Multi-criteria user equilibrium model considering travel time, travel time reliability and distance
Introduction
In transportation systems, travelers may be faced with several criteria to select the routes. Some travelers select routes to minimize their reliable travel time which consists of travel time and travel time reliability according to their past experience, such as the commuters, who travel regularly between the same O-D pair on a daily basis; and some travelers would like to select the routes with minimum travel time or travel distance using the vehicle navigation system (Lima et al., 2016, Wang et al., 2016). Statistical analysis was performed by Abdel-Aty et al. (1995) to examine which route attributes that lead to the choice of a route are considered important by travelers. Three most important factors derived from the analysis are: (1) shorter travel time (40% of respondents); (2) travel time reliability (32% of respondents); and (3) shorter distance (31% of respondents). Due to its theoretical and practical importance, modeling traffic assignment considering these factors is becoming an emerging research subject.
For single-objective traffic assignment model, the conventional Wardrop user equilibrium (UE) model (Wardrop, 1952), and the stochastic user equilibrium (SUE) model (Daganzo and Sheffi, 1977), only consider factor 1 (i.e., travel time). The UE model assumes that users try to minimize their travel time, while in the SUE model users are assumed to minimize their perceived travel time, which has a random error component. The traffic equilibrium models under uncertainty consider factors 1 and 2 (i.e., travel time and travel time reliability). To build the traffic equilibrium models under uncertainty, different theories were used, such as travel time budget (TTB) approach (e.g., Lo et al., 2006, Wu, 2015), game theory approach (e.g., Bell and Cassir, 2002), utility-based approach (e.g., Mirchandani and Soroush, 1987), expected residual minimization approach (e.g., Zhang et al., 2011), robust optimization approach (e.g., Ordóñez and Stier-Moses, 2010), prospect theory-based approach (e.g., Xu et al., 2011), schedule delay approach (e.g., Watling, 2006), and mean-excess travel time approach (e.g., Chen and Zhou, 2010).
For bi-objective traffic assignment model, Dial, 1996, Dial, 1997, Nagurney and Dong, 2002 and Larsson et al. (2002) propose the bi-objective user equilibrium assignment model based on factors 1 and 3 (i.e., travel time and travel distance which is directly related to vehicle operating cost for the trip). Recently, Wang et al. (2014) and Ehrgott et al. (2015) formulated a general travel time reliability bi-objective user equilibrium model which took factors 1 and 2 (i.e., travel time and travel time reliability) into consideration. This model assumes that users select routes to minimize their travel time and maximize their travel time reliability. The solution of this model is non-unique; and the solution algorithm is not given in their works.
Compared to single-objective traffic assignment model, bi-objective traffic assignment model is more realistic. However, it is difficult to find the set of route flow solutions in bi-objective traffic assignment model. Generalized route cost function was used by Dial, 1996, Dial, 1997 and Larsson et al. (2002) to deal with the bi-objective traffic assignment model. Nagurney and Dong (2002) proposed a weight method in which travelers perceive their generalized cost on a route by travel time and travel distance with different weight. Other bi-objective traffic assignment algorithms (Chen et al., 2010, Chen and Nie, 2013) also combined the two objective functions into a nonlinear generalized cost. All these methods convert the bi-objective problem to single-objective problem using linear combination of two different factors. In fact, combining the two factors into one implicitly assumes the existence of a linear (dis)utility function, and therefore presupposes a certain preference structure. As a result of this, there is the possibility that some reasonable solutions are never considered (Wang et al., 2014).
Considering both factors 1, 2 and 3 (i.e., travel time, travel time reliability and travel distance) in the route choice decision process, this paper presents a multi-criteria user equilibrium model considering travel time, travel time reliability and distance (MUE-TRD). This new model hypothesizes that for each user class and each O-D pair no traveler can reduce either his or her reliable travel time or travel distance or both without worsening the other objective by unilaterally changing routes. For computation of the reliable travel time which consists of travel time and travel time reliability, travel time budget (TTB), which is defined as a travel time reliability chance (or on-time arrival) constraint; such that the probability that travel time exceeds the budget is less than a predefined confidence level to represent the travel time reliability, is used in this paper. For addressing the non-uniqueness of the solution in MUE-TRD model, a maximum entropy multi-criteria user equilibrium (ME-MUE) model whose feasible set is the solution set of MUE-TRD model is proposed. This is the first attempt to use the principles of entropy maximizing to deal with the multi-criteria user equilibrium model. A route-based solution algorithm based on the partial linearization descent method (R-PLD) is developed to solve the ME-MUE model.
The rest of the paper is organized as follows. In Section 2, MUE-TRD model is presented. In Section 3, ME-MUE model whose feasible set is the solution set of MUE-TRD model is proposed. In Section 4, R-PLD is developed to determine the equilibrium flow pattern. In Section 5, numerical examples are presented to illustrate the essential ideas of the proposed models and the applicability of the solution algorithm. Finally, conclusions are provided.
Section snippets
Multi-criteria user equilibrium model
This section presents MUE-TRD, the MUE-TRD condition, the existence of solution set and an instance of MUE-TRD model.
Maximum entropy multi-criteria user equilibrium model
This section proposes the maximum entropy multi-criteria user equilibrium (ME-MUE) model, the existence of solution set and an instance of ME-MUE model.
Solution algorithm
The reliable travel time (i.e., TTB) in the proposed model is non-additive because it is not possible to decompose the route TTB into the sum of link-based generalized costs. And the route solution set of MUE-TRD model cannot be expressed using mathematical formulations, because of the complexity of multi-criteria programming. Therefore, a route-based solution algorithm based on the partial linearization descent method (R-PLD) is developed to solve the ME-MUE model as follows.
Step 1.
Numerical examples
In this section, a small network is presented to illustrate the essential ideas of the proposed model, and a medium network is presented to illustrate the applicability of the proposed solution algorithm.
Conclusions
This paper proposed MUE-TRD, a new model that assumes travelers minimize their reliable travel time (including travel time and travel time reliability) and their travel distance. The proposed model was then formulated as MUE-TRD conditions, by which, for each user class and each O-D pair no traveler can reduce either his or her reliable travel time or travel distance or both without worsening the other objective by unilaterally changing routes.
For obtaining the most likely traffic flows among
Acknowledgements
The authors are grateful to the referees for their constructive comments and suggestions to improve the quality and clarity of the paper in 7th International Conference on Green Intelligent Transportation System and Safety. This research was supported by the National Natural Science Foundation of China (No. 51578150, 51378119, and 51608115), the Scientific Research Foundation of Graduate School of Southeast University (No. YBJJ1679), the Natural Science Foundation of Jiangsu Province (No.
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