Journal of Advanced Transportation

Volume 2018, Article ID 9789316, 28 pages

https://doi.org/10.1155/2018/9789316

## Hybrid Random Regret Minimization and Random Utility Maximization in the Context of Schedule-Based Urban Rail Transit Assignment

School of Traffic and Transportation, Beijing Jiaotong University, Beijing, China

Correspondence should be addressed to Dewei Li; moc.361@utjb.ilwd

Received 28 July 2018; Revised 27 October 2018; Accepted 27 November 2018; Published 18 December 2018

Academic Editor: Oded Cats

Copyright © 2018 Dewei Li 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

Route choice is one of the most critical passenger behaviors in public transit research. The utility maximization theory is generally used to model passengers’ route choice behavior in a public transit network in previous research. However, researchers have found that passenger behavior is far more complicated than a single utility maximization assumption. Some passengers tend to maximize their utility while others would minimize their regrets. In this paper, a schedule-based transit assignment model based on the hybrid of utility maximization and regret minimization is proposed to study the passenger route choice behavior in an urban rail transit network. Firstly, based on the smart card data, the space-time expanded network in an urban rail transit was constructed. Then, it adapts the utility maximization (RUM) and the regret minimization theory (RRM) to analyze and model the passenger route choice behavior independently. The utility values and the regret values are calculated with the utility and the regret functions. A transit assignment model is established based on a hybrid of the random utility maximization and the random regret minimization (RURM) with two kinds of hybrid rules, namely, attribute level hybrid and decision level hybrid. The models are solved by the method of successive algorithm. Finally, the hybrid assignment models are applied to Beijing urban rail transit network for validation. The result shows that RRM and RUM make no significant difference for OD pairs with only two alternative routes. For those with more than two alternative routes, the performance of RRM and RUM is different. RRM is slightly better than RUM in some of the OD pairs, while for the other OD pairs, the results are opposite. Moreover, it shows that the crowd would only influence the regret value of OD pair with more commuters. We conclude that compared with RUM and RRM, the hybrid model RURM is more general.

#### 1. Introduction

Analysis of travelers’ route choice behavior is very important for daily urban rail transit operation. However, for a complex urban rail transit network, passengers’ route cannot usually be obtained directly, because only the departure and destination station are recorded in smart card data records in most urban rail transit networks. In this situation, passengers travel route choice can be estimated through a transit assignment model given OD demand data and train timetable. The accuracy of the estimation is highly dependent on the extent to which the model can reflect the realized passenger behaviors.

The majority of existing studies of traveler’s route choice behavior are based on the random utility maximization (RUM) [1]. These RUM models assume that when faced with a number of travel choice options, a traveler is rational enough and he or she will choose the one that has the highest utility value according to the information which was obtained by the traveler. However, it is difficult to fully get the accurate traffic information for the travelers. Moreover, the travelers’ route choices are affected by their preferences and attitudes, and the practical behavior of travelers’ route choice does not fully respect the axiomatic system of expected utility theory. So many scholars try to find a more realistic theory than RUM theory to explain and describe travelers’ route choice behavior. Among them, Loomes and Sudgen in 1982 and Bell in [2] independently proposed a regret theory, and they pointed out that the single factor’s utility function cannot explain the behavior of nonrational decision well. People would compare the actual situation and possible situations according to their decision-making factors. If they find the chosen one can get better results than other options, they tend to rejoice. Otherwise, they would feel regret. Based on the regret theory, Casper proposed a random regret minimization (RRM) model. RRM supposed that the satisfaction degree of a travel route depends not only on the utility of selected travel route, but also on the regret of other options [3].

Sociologists found that human behavior is far more complicated than a single standard [4]. Some people tend to maximize their utility while others would minimize their regret. This is also obvious from the travelers’ behavior in route choice. Passengers may not choose a faster but unfamiliar route, because they try to avoid regret from choosing the route. Moreover, for different user class, the degree of maximizing utility and minimizing the regret may be also different. Any single standard behavior assumption may not fully explain the complex route choice behavior of passengers. Based on this, this study tries to answer the research questions as follows: What is the outcome of hybrid of utility maximization and regret minimization? Which behavior assumption reflects the route choice behavior more accurately in urban rail transit?

In this study, the hybrid route choice behaviors of random utility maximization and random regret minimization are proposed and formulated to analyze the travelers’ route choice for an urban rail transit network. Although the regret theory has been applied to many fields, including road traffic assignment [5, 6], to the best of the author’s knowledge, this is the first research which applies the hybrid of utility theory and regret theory in the context of an urban rail transit. Different from cars on a road network where many routes can be selected, passengers in an urban rail transit network have less choices. Besides, the behaviors of drivers and passengers are different. Therefore, it is worth testing whether the hybrid of two kinds of behavioral assumptions is effective on an urban rail transit. The hybrid route choice was applied to a space-time expanded network. Till now, most transit assignment models considering the regret are based on the frequency and a physical network, which belongs to the frequency-based assignment. This research applies the regret to a schedule-based transit assignment by constructing a space-time expanded network from smart card data. Moreover, passengers’ heterogeneous choice behavior (e.g., regret, disappointment) is neglected in most previous studies. This paper incorporates regret minimization and utility maximization into a transit assignment model to characterize travelers’ route choice behavior. Two types of hybrid rules are considered, namely, attributes level hybrid rule and decision level hybrid rule. The effects of both of the hybrid rules are discussed. The model is applied to a real world case study in Beijing urban rail transit network. The effectiveness of the model is validated by the travel time estimated from smart card data. Compared with existing validation data relying on empirical investigation, using smart card data is more objective.

The remainder of the paper is organized as follows. Section 2 is the literature review. Section 3 proposes some basic assumptions and terminologies; Section 4 proposes the methodology. After that, Section 5 explains the MSA algorithm for solving the model. Furthermore, Section 6 applies the method to Beijing metro network, showing the effect of attributes level hybrid rule and decision level hybrid rule. Finally, Section 7 draws the conclusion.

#### 2. Literature Review

The existing transit assignment models can be classified into frequency-based assignment models [7] which are known as “line-oriented” and schedule-based model [8] which are known as “run or vehicle-oriented” [9]. The former assigns the traffic to a service transit lines, while the latter assigns the traffic to service runs. Due to strong capability of assignment of the dynamic assignment model, we are in particular interested in the latter one.

On frequency-based transit network, each train on transit lines is supposed to run with a constant headway, and the network is represented in a static manner. Generally the procedures of the assignments include 3 steps: path searching, path choice probability estimation, and traffic assignment. The first step searches effect and possible path between all O-D pairs of the network. In the second step, we compare each path in the effect and possible path set by the generalized cost function or utility function individually. With the cost distribution function, the path choice probability can be estimated. Finally, we assign the demand to paths with the path probability by O-D demand matrixes. Frequency-based transit assignment models always consider a constant demand and minimize the path cost [10] or optimize the strategy choice [11] of individual passengers, which is similar to user equilibrium assignment on network. Schmöcker et al. [12] proposed an assignment model based on frequency of departure. The model considered the probability of passengers finding seats in their perception of route cost. They introduced the probability of “fail-to-sit” at boarding points to calculate the travel cost and distribute the passenger flow. Zhang et al. proposed an assignment model based on frequency considering day-to-day evolution under oversaturated conditions and studied the impact of passenger comfort on the overload conditions and frequency of departure on passenger route selection [13]. Leurent et al. considered the capacity and provided a static and macroscopic traffic assignment model from the line submodel and the network [14]. As the frequency-based assignment model usually assigns the passenger based on a physical topology network, another important concept on frequency-based transit model is based on hyperpaths approaches. Hyperpaths form a directed acyclic graph with a flow distribution rule in network representation. Wu et al. [15] firstly proposed the hyperpath concept with strategy-based transit link cost function. Kurauchi et al. [16] also introduced traffic split in hyperpaths of transit network. For the majority of these models, schedule of transit system is assumed to be sufficiently reliable. Therefore the headway is calculated by the average frequencies of transit line in frequency-based network. And the waiting time and transfer time are implicitly estimated based on headway. Since the time dimension is not considered in frequency-based transit model, the assignment results in frequency-based models are the average value in the specified time period (e.g., the rush hour).

Unlike frequency-based type model, schedule-based models generally take into account explicitly timetable or schedule of the transit system, which means that the detailed departure or arrival times of vehicles in each transit lines are used in assignment procedures. On the other hand, schedule-based assignment model considers the temporal and spatial structure of travel demand on the network; therefore the assignment results would estimate explicitly the accurate number of passengers to each vehicle of the schedule. At present, this approach becomes hot spot to researchers. For an overview, readers are referred to [17, 18]. In a schedule-based network, passengers choose not only the optimal hyperpaths for their trip, but also the departure/arrival time and vehicles of different crowded levels by minimizing their generalized travel costs. In order to combine the time-dependent choice, a time-dependent transit network should be presented according to different schedule-based transit assignment model, which can be classified into these types: diachronic graph consisting of service subgraph, demand subgraph, and access/egress subgraph proposed by Nuzolo et al. [19]; dual graph proposed by Moller-Pedersen [20]; time-dependent graph formulation with line schedule information by Tong and Wong [8]; discrete space-time graph formulation with space-time nodes and space-time arcs proposed by Nguyen et al. [21], which is improved by Hamdouch et al. [22] to time-expanded network representation.

Modeling formulations of transit assignment constitute one of the schedule-based problems. Poon et al. [23] put forward a dynamic user equilibrium model considering the crowded environment in boarding stations at time-dependent demand distribution and taking passengers’ microbehavior on boarding link into account. With the numerical example in the study, dynamic user equilibrium mechanism is reasonably expressed with the factors of queueing at station due to congestion. Nieson [24] proposed a stochastic transit assignment model considering differences properties in passengers’ utility function as optimized problem. Tian et al. [25] improved the model proposed by Alfa and Chen [26]. The model considers the in-vehicle crowding and schedule delay in generalized cost function with departure time decisions of passengers and presents the theoretical properties of equilibrium status.

With considering the capacity in the transit assignment model, some factors related to the capacity are concerned with the RUM model in recent years. Hamdouch et al. considered the uncertainty of vehicle capacity. Passengers used a strategy to travel under the uncertainty of capacity. Using specific examples to analyze the impact of uncertainty on passengers’ travel strategies and departure time [22], Sumalee et al. considered one of the critical factors of capacity: sitting and standing capacities, and the treatment of seat allocation is considered as a random probability to get a seat or not [27]. Nuzzolo et al. presented a joint choice model by formulating departure/arrival time and train of different crowded level for maximizing the utility function. To solve the assignment model, a simulation procedure was put forward, taking congestion into account through explicit vehicle capacity. [28]. Han B et al. [29] proposed a stochastic user equilibrium model to solve transit assignment problem. This model was based on rail transit network schedule considering the travelers’ behavior assumption of train’s overload delay. The model was transformed into a dynamic schedule-based assignment model with splitting the origin-destination demands into the schedule-based network with time-space routes, using the Beijing urban rail transit (BURT) network as a case to verify the rationality of the model [29].

In terms of different behavior assumptions, the transit assignment model can be classified into the random utility maximization model and random regret minimization model.

The majority of the transit assignment models have used the random utility maximization (RUM) rooted in discrete choice analysis [1, 30]. The RUM model assumes that when a traveler is faced with a number of travel choices, he or she will choose the one with the highest utility. Poon et al. [23] assumed that a traveler can get all travel information including travel time and transfer time and constructed a generalized travel cost function. This function included in-vehicle travel cost, waiting cost, transfer cost, and transfer penalty and in-vehicle crowds. The transit assignment model was constructed based on the train schedule, and computer simulation is used to solve the model. Nuzzolo et al. [19] expressed the traffic network using diachronic graph and proposed a schedule-based assignment model considering the vehicle capacity limit. Considering the dynamics of passenger demand, Tong and Wong [8] established a stochastic transit assignment model, in which waiting time and walking time are defined as a density function, and employed Monte Carlo approach to solve the model. Nieson [24] proposes a stochastic transit assignment model considering differences preferences in passengers’ utility functions as optimized problem, which presents a framework for transit assignment based on a basic probit model.

Regret theory was presented decades ago [31], and, similar to prospect theory (PT), it originally assumed a decision-making process under uncertainty. RRM constitutes an alternative to both utility theory (UT) and PT. When an individual’s awareness perceives the product of the nonchosen alternative to be better than the result of the chosen alternative, it will build an emotion called regret [32]. The concept of regret as a determinant of decisions is often employed in areas such as psychology [33, 34], marketing [35], and finance [36]. The RRM model develops from the angle of bounded rationality and captures the scheme between multiple attribute trades-offs to the traveler’s choice of psychological and traffic behavior based on minimization of the perceived regret decision criteria [3]. Recently, regret-based choice models have gained in popularity in travel behavior research, as an alternative approach to modeling choice behavior, under conditions of both certainty and uncertainty [32, 37, 38]. The RRM model has been used to analyze and predict a wide variety of choices, such as departure time choices, route choices, mode-destination choices, activity choices, on-line dating choices, health-related choices, and policy choices [39, 40]. Chorus shows that RRM can be extended to the case of risky travel choice [32, 37]. Recently, he compared RUM with RRM in terms of theories and equations and showed their respective benefits of the scope of application [38]. Traditionally, the RUM model has dominated the travel choice behavior since it was accepted [3]. Studies to date suggest that the RRM is just as parsimonious as the standard RUM model, and it is unlike other models of contextual effects, which typically require the estimation of additional parameters [3].

Compared with the RUM, the RRM has two advantages:

(1) The RRM features logit choice probabilities and is easily estimated by using conventional discrete choice software packages in the research field [41, 42].

(2) The RRM model does not exhibit the property of independence from irrelevant alternatives (IIA), even with the assumption of independent and identically distributed (IID) error terms (Hensher et al. 2016). Therefore, RRM can be applied more widely.

Table 1 lists the typical literature of the transit assignment model, and a number of characteristics of this literature are shown, including the research subject, the assumption of choice behavior, whether it is scheduled based, whether it considers the capacity, and whether it uses the smart card data.