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This study provides a gradient projection (GP) algorithm to solve the combined modal split and traffic assignment (CMSTA) problem. The nested logit (NL) model is used to consider the mode correlation under the user equilibrium (UE) route choice condition. Specifically, a two-phase GP algorithm is developed to handle the hierarchical structure of the NL model in the CMSTA problem. The Seoul transportation network in Korea is adopted to demonstrate an applicability in a large-scale multimodal transportation network. The results show that the proposed GP solution algorithm outperforms the method of the successive averages (MSA) algorithm and the classical Evan’s algorithm.

Combined travel demand models (or combined models) have been used as a network equilibrium approach to resolve the inconsistency problem (i.e., flow) in the traditional four-step travel demand model (see, e.g., [

Various formulations, ranging from mathematical program (MP), nonlinear complementarity problem (NCP), variational inequality (VI), and fixed point (FP), have been provided to represent different combined models listed above. Please refer to [

For solving traffic assignment problems, numerous algorithms were developed based on link-based, path-based, and bush-based algorithms. The link-based algorithms operate in the space of link flows and do not require path storage, but it requires more computational efforts comparing to other groups of algorithms. On the other hand, path-based algorithms require explicit path storage in order to directly compute path flow equilibration and they have faster convergence comparing to link-based algorithms. The reason is that the link-based algorithm exhibits a zigzagging trajectory as it approaches the optimal solution. On the other hand, the path-based algorithm does not exhibit the zigzagging effect for the same initial solution. In the transportation literature [

Mitradjieva and Lindberg [

The disaggregated simplicial decomposition algorithm proposed by Larsson and Patriksson [

The bush-based algorithm sequentially solves a list of node-based subproblems defined on the bush rooted. The algorithm B adjusts flows between the longest paths and the shortest ones in the bush rooted path tree. A paired alternative segment (PAS) is defined as two completely disjoint path segments, and then the flows are adjusted in disjoint path segments.

In this paper, we are interested in developing solution algorithms for solving the MP formulation of the CMSTA problem that can represent large-scale multimodal transportation networks with both private and public modes and correlation among multiple public modes using the nested logit (NL) model under the network user equilibrium (UE) framework. Contrast to the simple structure of the multinomial logit (MNL) model, the NL model has a hierarchical structure that decomposes the choice probability into two levels represented by the marginal and conditional probabilities. A two-phase GP algorithm is developed to solve the CMSTA problem with a hierarchical nested modal split structure and UE traffic assignment, along with various improvement strategies to speed up the convergence of the path-based GP algorithm.

In the literature, a path-based GP algorithm has been adopted for solving various traffic assignment problems such as UE assignment problem, the nonadditive cost assignment problem, side constrained problem, stochastic user equilibrium assignment problem, transit assignment problem, system optimal assignment problem, and freight traffic assignment problem [

Our goal is to demonstrate that the improved GP algorithm can solve large-scale multiclass or multimodal traffic assignment problems in CMSTA with NL function. In addition, the proposed algorithm can apply to other logit models having a hierarchical structure such as the generalized extreme value model by [

This paper is organized as follows. After the introduction, a relevant background of the NL model for the modal split problem and UE conditions for the traffic assignment problem is introduced in Section

In this section, we provide some background of the nested logit (NL) model for the modal split problem and the UE model for the traffic assignment problem. A list of variable is provided first for convenience, followed by the NL model and multiclass UE model.

Consider a transportation network

List of variables.

Notation | Definition |
---|---|

Set | |

R | Set of origins, R |

S | Set of destinations, S |

RS | Set of O-D pairs |

Set of routes connecting O-D pair | |

Set of upper nests connecting O-D pair | |

Set of lower nests connecting O-D pair | |

Set of transit modes connecting O-D pair | |

Parameters and inputs | |

Given exogenous utility of mode | |

Expected perceived utility of mode | |

O-D specific parameter | |

Upper nest parameter of each O-D pair | |

Conditional probability | |

Marginal probability | |

Convergence error in the inner loop | |

Convergence error in the outer loop | |

Dwelling times on link | |

Equal to 1 for link | |

Length on link | |

Walking speed (i.e., 4 km/h) | |

Weighting time for the transit mode (bus and metro) on link | |

Transit fare connecting O-D pair | |

VOT | Inverse value of time for converting transit fare into equivalent time unit |

Total travel demand connecting O-D pair | |

Capacity on link | |

Decision variables | |

Route flow on the mode | |

Route cost on the mode | |

Minimum cost on the mode m connecting O-D pair | |

Route flow of the private transport mode on route | |

Route cost of the private transport mode route | |

Route cost of the private transport mode shortest route connecting O-D pair | |

Positive-definite scaling factor between route | |

Link flow of private transport mode on link | |

Travel time of private transport mode on link | |

Route-link indicator is 1 if link | |

Total demand of public transport mode connecting O-D pair | |

Demand from public transport mode | |

Generalized cost of public transport mode | |

Bus and metro route-link indicators: 1 if link | |

Abbreviations | |

GP | Gradient projection |

CMSTA | Combined modal split and traffic assignment |

NL | Nested logit |

UE | User equilibrium |

MSA | Method of successive averages |

CDA | Combined distribution and assignment |

MP | Mathematical program |

NCP | Nonlinear complementarity problem |

VI | Variational inequality |

FP | Fixed point |

MNL | Multinomial logit |

CNL | Cross-nested logit |

PCL | Paired combinatorial logit |

KKT | Karush–Kuhn–Tucker |

MAE | Mean absolute error |

RMSE | Root-mean-square error |

ATT | Average travel time |

IGP | Improved gradient projection |

The NL model is widely used to represent mode choice in the literature (see, e.g., [

Tree structure of the MNL and NL models: (a) multinomial logit; (b) nested logit.

The NL mode choice probability can be expressed as

With the tree structure, the NL probability can be decomposed as

Note that if the parameter (

For the road-based transportation mode such as passenger car and bus, we have the Karush–Kuhn–Tucker (KKT) conditions at equilibrium as follows:

Equations (

Without loss of generality, two types of transport modes (i.e., private vehicle and public transport) are considered in the upper nest while the lower nest can accommodate more than two modes (e.g., bus, metro, and bus-metro). Note that it can be extended to include more than two types of transport mode in the upper nest. Assuming that the number of public transport modes operated on a link in the road network is given (i.e., fixed number of buses on a link), the in-vehicle time of public transport mode (i.e., bus) is affected by the number of private vehicles on a link. In addition, the road capacity on a link is reduced as the number of buses increases. These assumptions enable to present the excess demand function analytically. Consider the following MP formulation:

The objective function consists of three terms. _{1} is the well-known Beckmann’s [_{2}−_{5} correspond to the NL model. Equation (

To set up the generalized travel costs (i.e., the lower nest modes), we adopt a flow-dependent travel time for the auto mode and bus mode and a flow-independent travel time for the metro mode as follows:

Travel times on each link are assumed to follow the Bureau of Public Roads (BPR) function.

The MP formulation in equation (

The Lagrangian of the equivalent minimization problem with respect to the equality constraints can be formulated as follows:

Given that the Lagrangian has to be minimized with respect to non-negative route flows and modal splits, the following conditions have to hold:

Note that equation (

From equation (

Rearranging equation (

From equation (

From

These probabilities are consistent with the NL model illustrated in the previous section for a case with two upper nests. This completes the proof.

The MP formulation in equation (

For the proof of Proposition

From equation (

Hence, when these two costs are equilibrated (i.e.,

Figure

Illustration of the mode choice equilibration under the nested logit modal split function; (a) network, demand, and cost functions; (b) upper level equilibrium (marginal prob.); (c) lower level equilibrium (conditional prob.); (d) travel time change by flows; (e) equilibrium solution; (f) probabilities verification; (g) cost verification.

The gradient projection (GP) algorithm is a well-known path-based algorithm for solving various traffic assignment problems. Before considering the gradient projection (GP) algorithm for solving the CMSTA model (or NL-UE model), we provide a brief review on the ordinary GP algorithm for solving the UE traffic assignment problem with a fixed demand. Equation (

Equation (

For a detailed description of the GP algorithm, readers should refer to [

Unlike the UE traffic assignment problem, the CMSTA problem considered in this paper (i.e., UE for route choice and NL for mode choice) requires not only equilibrating route flows of the auto mode for route choice but also equilibrating modal splits among multiple available modes (i.e., both private and public transport modes) following the NL model for mode choice. Based on the hierarchical choice structure given in Figure

Figure

Demand adjustment of the first phase equilibration (i.e., private car and public transit). (a) Demand adjustment on the auto mode according to _{.}

Once the upper nest is equilibrated in the first phase (i.e., the auto demand

The disaggregate simplicial decomposition (DSD) algorithm proposed by Larsson and Patriksson [

In this study, we apply a similar adaptive column generation scheme in the gradient projection algorithm. To improve the computational efficacy in the few iterations, we adopt the following condition in the inner loop (master problem) as the termination step:

First, the convergence error in the outer loop is computed after the column generation step. Then, the inner loop (equilibration) step is performed without the column generation step until it satisfies the termination condition in equation (

Another implementation improvement is the column dropping scheme. If the path set

Figure

Gradient projection algorithm with restricted equilibration.

Figure

Two-phase flow update strategy.

As described above, modified GP extends the equilibration of private vehicles (i.e., traffic assignment) to also consider equilibration of multiple transport modes (i.e., modal splits) by using a two-phase procedure which is the equilibration between public and private modes as the first phase and equilibration of modal splits among the public transport modes as the second phase.

In the column generation step, we adopt origin-based column generation to improve computation efficiency. See [

The transportation network in the city of Seoul, Korea, is used to investigate the performance of the proposed two-stage GP algorithm in solving the NL-UE model. The convergence characteristics are tested, and then parameter sensitivity analysis is examined.

The Seoul transportation network shown in Figure

Multimodal transportation network in Seoul, Korea: (a) auto network; (b) bus network, and (c) metro network.

Summary of the multimodal transportation network in Seoul, Korea.

Auto | Bus | Metro | Bus-metro | |
---|---|---|---|---|

Total length (km) | 2,790 | 1,202 | 467 | 1,669 |

Number of stations | — | 3,635 | 354 | 3,989 |

Number of lines | — | 1,765 | 14 | 1,779 |

Nested mode choice structure and its modal splits.

The occupancy of auto mode and bus mode is set 1.3 per passenger car and 19.27 per bus, respectively (MTA, 2015). The transit has the distance-based fare structure. Without loss of generality, the metro mode is assumed to provide enough operation capacity such that there is no congestion in this mode. One hundred maximum number of inner loop is set, and

The improved GP algorithm is coded in Intel Visual FORTRAN XE and runs on a 3.60 GHz processor and 16.00 GB of RAM.

Figure

Convergence characteristics.

In the following analyses, we conduct the sensitivity analysis with respect to the inner loop parameters

Computational efforts under different parameters: (a) computational efficacy with varying

This section examines the sensitivity of the IGP algorithm with respect to the nested logit parameters (i.e.,

Effect of logit parameters on modal splits.

Effect of logit parameter on flow allocation: (a) auto flow difference; (b) bus flow difference; (c) metro flow difference; (d) bus-metro flow difference.

Figure

Specifically, the figure shows the mean absolute error (MAE) and the root-mean-square error (RMSE) for assessing the link flow difference and average travel time (ATT) (i.e., total travel time/mode demand). Recall that the modal splits for case 1 are 72.19% and 27.81% for auto and transit mode, respectively, and the modal splits for case 2 are 75.30% and 24.70% for auto and transit mode, respectively. Because the modal splits are different using different parameters, the link flow pattern is also a different pattern as shown in the GIS map. The highest RMSE value is 164.16 in the metro mode, and the highest MAE is 99.51 in the bus mode. Albeit the link flow patterns are significantly different, the ATT values are similar values. This is because of the scaling effect in the large network. Although the ATT difference is less than 1.0 minutes between two cases, the total trip difference is 2,874,441 trips.

In this paper, we presented (1) the CMSTA model with the nested logit (NL) model for the mode choice and the user equilibrium (UE) model for the route choice and (2) a modified-and-improved gradient projection (IGP) algorithm to solve the proposed NL-UE model. The proposed NL-UE model considered the mode similarity through the NL model under the congested network obtained by the UE condition. Specifically, an equivalent MP formulation for the NL-UE model was provided using a modified access demand to incorporate the two-level nested tree structure. On the other hand, the IGP algorithm was developed in such a way to perform a two-level equilibration. In the first phase, the upper nest of the NL model is solved. Then, the lower nest of the NL model is applied with updated flows from the upper nest in the second phase. For the solution algorithm, the improved GP algorithm with inner loop equilibration procedure was introduced and shown the effectiveness in solving the combined NL modal split and UE traffic assignment (NL-UE) problem. This is because the excess demand cost function consists of two logarithm terms. These terms are sensitive to a small traffic flow change. A more stabilized path flow is required before generating new paths.

A real-size transportation network in Seoul, Korea, was used to demonstrate the applicability of the IGP algorithm for solving the proposed NL-UE model. The algorithm can conduct 1E-8 convergence criteria. The convergence time of the IGP algorithm seemed to be better than that of the ordinary GP algorithm. Further, the NL parameters have a significant impact on the convergence time and the mode share results.

For future research, the improved GP algorithm should be extended to consider mode (or vehicle) interactions with asymmetric cost functions. In addition, the route choice model for transit modes should be considered with transit information systems (e.g., arrival information). It would be interesting to see how the improved GP algorithm performs when more realistic features are incorporated into the CMSTA problem. In addition, a time-dependent model can be considered with travel activity patterns.

Equation (

Hence, equation (

The basic data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

Seungkyu Ryu collected and analyzed the data, interpreted the results, and prepared the manuscript draft. All authors contributed to study conception and design, reviewed the results, and approved the submitted version of the manuscript.

This research was supported by the Basic Science Research Program through the National Research Foundation (NRF) of Korea by the Ministry of Science (grant no. NRF-2016R1C1B2016254), the Ministry of Science, ICT, Republic of Korea (project no. K-21-L01-C05-S01), the Research Grants Council of the Hong Kong Special Administrative Region (project no. 15212217), the Research Committee of the Hong Kong Polytechnic University (project no. 1-ZVJV), and the Chiang Mai University.