Netw Spat Econ (2012) 12:589–608 DOI 10.1007/s11067-011-9168-5
A Cumulative Perceived Value-Based Dynamic User Equilibrium Model Considering the Travelers’ Risk Evaluation on Arrival Time Li-Jun Tian & Hai-Jun Huang & Zi-You Gao Published online: 4 November 2011 # Springer Science+Business Media, LLC 2011
Abstract This paper presents a cumulative perceived value-based dynamic user equilibrium model by applying the prospect theory to formulate the travelers’ risk evaluation on arrival time. The network uncertainty caused by link exit capacity degradation is incorporated into the analysis. The model which considers departure time and route choices simultaneously is expressed by a variational inequality in a discrete time space. Numerical results show that the travelers’ risk preference indeed has big influence on flow distribution. Our study constitutes a deepening of cognition in developing more realistic dynamic traffic assignment technologies. Keywords Dynamic user equilibrium . Cumulative prospect theory . Cumulative perceived value . Degradable network
1 Introduction The dynamic modeling and analyses of traffic on road networks have attracted much attention in both theoretical and practical communities during the past several decades. Compared with its static counterpart, the dynamic traffic assignment models are capable of estimating the time-varying traffic flow and revealing a profusion of complex phenomena on networks, thus providing valuable suggestions to improve traffic management. Peeta and Ziliaskopoulos (2001) reviewed the research progress of dynamic traffic assignment. L.-J. Tian : H.-J. Huang (*) School of Economics and Management, Beihang University, Beijing 100191, People’s Republic of China e-mail:
[email protected] Z.-Y. Gao School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, People’s Republic of China Present Address: L.-J. Tian School of Management, Fuzhou University, Fuzhou 350108, People’s Republic of China
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Merchant and Nemhauser (1978a, b) firstly formulated a dynamic traffic assignment model. Since then, a lot of efforts have been devoted to dynamic traffic modeling (Luque and Friesz 1980; Carey 1987; Friesz et al. 1989; Wie et al. 1990; Ran et al. 1992; Ran et al. 1996; Yang and Huang 1997; Wie and Tobin 1998; Mahmassani 2001; Nie and Zhang 2010; Zhang and Zhang 2010). In general, the challenges associated with dynamic traffic assignment arise from reasonable representation of traffic pattern and efficient solution algorithms. Friesz et al. (1993) developed an infinite-dimensional variational inequality (VI) formulation for this type of problems. Wie et al. (1995) extended the model to a discrete-time dimension in terms of unit path cost function and proposed a heuristic algorithm for obtaining an approximate solution. Huang and Lam (2002) proposed an equilibrium model considering the simultaneous path and departure time choices. Lim and Heydecker (2005) investigated a combined departure time and dynamic stochastic user equilibrium assignment problem. Ramadurai and Ukkusuri (2010) proposed a dynamic user equilibrium model which jointly considers the activity location, time of participation, duration and route choice decisions in a single unified dynamic framework referred to as activity-travel networks and, employs the cell-based transmission method to capture the traffic flow dynamics. Carey and Ge (2011) compared various methods for path inflow reassignment by embedding them in the DUE algorithm. In the above literatures, the dynamic traffic performance is basically modeled in a deterministic framework. By explicitly incorporating traffic dynamics and demand uncertainty, Chung et al. (2011) recently described a robust optimization approach for a network design problem. Besides the demand uncertainty, on the other hand, the network is daily inclined to degrade to some extent and then accompanies supply uncertainty (Lo and Tung 2003). In a general way, the travel time experienced by commuters is subject to different disturbances on the road (e.g., random accidents, routine road maintenance, natural disasters, or bad weather), and these relatively minor events occur quite frequently in our daily commutes, giving rise to stochastic link capacity degradations, which constitute the source of travel time variations. To represent the effect of travel time uncertainty, different approaches have been introduced to capture how travelers combine travel time uncertainty into their decision making process (Noland et al. 1998; Bates et al. 2001; Noland and Polak 2002; Watling 2002; Shao et al. 2006). Lam et al. (2008) and Lo et al. (2006) focused on the commuters’ path choice behavior under stochastic link capacity degradation by the notion of travel time budget required to arrive on time, while Shao et al. (2006) developed a novel reliability-based stochastic user equilibrium traffic assignment model in a multiclass transportation network. Also, active efforts have been made in the area of modeling travelers’ risk aversion behavior associated with travel time uncertainty (Bell and Cassir 2002; Lo et al. 2006; Szeto et al. 2006; Siu and Lo 2008). Siu and Lo (2008) proposed a new methodology to doubly model the uncertain transportation network with stochastic link capacity degradation and the stochastic demand, where the travel time budget varies among commuters according to their degrees of risk aversion and requirements on punctual arrival. The studies that include stochastic network degradation in dynamic transportation network analysis have, to the best of our knowledge, not been conducted before. By introducing an additional dimension in travel behavior modeling, the dynamic user
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equilibrium model can consider the simultaneous route and departure time choices, which has become a hot topic in recent years as its robust effect in exhibiting the basic characteristics in traffic scenario (Chen and Hsueh 1998; Chen et al. 2001; Heydecker and Addison 1998; Heydecker and Addison 2005; Chow 2009). Although providing valuable insights into commuters’ dynamic decision-making, these works do not identify the commuter’s responses to gains and losses arising from his/her actual feeling relative to reference points they may have in both travel utility and arrival time preference. How to integrate travelers’ departure time and path choices in the presence of travel time uncertainty and traveler’s risk preference definitely is an important topic for study. In this paper, hence, we will focus on the predictive dynamic user equilibrium modeling with consideration of the travelers’ risk evaluation on arrival time in a degradable network. Note that the predictive DUE means that travelers will make the route and departure time choice based on the actually experienced travel cost or utility. Apparently, the travel time herein will be a stochastic variable in degradable network, which gives rise to an uncertain arrival time. To prevent from arriving too early or too late, commuters can turn to adjusting their path choices or departure time (Noland and Small 1995; Jauffred and Bernstein 1996; Siu and Lo 2009) to minimize both the traffic congestion cost and the schedule delay cost associated with arrival time at destinations. Under the framework of cumulative prospect theory (Tversky and Kahneman 1992), we make an important step of introducing the notion of travelers’ cumulative perceived value, which is a synthetic outcome by calculating the traveling gain, the travel disutility with respect to travel time and the arrival time perceived value (Kahneman and Tversky 1979; Jou et al. 2008). Avineri and Prashker (2003) firstly made a simulation to the perception and evaluation of gains and losses in path choice decision process in the framework of the prospect theory. Then, the cumulative prospect theory has extensively been applied in transportation field (Avineri 2004; Senbil and Kitamura 2004; Avineri 2006; Henn and Ottomanelli 2006; Lu et al. 2008; Sumalee et al. 2009). Avineri (2006) examined the possibility of applying the prospect theory for modeling stochastic network equilibrium, and investigated the effect of reference point value on equilibrium solution. Jou et al. (2008) applied the prospect theory to commuter’s departure time choice and verified its explanatory power against survey data. Recently, Connors and Sumalee (2009) introduced a general modeling framework to represent the travelers’ path choice with travel time uncertainty, where the perceived value and perceived probabilities of travel time outcomes are obtained via nonlinear transformations of the actual travel times and their probabilities. In this paper, we attempt to formulate a cumulative perceived value-based dynamic user equilibrium model in a degradable network. The cumulative prospect theory, as a good paradigm for choice model, is herein adopted to transform both outcomes and corresponding probabilities into travelers’ perceived value for their decision-making criterion. The attraction of a given path involves evaluation of cumulative perceived value in terms of the total travel gain, the mean travel time outcomes, and the arrival time perceived value. The paper is organized as follows. Section 2 gives the basis model assumptions and necessary notations. Section 3 formulates the dynamic link travel time and path
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travel time functions. In Section 4, we first present the definition of cumulative perceived value and the transformation of arrival time perceived value, then formulate the cumulative perceived value-based dynamic user equilibrium model and succinctly investigate the existence of equilibrium solution, finally develop a solution algorithm. Numerical results are provided in Section 5. Summary remarks are presented in Section 6.
2 Notations and assumptions Consider a transportation network G(N, A), where N is the set of all nodes, and A the set of all directed links. A path P is defined by a set of acyclic ordered sequential links fa1 ; a2 ; ; am g from an origin r to destination s. Let Prs denotes the set of all feasible paths between the origin–destination (OD) pair (r, s). For simplicity, as done by Huang and Lam (2002), the studied time period T is divided into a finite set of discrete time intervals K ¼ f1; ; Kg. Let δ denotes the interval length, thus the relationship dK ¼ T holds. Note that the studied period [0, T] should be large enough so that it could cover the whole journeys of all commuters in the network. On the other hand, the value of δ should be small enough to enable the employed discrete-time simultaneous route and departure time choice equilibrium model close to its continuous-time counterpart. To facilitate our modeling and understanding, the essential notations and basic assumptions used throughout the paper are listed below unless otherwise specified. 2.1 Notations R S R S Q rs Grs k K δ K a p Prs 4 t 0a sa ta ðkÞ tprs ðkÞ fprs ðkÞ
the set of all origins in the network the set of all destinations in the network an origin, r ∈ R a destination, s ∈ S the commuting demand between OD pair (r, s) during the period [0, T] the commuting gain for unit demand between OD pair (r, s) a time interval the total number of time intervals the time interval length the set of all time intervals, K ¼ f1; ; Kg a link, a ∈ A a path connecting an origin and a destination the set of all feasible acyclic paths between OD pair (r, s) the set of all acyclic paths between all OD pairs the free-flow travel time on link a ∈ A the exit capacity of link a ∈ A the travel time on link a ∈ A for commuters entering this link during interval k the travel time on path p for commuters entering the network during interval k the inflow rate that enters the network through path p during interval k
A Cumulative Perceived Value-Based Dynamic Model
f W(f ) rs d apk ðlÞ uarsp ðkÞ ua ðkÞ varsp ðkÞ va ðkÞ z rsp a z rsp ba rsp VATP ðkÞ rsp 0 q f ðxÞ ¼ : ð2Þ : 0; otherwise 1
Clearly, X0 f ðxÞdx ¼ 1. Note that the scale parameter θ >0 should be calibrated by survey data. The smaller the value of θ is, the more frequently light degradations occur. With the use of exponential distribution on degradation degree, we can derive the mean and the variance of link travel time, as follows E½ta ðlÞ ¼ C0a þ
qa ðlÞ 1 1 1 qa ðlÞ 1 expðx=EÞdx ¼ C0a þ X ; %sa 0 expðha xÞ E %sa 1 Eha
var½ta ðlÞ ¼
qa ðlÞ %sa
2
1 qa ðlÞ 2 ðEha Þ2 var : ¼ ra ðxÞ %sa ð1 2Eha Þð1 Eha Þ2
ð3Þ
ð4Þ
It is noteworthy that Eh a < 0:5 must hold for guaranteeing the non-negativity of the variance. We can see from Eqs. (3) and (4) that both mean and variance of link travel time increase with the queuing length. This is consistent with our expectation that the experienced travel time and the corresponding variability in heavy traffic are higher than those in light traffic. When queuing disappears, the variance becomes zero. In addition, the mean link travel time under network degradation is certainly larger than that without degradation since 0 < Eh a < 0:5 holds. Furthermore, we have ¯E ½ta ðlÞ ¯var½ta ðlÞ ¯E ½ta ðlÞ ¯var½ta ðlÞ > 0; > 0; > 0; > 0: ¯E ¯E ¯h a ¯h a
ð5Þ
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The above results show that both mean and the variance of link travel time increase with the values of θ and ηa. This is consistent with our initial cognizance about θ and ηa. In a dynamic network, the inflow rate ua ðlÞ and departure rate va ðlÞ have to be governed by the following relationship with requirement of the commuters’ first-infirst-out movement (Huang and Lam 2002) ua ðlÞ ¼ va ðl þ ta ðlÞÞð1 þ ta ðlÞ ta ðl 1ÞÞ:
ð6Þ
Considering the link capacity degradation, the above equation becomes qa ðlÞ qa ðl 1Þ ua ðlÞ ¼ va ðl þ ta ðlÞÞ 1 þ ; ra ðxÞ%sa where
( va ðl þ ta ðlÞÞ ¼
ð7Þ
sa ; if ta ðlÞ > C0a or ua ðlÞ > ð1 Eh a Þsa ua ðlÞ; otherwise
ð8Þ
The queuing length on link a during interval l can be explicitly expressed as qa ðlÞ ¼ maxfqa ðl 1Þ þ %ðua ðlÞ ð1 Eha Þsa Þ; 0g:
ð9Þ
Equations (1) and (9) are used to compute the link travel times, where the link inflow rates have to be determined accordingly. The inflow rate of link a during interval l can be decomposed according to origins, destinations and paths contained. We then have XX X ua ðlÞ ¼ uarsp ðlÞ; ð10Þ r2R s2S p2Prs
where rsp rsp uarsp ðlÞ ¼ fprs ðlÞKrsp a þ vb ðlÞKba ; b is the predecessor of a on path p;
8 rsp 0 > < ub l Cb ; if the queue is null at interval l
vrsp ursp ðiÞ
b ðlÞ ¼ > ; otherwise; where i þ tb ðiÞ ¼ l : sb b ub ðiÞ
ð11Þ
ð12Þ
rsp Where ursp a ðlÞ is the flow rate entering link a on path P during interval l; z a ¼ 1 rsp rsp if link a is the first link on path P, z a ¼ 0 otherwise; vb ðlÞ the flow rate exiting from link b on path P during interval l; z rsp ba ¼ 1 if link b is the predecessor of a on path P, z rsp ¼ 0 otherwise. Equation (12) indicates that the path-specified link exit ba rate is either the entry rate at interval l C0b (if there is no queue) or a portion of the link capacity. For the latter case, the portion given in Eq. (12) results from applying the first-in-first-out condition (6) for all commuters on the same path. When ub ðiÞ ¼ 0, vrsp b ðlÞ takes an average of sb on all paths that use link b at interval i. Still, another important point should be mentioned, the link travel time calculated by Eq. (1) maybe not an integer, leaving the expression i þ tb ðiÞ in Eq. (12) non-integer. Therefore, some technologies should be developed to smooth out the exit flow (Ban
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et al. 2008; Wie et al. 2002). For simplicity, we adopt the method in Wie et al. (2002), in this a manner, juj, representing the maximal integer less than or equal to υ, is introduced to approximately replace the value of υ. After formulating the link travel time, we now turn to establish the path travel time in detail and illustrate how flows spread through the network dynamically. For a given traffic flow pattern f, there are corresponding path travel times tprs ðk; f Þ, p ∈ Prs, r ∈ R, s ∈ S, k ∈ K. The time required to traverse a specific path p ¼ fa1 ; a2 ; . . . ; am g for commuters entering the network during interval k, is a nested function ( Huang and Lam 2002) as follows:
tprs ðkÞ ¼ ta1 ðkÞ þ ta2 k þ ta1 ðkÞ þ þ tam k þ ta1 þ ta2 þ þ tam1 ; ð13Þ
where ta1 ¼ ta1 ðkÞ, ta2 ¼ ta2 k þ ta1 ðkÞ ; , for short. Equation (13) can be rewritten as X X tprs ðkÞ ¼ ta ðlÞ%rs ð14Þ apk ðlÞ; a
lðkÞ2K
%rs apk ðlÞ
where is equal to 1, if the flow on path P of OD pair (r, s) entering the network at interval k arrives link a at interval l; otherwise, 0. This is detailed as 8
< 1; if
k þ ta þ ta þ þ ta
¼ l 1 2 i1 ð15Þ %rs ai pk ðlÞ ¼ : 0; otherwise and for any link a on path P, clearly X
%rs apk ðlÞ ¼ 1:
ð16Þ
l ðk Þ2K
Thus, the path travel time is simply the sum of all the link travel times on the path, and the link travel times are computed based on the link traffic conditions when commuters enter these links. The difficulty of dynamic equilibrium assignment arises because d rs apk ðlÞ depends on link travel times which in turn depend on link inflows. Consequently, the path travel times computed from Eq. (14) are essentially non-linear and non-convex. It is clear that accurately determining the link entering time of a commuter is essential for computing his/her link travel time and then path travel time. Hence, we cannot treat all link travel times along a path as random parameters. For this reason, we here introduce an assumption that the exit rate degradation merely occurs on the rearmost link of a path, i.e., link am. To make it clear, only the links connecting the destinations degrade. With this assumption, we can derive the mean and variance of the path travel time as follows h i X X E tprs ðkÞ ¼ ta ðlÞ%rs ð17Þ apk ðlÞ; a
l ðk Þ2K
h i
var tprs ðkÞ ¼ var½tam ðlÞ; l ¼ k þ ta1 þ ta2 þ þ tam1 :
ð18Þ
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The above two equations state that the mean of the path travel time is simply the sum of the means of all link travel times on the path, while the variance is given by the variance of the last link’s travel time. Based on the Eqs. (13) and (14), tprs ðkÞ is comprised by the link travel time tam, which is a function of degradation degree x. Therefore, tprs ðkÞ is also a function of degradation degree x. Let tp0rs ðkÞ ¼ ta1 þ ta2 þ þ tam1 þ C0am . The cumulative density function (CDF) and probability density function (PDF) of tprs ðkÞ can be calculated as follows, 8 11=Eh a m 0 > rs 0rs > t ðkÞ t ðkÞ %sa > p p < A ; tprs ðkÞ > * 1@ rs ð19Þ F tp ðkÞ ¼ qam ðlÞ > > > : 0; otherwise 8 > > > >
* f tprs ðkÞ ¼ Eh t rs ðkÞ t 0rs ðkÞ qam ðlÞ am p > p > > > : 0; otherwise 1
1 X0 f
ð20Þ
where * ¼ tp0rs ðkÞ þ qam ðlÞ=ð%sa Þ. It is easy to get tprs ðkÞ dtprs ðkÞ ¼ 1. So far, we have formulated all link and path travel times and their means as well as variances. In the following, we will investigate the dynamic network equilibrium using the concept of cumulative perceived value (CPV).
4 CPV-based dynamic user equilibrium 4.1 Cumulative perceived value and arrival time perceived value Here, we first introduce a concept so-called cumulative perceived value (CPV). We define CPV as the value perceived by a commuter, which consists of three parts: (a) the constant representing the gain from commuting; (b) the travel disutility (i.e., the expected mean travel cost); and (c) the arrival time perceived value. This definition is somewhat similar to that used in studying the simultaneous departure time and route choice problem in which the schedule delay costs have to be considered (Vickrey 1969; Watling 2006). In our problem, however, all path travel times are random variables with a predefined probability density function (PDF), the evaluation by commuters will be based on the synthetic value comprised of the mean and variance. Hence, how to integrate the path travel time variability into the departure time and route choice model becomes the kernel of this study. The CPV for commuters entering the network through path p during interval k, when the path flow pattern is f, is formulated below h i rsp rsp !3 "3 , !3 "3 > !2 "2 , !1 "1 > !2 "2 and !4 "4 > !1 "1 have to be satisfied. In other words, an incremental gain or loss in the late-side is psychologically larger than that in the early-side. For simplicity, let F(t) and f(t) denote the cumulative density function (CDF) and the probability density function (PDF) of actual arrival time, respectively. Using the approach in Connors and Sumalee (2009), and regarding tp in Fig. 1 as the watershed, we have dwþ dw e F tp FðtÞ e ðFðtÞÞ dt; dt and : ¼ ðtÞ ¼ e dt dt
ð28Þ
dwþ dw l FðtÞ F tp l ð1 FðtÞÞ dt: dt and : ¼ l ðtÞ ¼ dt dt
ð29Þ
:þ e ðtÞ
:þ l ðtÞ
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Then, considering the continuous version of Eqs. (23) and (24), the ATPV for the side of ascending value is dwþ te dwe ðFðtÞÞ tp e F tp FðtÞ þ gðtÞdt þ Xte gðtÞdt; ð30Þ Ve ¼ Ve þ Ve ¼ X 1 dt dt and that for the side of descending value is þ tw dwl FðtÞ F tp þ1 dw ð1 FðtÞÞ ~ þ ~ Vl ¼ Vl þ Vl ¼ Xtp gðtÞdt: ð31Þ gðtÞdt þ Xtw l dt dt So, the total ATPV is rsp ðkÞ ¼ Ve þ Vl ; 8p 2 Prs ; r 2 R; s 2 S; k 2 K; VATP
ð32Þ
where Ve and Vl are computed using Eqs. (30) and (31) with respect to specific path, OD pair and departure interval. 4.2 Conditions for CPV-based dynamic user equilibrium The CPV-based dynamic traffic assignment assumes that travelers choose their departure times and paths for maximizing their cumulative perceived values. In the equilibrium state, for each OD pair, the CPV experienced by travelers is equal and maximal, i.e., no travelers would be better off by unilaterally changing his/her departure time or path. Mathematically, the CPV-based dynamic user equilibrium flow pattern, f*, should satisfy the following conditions 8 » » » < ¼ 0 max f rsp
