# Bilevel Programming Model of Urban Public Transport Network under Fairness Constraints

Authors: Jingjing Hao; Xinquan Liu; Xiaojing Shen; Nana Feng
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1] analyzed the manifestation of transportation equity and its interaction with urban planning, traffic planning, and traffic development strategies systematically. Vasconcellos [2] studied the negative externalities of different social groups due to travel by using disaggregate model and pointed out that low-income groups only obtain lower levels of mobility when they spend more on transportation costs in a selected case of São Paulo. Preston et al. [3] investigated the social exclusion phenomenon related to transportation from the perspective of accessibility and mobility and constructed a social-spatial model that caused social exclusion. Ahmed et al. [4] conducted a comparative analysis of Beijing and Karachi cities as examples and illustrated the development trend of transportation equity in the urbanization from a macrolevel; Nuworsoo et al. [5] analyzed the impact of different public transport charging schemes on transportation equity through resident surveys and assessed the actual impact on low-income groups specifically; Brocker et al. [6] studied the actual impact of transportation infrastructure investment and construction on regional transportation equity and established a spatial equilibrium model to evaluate the transportation infrastructure investment policy; Ying et al. [7] used the Logit model to describe the competition and development of different traffic modes in urban traffic networks and evaluated the fairness of road resource allocation under various scenarios via the Lotka-Volterra model.The improvement of transportation equity has also attracted the attention of researchers. Farrington et al. [8] analyzed the impact mechanism of accessibility and transportation system on social exclusion in suburban and rural areas and discussed policy design to improve accessibility from the perspective of social equity; Olvera et al. [9] provided policy recommendations for enhancing residents' travel environment and social equity based on the analysis of trends in different household transportation expenditures. Duvarcia et al. [10] discussed the suppressed travel demand of traffic vulnerable groups and their equality with ordinary groups and put forward corresponding suggestions from the technical aspects of urban traffic policy and traffic planning; Lucas [11] took South Africa as an example to analyze the typical social exclusion phenomenon in urban transportation systems in developing countries and proposed specific suggestions for improving the traffic environment of low-income groups from the perspective of traffic policy; Ferguson et al. [12] exemplified the important impact of the accessibility of public transport services on the employment opportunities for low-income groups and tried to embed the evaluation indicators of transportation equity into the evaluation process of the public transport system.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 [13–20], excluding the transportation equity. Therefore, in view of uneven urban transportation resources and differentiated spatial accessibility, this paper intends to use the Gini coefficient of road area and relative deprivation coefficient of travel cost to evaluate the utility of public transit network optimization scheme. The bilevel optimization model of public transport network considering transportation equity constraints is established, and NSGA-II algorithm is designed to solve the problem. The main work in this paper is as follows:(i)
The complex travel behavior of urban residents is analyzed systematically by constructing multimode and multiuser traffic network.(ii)
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.(iii)
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.


## 2. Urban Traffic Network of Multimode and Multiuser

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 Figure1. It consists of 2 traffic zones, 16 nodes and 24 road segments, where O and D indicate the origin and destination zones, respectively.Figure 1
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,Gik represents the network formed by the i-th group choosing the k-th traffic mode. Among them, i =1, 2, 3 represents three groups of high, medium and low income; k =1, 2, 3 represents three travel modes of car, bus and bicycle.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:(1)∑iqiw=qw∀i,w(2)∑kqi,kw=qiw∀i,k,wwhere qw denotes the total travel amount of OD pair w; qiw denotes the travel demand of the i-th group OD pair w; qi,kw denotes the travel demand of thei-th group choosing the traffic mode k on OD pair w.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:(3)xi,ka=∑w∑rfi,kw,rδi,kw,r,a∀w,a(4)∑rfi,kw,r=qi,kw∀w,awhere xi,ka denotes the travel demand on the segment a in the subnet Gik; fi,kw,r denotes the travel demand of the OD pair w on the route r in the subnet Gik; δi,kw,r,a denotes the Boolean variable associated with routes and segments. If the segment a is on the route r that between OD pair w, its value is 1; otherwise it is 0.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(5)vi,ka=xi,kaEkNk∀k,awhere vi,ka denotes the road segment traffic on the segment a in the subnet Gik; Ek denotes the equivalent car conversion coefficient of the traffic mode k; Nk denotes the average number of passengers in the traffic mode k. In the subnet Gik, hi,kw,r denotes the road traffic on the route r and satisfies the following path-segment relationship:(6)vi,ka=∑w∑rhi,kw,rδi,kw,r,a∀w,a(7)hi,kw,r≥0Since the standard BPR function does not consider the interaction of different traffic modes, it is not suitable for multimode traffic networks. Thus, this paper uses the following improvements:(8)ti,ka=ti,ka0∏k1+0.15∑i=1vi,kaCka4∀k,awhere ti,ka represents the travel time on the segment a in the subnet Gik; ti,ka(0) represents the free flow time on the segment a in the subnet Gik; Cka represents the travel capacity of the k-th travel mode for all groups on the segment a. In the subnet Gik, the route travel time and the segment travel time satisfy the following relationship:(9)di,kw,r=∑αti,kaδi,kw,r,a∀w,awhere di,kw,r represents the travel time on the route r in the subnet Gik.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 thei-th travel group selects the pathr in the subnetwork Gik is(10)Pi,kw,r=exp⁡-θi·di,kw,r∑lexp⁡-θi·di,kw,lr≠lwhere Pi,kw,r represents the probability that the i-th travel group chooses the route r in the subnet Gik; θi reflects the familiarity of different travel groups with the road network.


## 3. Model Construction and Solution

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 Figure2.Figure 2
Model structure.


### 3.1. Upper Model

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.


#### 3.1.1. Travel Cost Relative Deprivation Coefficient

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.(11)miny1=∑k∑i,jHj,kw-Hi,kwi≠j(12)s.t.Hi,kw=τi·Ti,kw+Lk∀w,i,jwhere y1 denotes the relative deprivation of travel costs incurred by all travelers in the multimode transportation network, and the smaller the y1, the lower the relative deprivation of all travelers; Hi,kw denotes the travel cost of selecting the k-th way for the OD pair w in the subnet Gik to the i-th resident group; Hj,kw denotes the travel cost of selecting the k-th way for the OD pair w in the subnet Gjk to the j-th group of residents; Ti,kw denotes the travel time of the k-th mode in the subnet Gik to the i-th resident group for the OD pair w, and its value can be expressed by the travel time in the subnetwork balance state; τi denotes the time value of the i-th resident group; Lk denotes the cost of vehicle use for selecting the k-th traffic mode, such as car fuel consumption, bus fare.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:(13)Qi,kw=exp⁡-λ·Hi,kw∑πexp⁡-λ·Hi,πw∀k,π,k≠πwhere Qi,kw denotes the probability that the i-th group selects the traffic mode k on OD pair w; λ denotes the correction parameter.


#### 3.1.2. Road Area Gini Coefficient

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 Figure3, the Lorenz curve is OD1D2A, and the absolute fair line for road resource allocation is OA. The fairness of distribution is determined by the distance between the Lorenz curve and the absolute fair line. That is, the smaller the area enclosed by the Lorenz curve and the absolute fair line, the smaller the Gini coefficient, and the more equitable the actual distribution of traffic resources.Figure 3
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 groupU1 and the ratio of road resources occupied by the group Z1, the ratio of medium income group U2 and the ratio of road resources occupied by the group Z2, and the ratio of high income group U3 and the ratio of road resources occupied by the group Z3. The proportion of road resources occupied by different groups can be expressed as(14)Zi=∑kqi,kw·Sk/Nk∑j∑kqj,kw·Sk/Nk∀i,j,kwhere Sk denotes the road area occupied by the single vehicle of the k-th travel mode; Nk denotes the standard passenger number of the single vehicle of the k-th travel mode.Therefore, it can be determined that the areaM2 enclosed by the line of points O, D1, D2, A, B is(15)M2=12Z1U1+2U2+U3+Z2U2+U3+U3Then the Gini coefficient evaluation equation is(16)miny2=M1M1+M2=1-2M2(17)s.t.equation14,15where y2 represents the Gini coefficient of the road area; M1 represents the area enclosed by the Lorenz curve and the absolute equality line.


### 3.2. Lower Model

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:(18)miny3=1θi∑i∑k∑w∑rhi,kw,rln⁡hi,kw,r+∑i∑k∑α∫0vi,kαti,kαxdx(19)s.t.equation1-4,6,7where y3 is the sum of travel time.


### 3.3. Model Solution

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 [21]. The specific steps of the algorithm are as follows.Step 1.
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.Step 2.
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.Step 3.
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.Step 4 (NSGA-II algorithm core operation).
(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.Step 5 (the termination condition was determined).
If the maximum number of iterations is reached, the algorithm terminates, producing a Pareto optimal solution set. Otherwise, turn to Step2.


## 3.1. Upper Model

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.


### 3.1.1. Travel Cost Relative Deprivation Coefficient

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.(11)miny1=∑k∑i,jHj,kw-Hi,kwi≠j(12)s.t.Hi,kw=τi·Ti,kw+Lk∀w,i,jwhere y1 denotes the relative deprivation of travel costs incurred by all travelers in the multimode transportation network, and the smaller the y1, the lower the relative deprivation of all travelers; Hi,kw denotes the travel cost of selecting the k-th way for the OD pair w in the subnet Gik to the i-th resident group; Hj,kw denotes the travel cost of selecting the k-th way for the OD pair w in the subnet Gjk to the j-th group of residents; Ti,kw denotes the travel time of the k-th mode in the subnet Gik to the i-th resident group for the OD pair w, and its value can be expressed by the travel time in the subnetwork balance state; τi denotes the time value of the i-th resident group; Lk denotes the cost of vehicle use for selecting the k-th traffic mode, such as car fuel consumption, bus fare.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:(13)Qi,kw=exp⁡-λ·Hi,kw∑πexp⁡-λ·Hi,πw∀k,π,k≠πwhere Qi,kw denotes the probability that the i-th group selects the traffic mode k on OD pair w; λ denotes the correction parameter.


### 3.1.2. Road Area Gini Coefficient

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 Figure3, the Lorenz curve is OD1D2A, and the absolute fair line for road resource allocation is OA. The fairness of distribution is determined by the distance between the Lorenz curve and the absolute fair line. That is, the smaller the area enclosed by the Lorenz curve and the absolute fair line, the smaller the Gini coefficient, and the more equitable the actual distribution of traffic resources.Figure 3
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 groupU1 and the ratio of road resources occupied by the group Z1, the ratio of medium income group U2 and the ratio of road resources occupied by the group Z2, and the ratio of high income group U3 and the ratio of road resources occupied by the group Z3. The proportion of road resources occupied by different groups can be expressed as(14)Zi=∑kqi,kw·Sk/Nk∑j∑kqj,kw·Sk/Nk∀i,j,kwhere Sk denotes the road area occupied by the single vehicle of the k-th travel mode; Nk denotes the standard passenger number of the single vehicle of the k-th travel mode.Therefore, it can be determined that the areaM2 enclosed by the line of points O, D1, D2, A, B is(15)M2=12Z1U1+2U2+U3+Z2U2+U3+U3Then the Gini coefficient evaluation equation is(16)miny2=M1M1+M2=1-2M2(17)s.t.equation14,15where y2 represents the Gini coefficient of the road area; M1 represents the area enclosed by the Lorenz curve and the absolute equality line.


## 3.1.1. Travel Cost Relative Deprivation Coefficient

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.(11)miny1=∑k∑i,jHj,kw-Hi,kwi≠j(12)s.t.Hi,kw=τi·Ti,kw+Lk∀w,i,jwhere y1 denotes the relative deprivation of travel costs incurred by all travelers in the multimode transportation network, and the smaller the y1, the lower the relative deprivation of all travelers; Hi,kw denotes the travel cost of selecting the k-th way for the OD pair w in the subnet Gik to the i-th resident group; Hj,kw denotes the travel cost of selecting the k-th way for the OD pair w in the subnet Gjk to the j-th group of residents; Ti,kw denotes the travel time of the k-th mode in the subnet Gik to the i-th resident group for the OD pair w, and its value can be expressed by the travel time in the subnetwork balance state; τi denotes the time value of the i-th resident group; Lk denotes the cost of vehicle use for selecting the k-th traffic mode, such as car fuel consumption, bus fare.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:(13)Qi,kw=exp⁡-λ·Hi,kw∑πexp⁡-λ·Hi,πw∀k,π,k≠πwhere Qi,kw denotes the probability that the i-th group selects the traffic mode k on OD pair w; λ denotes the correction parameter.


## 3.1.2. Road Area Gini Coefficient

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 Figure3, the Lorenz curve is OD1D2A, and the absolute fair line for road resource allocation is OA. The fairness of distribution is determined by the distance between the Lorenz curve and the absolute fair line. That is, the smaller the area enclosed by the Lorenz curve and the absolute fair line, the smaller the Gini coefficient, and the more equitable the actual distribution of traffic resources.Figure 3
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 groupU1 and the ratio of road resources occupied by the group Z1, the ratio of medium income group U2 and the ratio of road resources occupied by the group Z2, and the ratio of high income group U3 and the ratio of road resources occupied by the group Z3. The proportion of road resources occupied by different groups can be expressed as(14)Zi=∑kqi,kw·Sk/Nk∑j∑kqj,kw·Sk/Nk∀i,j,kwhere Sk denotes the road area occupied by the single vehicle of the k-th travel mode; Nk denotes the standard passenger number of the single vehicle of the k-th travel mode.Therefore, it can be determined that the areaM2 enclosed by the line of points O, D1, D2, A, B is(15)M2=12Z1U1+2U2+U3+Z2U2+U3+U3Then the Gini coefficient evaluation equation is(16)miny2=M1M1+M2=1-2M2(17)s.t.equation14,15where y2 represents the Gini coefficient of the road area; M1 represents the area enclosed by the Lorenz curve and the absolute equality line.


## 3.2. Lower Model

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:(18)miny3=1θi∑i∑k∑w∑rhi,kw,rln⁡hi,kw,r+∑i∑k∑α∫0vi,kαti,kαxdx(19)s.t.equation1-4,6,7where y3 is the sum of travel time.


## 3.3. Model Solution

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 [21]. The specific steps of the algorithm are as follows.Step 1.
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.Step 2.
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.Step 3.
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.Step 4 (NSGA-II algorithm core operation).
(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.Step 5 (the termination condition was determined).
If the maximum number of iterations is reached, the algorithm terminates, producing a Pareto optimal solution set. Otherwise, turn to Step2.


## 4. Example Analysis

Taking the simple road network shown in Figure1 as an example, the optimization model and algorithm of public transport network are verified. Due to space constraints, this paper lists certain initial data of the public transport network, as shown in Table 1. It is assumed that the road network between origin and destination has sufficient capacity to allow multiple travel lines in the network without considering the transfer mode and the interests of the operators. In this paper, the existing two bus lines are optimized, and direct bus lines are rearranged between origin and destination, in order to improve the fairness of the public traffic line network scheme.Table 1
Relevant data of the sample of public transport network.
Segment 1 2 3 4 5 6 7 8 9 10 11 12 t i , 2 a ( 0 )/h 0.196 0.186 — — 0.165 0.164 — — — — — 0.157  C 2 a(veh⋅h−1) 900 600 — — 800 700 — — — — — 700  Segment 13 14 15 16 17 18 19 20 21 22 23 24  t i , 2 a ( 0 )/h 0.196 — — — — — 0.195 0.206 — — 0.197 0.187  C 2 a(veh⋅h−1) 900 — — — — — 600 700 — — 900 700Takeqw =8000people·h−1, λ=0.1, θ1=1, θ2=2, θ3=5. The relevant data of different travel groups are shown in Table 2, and the relevant data of traffic modes are shown in Table 3.Table 2
Relevant data of different travel groups.
Travel Groups Percentage of total travel Car ownership ratio Time valueτi (yuan/h) High income group 25% 90% 25  Medium income group 40% 45% 16  Low income group 35% 5% 6Table 3
Relevant data of different traffic modes.
Traffic Modes Car unit conversion factorEk Average number of passengers per vehicleNk (person / vehicle) Vehicle usage feeLk (yuan) Road area occupied by vehicleSk (m2) Car 1.00 2 12 30  Bus 1.50 30 2 45  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 Figure4. Table 4 lists the six groups of noninferior solutions obtained by NSGA-II algorithm. Each group of noninferior solution corresponds to the optimization scheme of public transport network under traffic fairness constraints. The decision makers select the best scheme among these six noninferior solutions, considering the construction and operation cost, the service level, and other factors of bus lines. The optimization calculation results are shown in Figures 5 and 6. Through the analysis of six groups of data related to optimization scheme, the following conclusions are drawn:Table 4
Calculation results.
Optimization Bus lines Target value Optimization Bus lines Target value y 1 y 2 y 1 y 2 1 1→2→3→4→8→12→16 1→2→6→10→11→15→16 1→5→9→10→14→15→16 1→5→9→13→14→15→16 22.35 0.061 4 1→2→6→7→11→15→16 1→5→6→7→8→12→16 1→5→6→7→11→12→16 26.13 0.022  2 1→2→6→7→11→15→16 1→5→6→10→11→12→16 1→5→9→10→14→15→16 23.64 0.050 5 1→2→6→7→8→12→16 1→5→6→7→11→12→16 1→5→6→10→11→15→16 1→5→9→13→14→15→16 29.78 0.004  3 1→2→6→7→11→12→16 1→5→6→7→8→12→16 1→5→6→10→11→12→16 1→5→9→10→11→15→16 1→5→9→13→14→15→16 24.24 0.043 6 1→5→6→10→11→12→16 1→2→3→4→8→12→16 1→5→9→13→14→15→16 32.70 0.003Figure 4
Distribution of Pareto optimal solutions and other solutions.Figure 5
Compared the different traffic mode ridership.Figure 6
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 Table4, the travel cost relative deprivation coefficient of the six optimized schemes averages 26.51, which is 20.68% lower than that before optimization; the Gini coefficient of road area averages 0.030, which is 87.76% lower than before. Thus, the model optimized the public transit network effectively. While meeting the travel needs of residents, it shortened the difference in travel costs between different groups and improved the fairness of transportation resource allocation.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 Figure5, certain high and medium income groups who used to travel by car begin to choose bus and bicycle modes, while the low-income group choose a higher rate of public transportation. Besides, both car and bicycle usage rates are declining. This trend indicates that the optimized public transport network has significantly increased the attraction for each group. The reason is known from Figure 6. It is mainly because the travel time of all groups by car has increased, the travel time by bicycle is basically the same, and the travel time by bus has decreased, after the public transit network optimization. The change of travel time not only restrains the demand of each group to choose cars, but also improves the convenience and accessibility of public transportation, which makes more travelers willing to choose bus travel.


## 5. Conclusions

(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.


## Variables

See Table5.Table 5
Variable list description.
Symbol Symbolic Meaning Subscript and Superscript Meaning 1 Urban traffic network of multi-mode and multi-user  G i k the network formed by thei-th group choosing the k-th traffic mode thei-th group; traffic mode k  q w the total travel amount of OD pairw Any OD pairw  q i w the travel demand of thei-th group OD to w thei-th group  q i , k w the travel demand of thei-th group on OD pair w choosing the traffic mode k traffic modek  x i , k a the travel demand on the segment a in the subnetGik segmenta; the i-th group; traffic mode k  f i , k w , r the travel demand of the OD pairw on the route r in the subnet Gik OD pairw; route r; the i-th group; traffic mode k  δ i , k w , r , a Boolean variables associated with routes and segments OD pairw; route r; segment a; the i-th group; traffic mode k  v i , k a the road segment traffic on the segmenta in the subnet Gik segmenta; the i-th group; traffic mode k  E k the equivalent car conversion coefficient of the traffic modek traffic modek  N k the average number of passengers in the traffic modek traffic modek  h i , k w , r the road traffic on the router OD pairw; route r; the i-th group; traffic mode k  t i , k a the travel time on the segmenta in the subnet Gik segmenta; the i-th group; traffic mode k  t i , k a ( 0 ) the free flow time on the segmenta in the subnet Gik segmenta; the i-th group; traffic mode k  C k a the travel capacity of thek-th travel mode for all groups on the segment a segmenta; traffic mode k  d i , k w , r the travel time on the router in the subnet Gik OD pairw; route r; the i-th group; traffic mode k  P i , k w , r the probability that thei-th travel group chooses the route r in the subnet Gik OD pairw; route r; the i-th group; traffic mode k  θ i the familiarity of different travel groups with the road network thei-th group  2 Model Construction and Solution  y 1 the relative deprivation of travel costs incurred by all travelers in the multi-mode transportation network Serial number  H i , k w the travel cost of selecting thek-th way for the i-th resident group on the OD pair w in the subnet Gik OD pairw; the i-th group; traffic mode k  H j , k w the travel cost of selecting thek-th way in the subnet Gik of the i-th resident group on the OD pair w OD pairw; the j-th group; traffic mode k  T i , k w the travel time of thek-th mode in the sub-Gik of the i-th resident group on the OD pair w OD pairw; the i-th group; traffic mode k  τ i the time value of thei-th resident group thei-th group  L k the cost of vehicle use for selecting thek-th traffic mode traffic modek  Q i , k w the probability that thei-th group selects the traffic mode on OD pair w OD pairw; the i-th group; traffic mode k  λ the correction parameter  U 1 the ratio of low income group Serial number  U 2 the ratio of medium income group Serial number  U 3 the ratio of high income group Serial number  Z 1 the ratio of road resources occupied by the low income group Serial number  Z 2 the ratio of road resources occupied by the medium income group Serial number  Z 3 the ratio of road resources occupied by the high income group Serial number  S k the road area occupied by the single vehicle of thek-th travel mode traffic modek  N k the standard passenger number of the single vehicle of thek-th travel mode traffic modek  M 2 the areaM2 enclosed by the line of points O, D1, D2, A, B Serial number  M 1 the area enclosed by the Lorenz curve and the absolute equality line Serial number  y 2 the road area Gini coefficient Serial number  y 3 The sum of travel time Serial number



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*Source: 2930502-2019-03-12.xml*