing the travel cost relative deprivation coefficient and the road area Gini coefficient, and the lower level optimization was a stochastic equilibrium traffic assignment model of multimode and multiuser. Furthermore, the complex selection behaviors of different groups to different travel modes were analyzed.
(2) The example analysis showed that the public transport network model considering the traffic fairness constraints shortened the travel cost difference among different groups effectively and distribute road resources more equitably on one hand; it guided all travel groups to choose more bus trips, improve the fairness of the public transit network optimization schemes, and reduced the chances of high income groups choosing cars on the other hand.
See Table
Variable list description.
Symbol | Symbolic Meaning | Subscript and Superscript Meaning |
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| the network formed by the | the |
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| the total travel amount of OD pair | Any OD pair |
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| the travel demand of the | the |
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| the travel demand of the | traffic mode |
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| the travel demand on the segment a in the subnet | segment |
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| the travel demand of the OD pair | OD pair |
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| Boolean variables associated with routes and segments | OD pair |
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| the road segment traffic on the segment | segment |
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| the equivalent car conversion coefficient of the traffic mode | traffic mode |
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| the average number of passengers in the traffic mode | traffic mode |
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| the road traffic on the route | OD pair |
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| the travel time on the segment | segment |
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| the free flow time on the segment | segment |
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| the travel capacity of the | segment |
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| the travel time on the route | OD pair |
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| the probability that the | OD pair |
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| the familiarity of different travel groups with the road network | the |
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| the relative deprivation of travel costs incurred by all travelers in the multi-mode transportation network | Serial number |
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| the travel cost of selecting the | OD pair |
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| the travel cost of selecting the | OD pair |
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| the travel time of the | OD pair |
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| the time value of the | the |
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| the cost of vehicle use for selecting the | traffic mode |
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| the probability that the | OD pair |
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| the correction parameter | |
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| the ratio of low income group | Serial number |
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| the ratio of medium income group | Serial number |
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| the ratio of high income group | Serial number |
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| the ratio of road resources occupied by the low income group | Serial number |
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| the ratio of road resources occupied by the medium income group | Serial number |
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| the ratio of road resources occupied by the high income group | Serial number |
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| the road area occupied by the single vehicle of the | traffic mode |
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| the standard passenger number of the single vehicle of the | traffic mode |
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| the area | Serial number |
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| the area enclosed by the Lorenz curve and the absolute equality line | Serial number |
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| the road area Gini coefficient | Serial number |
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| The sum of travel time | Serial number |
The example data used to support the finding of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The research is funded by the National Natural Science Foundation of China (no. 71462005; no. 71563023) and the Basic Competence Improvement Project for Young and Middle-Aged Teachers in Higher Education of Guangxi (no. 2018KY0341).