A Link-Node Nonlinear Complementarity Model for a Multiclass Simultaneous Transportation Dynamic User Equilibria
چکیده
Static transportation network equilibrium models have evolved from traditional sequential models to simultaneous (combined) models, and then to the multiclass simultaneous models to improve prediction of traffic flow. Most Dynamic Traffic Assignment (DTA) models, however, still deal only with the trip assignment step (traveler route choice) that is one of several steps in the transportation planning process. In this paper, the authors combine a dynamic link-node based discrete-time Nonlinear Complementarity Problem (NCP) DTA model with a static Multiclass Simultaneous Transportation Equilibrium Model (MSTEM) in a unified dynamic link-node based discrete-time NCP Dynamic Multiclass Simultaneous Transportation Equilibrium Model (DMSTEM) model. The new model improves the prediction process and eliminates inconsistencies that arise when the DTA or Dynamic Traffic Assignment with Departure Time (DTA-DT) is embedded in a more comprehensive transportation planning framework. An iterative solution algorithm for the proposed DMSTEM model is proposed by solving several relaxed NCPs in each iteration of the algorithm. DOI: 10.4018/joris.2013010102 28 International Journal of Operations Research and Information Systems, 4(1), 27-48, January-March 2013 Copyright © 2013, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. tion of time-dependent traffic flow of a traffic network assuming travelers follow rational route choices (such as to minimize travel time or other forms of cost), is one of the most challenging. Two distinct approaches have dominated the methodologies applied to DUE research: the simulation-based (microscopic/mesoscopic) approach and the analytical (macroscopic) approach. A review of these two approaches can be found in Ran and Boyce (1996); Peeta and Ziliaskopoulos (2001). Among the analytical modeling techniques used for DUE, Variational Inequality (VI) and Nonlinear Complementarity (NC) have been shown to be the most effective approaches (Friesz et al., 1993; Ran & Boyce, 1996; Lo et al., 1996; Lo & Szeto, 2002; Chen, 1999; Heydecker & Verlander, 1999; Peeta & Ziliaskopoulos, 2001; Akamatsu, 2001; Wie et al., 2002). Ban et al. (2008) recently applied NCP to model DUE with exact flow propagation based on a specially designed discretization scheme. Friesz and Mookherjee (2006) developed a differential variational inequality (DVI) technique to model DUE in continuous time. The DVI framework has been used more recently by Friesz et al. (2010) to model both the within-day and day-today traffic dynamics, and by Pang et al. (2009) to model multi-class single bottleneck dynamic departure time choice problems. In the DUE literature, elasticity of the dynamic demand (input to DUE models) is usually modeled as departure-time choices for a given total fixed demand flow between each O-D pair during the whole time horizon (Friesz et al., 1993). Multiclass DUE models have also been studied with these analytical approaches (Ran & Boyce, 1996; Belimer et al., 2003) or simulationbased approaches (Ben-Akiva et al., 2001; Mahmassani et al., 2001). The early studies, however, focused on different vehicle classes and a few social-economic factors of drivers, such as value of times (VOD) preferences (Lu et al., 2008). With this goal, Ramadurai and Ukkusuri (2010) integrate activity-based models for demand analysis into DUE. The model is solved using a super-network representation of activities and traffic networks. By investigating further how socioeconomic factors of drivers may affect trip generation, trip distribution, mode choice, and departure-times in a dynamic (i.e. timedependent) fashion, and integrating the findings into DUE modeling, we expect to improve the practicality and accuracy of DUE models. With this objective, we combine the NCP-based DUE model in Ban et al. (2008) and the Multiclass Simultaneous Transportation Equilibrium Model (MSTEM) in Hasan and Dashti (2007) to develop a multiclass combined Trip Generation (TG)-Trip Distribution (TD)-Nested Modal Split (NMS)-Modal Split (MS)-Dynamic Trip Assignment (DTA)-Departure Time (DT) Model (NCPDUE-TG-TD-NMS-MS-DTA-DT). In the next paragraphs we provide the necessary background on static multiclass combined models as employed in the Multiclass Simultaneous Transportation Equilibrium Model (MSTEM) that we utilize in this paper. Single-class static combined equilibrium models consider all travelers as one homogeneous group with respect to their travel-decision characteristics (particularly their money-value of time and sensitivity to travel times in choosing their origin, destination and mode of travel, etc. To be more realistic, travelers must often be divided into classes -either by socio-economic attributes or purpose of travel. Travel-decision characteristics are thus the same within each class, but differ among classes. The introduction of multiple classes increases the mathematical complexity of travel forecasting models. Travel costs in single class models are often separable and symmetric, allowing for convex optimization formulation. In multiclass models, travel costs of one class are affected by decisions of other classes; hence the cost structure is not separable, asymmetric, and not amenable to a convex optimization formulation (Patriksson, 2003). Boyce and BarGera (2004) review the most recent multiclass combined models and provide a good detailed comparison of the implementation of four multiclass combined models. Boyce (2004) presents the most recent comprehensive review and prospects for network equilibrium models. International Journal of Operations Research and Information Systems, 4(1), 27-48, January-March 2013 29 Copyright © 2013, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. As summarized by Hasan and Dashti (2007), researchers in Chile began to implement multiclass combined models in 1986 -emphasizing route choices in a congested transit network with several combinations of transit modes, as found in Santiago (de Cea et al., 2003). This research led to the development of ESTRAUS and related software, which has been extensively applied to Santiago as well as other Chilean cities. Florian, We and He (2002) proposed a variant of ESTRAUS intended to be more efficient computationally. Although several works concerning departure time choices are reported in the technical literature, this travel decision has not been integrated yet to supply-demand equilibrium models. Dekock (2001) and Dekock et al. (2001) have described a simultaneous equilibrium model considering trip distribution, modal split and departure time choices. The basic idea of this model is that even if trip generations (and attractions) are fixed for a given period of time, users can choose the sub-period in which they travel, according to a Logit model. The ESTRAUS combined model considers a doubly constrained entropy-maximizing model, while modal split and departure time choices are modeled with a hierarchical Logit model. Based on these works, two different models, depending on the relative values of the calibration parameters of the Logit model, were developed but not used yet in any implementation.. All existing multiclass combined models, consider that the total originating and terminating flows are known, i.e., the trip generation step of transportation planning process is exogenous to the combined prediction process. This deficiency wass addressed in the STEM model which is the only model that combined the trip generation in the prediction process, but it is not a multiclass model. The STEM model was extended to incorporate multiple user classes (socio-economic groups), trip purposes, and both pure and combined transportation modes, interacting over a physically unique network. The resulting Multiclass Simultaneous Transportation Equilibrium Model (MSTEM) also combined explicitly the departure time as one of the main components of the prediction process for the first time and will be considered as a new generation of new Case 6 of the multiclass model classification of Boyce and Bar-Gera (2004). The MSTEM included all the features of ESTRAUS in addition to the other features mentioned above and a more flexible structure for demand models where the trip generation can depend upon the system’s performance through an accessibility measure that is based on the random utility theory of users’ behavior (instead of being fixed as in ESTRAUS). The trip distribution is determined by a more behaviorally richer Multinomial Logit (MNL) model based on the random utility theory (instead of being given by an entropy maximization model in ESTRAUS), and modal split and departure time are given by Multinomial Logit (MNL) models based on the random utility theory (instead of hierarchical Logit for modal split used in ESTRAUS). In the next section we introduce the discrete-time NCP-based DUE model in Ban et al. (2008).Section 3 includes the brief description of the Multiclass Simultaneous Transportation Equilibrium Model (MSTEM) of Hasan and Dashti (2007). Section 4 gives a detailed development of discrete-time domain finite-dimensional Nonlinear Complementarity DUE Multiclass Simultaneous Transportation Equilibrium Model (DMSTEM). Section 5 covers in detail the solution algorithm of the NCP that was formulated in section 4. The where the appendix gives a detailed description of the relaxed sub-problems that should be solved by the algorithm. In Section 6 the main conclusions of the paper are presented along with some directions of future research. THE NONLINEAR COMPLEMENTARITY FORMULATION FOR DUE In this paper, we adopt the NCP model in Ban et al. (2008). In particular, we show how to extend it to DUE with multiclass trip generation, trip distribution, modal split, and departure time choices. Our focus here is the finite-dimension30 International Journal of Operations Research and Information Systems, 4(1), 27-48, January-March 2013 Copyright © 2013, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. al NCP model in discrete-time domain. In order to obtain the discrete-time model, Ban et al. (2008) evenly divided the entire study period into K ' time intervals by introducing the length of each time interval ∆ such that K T ' ' ∆ = . We use index k to denote the k-th time interval [( ) , ) k k −1 ∆ ∆ for1≤ ≤ k K ' . The notation for the discrete-time model is first listed as follows (Ban et al., 2008): u u k as k as = − (( ) ) 1 ∆ : The inflow rate to link a towards destinations at the beginning of time intervalk , which is also assumed to be constant during the entire time interval k v v as k a s k , ( ) , , = ∀ . v v k as k as = − (( ) ) 1 ∆ : The exit flow rate from link a towards destination s at the beginning of time interval k, which also assumed to be constant during the entire time interval k v v as k a s k , = ∀ ( ) , , . u as k ∆ : The inflows to link a towards destination s during time interval k. v as k ∆ : The exit flows to link a towards destination s during time interval k. x x k as k as = − (( ) ) 1 ∆ : The flows of link a towards destination s at the beginning of time interval k, which also assumed to be constant during the entire time interval k x x as k a s k , = ∀ ( ) , , . u u s k as k s S = ∈ ∑ : The aggregated inflow rate to link a at the beginning of time interval k, which assumed to be constant during the entire time interval k u u A a k a k , = ∀ ( ) , . v v a k as k s S = ∈ ∑ : The aggregated exit flow rate from link a at the beginning of time interval k, which assumed to be constant during the ent i re t ime interval k v v A a k a k , = ∀ ( ) , . x x a k as k s S = ∈ ∑ : The aggregated flows of linka towards destination s at the beginning of time interval k, which assumed to be constant during the entire time interval k x x A a k a k , = ∀ ( ) , . d d k is k is = − (( ) ) 1 ∆ : The demand rate generated from node i towards destination s at the beginning of time intervalk , which assumed to be constant during the entire time intervalk d d is k i s k i s , = ∀ ≠ ( ) , , , . d is ∆ : The demands generated from node i to destination s during the time interval k Ca k u Ca k ( ) ( ) = ∆ : The travel time of link a at the end of time interval k, a function of u,C Ca k a k = ∀ ( ) , . π π is k is k = ( ) ∆ : The minimum travel time from node i to destination s at the end of time intervalk is k i s k i s , ( ) , , , π π = ∀ ≠ . ea k u k Ca k u ( ) ( ) ( ) ≡ − + − 1 1 ∆ : The exit time for vehicles entering link a at the beginning of time interval k, a function of u e ea k a k , ( ) , = ∀ . Note that in the above notation, u, v, x, and d are defined at the beginning of a time interval, while C and π are defined at the end of the time interval. This is to associate the inflow to a link at the beginning of any time interval k with the cost (travel time) at the end of the time interval in DUE route choice. As already discussed in Heydecker and Verlander (1999), this way of associating inflow and cost will lead to predictive DUE that has more plausible solution properties compared with reactive DUE. For more details of the concepts of predictive and reactive DUE, one can refer to Heydecker and Verlander (1999), Han (2000), and Han (2003). Ban et al. (2008) showed that the above discretization scheme leads to the following NCP formulation for DUE with fixed demand: find ( , ) u π such that: 0 0 ≤ ⊥ ≥ u u u Φ ( , ) π 0 0 ≤ ⊥ ≥ π π π Φ ( , ) u International Journal of Operations Research and Information Systems, 4(1), 27-48, January-March 2013 31 Copyright © 2013, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. where the two functions Φ u u ( , ) π and Φπ π ( , ) u are defined as seen in Box 1.where e k C a k a k u u ( ) = − ( ) + ( ) − 1 1 ∆ ( ) is the exit time of vehicles entering a at the beginning of the k -th interval, which is a function of the inflow vector u since C a k ( ) −1 is so. He also define three types of indicator functions on u : λ a k 1, ′ ( ) u , λ a k k 2, , ′ ( ) u and λa k q 3, , u ( ) . First, flow propagation constraints can be represented by , λ a k 1 ′ ( ) u , which is defined as: