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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.

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 [

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 [

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 [

Researches on other areas of transportation network optimization typically adopted two methods to address the uncertain O-D demand [

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.

The reserve capacity was proposed as the largest multiplier

The classical model of reserve network capacity (RNC) is defined as follows:

In the above model, the upper-level model maximizes the O-D matrix multiplier without violating the capacity constraints (

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 [

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

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 [

Robust solution of network capacities with different demand pattern.

In this study, we extended the reserve capacity model by considering the existing O-D demands as uncertain parameters within a certain bounded region. Robust solutions to the network capacity can be conducted using the robust optimization. We utilize the classical reserve capacity model to conduct the robust network capacity for two reasons: (i) the reserve capacity is easy to solve, and the O-D travel demand is allowed either increasing or decreasing by applying an O-D matrix multiplier greater than one or less than one; (ii) as the existing O-D matrix is extended to be an uncertain parameter in the reserve capacity model, the O-D distribution is no longer fixed but a variable pattern within some range given by the uncertainty set.

In this section, we assume that the prescribed O-D trip demand is unknown but bounded within an uncertainty set

In this study, three typical uncertainty sets were constructed for the existing travel demands.

Note that the shape of uncertainty set affects the efficiency and robustness of network capacity value. Ben-Tal and Nemirovski [

The above model is referred to as the robust counterpart of the original reserve network capacity problem. The solution of the robust counterpart results in a maximum total travel demand scheme under the corresponding worst-case demand pattern.

A heuristic algorithm is proposed to solve the above robust optimization model. It takes a similar framework as the procedure presented in [

Give the initial values of the O-D demand

Set the iteration counter

Compute

If the objective value of the WCS problem

In the above steps, the WCS problem is formulated to find a solution of

The second inner problem is a standard RNC model when the existing O-D demand is determined. The RNC can be solved efficiently by applying the SAB algorithm [

In this study, we used the restriction approach for the sensitivity analysis of the lower-level UE model. The restriction approach was proposed by Tobin and Friesz [

For the reserve capacity model, the link flows in upper-level,

From the results in Tobin and Friesz [

Thus, the derivatives of the link flows to the multiplier are obtained by

The WCS problem is also formulated as a bilevel programming using equilibrium constraints, so the SAB method can also be modified for its solution. The implicit relationship

The derivatives of the route flows,

Because in this inner problem the value of

Computational experiments are presented in this section to illustrate the results of the robust network capacity model. The example is based on a road network which is adopted from Nguyen and Dupuis [

Link characteristics of the example network.

Link number |
Free-flow time |
Capacity |
---|---|---|

1 | 7.0 | 800 |

2 | 9.0 | 400 |

3 | 9.0 | 200 |

4 | 12.0 | 800 |

5 | 3.0 | 350 |

6 | 9.0 | 400 |

7 | 5.0 | 800 |

8 | 13.0 | 250 |

9 | 5.0 | 250 |

10 | 9.0 | 300 |

11 | 9.0 | 550 |

12 | 10.0 | 550 |

13 | 9.0 | 600 |

14 | 6.0 | 700 |

15 | 9.0 | 500 |

16 | 8.0 | 300 |

17 | 7.0 | 200 |

18 | 14.0 | 400 |

19 | 11.0 | 600 |

An example network.

We applied the proposed approach for robust network spare capacity estimation with the three typical uncertainty sets which are described in this paper. Assume that the intervals for the O-D travel demands are

Robust reserve capacities with interval constraints under different maximal total existing demand.

Robust reserve capacities with ellipsoidal region and polyhedral region.

Ellipsoid | Polyhedron | ||||
---|---|---|---|---|---|

Parameter |
Max |
Robust capacity | Parameter |
Max |
Robust capacity |

0.0 | 0.1795 | 392.63 | 0.0 | 0.1944 | 388.89 |

0.2 | 0.1727 | 391.03 | 0.25 | 0.1892 | 378.38 |

0.5 | 0.1633 | 388.71 | 0.5 | 0.1843 | 368.54 |

1.0 | 0.1498 | 385.46 | 0.75 | 0.1795 | 359.01 |

2.0 | 0.1286 | 376.68 | 1.0 | 0.1753 | 350.54 |

Table

Robust reserve capacity estimations and the performances.

Network: Nguyen-Dupuis | Nominal | Conservative | Robust-e | Robust-p |
---|---|---|---|---|

Reserve capacity estimation | 388.89 | 363.42 | 385.46 | 368.54 |

Travel demand multiplier | 0.1944 | 0.1346 | 0.1498 | 0.1843 |

Percentage of meeting capacity constraints (%) | 17.2 | 100.0 | 98.4 | 72.4 |

Percentage of above-robust-estimation (%) | 51.8 | 100.0 | 54.6 | 98.8 |

To evaluate the four estimations of the network capacity, we randomly generate 500 samples of O-D demand as the possible realizations of the existing travel demand pattern. The samples are from the normal distribution,

Firstly, the user equilibrium assignment associated with each reserve network capacity estimation (i.e., the largest

Furthermore, considering that the demand multiplier in essence is a relative value, the reserve capacity results are derived. Therefore, the reserve capacity problem with the every random existing O-D demand is solved in our test. For each estimation value of the reserve capacity: the number of successes is counted whenever the reserve capacity value of the sample exceeds the robust capacity estimation. Consequently, the successful rates are computed as [total number of successes/number of samples]. These results are also presented in Table

Experiments are further presented on the Sioux-Falls network [

Table

Network capacity estimations on Sioux-Falls.

Network: Sioux-Falls | Nominal | Conservative | Robust-e | Robust-p |
---|---|---|---|---|

Reserve capacity estimation | 209,029.1 | 195,065.8 | 201,366.7 | 176,850.3 |

Travel demand multiplier | 0.5797 | 0.4291 | 0.5588 | 0.4904 |

We selected the conservative, robust-e, and robust-p solutions to further inspect the link flow patterns at the maximum travel demand situations (i.e., the reserve capacity). The link flow patterns are shown in Figures

Link flow pattern associated with different maximum travel demands (a, b, and c) and probabilities that links are congested (d).

Conservative solution

Robust solution with ellipsoid (

Robust solution with polyhedron (

Simulation results

In this study, a robust network capacity model with uncertain demand has been proposed. The robust optimization is formulated using the min–max model with a bounded uncertainty set of the existing O-D travel demands. With the uncertainty set, the low-probability realizations of the travel demand pattern are excluded, and thus the robust model can produce a proper estimation of network capacity which can be achieved with a large probability. Then, a heuristic algorithm has been proposed for the proposed robust model. It solves two inner problems iteratively: one is the worst-case scenario problem; and the other is the relaxed robust optimization, namely, the standard reserve capacity model. At each iteration, the cutting plane method has been adopted to generate the worst-case demand scenario, and the sensitivity analysis based approach has been developed for the solution of the worst-case model and the reserve capacity model. The validity and performance of the proposed robust model have been demonstrated in the computational experiments. Different results under three typical uncertainty sets, say interval, ellipsoidal, and polyhedral region, have been conducted and compared. The interval set is simple but easy to produce too conservative results; the ellipsoidal set is a good approximation to the uncertain region and produces results with moderate robustness, but its solution is more complicated due to the nonlinear constraints; the polyhedral set is considered if a high level of robustness is required, and its linear formulation makes the robust model easier to solve. Furthermore, by conducting computational experiments on the Sioux-Falls network, robust solutions shown can provide more practical results of the link flow patterns. In applications, these uncertainty sets and their parameters should be selected according to the desired level of robustness.

The robust model based on the reserve capacity model has been proposed and explored in this study. Future researches should focus on more efficient solution approaches for the robust problem with the min–max model. The experiments on large-scale networks are also needed. Besides, the demand uncertainties existing in other network capacity models are also expected to be detected and discussed. Alterative traffic assignment model, such as the stochastic user equilibrium, could also be discussed for the network capacity problems. The robust solution of network capacity gives a lower bound to the possible schemes of the maximum demand in a transportation network. These possibilities constitute a range where the robust solution can be most likely to be reached in reality. Therefore, the robust solution to network capacity problems needs to receive more attentions in transportation planning applications.

An initial version of this paper was presented at the Transportation Research Board (TRB) 96th Annual Meeting. It therefore also appears in the proceedings of the TRB 96th Annual Meeting Compendium of Papers.

The authors declare that they have no conflicts of interest.

This research is supported by “the National Natural Science Foundation of China (no. 51508161),” “the Natural Science Foundation of Jiangsu Province (no. BK20150817),” and “the Fundamental Research Funds for the Central Universities (no. 2017B12414).”