Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 5049657, 13 pages

https://doi.org/10.1155/2017/5049657

## Bilevel Traffic Evacuation Model and Algorithm Design for Large-Scale Activities

Collage of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Correspondence should be addressed to Danwen Bao; nc.ude.aaun@newnadoab

Received 28 August 2016; Revised 14 December 2016; Accepted 11 May 2017; Published 13 June 2017

Academic Editor: Gennaro N. Bifulco

Copyright © 2017 Danwen Bao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper establishes a bilevel planning model with one master and multiple slaves to solve traffic evacuation problems. The minimum evacuation network saturation and shortest evacuation time are used as the objective functions for the upper- and lower-level models, respectively. The optimizing conditions of this model are also analyzed. An improved particle swarm optimization (PSO) method is proposed by introducing an electromagnetism-like mechanism to solve the bilevel model and enhance its convergence efficiency. A case study is carried out using the Nanjing Olympic Sports Center. The results indicate that, for large-scale activities, the average evacuation time of the classic model is shorter but the road saturation distribution is more uneven. Thus, the overall evacuation efficiency of the network is not high. For induced emergencies, the evacuation time of the bilevel planning model is shortened. When the audience arrival rate is increased from 50% to 100%, the evacuation time is shortened from 22% to 35%, indicating that the optimization effect of the bilevel planning model is more effective compared to the classic model. Therefore, the model and algorithm presented in this paper can provide a theoretical basis for the traffic-induced evacuation decision making of large-scale activities.

#### 1. Introduction

China has had an increased influence in politics, culture, economy, and sports in recent years due to its increased international status. There are a number of international large-scale activities of various forms held in China, with recent examples including the Beijing Olympic Games, Shanghai World Expo, and Guangzhou Asian Games. These activities enhance the attraction of cities and bring considerable gains but also result in opportunities and challenges. Large-scale activities held frequently in various forms improve the construction of infrastructure but impact and influence city traffic. It is important to develop a reasonable and effective traffic evacuation strategy to cope with large-scale activities that are characterized by strong agglomeration, suddenness, and short-term durations.

Research on the organization and management of large-scale activities has increased with the increasing prevalence of international events; in particular, the recent Olympic Games provide sufficient data and experience for such research. Amodei et al. [1–4] studied the experiences in traffic organization and management from the Atlanta Olympic Games in 1996, the Sydney Olympic Games in 2000, the Salt Lake City Winter Olympics in 2002, and the Athens Olympic Games in 2004. The studies focused on the management of traffic demand, planning of public transport, and planning of park and ride. The Federal Highway Administration (FHWA) [5] issued a research report on the traffic management of large-scale activities. It provided a reference but lacked details for establishing a traffic organization method and relevant theories. Liu et al. [6] studied the traffic planning of the Beijing Olympic Games in 2008 and published a report entitled “Traffic Planning for Olympic Games.” Chen et al. [7, 8] investigated the planning and organization of traffic for the Shanghai World Expo in 2010 and proposed relevant traffic plans. Ma et al. [9, 10] proposed detailed planning for the construction and organization of traffic for the Guangzhou Asian Games in 2010. Traffic planning models were established to make quantitative predictions of venue accessibility. The research focused on the macropolicy of traffic organization and principles of traffic management; few references have provided an induced evacuation model of traffic and theoretical foundations. After hurricane Katrina swept across New Orleans in 2005, research on that aspect has gradually become thorough. Many scholars applied the traffic flow distribution to the building of an evacuation model. Liu et al. [11] proposed a method that applied a bilevel planning model to evacuation. Chiu et al. [12] introduced a no-notice mass evacuation model. Liu et al. [13] designed a real-time control evacuation system by introducing adaptive control theory. Yazici et al. [14] introduced stochastic road capacity to propose a more robust evacuation method. Yan et al. [15] established a bilevel planning model to study a one-way passing traffic network for the evacuation of large-scale activities and rush hour daily commute. The solution algorithm was also proposed. Doan and Ukkusuri [16] studied the traffic flow blocking phenomenon. As an innovation algorithm, Li et al. [17] proposed an approximate solution method. Robustness, optimization, and randomness were also introduced [18]*.* In addition, meaningful research results were obtained through simulation, such as the research conducted by Naghawi and Wolshon [19] and Zhao et al. [20]*.* Pel et al. [21] reviewed traffic evacuation problems by using simulation methods. Use of the theory of traffic flow distribution to solve evacuation problems has become a research topic of increasing interest. With the improvement of computer power, these theories have become closer to the actual demands of traffic management. However, the research has been focused on evacuation problems resulting from disaster scenarios that tend to give priority to the victims’ survival and property safety. In view of the general growth of large-scale activities, it is worthwhile to study evacuation from the perspective of how to reduce the impact on the surrounding road network and improve traffic efficiency.

A number of studies have focused on traffic organization and evacuation for large-scale activities. There are three main shortcomings. First, the available research on traffic evacuation focuses on management by using simulation software to assess the traffic evacuation plans. However, these empirical plans are not always the optimized plan, and, thus, it is necessary to use mathematical modeling methods to verify their performance. Second, as the bilevel model has been widely used in traffic network design, the upper-level model is always established based on the practical case and is generally a two-value and continuous problem. Few studies have focused on the multivalue discrete problem. The computational efficiency of the available algorithms is low when there are a large number of network points and large demand for evacuation. There are problems with being trapped in a local optimal solution and premature convergence; thus, the solution algorithm must be optimized. Third, there are many unknowns during large-scale activities that may cause a series of reactions and emergencies. The available models always focus on traffic evacuation for the activity itself rather than the induced emergencies.

This paper establishes a bilevel planning model with a single master and multiple slaves. The minimum saturation of the evacuation network is used as the objective function in the upper-level planning model, whereas the minimum evacuation time for the large-scale activity and induced emergencies is used as the objective function in the lower-level planning model. Traffic evacuations for large-scale activities and induced emergencies are both studied. An improved particle swarm optimization method is proposed to solve the bilevel planning model and enhance its computing efficiency. A case study is performed using the Nanjing Olympic Sports Center to further verify the accuracy and feasibility of the proposed model and solution algorithm. This study intends to provide a reference for the traffic evacuation of large-scale activities.

#### 2. Establishment of the Model

##### 2.1. Construction of the Bilevel Planning Model

For a given topological space structure of road network, is the set of road nodes and is the set of directed arcs, that is, the set of roads in the network. The modeling objective is to obtain the minimum saturation of the evacuation network. For a large-scale activity, travelers may choose the shortest evacuation path to achieve the shortest individual evacuation time. If emergencies occur during the activity, emergency plans should be initiated to avoid the occurrence of additional accidents. Emergency plans target the shortest total evacuation time, which is achieved by choosing evacuation paths based on system optimization.

A bilevel planning model is proposed using the minimum saturation of the evacuation network as the objective function for the upper-level planning model and the minimum evacuation time as the objective function for the lower-level planning model. The objective function and constraint conditions of the upper-level model depend on the optimal solution of the lower-level planning model, which is in turn affected by the strategy variables in the upper-level model [22]. Based on the dynamic traffic evacuation characteristics for large-scale activities, the upper-level model mainly focuses on the effect of saturation on the practical operation of traffic, whereas the lower-level model mainly considers the shortest total evacuation time of all vehicles. The objective function of the lower-level model is to determine which evacuation model to use under various traffic conditions to obtain the shortest evacuation time. The upper-level model can be expressed aswhere represents regional road network saturation; represents the number of vehicles on evacuation road at time ; represents the capacity of evacuation road ; represents the traffic demand at time on path between and of OD point pairs in units of pcu; represents the length of evacuation road in units of m; represents the road capacity of evacuation road in units of pcu; represents the average length of evacuation vehicles running on road ; represents the total evacuation time; represents the rate of inflow on evacuation road at time in units of pcu/s; represents the rate of outflow on evacuation road at time in units of pcu/s; represents the space-time resource on evacuation road ; represents the exhaustion of time and space on evacuation road ; and represents the influence coefficient that affects road capacity and is determined by the Highway Capacity Manual (HCM2000).

Saturation is an important index to reflect traffic conditions. In the evacuation of large-scale activities, high traffic saturation may cause traffic jams that are adverse to traffic evacuation. Therefore, an acceptable saturation level should be ensured in the upper-level model and evacuation routes provided in the lower-level model under this restriction. The section flow conservation equation, capacity equation of evacuation routes, and space and time resource equation of evacuation routes are provided in (2) to (4), respectively.

The lower-level model 1 for large-scale activities can be established aswhere represents the total evacuation time for normal large-scale activities; (*t*) represents the transient impedance on evacuation road at time ; represents the capacity of evacuation road in units of pcu; represents the rate of outflow on evacuation road at time in units of pcu/s; represents the number of vehicles on evacuation road at time ; represents the flow on path between and of OD point pairs in units of pcu; indicates whether road belongs to path between and of OD point pairs, and if , then ; otherwise, ; represents the traffic demand on path between and of OD point pairs in units of pcu; represents the traffic volume between and of OD point pairs in units of pcu; (*t*) represents the average travel time through evacuation road at time ; represents the travel time through evacuation road under free flow; and and are both travel time parameters, with recommended values of 0.15 and 4, respectively, in highway network applications.

Equations (6) to (8) are section flow conservation equations. The sum of flows between all the OD pairs is the total number of evacuation vehicles. Equations (9) and (10) represent the section travel time equations, and (11) and (12) are the nonnegative constraint equations.

The lower-level model 2 for emergencies in large-scale activities can be established aswhere represents the total evacuation time under induced emergencies; (*t*) represents the rate of inflow at time on evacuation road in units of pcu/s; (*t*) represents transient impedance at time on evacuation road ; represents the road capacity at time on evacuation road in units of pcu; represents the total number of evacuation vehicles on road ; represents unit interval in units of s; represents the rate of outflow at time on evacuation road in units of pcu/s; represents the minimum travel speed on evacuation road ; and represents the free-flow travel speed on evacuation road* a*.

The evacuation time of individual vehicles is the sum of the waiting time at the starting point of evacuation and the travel time from the starting point to the ending point. The waiting time depends on the inflow rate . If is larger, the waiting time will be shorter and vice versa. The inflow rate depends on the road volume, outflow rate, and traffic capacity. Therefore, the solution for the shortest evacuation time of the objective function is transformed to find the shortest travel time in the evacuation route when is as large as possible.

##### 2.2. Optimizing Condition of the Model

The Pontryagin maximum principle in optimizing control theory was used to solve the bilevel evacuation model. The general form of the conditions for the optimal solution of the upper-level model is expressed as

The Hamiltonian function was established based on the upper-level level as

The Euler-Lagrange equation was constructed aswhere are the Lagrange multipliers.

Thus, the objective function of the upper-level model can be written as

Assuming as the optimal solution of the upper-level model, there exists satisfying

The lower-level model can be uniquely determined according to the optimal solution of the upper-level model. The optimizing conditions of the lower-level model can be deduced from the optimizing conditions of the upper-level model as

#### 3. Algorithm Design

The improved particle swarm optimization method was used to solve the bilevel planning model. The particle swarm optimization (PSO) method was first proposed by Kennedy and Eberhart in 1995. The PSO method is a group evolution-based algorithm. The optimal solution for a complex problem is obtained based on the collaboration and competition of individuals. The classical PSO algorithm is to randomly form an initial particle swarm in the feasible solution space. Each particle represents a feasible solution. The objective function composes the fitness of the particle. During the evolution of particles, an individual particle will optimize itself only when its current position is better than the optimized individual. If the fitness of several particles is not better than the current optimum, there will be no update to individual optimization, which may reduce the convergence of the entire group, thus leading to premature convergence problems. Therefore, the traditional PSO method was improved in this study to make it more applicable for the traffic evacuation problem.

The electromagnetism-like mechanism (EM) was introduced in the PSO method (i.e., PSO-EM) to accelerate the convergence from the individual optimal to group optimal solution and reduce the possibility of stagnation of the individual optimal solution. Individual optimal solution of adjacent particles and group optimal solution act directly on the process of individual optimal solution through the absorption-rejection mechanism. The flowchart of the proposed PSO-EM algorithm is shown in Figure 1.