Because of the limitation of budget, in the planning of road works, increased efforts should be made on links that are more critical to the whole traffic system. Therefore, it would be helpful to model and evaluate the vulnerability and reliability of the transportation network when the network design is processing. This paper proposes a bilevel transportation network design model, in which the upper level is to minimize the performance of the network under the given budgets, while the lower level is a typical user equilibrium assignment problem. A new solution approach based on particle swarm optimization (PSO) method is presented. The topological effects on the performance of transportation networks are studied with the consideration of three typical networks, regular lattice, random graph, and smallworld network. Numerical examples and simulations are presented to demonstrate the proposed model.
The network design problem (NDP) involves the optimal decision on the expansion of an urban street and highway system in response to a growing demand for travel. It has emerged as an important area for progress in handling effective transport planning, because the demand for travel on the roads is growing at a rate faster than our urban transport systems can ever hope to accommodate, while resources available for expanding the system capacity remain limited. The objective of NDP is to optimize a given system performance measure such as to minimize total system travel cost, while accounting for the route choice behavior of network users [
Studies have been overwhelmingly focused on the continuous network design problem (CNDP) and substantial achievements in algorithmic development have been made. Abdulaal and LeBlanc [
In the planning of road works, there should be awareness about the impacts of the increased capacity of a link on the whole network. Because the improvement of road capacity will attract the limitation of budget, the prioritization for road maintenance, repair, and contingency planning should be considered carefully in NDP. Therefore, increased efforts should be made on links that are more critical to the system. It would be thus helpful to model and evaluate the vulnerability and reliability of the transportation network when the network design is processing.
In a series of papers, the measures to quantifiable efficiency/performance of a network have been developed. For example, Latora and Marchiori discussed the network performance issue by measuring the “global efficiency” in a weighted network as compared to that of the simple nonweighted smallworld network [
The flow on a network is an additional important indicator of network performance as well as network vulnerability. Indeed, flows represent the usage of a network and which paths and links have positive flows and the magnitude of these flows are relevant in the case of network disruptions [
Recently, Jenelius et al. [
Although previous work for the traffic network design has been focused on minimizing the total system cost or maximizing the total profits of the network, little attention has been given to the structure factors underlying vulnerabilities and the robustness. In the traffic network, if the Hub link is attacked, the whole network may be broken down soon. Therefore, the purpose of network designs is not only to decrease the travel cost of the system, but also to increase the robustness and reduce the vulnerabilities. Different with most previous works on CNDP, the performance measure is adopted to test the network design strategies in this paper. Our results give a better understanding of the network structure effects on its performance.
This paper is organized as follows. A bilevel model of CNDP is given in Section
The transportation CNDP can be represented as a leaderfollower game where the transportation planning departments are leaders, and the users who can freely choose the path are the followers [
The transportation planners, the upper level, decide the capacity of each road to maximize the system performance based on traffic flows. However, the lower level reflects the choice behaviors of drivers with user equilibrium assignment. In this model, the system performance and link flows are considered all together.
It is worth emphasizing that the network design problem must be solved with the network flow pattern constrained to be a user equilibrium problem. In general, improvement of road network characteristics will definitely induce changes in traffic flow over the network. More importantly, addition of a new road segment or capacity enhancement to a congested network, without considering the response of network users, may actually increase networkwide congestion. This wellknown phenomenon has been demonstrated by the ostensible Braess paradox. Therefore, prediction of traffic patterns via a comprehensive behavior model is essential to the network design process.
Traditionally, CNDP models hypothesize that the demand is given and fixed, and the users’ route choice is characterized by the user equilibrium assignment problem. Let
In this model, the users at the lower level are assumed to follow the userequilibrium principle of Wardrop under the given network. Constraints (
Recently, the vulnerability of a network has attracted many interests in the urban traffic system. As an important performance index, it can be used to assess the efficiency of a network in the case of either fixed or elastic demands and capture flow information and behavior, allowing one to determine the criticality of various nodes (as well as links) through the identification of their importance and ranking [
The network performance/efficiency measure
In this paper, it is an optimized objective in the upper level model. The network planners of the upper level are assumed to make the decisions about the improvement of link capacities and investments in order to maximize the performance of the whole system in the range of budgets formulated by the government. Therefore, the upper level of the bilevel model (
The objective of upper level model is to maximize the performance. Constraint (
Since the bilevel programming is a NPhard problem, in general, it is difficult to solve with optimization algorithms. Although many solution algorithms such as the sensitivity analysis based on an algorithm (SAB) [
Bird flocking optimizes a certain objective function. Each particle knows its personal best position (
In this paper, a particle position is denoted by a feasible link capacity enhancement
Fourie and Groenwold [
The major steps of the PSO method for solving the CNDP are summarized as follows.
Initialize parameters
For each particle, we translate it into a feasible solution. Then, we solve the corresponding lower level problem by FrankWolfe method [
For each particle, update its
Generate the new position with (
All the new positions are tentative particles. Discard the new position
If the iteration number reaches the maximum iteration number or other termination criteria, then stop. Get the last
To illustrate the effectiveness of the proposed approach, consider the 5node and 6link graph given in Figure
Data for the test network.
Link 





1→2  1.0  0.15  0.5  40 
1→3  2.0  0.15  0.5  40 
2→4  1.0  0.15  0.5  40 
2→5  1.5  0.15  0.5  40 
3→4  1.0  0.15  0.5  40 
3→5  1.5  0.15  0.5  40 

Test network.
The investment function is given as
Parameters used in Example
Parameters 










Values  10.0  0.98  15.0  0.05  0.2  0.8  2.1  0.9  1.0 
Comparison results of different budgets.
Budget  Capacity  

Link  
1→2  1→3  2→4  2→5  3→4  3→5  
150  4.60  2.46  3.12  2.41  1.72  1.26 
250  5.73  2.97  4.42  3.28  2.96  0.63 
350  6.18  4.45  4.85  0.11  4.03  1.58 
Tables
TSC of the network with different OD demands and budgets.
Budget  TSC  

OD  
10020  10040  10060  10080  100100  
150  432.44  584.10  865.89  1213.67  1853.30 
250  431.25  579.45  793.35  1159.37  1636.84 
350  410.91  560.11  743.75  1112.40  1509.32 
PER of the network with different OD demands and budgets.
Budget  Performance  

OD  
10020  10040  10060  10080  100100  
150  0.3992  0.2799  0.1795  0.1272  0.0798 
250  0.4188  0.2837  0.1969  0.1356  0.0954 
350  0.4447  0.3289  0.2265  0.1446  0.1057 
To illustrate the efficiency of the solution algorithm further, a convergence test is given in the numerical example. Figure
Illustration of the convergences for the algorithm.
The structure properties on traffic networks have attracted a tremendous amount of recent interest which shows different network structures have important effects on their performance. It has been known well that there are three typical structures in traffic networks: random graph (RG), smallworld network (SW), and regular lattice (RL). We start by constructing networks according to regular ER [
In this example, to investigate the effects of topologies on NDP, the network design decisions for different network structures with the same nodes and edges are given. Three typical network topologies are showed in Figure
Three different network topologies.
The number of the brackets represents the initial performance before the network design. Table
The optimization results for different network topologies.
Budget  Topology  

SW  ER  RL  
200  0.5483  0.5228  0.4604 
250  0.5486  0.5231  0.4650 
300  0.5486  0.5234  0.4659 
350  0.5487  0.5238  0.4666 
400  0.5488  0.5241  0.4667 
The transportation network performance is an important indicator to evaluate its reliability. In order to improve the network vulnerability, the performance should be considered in network design decisions. This paper studies a new form of transportation network design problem by optimizing the network performance, and a bilevel programming model is proposed to describe this problem. In the model, the upper level is to optimize the system performance within limited budgets, while the lower level is user equilibrium assignment problem. Finally, numerical examples show that, by redesigning the network, the performance (the total system cost) will increase (reduce) greatly. In addition, the network structure has profound effects on its performance. Compared with the other two topologies, the form of smallworld network is the best one.
In this paper, it is assumed that the OD matrix is fixed. In reality, OD demand would fluctuate every day, even every hour. Therefore it would be interesting to examine the dynamic OD demand in network design problem further.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper is partly supported by National Basic Research Program of China (2012CB725400), NSFC (71271024), FANEDD (201170), the Program for New Century Excellent Talents in University (NCET120764), and the Fundamental Research Funds for the Central Universities (2012JBZ005).