Journal of Advanced Transportation

Volume 2017 (2017), Article ID 9814909, 11 pages

https://doi.org/10.1155/2017/9814909

## Robust Evaluation for Transportation Network Capacity under Demand Uncertainty

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

Correspondence should be addressed to Muqing Du

Received 28 April 2017; Accepted 17 July 2017; Published 10 September 2017

Academic Editor: Dongjoo Park

Copyright © 2017 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

As more and more cities in worldwide are facing the problems of traffic jam, governments have been concerned about how to design transportation networks with adequate capacity to accommodate travel demands. To evaluate the capacity of a transportation system, the prescribed origin and destination (O-D) matrix for existing travel demand has been noticed to have a significant effect on the results of network capacity models. However, the exact data of the existing O-D demand are usually hard to be obtained in practice. Considering the fluctuation of the real travel demand in transportation networks, the existing travel demand is represented as uncertain parameters which are defined within a bounded set. Thus, a robust reserve network capacity (RRNC) model using min–max optimization is formulated based on the demand uncertainty. An effective heuristic approach utilizing cutting plane method and sensitivity analysis is proposed for the solution of the RRNC problem. Computational experiments and simulations are implemented to demonstrate the validity and performance of the proposed robust model. According to simulation experiments, it is showed that the link flow pattern from the robust solutions to network capacity problems can reveal the probability of high congestion for each link.

#### 1. Introduction

The capacity of transportation network reflects the supply ability of its infrastructure and service to the travel demand which is generated from the zones covered by the transportation system in a specific period. For many years, transportation planners and managers wanted to understand how many trips can be accommodated at the most by the current or designed network in a certain period of time. This need is more necessary in those developing regions which are confronted with rapid growth of private vehicles and increased urban congestion. Meanwhile, the researchers made a long-term effort to model and estimate the maximum throughput of transportation networks. The achievements include max-flow min-cut theorem [1], incremental assignment approach [2], and later bilevel programming models [3–5].

For the network capacity model, the most popular formulation in passenger transportation system is the bilevel model, which maximizes the traffic flows under the equilibrium constraints. Wong and Yang [3] first incorporated the* reserve capacity* concept into a traffic signal control network. The reserve capacity is defined as the largest multiplier applied to a given O-D demand matrix without violating capacity constraints, so the solution is significantly affected by the predetermined O-D matrix. Ziyou and Yifan [6] extended the reserve capacity model by considering O-D specific demand multipliers, and all demand multipliers should be ensured not lower than a predetermined minimum value. In order to avoid assuming that all O-D flows increase in a same rate, another concept of* ultimate capacity* was proposed [5]. But it assumes that the O-D distribution is totally variable, which may produce unrealistic results that cause the trip productions at some origins below their current levels. Furthermore, Yang et al. [4] suggested that the new increased O-D demand pattern should be variable in both level and distribution, while the current travel demand is fixed. Later, Yang’s model was also referred to as the* practical capacity* by Kasikitwiwat and Chen [5]. In summary, although unrealistic, the reserve capacity model is more easy-to-use and has been adopted widely in many researches [7–10]. The ultimate capacity and practical capacity model are more practical but have more parameters to be calibrated when applied, and the formulated models are still difficult to solve [11].

While the deterministic network capacity problem has been explored extensively, few studies have investigated the issue of uncertainties in demand data associated with this problem. The ultimate capacity and practical capacity model are only concerned with the uncertainties related to the new increased travel demand by using combined models [5], while the uncertainties in the current (or existing) demand are not considered. In reality, travel demands in transportation system are always fluctuant day by day, even hour by hour. Besides, errors of survey data also affect the accuracy of the existing O-D matrix. As a consequence, the existing travel demands are usually difficult to be obtained in actual transportation projects and then are not easy to be represented using fixed values. As the existing O-D matrix is usually used as the reference matrix in reserve capacity or practical capacity model and its pattern significantly influences the result of the models, we first consider it as an uncertain variable in this study. And thus the network capacity model is extended to be an optimization with parameter uncertainty.

Researches on other areas of transportation network optimization typically adopted two methods to address the uncertain O-D demand [12]: (i) stochastic optimization aims at maximizing the expected profit by assuming that the demand follows a known probability distribution; (ii) robust optimization aims at maximizing the profit with the worst-case scenario of the demand pattern. Considering the exact probability distribution of the O-D demand is still hard to be obtained, the robust optimization is more effective in dealing with this problem. If a limited number of discrete scenarios of O-D demand patterns are detected, the scenario-based robust optimization [13] is conducted, which is a practical approach usually implemented in transportation projects. It is more general to assume the possibility of the travel demand to be a continuous variable within a bounded set, and the set-based robust optimization can be used for decision-making [14]. The uncertainty set is constructed to include most of possible values of the travel demand. The decision-makers’ attitudes to risk should be considered as well when deciding the shapes and size of the uncertainty sets. It is important to make a trade-off between the system performance and the level of robustness achieved [13].

In this study, we propose a robust optimization model for the network capacity problem by using the existing O-D travel demands as uncertain parameters. The existing demand between each O-D pair is assumed to be variable between its upper and lower limits. Besides, three typical uncertainty regions are introduced to provide a bounded set for the uncertain demand. A heuristic solution is developed for the solution to the robust network capacity model. In the next section, the concept of network spare capacity is revisited based on the reserve capacity model. Then, the robust model for network capacity estimation is presented, and the three typical uncertainty sets of existing travel demand are defined. After that, the solution algorithm is described. Computational experiments show the validation and justification of the robust model. Conclusions and perspectives for further research are provided in the last section.

#### 2. Network Spare Capacity and Its Flexibility

The reserve capacity was proposed as the largest multiplier applied to a given existing O-D demand matrix that can be allocated to a transportation network without violating any individual link capacity [3]. The product of the largest multiplier and the existing O-D demand (represented by vector** q**) gives the maximum travel demand which can be loaded to the network. For clarity sake we refer to the maximum travel demand as the value of* network capacity* in rest of this paper. For passenger network, it is well known that multiple O-D pairs exist and demands between different O-D pairs are not exchangeable or substitutable. Thus, the* travel demand pattern* or* matrix* reflects both its quantity and spatial distribution. The method of reserve capacity assumes that the existing O-D demand is scaled with a uniform O-D growth. The largest value of indicates whether the current network has spare capacity or not. So the* network spare capacity* is generally explained as follows: if > 1, then the network can be loaded more travel demand and the additional demand can be accommodated by the network which is ; otherwise, that is, < 1, the network is overloaded and the existing O-D demand should decrease by to satisfy the capacity constraints [10]. In some researches, the demand multiplier is regarded as the uncertainties in the future O-D demand [9, 15].

The classical model of reserve network capacity (RNC) is defined as follows:where is obtained by solving the following user equilibrium problem:where is the O-D demand multiplier to all O-D demands; is the set of all routes in the network; is the origin index, , and is the set of all origin nodes; is the destination index, , and is the set of all destination nodes; is the capacity of link ; is the flow on link , ; is the vector of all link flows; is the existing trip demand between O-D pair ; is the vector of all O-D demand; is the flow on route , , between O-D pair associated with ; is the vector of flows of all route in ; is the link-route incidence indicator: 1 if link is on route between O-D pair and 0 otherwise; is the travel cost function for link .

In the above model, the upper-level model maximizes the O-D matrix multiplier without violating the capacity constraints (2) for every individual link. The parameter gives the prescribed O-D travel demand in the network, which can be obtained according to the current trip demand or a predicted demand pattern accordingly. Route choice behavior and congestion effect are considered in the user equilibrium (UE) model as the lower-level model in (3)–(6). Generally, other traffic assignment methods, such as stochastic user equilibrium (SUE) model, can be used in place of the above deterministic UE model as required [10].

The result of the reserve capacity model which is considered may underestimate the capacity of the passenger network, because only the existing O-D demand pattern that is more congruous with the network topology would achieve a higher value of network capacity [16]. Basically, the reserve capacity depends on the initial O-D demand patterns and route choice behavior of the users. Given the lower-level traffic assignment method, the existing O-D demand should be the only determinant to the result of the above model. It means that if the given O-D matrix is not consistent with the network, the reserve capacity model will produce a result having a low level of maximum demand. Otherwise, if the O-D pattern is determined according to the network spatial structure, the travel demand can grow to a very high amount.

Directly applying the result of the reserve capacity may have the following problems. (i) It is hard to decide an exact existing (or predetermined) O-D matrix, because the real travel demand pattern is changing at different hours every day and different days every week. Also, it is still very difficult to obtain the full data of the O-D demands covering many different hours. (ii) In real-world applications, decision-makers tend to be risk averse and may be more concerned with the worst cases. Using only a few situations of the O-D demand pattern may not provide a* robust* answer to the network capacity estimation. Conversely, as long as the system performance reaches an acceptable level, it does not matter how much it changes above that level. Thus, it may be more desirable to have an optimization result that performs better in the worst case.

When estimating the capacity of transportation systems, decision-makers are not only concerned with the extreme results that the total trips can be allocated to a transportation network but also need to evaluate the unknown situations resulted from the fluctuation of the travel demand. Thus, to measure the ability of transportation networks that can deal with the variation of travel demand, Chen and Kasikitwiwat [16] discussed the concept of the* network capacity flexibility* using three typical network capacity models. The network capacity flexibility is defined as the ability of a transport system to accommodate changes in traffic demand while maintaining a satisfactory level of performance [16, 17]. In this study, integrated with the uncertainties from the existing demand in transportation networks, the network capacity flexibility is further illustrated in Figure 1. On the basis of this, the robust estimation of network capacity is defined as* the maximum travel demand can be allocated to a transportation network when satisfying all the possibilities of the uncertain changes in the quantitative and spatial demand pattern.* The robust value of the network capacity is also illustrated in Figure 1.