Discrete Dynamics in Nature and Society

Volume 2015, Article ID 827094, 13 pages

http://dx.doi.org/10.1155/2015/827094

## Estimating the Capacity of Urban Transportation Networks with an Improved Sensitivity Based Method

^{1}College of Civil and Transportation Engineering, Hohai University, 1 Xikang Road, Nanjing 210098, China^{2}School of Transportation, Southeast University, 35 Jinxianghe Road, Nanjing 210096, China

Received 7 July 2014; Accepted 18 August 2014

Academic Editor: Yongjun Shen

Copyright © 2015 Muqing Du 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

The throughput of a given transportation network is always of interest to the traffic administrative department, so as to evaluate the benefit of the transportation construction or expansion project before its implementation. The model of the transportation network capacity formulated as a mathematic programming with equilibrium constraint (MPEC) well defines this problem. For practical applications, a modified sensitivity analysis based (SAB) method is developed to estimate the solution of this bilevel model. The high-efficient origin-based (OB) algorithm is extended for the precise solution of the combined model which is integrated in the network capacity model. The sensitivity analysis approach is also modified to simplify the inversion of the Jacobian matrix in large-scale problems. The solution produced in every iteration of SAB is restrained to be feasible to guarantee the success of the heuristic search. From the numerical experiments, the accuracy of the derivatives for the linear approximation could significantly affect the converging of the SAB method. The results also show that the proposed method could obtain good suboptimal solutions from different starting points in the test examples.

#### 1. Introduction

As the rapid growth of urbanization, the population and economy of most cities in the developing countries or regions are going through significant changes. The fast development of transportation infrastructures in these areas gives rise to a quick change on the travel behaviors. Along with the city’s expansion, an inevitable thing is to design new transportation system which is capable of meeting the future development in land-using and population growth. However, under fast-changing of travel demand, the conventional forecasting methods (e.g., “four-step”) could not provide a straightforward evaluation to the new designed transportation network such as on whether the new network is suitable for the travel demand in the target year or how many trips can be accommodated. Thus, to help the government to make decision on the expansion or construction project or preassessment of social benefits, the throughput, or “capacity,” of the given transportation network is of practical meaning to be estimated before planning implementation.

Capacity is a commonly used property to represent the maximum flows that can pass through the link or node in transportation system. In an attempt to address the question of what is the maximum attainable throughput of the given network, the concept of capacity is employed as an important measurement for transportation system evaluation. It is able to reflect how much traffic demand can be accommodated by a given transportation system. Thus, efficient policy for land use or traffic restraint and growth can be established in advance. According to the conventional network flow theory, the capacity problem is stated to find the maximum flows that can be sent from a specified source node to another specified sink node without exceeding the capacity of any link. This well-known problem is extended to the multicommodity and is widely used in freight transport. However, when modeling the capacity of urban transportation network, the problem becomes quite complex. Noted by Yang et al. [1], travelers in urban transportation network can choose their routes and their trip costs increase with increasing flow as a result of congestion. Besides, multiple origin and destination (O-D) pairs exist and the flows between distinct O-D pairs cannot be exchanged in passenger transportation system. These differences make the modeling of transportation network capacity complex, and the intriguing problem is also hard to solve.

The most popular formulation of the transportation network capacity is the bilevel programming model, which maximizes the traffic flows under the equilibrium constraint. Wong and Yang [2] first incorporated the* reserve capacity* concept into a traffic signal control network. This concept was widely extended in the study of signal controlled networks [3–5]. The reserve capacity is defined as the largest multiplier applied to a given O-D demand matrix that can be allocated to a network, so the solution is significantly affected by the predetermined O-D matrix. However, it is unrealistic to assume that all O-D flows increase in the same rate, especially for the areas under rapid changing. If the predetermined distribution proportion is far from the future tendency, the solution will be of little use. Consequently, in order to reflect the differences of the future development of each urban subarea, Yang et al. [1] considered that the new increased O-D demand pattern should be variable both in level and in distribution, while the distribution of the current trips would be relatively fixed. Thereby they introduced the equilibrium trip distribution/assignment model with variable destination costs (ETDA-VDC) [6] to capture this characteristic for network capacity estimation. Based on this model, Kasikitwiwat and Chen [7] proposed the concepts and models of the* ultimate* and* practical capacity*. The former is used for the new network without any current flow, while the latter is the same as Yang’s model. Then, Chen and Kasikitwiwat [8] used the practical network capacity model to describe the limited flexibility of transportation networks. According to the literatures, the concept of the practical capacity model is more fully functional and preferred, as it takes both the current demand pattern and the variability of future growth into consideration.

In order to solve the various bilevel capacity models, the SAB algorithm is generally employed. This SAB algorithm for bilevel programing was first presented in Friesz et al. [9]. It is heuristic and depends absolutely on the derivative information produced by the sensitivity analysis of the lower-level problem [10]. Benefiting from the rich achievements in the study of the sensitivity analysis for equilibrium models [11–18], SAB algorithm has been widely utilized in the optimization problems of equilibrium network flows, such as traffic signal control [3, 5, 19, 20] and network design [21–23], as well as network capacity [1, 3–5, 24]. But due to the difficulty of the sensitivity analysis for ETDA-VDC model, Yang et al. [1] used an iterative estimation-assignment (IEA) algorithm [11] instead to solve the transportation network capacity problem. Later, Kasikitwiwat and Chen [7] and Chen and Kasikitwiwat [8] selected using a genetic algorithm to solve the problem in a very small network. However, since the complexity of the network capacity problem, the global optimization algorithms (e.g., genetic algorithm or simulated annealing) can hardly find the exact solution to the capacity problem in larger networks, and the computation time could be intolerably long. By contrast, SAB algorithm has the property of fast convergence which makes the computation terminate at a local optimum within a considerable time. Nevertheless, the calculation issue of the matrix inversion still limits the applications of the SAB method. To address this problem, we developed an effective method by simplifying the matrix inversion in the sensitivity analysis approach, which will take much less memory space, so the capacity of the real transportation networks could be estimated.

In this study, Yang’s formulation [1] for the transportation network capacity model is employed to describe the practical capacity of the urban road system. In an attempt to estimate the capacity of the real road networks, a series of improvements are taken to the SAB method to make the heuristic search successfully converge to a relatively better suboptimal solution. Firstly, the OB algorithm [25] is modified for the solution of the lower-level ETDA-VDC model. Then, the restriction sensitivity analysis approach for the ETDA-VDC model [18] is employed in and improved on the expressions so as to deal with the large-scale problems. Besides, the solution update strategy is modified on the step-size adaption, which ensures the entire heuristic search to converge to a local optimum. Finally, numerical experiments are implemented to show the efficiency and capability of the proposed SAB method.

#### 2. Road Network Capacity Model

It should be noted that the boldface type of the Notation section represents the corresponding column vectors in the remainder of this paper.

##### 2.1. Model Formulation

Conversional methods, like the reserve capacity model, evaluate the capacity of transportation networks by assuming that the travel demand increases with a determined distribution proportion, which is usually far from the regularity and underestimates the results. In order to evaluate zonal development potential and equilibrium network capacity more appropriately, Oppenheim’s definitions on the behaviors of the existing travel demand and the additional demand are introduced as follows.(i)*The existing demand*, denoted by , has predetermined origins and destinations. The pattern of the existing demand is formed during the past long term, so its distribution is going to be relatively stable and can be regarded as fixed. The existing demand only changes routes to optimize the travel cost.(ii)*The additional demand*, , is variable. The new generated demands from residential area can decide their daily travels without the constraints of either destination or route choices. But the behavior of the additional demand still follows the rule that selects the destinations which maximize the “utility” of the trips.

The utility could include the destination attractiveness, the cost along traveling route, and other factors. The attractiveness of destination is determined by the congestion at destination and the expenses for the activity in that area. In network capacity model, the utility from origin to destination is formulated as , in which is the travel cost from to , and denotes the destination cost which could be a decrease function of the total additional trip attraction, , at destination . Besides, the destination choices of the travelers at each origin are assumed to have certain randomness. Thus, the conditional probability that an individual will choose destination is derived by using the standard logit function, so the O-D travel demand is conducted by

Thus, with the objective to maximize the additional demand under the above travel behavior regularity and certain physical constraints, the typical road network capacity model is formulated as the following bilevel programming problem.

*Upper-level problem *is as follows:where and are obtained by solving the ETDA-VDC problem in lower-level problem.

*Lower-level problem, ETDA-VDC model, *is as follows:

*The upper-level problem* defines a maximal trip production model. The objective is to maximize the summation of the additional trip production at origins. Equation (3) represents that the traffic flow on every link should not exceed its capacity. Constraints (4) and (5) are the limitation of the zonal trip production and attraction. They mean the number of trips generated and attracted at each traffic zone should be limited by some upper bounds, namely, and , respectively.

*The lower-level problem* is the ETDA-VDC model. The objective function (7) indicates the choice behavior of both the existing and additional travel demand. Constraint (9) shows that the amount of the existing flows is fixed for each O-D, while constraints (8) and (10) show that the additional flows are only restrained at the origin productions. The relationship between the link flow and route flow is represented in (11). All the variables must be nonnegative, that is, constraints (12)–(14). The lower-level problem is a combined distribution and assignment model.

This bilevel model was first presented in work by Yang et al. [1]. Because of the advantages on the formulation of the travel demand growth, it was continually used in later researches as a typical model for the road network capacity concept. The remaining part of this study focuses on the solution of this model in the real-sized road networks.

#### 3. Sensitivity Based Heuristic Algorithm

This section presents an improved version of the SAB algorithm for the solution of the road network capacity model. To overcome the drawbacks of the conventional SAB algorithm [9] that cannot be applied to any real-sized network for capacity estimations, the following improvements are carried out.(i)The lower-level ETDA-VDC model is fast solved by a modified OB algorithm to produce a high accurate solution.(ii)The rectified sensitivity analysis method for the ETDA-VDC model is simplified on the calculation of the matrix inverse to be applicable for large-scale problems.(iii)The solution update of the heuristic search is improved by step-size adaption in order to ensure that the SAB algorithm can converge to a local optimum.

Correspondingly, a series of techniques is proposed in this section, so our modified SAB algorithm will be capable to solve the bilevel road capacity model efficiently.

##### 3.1. Origin-Based Algorithm for the ETDA-VDC Model

In the standard SAB search, the lower-level ETDA-VDC model should be solved in every iteration to conduct an equilibrium traffic flow pattern, namely, the solution to the lower-level problem. According to recent researches on the traffic assignment problems, the OB algorithm is demonstrated to be one of the state-of-the-art algorithms [26]. In addition, from the results of the OB algorithm, the set of all equilibrated routes can be easily extracted, which will be a precondition for the restriction sensitivity analysis approach in next step of SAB algorithm.

The OB algorithm uses the* origin-based approach proportions*, , as its main solution variables, where represents the proportion of flow that comes from origin through link . The approach proportions are updated by shifting flows within the restricting subnetwork, . The details of the OB algorithm can be referred to in Bar-Gera [26]. Using the approach proportions, the OB algorithm is able to store the route flow information with significantly less memory than the route-based algorithms and achieve high-accuracy solutions compared to the link-based algorithms.

In the process of the OB algorithm, it starts with trees of minimum cost routes from origins as restricting networks. Then steps of updating restricting network and approach proportions are implemented for each origin separately. An “inner loop” is performed subsequently to accelerate convergence, in which the origin-based link flows are updated while keeping the restricting subnetworks fixed. The inner loop is useful, because the restricting subnetworks tend to stabilize fairly quickly but updating the restricting subnetworks requires more computational effort [26, 27].

The OB algorithm for the combined trip distribution and assignment problems requires an additional step to update the O-D flows while keeping the route proportions fixed. Taking Evans’ algorithm as a reference, we modified the original OB algorithm for the solution of the ETDA-VDC model, in which the step size of the O-D flow update is obtained by solving a one-dimension search problem. The proposed algorithm is summarized as shown in Algorithm 1.