A semi-analytical approach for solving the bottleneck model with general user heterogeneity

Transportation Research Part B 71 (2015) 56–70

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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

A semi-analytical approach for solving the bottleneck model with general user heterogeneity Yang Liu a, Yu (Marco) Nie b,⇑, Jonathan Hall c a

Department of Industrial and Systems Engineering, National University of Singapore, Singapore Department of Civil and Environmental Engineering, Northwestern University, Evanston, Illinois, United States c Department of Economics and School of Public Policy and Governance, University of Toronto, Toronto, Canada b

a r t i c l e

i n f o

Article history: Received 13 March 2014 Received in revised form 18 September 2014 Accepted 19 September 2014

Keywords: Bottleneck model General heterogeneity Dynamic user equilibrium Variational inequality problem P-property

a b s t r a c t This paper proposes a novel semi-analytical approach for solving the dynamic user equilibrium (DUE) of a bottleneck model with general heterogeneous users. The proposed approach makes use of the analytical solutions from the bottleneck analysis to create an equivalent assignment problem that admits closed-form commute cost functions. The equivalent problem is a static and asymmetric traffic assignment problem, which can be formulated as a variational inequality problem (VIP). This approach provides a new tool to analyze the properties of the bottleneck model with general heterogeneity, and to design efficient solution methods. In particular, the existence and uniqueness of the DUE solution can be established using the P-property of the Jacobian matrix. Our numerical experiments show that a simple decomposition algorithm is able to quickly solve the equivalent VIP to high precision. The proposed VIP formation is also extended to address simultaneous departure time and route choice in a single O–D origin-destination network with multiple parallel routes. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The analysis of dynamic commute travel patterns, originated by Vickrey (1969) and refined subsequently in Hendrickson and Kocur (1981), Smith (1984), Daganzo (1985), Newell (1987), Arnott et al. (1990, 1994), Yang and Huang (1997) has been extensively studied in the literature. The underlying assumption in these analyses is that travelers make trade-off between the anticipated costs of travel time and schedule delay (incurred when travelers cannot arrive at their destination at a desired time). Accordingly, the general pattern of commuters’ departure time choices is explained as a dynamic user equilibrium at which nobody can reduce his/her own commute cost by unilaterally shifting his/her route and departure time choice. Vickrey’s original bottleneck model does not consider users who differ in their valuation of travel time and schedule delay. Such heterogeneity has since attracted much attention because of its potential impacts on the equilibrium solutions as well as the welfare effects of demand management policies (Small, 1982; Cohen, 1987; Arnott et al., 1994; Lindsey, 2004; Small et al., 2005; van den Berg and Verhoef, 2011b; Liu and Nie, 2011; Hall, 2013). It is well known that analytical solutions for the bottleneck model with user heterogeneity exist in special cases, such as when restrictions are imposed on how the value of time (a) and unit schedule cost (b for early arrival and c for late arrival) may be correlated. Vickrey (1973) studied the case where a is proportional to b and c. Cohen (1987) considered two typical groups of commuters: low-income commuters who have low value of time but rigid work schedule and high-income ⇑ Corresponding author. Tel.: +1 847 467 0502. E-mail address: [email protected] (Y. Nie). http://dx.doi.org/10.1016/j.trb.2014.09.016 0191-2615/Ó 2014 Elsevier Ltd. All rights reserved.

Y. Liu et al. / Transportation Research Part B 71 (2015) 56–70

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commuters who value their time higher and have more flexible work schedule. His analysis requires the ratio of b/c to be a constant. Arnott et al. (1988, 1994) generalized Cohen’s analysis by also considering other dimensions of user heterogeneity (e.g., desired arrival time). van den Berg and Verhoef (2011a) assume that b and c are fixed and identical for all users while a varies across users. Palma and Lindsey (2002a) assume that a is log-normally distributed and all the users have the same ratios of b=a and c=b. van den Berg and Verhoef (2011b) examined welfare effects of Vickrey’s time-varying toll using a similar heterogeneity structure as Cohen’s. They generalize the model to handle arbitrary number of user groups. Qian and Zhang (2013) studied the morning commute problem with infinite number of user groups by also assuming the constant ratio of b over c and extended the model into two-parallel routes network. Recently, Hall (2013) allows user preferences to be continuously distributed in all three dimensions, but the restriction of constant b=c ratio is still imposed. The heterogeneity setting of Cohen (1987) was also adopted to analyze simple network bottleneck model with two parallel routes by Arnott et al. (1992) and Liu and Nie (2011). Several studies have addressed the analytical properties of the bottleneck model, such as the existence and the uniqueness of the equilibrium solution (e.g., Hendrickson and Kocur, 1981; Smith, 1984; Newell, 1987; Daganzo, 1985; Lindsey, 2004). Most of these studies considered user heterogeneity only in the dimension of the desired arrival times, with the exception of Newell (1987) and Lindsey (2004). Newell also made use of the assumption that the ratio of b and c is constant to simplify his analysis. Lindsey (2004) considered a bottleneck model with a general heterogeneity structure (i.e., all user preferences are allowed to vary independently) and proved that it admits one and only one user equilibrium under mild conditions. His result is theoretically significant, albeit it does not prescribe a solution method for the dynamic user equilibrium under general heterogeneity. In fact, if a general joint distribution of user preferences is considered, obtaining an equilibrium solution for bottleneck model seems analytically intractable. Vickrey’s model also inspired a large body of literature under the umbrella of dynamic traffic assignment (DTA) (e.g., Merchant and Nemhauser, 1978; Friesz et al., 1993; Ran et al., 1993; Lu et al., 2006; Nie and Zhang, 2007), which seeks to forecast equilibrium traffic patterns in more general network settings. Because the DTA models aim at representing realistic traffic phenomena (e.g., physical queue, traffic controls), the commute cost is typically evaluated through traffic simulation (also known as dynamic network loading) instead of closed-form formulae. Accordingly, the equilibrium problem may only be solved approximately in most cases. The reader is referred to Peeta and Ziliaskopoulos (2001) for a comprehensive review of the DTA literature. Recently, Ramadurai et al. (2010) and Pang et al. (2012) tackled the Vickrey’s bottleneck model with heterogeneous users using a DTA approach. Specifically, they formulated and solved the problem as a general linear complementarity system, in which another important dimension of travel preferences, i.e., the desired arrival time, is considered. Note that time is discretized in Ramadurai et al. (2010). Later, Pang et al. (2012) proposed to use the time-stepping numerical technique to approximate the discrete-time model. The approach proposed in this paper differs from those in the classical bottleneck analysis and the DTA research. It makes use of the analytical solutions from the bottleneck analysis to create an equivalent assignment problem that explicitly admits closed-form commute cost functions. More specifically, the underlying equivalent assignment problem is a static and asymmetric traffic assignment problem, which may be formulated as a variational inequality problem. We call this approach ‘‘semi-analytical’’ because it blends analytical and numerical methods. This allows us to analyze the analytical properties of the underlying problem since we have a closed-form commute cost function. In particular, we prove the existence and the uniqueness of DUE by examining the P-property of VIP’s Jacobian matrix. Perhaps more important for practical purposes, the proposed VIP can be solved to high precision with simple assignment algorithms, which makes it a useful instrument to perform numerical analysis for congestion management policies using bottleneck model with general user heterogeneity. For the reminder, Section 2 introduces a single bottleneck model with a fixed number of heterogeneous commuters. In Section 3, a semi-analytical approach is developed to transform the DUE problem of the bottleneck model into a static traffic assignment problem, which is then formulated as a variational inequality problem. Section 4 proves the existence and the uniqueness of DUE solution. In Section 5, we extend the variational inequality formulation to solve DUE in a single origin– destination corridor network with multiple parallel routes. Section 6 reports results of several numerical experiments, including one constructed using empirical data from California State Route 91. Section 7 concludes our findings. 2. Model setting Consider a fixed number of travelers who commute from home to work through a corridor during the morning peak-hour. Without loss of generality, we assume that a bottleneck is located at the exit of the corridor, such as an off-ramp leading to downtown. When the demand (the departure rate) exceeds the capacity of the bottleneck, denoted as s, a queue forms and consequently commuters experience queuing delays. Therefore, the travel time along the corridor consists of two parts: (1) the fixed free flow travel time T, i.e., the time needed to traverse the corridor when there is no congestion, and (2) queuing delay. Since T does not affect the analysis in the case of single bottleneck, it is assumed to be zero except in Section 5 where the route choice is discussed. Note that travel time and queuing delay are equivalent when T ¼ 0. When commuters arrive at their work place, their schedule delay will be the difference between the actual and desired arrival times. Each commuter chooses a departure time t (from the bottleneck) so as to minimize his/her commute cost cðtÞ, which consists of the costs of travel delay and schedule delay as in the classic bottleneck analysis (Vickrey, 1969). Specifically, the following piecewise linear function is adopted in this paper

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Y. Liu et al. / Transportation Research Part B 71 (2015) 56–70

 cðtÞ ¼

axðtÞ þ bðt  tÞ; t < t early arrival auðtÞ þ cðt  t Þ; t > t late arrival

ð1Þ

where a (unit cost of travel time), b (unit cost of early arrival) and c (unit cost of late arrival) are positive scalars, and xðtÞ and uðtÞ is the travel time corresponding to a departure time t for early and late arrivals, respectively. To consider user heterogeneity, the population is divided into n groups of commuters. For simplicity, all groups are assumed to have an identical desired arrival time t  . Group i includes N i ði ¼ 1; . . . ; nÞ commuters, who have identical ai , bi and ci . Because when bi > ai a traveler can choose to sit in their vehicle instead of going into work, we assume that the cost of being early to work must be less than the cost of travel time, i.e., bi < ai ; 8i. This assumption was proved by Lindsey (2004) as a necessary condition for the existence of equilibrium when heterogeneous users are modeled in a discrete-group setting. To formulate the equilibrium problem, four one-to-one mappings are created between the set of group IDs i and the set of departure orders k. Arnott et al. (1994) has shown that the departure order k of the group i in the dynamic user equilibrium of a bottleneck is determined by the relative schedule delay and travel time costs, i.e., bi =ai for early arrival and ci =ai for late arrival. When commuters choose their departure time, they are trading off the travel delay and the schedule delay. The farther from the center of rush hour a group travels, the lower travel time and the higher schedule delay is. Therefore, those with a higher ratio of schedule delay to travel time (bi =ai and ci =ai ) will prefer to travel closer to the peak. Arnott et al. (1994) also proved that each group’s early/late departure interval is a connected set, and there is no mixing of groups as long as the ratios bi =ai and ci =ai are different across groups. Accordingly, it is assumed here that bi =ai – bj =aj ; ci =ai – cj =aj ; 8i – j. As shown in Table 1, group i is the f ðiÞ-th departing from the beginning of the rush hour who arrives at the work place earlier than the desired arrival time t ; group EðkÞ is the k-th departing from the beginning of the rush hour who arrives at the work place earlier than the desired arrival time t . Because it is assumed that the queue at bottleneck follows the First-InFirst-Out (FIFO) principle, the groups arrive at work in the order in which they depart from home. Similarly, we have two mappings, gðiÞ and AðkÞ, for the case of late arrival. The mapping f ðiÞ and gðiÞ can be obtained by ranking the ratios of bi =ai and ci =ai , respectively, and mapping EðkÞ and AðkÞ are the inverses of mapping f ðiÞ and gðiÞ, respectively. The assumptions on the input parameters are now summarized as below: Assumption 1. bi < ai ; 8i Assumption 2. All groups have an identical desired arrival time ti ¼ t  ; 8i Assumption 3. bi =ai – bj =aj ; ci =ai – cj =aj ; 8i – j Assumption 1 is a necessary condition for the existence of meaningful equilibrium (Lindsey, 2004); Assumption 2 simplifies the relationship between queuing delay and equilibrium flows which will be shown in this section; and Assumption 3 is necessary to have a unique departure order for each commute group. Let N ia denote the number of commuters in group i who arrive later than their desired arrival time t  and N ie denote the number of commuters in group i who arrive earlier than their desired arrival time t . The following demand constraints must be satisfied:

Nie þ Nia ¼ Ni ; 8i Nie ; N ia P 0; 8i

ð2Þ ð3Þ

Without loss of generality, let us assume that each group has both arrival early and arrival late intervals. If all the commuters in group i choose to arrive early (or late), then N ia ¼ 0 (or N ie ¼ 0). Therefore, there is a total of 2n departure intervals. We also select a commuter from each departure interval to represent the commute cost of the departure interval: the last commuter in each early-arrival interval and the first commuter in each late-arrival interval. The reason we select in this way is that the commute cost of the first and the last commuter of each departure interval can be formulated as an affine map of equilibrium flows (N ia and N ie in Table 2), which will be demonstrated in Section 3. Let xk and yk denote the travel time and the schedule delay of the last commuter in the k-th early-arrival group from the beginning of congestion. Similarly, let uk and v k denote the travel time and the schedule delay of the first commuter in the k-th late-arrival group from the end of congestion. The commute cost of the representative commuter in each departure interval, i.e., the sum of the costs of travel delay and schedule delay, can be formulated as:

Table 1 Mappings between group IDs and departure order IDs. f ðiÞ gðiÞ EðkÞ AðkÞ

Group Group Group Group

i’s departure order of the early arrival from the beginning of the rush hour i’s departure order of the late arrival from the end of the rush hour ID of the k-th group arriving early from the beginning of the rush hour ID of the k-th group arriving late from the end of the rush hour

Y. Liu et al. / Transportation Research Part B 71 (2015) 56–70

59

cie ðNÞ ¼ ai xf ðiÞ þ bi yf ðiÞ

ð4Þ

cia ðNÞ ¼ ai ugðiÞ þ ci v gðiÞ

ð5Þ

where cie represents the commute cost of an early-arrival commuter in group i, i.e., the f ðiÞ-th early-arrival group from the beginning and cia represents the commute cost of a late-arrival commuter in group i, i.e., the gðiÞ-th late-arrival group from the end. Note that subscripts f ðiÞ and gðiÞ provide the departure order mapping of group i in the early and late arrival intervals. The rest of this section will build an affine map from equilibrium flow

N ¼ ðNEð1Þe ; NEð2Þe . . . ; NEðnÞe ; NAð1Þa ; NAð2Þa ; . . . ; NAðnÞa Þ to commute cost

C ¼ ðcEð1Þe ; cEð2Þe . . . ; cEðnÞe ; cAð1Þa ; cAð2Þa ; . . . ; cAðnÞa Þ i.e., CðNÞ ¼ J 0 N; J 2 R2n2n . In order to keep the commute cost identical across the commuters within the same group, the queuing time (the additional travel time caused by congestion) for group i will increase at a rate bi =ai within the early-arrival interval and decrease at a rate ci =ai within late-arrival interval (Vickrey, 1969). It imposes the user equilibrium condition within each departure interval, i.e., the commute cost across commuters within the same departure interval will remain the same. To see this, note that commuters in group i value their travel time by ai and value their unit schedule delay by bi and ci . As a consequence, we have the following relationship between travel time (equals queuing delay because T ¼ 0) and equilibrium flows (N ie and N ia ):

NEð1Þe bEð1Þ s aEð1Þ NEðkÞe bEðkÞ xk ¼ xk1 þ ; k ¼ 2; . . . ; n s aEðkÞ NAð1Þa c1 u1 ¼ s a1 NAðkÞa cAðkÞ uk ¼ uk1 þ ; k ¼ 2; . . . ; n s aAðkÞ

x1 ¼

ð6Þ ð7Þ ð8Þ ð9Þ

Note that Eqs. (6)–(9) partially impose the equilibrium condition under Assumption 1–3 aforementioned. They do not guarantee the same commute cost for both the early-arrival and late-arrival intervals of group i. Additional constraints will be added to address this issue in the next section. Moreover, we have the following relationship between schedule delay and equilibrium flows

NEðnÞe þ NEðn1Þe þ . . . þ NEðkþ1Þe ; k ¼ 1; . . . ; n  1 s yn ¼ 0 N þN þ . . . þ NAðkþ1Þa v k ¼ AðnÞa Aðn1Þa ; k ¼ 1; . . . ; n  1 s vn ¼ 0 yk ¼

ð10Þ ð11Þ ð12Þ ð13Þ

These results can be derived from the fact that the schedule delay of a commuter is equivalent to the time required to serve all the commuters who arrive between this commuter’s arrival time and t  . Substituting Eqs. (6)–(13) into commute cost function (Eqs. (4) and (5)) gives an affine map from flows to commute costs ðN ! CÞ. 3. Equivalent assignment formulation Now, with an explicit formulation of the commute cost function CðNÞ, we are ready to formulate the dynamic user equilibrium problem of the bottleneck model with heterogeneous commuters. A semi-analytical approach is proposed here that

Table 2 Unknown variables in the equilibrium problem.

Equilibrium flow of group i Travel time of the k-th group Schedule delay of the k-th group

Early arrival

Late arrival

N ie xk (the last commuter) yk (the last commuter)

N ia uk (the first commuter) v k (the first commuter)

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Y. Liu et al. / Transportation Research Part B 71 (2015) 56–70

transforms the dynamic equilibrium problem into a static, multi-class, asymmetric traffic assignment problem, which is subsequently formulated and solved as a variational inequality problem (VIP). According to the definition of the user equilibrium, the commute costs should be the same across commuters within a group no matter whether they arrive before or after the desired arrival time t . In Section 2, the class-specific commute cost is defined based on the assumption that the queuing delay increases at the rate of bi =ai and decreases at the rate of ci =ai (Eqs. (6)–(13)). Effectively, this forces all commuters within the same departure interval to have the same commute cost. To determine the user equilibrium, additional constraints are needed since each group has two departure intervals. If the early-arrival and late-arrival intervals are each treated as one of two parallel routes connecting a single origin–destination pair, the dynamic user equilibrium problem is equivalent to assigning demand of each group to these two ‘‘imaginary’’ routes. The above choice can be stated as follows

ðcie ðNÞ  li ÞNie ¼ 0;

cie ðNÞ P

ðcia ðNÞ  li ÞNia ¼ 0;

cia ðNÞ P

li ; i ¼ 1; . . . ; n li ; i ¼ 1; . . . ; n

Nie þ Nia ¼ Ni ; i ¼ 1; . . . ; n Nie ; N ia P 0; i ¼ 1; . . . ; n

ð14Þ ð15Þ ð16Þ ð17Þ

Specifically, Fig. 1 demonstrates that for each group i, commuters have their own origin i and destination i þ 1 and choose between the two routes: the early arrival route and the late arrival route. In Section 2, a closed-form commute cost function has been developed (Eqs. (4) and (5)).Thus, the dynamic user equilibrium can be determined by solving the following equivalent static traffic assignment problem. Equivalent static traffic assignment problem: Consider a network with a set of n þ 1 nodes and a set of 2n routes, as shown in Fig. 1. There are n origin–destination pairs, and commuters entering the network at node i always exit at node i þ 1. The demand between origin i and destination i þ 1 is denoted by N i . There are two routes connecting a pair of nodes i and i þ 1. Therefore, commuters between origin i and destination i þ 1 choose between the two routes, so that the flows on the two routes, denoted as N ie and N ia , are at a user equilibrium. The commute cost functions (cie and cia ) are given in Eqs. (4) and (5). Important to note is that the commute cost functions (4) and (5) are not separable, i.e., the commute cost on one route depends on not only the flow on its own, but also the flows on other routes. To see this, note that the route interactions, i.e., the marginal effect of one additional route flow on the commute cost of any other route, can be derived as:

@C ie @C ia ¼ 0; ¼0 @Nja @Nje 8 a bj @C ie < aij s if f ðjÞ < f ðiÞ ¼ @Nje : bi if f ðjÞ P f ðiÞ s ( a cj i @C ia a s if gðjÞ < gðiÞ ¼ cj i @Nja if gðjÞ P gðiÞ

ð18Þ ð19Þ

ð20Þ

s

This reveals the asymmetric link interactions between the early-arrival (or late-arrival) commute costs and early-arrival (or late-arrival) flows, i.e.:

@C ie @C je @C ia @C ja – and – ; for i – j @Nje @Nie @Nja @Nia

ð21Þ

The Jacobian matrix is formed by arranging the derivatives of all route commute functions C in the order of departure C ¼ ðcEð1Þe ; cEð2Þe . . . ; cEðnÞe ; cAð1Þa ; cAð2Þa ; . . . ; cAðnÞa Þ (from the first-departure group to the last-departure group for early arrival, then from the last-departure group to the first-departure group for late arrival), with respect to route flows N.

Fig. 1. Equivalent static traffic assignment problem.

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Y. Liu et al. / Transportation Research Part B 71 (2015) 56–70

0 @C

Eð1Þe ðNÞ

B @NEð1Þe B @CEð1Þe ðNÞ B B @NEð2Þe B B . B .. B B @CEð1Þe ðNÞ B @N B EðnÞe J ¼ rN C ¼ B B 0 B B B B 0 B B B B 0 @ 0

@C Eð2Þe ðNÞ @N Eð1Þe

 

@C EðnÞe ðNÞ @N Eð1Þe

0

0

0

@C Eð2Þe ðNÞ @N Eð2Þe

 

@C EðnÞe ðNÞ @N Eð2Þe

0

0

0

0 0

0 0

0 0

.. . @C Eð2Þe ðNÞ @N EðnÞe

..

.  

.. . @C EðnÞe ðNÞ @N EðnÞe

0

0

0

@C Að1Þa ðNÞ @C Að2Þa ðNÞ @N Að1Þa @N Að1Þa

0

0

0

@C Að1Þa ðNÞ @C Að2Þa ðNÞ @N Að2Þa @N Að2Þa

0 0

0 0

0 0

.. .

.. .

@C Að1Þa ðNÞ @C Að2Þa ðNÞ @N AðnÞa @NAðnÞa

    ..

.  

1

0 b aEð2Þ abEð1Þ C Eð1Þ B Eð1Þ C B 0 C B bEð1Þ bEð2Þ C B C B C B . .. B .. 0 C C . B C B 0 C 1Bb bEð2Þ Eð1Þ C B C¼ B @C AðnÞa ðNÞ C sB 0 0 @N Að1Þa C B C B @C AðnÞa ðNÞ C B 0 C B 0 @N Að2Þa C B C B .. C B 0 0 . C @ A @C AðnÞa ðNÞ 0 0 @N 0

AðnÞa

b

0

0

0

b

0

0

0

0 0

0 0

0 0

   aEðnÞ aEð1Þ Eð1Þ    aEðnÞ aEð2Þ Eð2Þ ..

.. .

. 

bEðnÞ

0

0

cAð1Þ aAð2Þ aAð1Þ   Að1Þ

0

0

cAð1Þ

cAð2Þ

 

0

0

.. .

.. .

..

0

0

cAð1Þ

cAð2Þ

 

c

.

0

1

C C C C C C C 0 C C C 0 C cAð1Þ C aAðnÞ aAð1Þ C C c C C aAðnÞ aAð2Þ Að2Þ C C C .. C . A 0

ð22Þ

cAðnÞ

With the Jacobian matrix, we can write the commute function as:

CðNÞ ¼ J0 N; J 2 R2n2n

ð23Þ

0

where J represents the transpose of J. When the Jacobian matrix of the commute cost functions is symmetric, i.e., the link interactions are symmetric, it is possible to find an equivalent convex minimization problem (Beckmann et al., 1956; Dafermos, 1971; Sheffi, 1985). However, in our model the link interactions and Jacobian matrix are asymmetric, and therefore there is no known equivalent minimization program. That said, asymmetric traffic assignment problems can be formulated as a variational inequality problem (see Smith, 1979; Dafermos, 1980). We state the formulation as follows: A route flow pattern N ¼ ðN Eð1Þe ; N Eð2Þe . . . ; N EðnÞe ; N Að1Þa ; N Að2Þa ; . . . ; N AðnÞa Þ is a user equilibrium solution of the equivalent static traffic assignment problem if and only if it solves the variational inequalities problem VIðK; CÞ:

D

T

CðN Þ ; N  N

E

P 0

ð24Þ

for any N 2 K. Here, h; i denotes the inner product in 2n-dimensional Euclidean space, and the feasible set Kfð   ; N ie ;    ; N ia   Þ j N ie þ N ia ¼ N i ; N ie ; N ia P 0; i ¼ 1; 2;    ; ng. The equivalence between the VI formulation and the equilibrium conditions (14) is well known in the literature, and hence the proof is omitted here for brevity. The reader is referred to Harker and Pang (1990) for a survey of theory, algorithms and applications of VIP. We close this section by noting that the system optimum problem corresponding to the above user equilibrium problem may be treated similarly. However, in this paper, we simply solve it as a minimization problem because it is more straightforward. Note that at system optimum: (1) the user departure order will be given by ranking bi for early arrival and ci for late arrival which will be different from the departure order at no-toll equilibrium, (2) all queuing delays (xk and uk in Table 2) will be eliminated by a time-varying toll, and thereby (3) the total system cost will only consist of the total cost of schedule SO delay (yk and v k in Table 2). Let f ðiÞ, g SO ðiÞ; ESO ðkÞ, and ASO ðkÞ denote the corresponding mapping (Table 1) between deparSO ture order and group ID at System Optimum. Using the predetermined mappings (i.e., f ðiÞ; g SO ðiÞ; ESO ðkÞ, and ASO ðkÞ), the system optimum can be formulated as the following minimization problem:

min

TC ¼

n X ðcie ðNÞ þ cia ðNÞÞ

ð25Þ

i¼1

s:t: :

xk ¼ 0

ð26Þ

uk ¼ 0

ð27Þ

ð4Þ; ð5Þ; ð10Þ—ð13Þ; ð16Þ; ð17Þ

ð28Þ

4. Existence and uniqueness Lindsey (2004) has proved the existence and uniqueness of the dynamic user equilibrium solution of a single bottleneck with general heterogeneous users. The variational inequality formulation proposed in the previous section offers a new perspective to reexamine these properties. In particular, thanks to the transformation, we can focus on the existence and uniqueness of the solution to the VIP formulation (24). Proposition 1 (Existence). Under Assumptions 1–3, there exists a solution to VIðK; CÞ as defined in (24).

Proof. Note that K is a polyhedral, hence a nonempty, compact and convex subset of R2n . Also C is linear, and thus is a continuous mapping from K to R2n . The solution existence of VIP (24) then follows from, for example, Theorem 3.1 in Harker and Pang (1990).  Note that Assumptions 1–3 are not used directly when proving existence. However, they are needed to formulate the VIP (24).

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In order to prove the uniqueness of the equilibrium solution, let us first make some necessary definitions. Fiedler and Miroslav (1962) defined a P-matrix as follows: Definition 1. A matrix A is a P-matrix if it fulfills one of the following conditions:  all principal minors of A are positive;  to every vector x – 0, there exists an index k such that xk yk > 0 where y ¼ Ax.

Lemma 1. The following two properties of a matrix A are equivalent:  all leading principal minors are positive;  there exists a lower triangular matrix L and an upper triangular matrix U both with positive diagonal elements such that A ¼ LU.

Proof. see Theorem 3.1 (Fiedler and Miroslav, 1962).  To facilitate the following analysis, we first define a class of real square matrices A in the following form:

3

2

a1 h1 a2 h1    ai h1    an h1 6a h a h  a h  a h 7 2 2 n 27 i 2 6 1 1

6 . 6 . 6 . A¼6 6a h 6 1 1 6 . 6 . 4 .

.. .

..

.

a2 h2    ai hi    .. .

..

.

7 7 7 7 a n hi 7 7 7 7 5

ð29Þ

a1 h1 a2 h2    ai hi    an hn where ai > 0; hi > 0; 8i, and h1 < h2 <    < hi <    < hn . Below, we first prove that any matrix in the form of A is a P-matrix. As it turns out, the P property is essential to the uniqueness of equilibrium. Lemma 2. A matrix A of form (29) is a P-matrix if ai > 0; hi > 0; 8i, and h1 < h2 <    < hi <    < hn . Proof. See Appendix A.



Next, we note that, under Assumptions 2 and 3, the Jacobian matrix J of the commute cost function C is in the form of   A1 0 , where both A1 and A2 are P-matrices. 0 A2   A1 0 Lemma 3. Matrix J ¼ is a P-matrix if both A1 and A2 are P-matrices. 0 A2 Proof. See Appendix A.  Thus, the Jacobian matrix J of the commute cost function C is also a P-matrix. Now, let us define a class of functions, Pfunction (Definition 2.5 in Moré and Rheinboldt (1973)), and we will prove that CðNÞ (Eq. (23)) is a P-function. Definition 2. A mapping F : Q  Rn ! Rn is a P-function if for any x; y 2 Q; x – y, there is an index k ¼ kðx; yÞ 2 N such that ðxk  yk Þ½f k ðxÞ  f k ðyÞ P 0 and xk – yk . Proposition 2. The affine mapping CðNÞ ¼ J0 N in the problem VIðK; CÞ is a P-function when J is a P-matrix. Proof. According to Moré and Rheinboldt (1973) (Theorem 5.2), if C is differentiable on a rectangle Q and its Jacobian matrix C0 is a P-matrix for each point in Q; C is a P-function on Q. Lemma 3 proved that J is a P-matrix. According to Definition 2, thus, for any x; y 2 Q; x – y, there is an index k ¼ kðx; yÞ 2 N such that ðxk  yk Þ½f k ðxÞ  f k ðyÞ P 0 and xk – yk . Q Let Q be a large rectangle Q ¼ 2n i¼1 ½0; r; r N i ; i ¼ 1; . . . ; n, so that K  Q. It can be verified that C is still a P-function on K  Q, because for any x; y 2 K  Q; x – y, there is an index k ¼ kðx; yÞ 2 N such that ðxk  yk Þ½ck ðxÞ  ck ðyÞ P 0 and xk – yk . . Proposition 3 (Uniqueness). Under Assumption 1–3, the problem VIðK; CÞ has a unique solution.

Y. Liu et al. / Transportation Research Part B 71 (2015) 56–70

63

Proof. We invoke Theorem 2.3 in Moré (1974), which states that that there is at most one solution to VIðRnþ ; CÞ if C is P-function. Because K  Rnþ and C is P-function as per Proposition 2, there is at most one solution to VIðK; CÞ. Proposition 1 established that there exists at least one solution to VIðK; CÞ. Hence, we conclude that VIðK; CÞ has a unique solution.  Remark. It is worth noting that the uniqueness of the VIP solution can also be guaranteed if: (a) the Jacobian matrix of commute cost functions is positive definite; or (b) CðN Þ is strictly monotone, i.e.,

ðCðN1 Þ  CðN2 ÞÞT ðN1  N2 Þ > 0; 8N1 ; N2 2 K; N1 – N2 : Note that the class of positive definite matrices is a subset of P-matrices. An asymmetric Jacobian matrix J is positive definite if and only if matrix H ¼ ðJT þ JÞ=2 is positive definite because hx; Hxi ¼ hx; Jxi always holds. However, we note that the Jacobian matrix in our problem may not be positive definite and the commute cost function CðN Þ may not be strictly monotone. This is not a surprise since Conditions (a) and (b) are both sufficient conditions for the uniqueness. In other words, violating these conditions does not necessarily lead to non-uniqueness. To better demonstrate it, let us establish a two-classes example with a Jacobian matrix that is not positive semi-definite and thereby its variational inequality is not monotone. The demand consists of two commuters groups: group 1 (a1 ¼ 6$=h; b1 ¼ 5$=h; c1 ¼ 15$=h with demand N 1 ¼ 1500) and group 2 (a2 ¼ 0:5$=h; b2 ¼ 0:45$=h; c2 ¼ 1$=h with demand N 2 ¼ 1140). The corresponding Jacobian matrix is:

0

1 5:0 0:42 0 0 B 5:0 0:45 0 0 C B C J¼B C @ 0 0 1:0 12:0 A 0 0 1:0 15:0 0 5:0 2:71 0 B 2:71 0:45 0 B ) H ¼ ðJT þ JÞ=2 ¼ B @ 0 0 1:0 0

0

0

1

0 C C C 6:5 A

6:5 15:0

which is not positive definite. And we can easily find N1 ¼ ½1 3 1137 1499 and N2 ¼ ½2 1 1139 1498 such that: T

ðHT N1  HT N2 Þ ðN1  N2 Þ < 0 Namely, the variational inequality problem is not monotone. However, as proved in Proposition 3, it has a unique DUE solution. 5. Model with route choice It is straightforward to extend the above semi-analytical approach to address the route choice in a single origin–destination network with a number q of parallel routes. For each group, each route will be treated as two ‘‘imaginary’’ routes corresponding to early and late arrivals, thus there are a total of 2nq routes for each group in the equivalent static traffic assignment problem. Let S ¼ ðs1 ; s2 ; . . . ; sq Þ be the capacity; T ¼ ðT 1 ; T 2 ; . . . ; T q Þ be the free flow travel time of a set of routes; N jie (or N jia ) be the flow of an early-arrival (or a late-arrival) group i on route j; cjie (or cjia ) be the commute cost of an earlyarrival (or late-arrival) commuter in group i on route j; C ¼ ðC1 ; C2 ; . . . ; Cq Þ be the commute cost function, where Cj ¼ ðcjEð1Þe ; cjEð2Þe ; . . . ; cjEðnÞe ; cjAð1Þa ; cjAð2Þa ; . . . ; cjAðnÞa Þ; N ¼ ðN1 ; N2 ; . . . ; Nq Þ denote the flow, where Nj ¼ ðN jEð1Þe ; N jEð2Þe ; . . . ; N jEðnÞe ; N jAð1Þa ; N jAð2Þa ; . . . ; N jAðnÞa Þ. C can be shown as an affine mapping of N with a constant vector x:

CðNÞ ¼ J0 N þ x; J 2 R2nq2nq ; x 2 R2nq2nq where

0

J2 .. .

... .. . .. .

1 0 .. C . C C C C 0A

...

0

Jq

J1

0

B B0 B J¼B B .. @ . 0

ð30Þ

and Jj can be obtained by replacing s of the Jacobian matrix in Section 3 by sj , and the constant vector x depends on the value of time vector a ¼ ða1 ; a2 ; . . . ; an Þ :

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Y. Liu et al. / Transportation Research Part B 71 (2015) 56–70

x ¼ ðT 1 a; T 1 a; T 2 a; T 2 a; . . . ; T q a; T q aÞ Similarly, the variational inequality problem with route choice will be: T

CðN Þ ðN  N Þ P 0 where N denotes flow pattern that lies in a convex set K of vectors in R

Xq

Nj j¼1 ie

Njie ; N jia

þ

Njia

ð31Þ 2nq

satisfying the following demand constraints:

¼ Ni ; i ¼ 1; 2;    ; n

P 0; i ¼ 1; 2;    ; n

ð32Þ ð33Þ

Existence of the user equilibrium with route choice can also be verified by showing that K is a nonempty, compact and convex subset of R2nq and C is a continuous mapping from R2nq to itself. The proof is omitted here. This goes beyond Lindsey (2004) which only proves existence and uniqueness for a single route. Proposition 4 (Uniqueness). Under Assumptions 1–3, the problem VIðK; CÞ as defined by (31)–(33) has a unique solution.

Proof. Because each Jj ; 8i is a P-matrix which was proved in Section 4, we can invoke Lemma 3 and prove that Jacobian matrix in Eq. (30) is a P-matrix. Similarly, as in Proposition 3, one can show C is P-function and there is at most one solution 2nq to R2nq  þ . Because K  Rþ , there is at most one solution to ðK; CÞ. The detailed proof is omitted here. It is worth noting that if a route is not competitive for a commuter group, its corresponding class-specific route flow will become zero in the VI solution. In other words, the VI formulation allows the situation where some routes are not used by certain commuter groups. 6. Numerical results 6.1. Solution algorithm A simple assignment algorithm is developed based on the notion of Gauss–Seidel decomposition (Pang, 1985). Simply speaking, the idea is to iteratively solve a sequence of n ‘‘mini’’ assignment problems that only involves two parallel routes, i.e., a single pair in Fig. 1. The key is that the interactions with other routes are fixed when each subproblem is solved (so called ‘‘diagonalization’’). At each iteration k, we solve these n mini problems sequentially and immediately update the class flows afterwards. Mathematically, the problem in step i; ði ¼ 1 . . . ; nÞ at iteration k may be represented with a VIðK;  ckþ1 Þ i  kþ1 kþ1  kþ1  where ci ¼ cie ; cia , and

  kþ1 kþ1 k k kþ1 kþ1 k k ckþ1 ie ðN ie ; N ia Þ ¼ c ie N 1e ; . . . ; N ði1Þe ; N ie ; N ðiþ1Þe ; . . . ; N ne ; N 1a ; . . . ; N ði1Þa ; N ia ; N ðiþ1Þa ; . . . ; N na   kþ1 kþ1 k k kþ1 kþ1 k k ckþ1 ia ðN ie ; N ia Þ ¼ c ia N 1e ; . . . ; N ði1Þe ; N ie ; N ðiþ1Þe ; . . . ; N ne ; N 1a ; . . . ; N ði1Þa ; N ia ; N ðiþ1Þa ; . . . ; N na

where cie and cia are given in Eqs. (4) and (5). Solving the above VI problem is trivial because it amounts to equilibrating flows on two routes with linear cost functions. The solution to this problem is the new class flow for pair i at iteration k þ 1, i.e., kþ1 kþ1 N ie ; N ia . The procedure is repeated until a satisfactory convergence is achieved, which is measured by the distance between class flow vectors in two consecutive iterations, i.e., kN kþ1  N k k. The initial solution N 0 is obtained by solving equilibrium without link interaction, i.e., the Jacobian matrix has the same diagonal elements as Eq. (22) but zero off-diagonal elements:

J ¼ diagðbEð1Þ ; . . . ; bEðnÞ ; cAð1Þ ; . . . ; cAðnÞ Þ

ð34Þ

With Jacobian matrix above, to equilibrate the arrival-early and the arrival-late routes for each group i, the demand will be distributed between the two routes by the ratio ci =bi . The algorithm is coded in MATLAB and all tests are conducted on a desktop PC with Intel Core (@1.90 GHz, 1.90 GHz) and 4 GB of RAM. 6.2. Four-class example We first construct and test a four-class example to verify the proposed methodology. The experiment examines the welfare effects of Vickrey’s marginal cost toll under general user heterogeneity, by comparing the user equilibrium and system optimum solutions. The input parameters for the four classes are as follows: value of time a ¼ ½6:4 2:5 2:0 1:7, unit cost of schedule delay b ¼ ½3:9 1:95 1:8 1:5; c ¼ ½15:21 4:5 3:5 5,1 and demand is N ¼ ½1500 1140 800 500. The capacity of the bottleneck is 1500 vehicles per hour (vph). 1 It is worth noting that the values 6.4, 3.9 and 15.21 from Small (1982) were actually relative estimates. Yet, in this paper, as in many existing bottleneck studies, they were directly used as dollar estimates.

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The no-toll equilibrium (NTE) and system optimum (SO) flows as solved by the proposed method are given in Table 3. It took 0.2 s of CPU time and 47 iterations to solve the UE problem to a precision of 104 (measured by the Euclidean distance between the flows of the two consecutive solutions). At no-toll equilibrium, the departure order is determined by the relative ratio bi =ai for early arrival and ci =ai for late arrival. As visualized in Fig. 2 (a), all the group 3 commuters arrive later than their desired arrival time t  and all the group 1 commuters arrive earlier than their desired arrival time t  . This is because group 3 has a higher relative cost of early arrival to travel time (i.e., b=a) than the other groups and group 1 has a higher relative cost of late arrival to travel time (i.e., c=a) than the other groups. The system optimum, i.e., minimizing the total system cost, can be obtained by solving a minimization problem (Eqs. (25)–(28)). In this example, the queuing delay (xk and uk ) is eliminated by a time-depending toll, as shown in Fig. 2(b). Group 3 is absent from early-arrival route and group 4 is absent from late-arrival route. The departure order in Fig. 2(b) is very different from the departure order in Fig. 2(a). For the case with the homogeneous user preference, Vickrey (1969) demonstrated that a time-depending toll can successfully eliminate the congestion without changing commuters’ travel cost (i.e., everyone breaks even after pricing). When heterogeneous user preference is considered, the welfare effects of congestion pricing are complicated in general networks (Liu and Nie, 2012). This example demonstrates a non-monotonic pattern, in which the individual welfare gain does not always increase with his/her value of time. By comparing the commute cost of each group before and after pricing, the benefit from SO toll is provided in the last column in Table 3. Only group 1, who value their schedule delay highest among the four groups, benefit from SO toll, and the other three classes are worse off. The group 3, whose value of time is in the middle, have the highest increase in travel cost, and suffer the greatest loss in this example. While not reported due to space limit, we have also found examples in which the class with either the highest or the lowest value of time was affected the most by the toll. These finding are consistent with those in the literature (see e.g., van den Berg and Verhoef, 2011b; Liu and Nie, 2012). 6.3. California State Route 91 To demonstrate the utility of the proposed method as a numerical analysis tool, we test a large example constructed based on empirical data on California State Route 91. To do so we will first need to estimate the distributions of a; b and P c, the length of rush hour, Ni =s, free flow travel times, T, and users’ desired arrival time, t . We will combine our estimates of the marginal distributions to form the joint distribution of agent preferences, and then will solve for dynamic user equilibrium using our proposed method. We assume that the value of time for journey to work a follows a log-normal distribution because the literature suggests that a may be represented as a fraction of the wage rate (see e.g., page 53, Small and Verhoef, 2007), which can be fitted by log-normal distribution reasonably well (see e.g., Roy, 1950). To fit the distribution for a, we use the mean ($21) and the variance ($110) from the study on SR-91 commuters (Lam and Small, 2001). To estimate the remaining parameters we will use the data and methodology of Hall (2013). We cannot simply use his results because our model differs from his in that we assume all drivers have the same desired arrival time while he allows drivers’ desired arrival times to be uniformly distributed. To use his methodology we will also need to make some simplifying assumptions. First, we assume that c=b is constant for all users. This is necessary because we do not have repeated observations of individual drivers and so cannot estimate the relationship between a driver’s willingness to be early and his willingness to be late. Second, we find we are best able to match the data when we assume b=a is distributed uniformly over ½0; ^ d with a point mass at ^ d. 2 Finally, because our data on value of time is coming from a different source than the data we use to estimate b=a, we must assume there is no correlation between b=a and a. Following Hall (2013) we use road detector data from the California Department of Transportation’s Performance Measurement System (PeMS) (California Department of Transportation, 1999) to calculate travel times in the general purpose lanes3 for each arrival time for every non-holiday weekday4 in 2004 for trips on California State Route 91 westbound from the middle of Corona to I-605.5 We do this for the start of every five minute interval between 4:00 a.m. and 10:00 a.m. The PeMS dataset contains historical road detector data, primarily loop detectors, from almost all California highways. The loop detectors are able to estimate the speed at which traffic is moving above them. The raw data is reported for 30 s intervals. Caltrans then checks the data for a variety of errors, applies statistical corrections when needed, and aggregates the data to 5 min intervals. We use the 5 min frequency data. For the 33 mile long section of highway we have 39 loop detectors.

2 The associate editor who handles this paper pointed out that when drivers differ in their preferred arrival times, the departure rate at equilibrium may not depend on the ratios of b=a (or c=a) above some fractile of the distribution (Newell, 1987; Lindsey, 2004; de Palma and Lindsey, 2002b) Since heterogeneity in desired arrival time is not explicitly considered here, introducing a mass point might help fit the data better. 3 We note that Lam and Small (2001) estimated the value of time using the data from both the general purpose lanes and the express lanes. Thus, the mean and variance of VOT derived from their study may be slightly different from those of the drivers considered in this study. 4 We define holidays as the ten United States Federal Holidays. 5 This roughly represents the typical commute for those living in Corona and using State Route 91; which we calculated using data from Sullivan (1999).

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Table 3 Four-class example: NTE and SO flows. Group

No-toll equilibrium

1 2 3 4

System optimum

Benefit from toll($)

Early

Late

Early

Late

1500 873 0 459

0 267 800 41

1269 891 0 500

231 249 800 0

0.9

1.5 0.03 0.25 0.24

4

0.8

0.7

3 0.6

0.5

2

0.4

0.3

1

0.2

0.1

0 −1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0 −1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Fig. 2. No-toll equilibrium of four-class example.

We then use the generalized method of moments (Hansen, 1982) to estimate parameters of the distribution of b=a,6 the length of rush hour, free flow travel times, T, and users’ desired arrival time t  that best fit the empirical travel time profile. Specifically, our moment conditions are that for the start of each 5 min interval the model predicted travel times match the empirical average travel times. The model predicted travel times are calculated using (E.4) of Hall (2013). We then minimize the sum of squared errors in the moment conditions using the interior/direct algorithm from the Knitro optimization software package (Byrd et al., 2006). It took 0.001 s of CPU time and 10 iterations to find the minimum to a precision of 106 (measured by the Euclidean distance between two consecutive solutions). The estimation results indicate that b=a follows a uniform distribution between ½0; 0:23 with a mass point at b=a ¼ 0:23 with density 0.59 and the constant b=c ¼ 0:40. Furthermore, we find that rush hour is 7.5 h long, running from 4:35 a.m. to 12:19 p.m., meaning total demand is 7.5 times of the capacity in the morning, and that users’ desired arrival time t  is 6:50 a.m. Our estimate that rush hour is 7.5 h long stems from our definition of rush hour and our focus on the morning peak in isolation. Our definition of rush hour is the period of time when travel times are above free flow conditions, not just when congestion is particularly bad. That is, if in free-flow conditions travel times are 37 min, then rush hour begins once travel times climb to 37 min and 1 s. This means rush hour under our definition is significantly longer than it would be under the standard definition. Furthermore, we are focusing on the morning peak in isolation, that is, assuming that after the morning peak is over travel times have returned to free flow conditions, but problematically on SR-91 travel times do not return to free flow conditions during the lull between the morning and afternoon peaks. This leads to the decision to only match the model to the data between 4:00 a.m. and 10:00 a.m., ignoring the data afterwards. That said, we are choosing model parameters as though the morning peak occurred in isolation, and so we find a length of rush hour that, along with the other parameters, best explains the travel times we see between 4:00 a.m. and 10:00 a.m. Doing so leads us to predict that rush hour would not end until shortly after noon. The estimated cumulative density function of a and b=a is depicted in Fig. 3. Each dimension of distribution, a and b=a, is discretized into 10 ranges by identical steps and the weighted a and b=a of each range represent the value of time and relative cost of schedule delay to value of time of the commuters within each range. Therefore, the demand is discretized into 10 by 10, i.e., 100, groups. We also discretize a dimension of the mass point at b=a ¼ 0:23 into 10 ranges. Therefore, there are 110 classes. The demand d of a class within each joint interval a 2 ½a1 ; a2  and b=a 2 ½b1 ; b2  can be calculated by:

d¼

Z

a2

a1

6

Z

b2

f ða; bÞdadb

b1

Specifically ^ d and the size of the point mass.

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Y. Liu et al. / Transportation Research Part B 71 (2015) 56–70

0.4

0.35 0.45 0.30 0.40 0.35

0.25

0.30 0.20

0.25 0.20

0.15 0.15 0.10

0.1

0

10 0.05

20 30

0 0.25

0.05

40 50

0. 2

0.15

0.1

60 0.05

70

0

0

Fig. 3. Cumulative density function of commuters on SR-91.

where f ða; bÞ denotes the density function. Now that we have a realistic example of the distribution of agent preferences as well as the ratio of demand to road capacity, we can test our decomposition algorithm on it. Fig. 4 shows how the DUE solution solved by our VI approach fit the actual travel time from PeMS data. It provides the curve which demonstrates the trade-off between schedule delay and travel time at user equilibrium solution. The closer to the center of rush hour a commuter is, the higher the queuing delay she/he will suffer. Because of the constant ratio bi =ci 0:40 here, our results are verified by comparing with the analytical solution in Arnott et al. (1994). Interestingly, the decomposition algorithm found the exact equilibrium solution in the first iteration in this example. The reason is that the initial Jacobian matrix adopted in (34) (which ignores the interactions) dictates that the flow distributes between the two imaginary routes according to the ratio of ci =bi , which happens to give the analytical solution in a single-route problem.

60

PeMS Data

Travel time (min)

55

DUE Solution

50

45

40

4:00

6:00

8:00

Arrival time (am)

Fig. 4. DUE travel time vs. actual travel times.

10:00

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Y. Liu et al. / Transportation Research Part B 71 (2015) 56–70

Queuing Delay (minute)

25 20 Classes 10 Classes 5 Classes

20

15

10

5

0 4

5

6

7

8

9

10

11

12

(Morning) Fig. 5. Queuing delay and departure time of DUE in three scenario.

Sensitivity analysis is conducted to show how the discrete setting will affect the estimated traffic congestion evolution over time at DUE. DUEs under three scenarios are solved, respectively: 5 classes, 10 classes and 20 classes. Fig. 5 demonstrates the queuing delay (Y in the plot) a commuter encounters if departs at a time (X in plot). As we can see, three scenarios produce very similar results. 6.4. Multi-route example We proceed to test the proposed algorithm on a single origin–destination network with two routes. For the purpose of verification, the same two-route example in Liu and Nie (2011) is used. The parameters of the two-route numerical experiment are provided in Table 4. Each route is associated with a free flow travel time T i and capacity si . There are two commuter classes, whose travel preferences are also given in Table 4. All the commuters have identical desired arrival time, i.e., 8 a.m. The convergence performance of the algorithm is reported in Fig. 6, in which X represents the number of iterations and Y represents the convergence measure (the distance between two consecutive iterations). The convergence curve shows that the algorithm converged quickly with sharp drop in the first 20 iterations. It took 8.8 s of CPU time to reach a convergence measure of 104 . Importantly, the numerical solution obtained by the decomposition algorithm converged to the analytical solution provided   in Liu and Nie (2011), i.e., N ¼ N 11e ; N 12e ; N 11a ; N 12a ; N 21e ; N 22e ; N 21a ; N 22a ¼ ½840; 907; 216; 233; 35; 605; 9; 155. Table 4 Parameters adopted in the multi-route example. Travel time (min)

Parameter Value

T1 18 min

VOT parameters $/h T2 30 min

a1

a2

6.4

2.5

Capacity veh/h b1 3.9

b2 1.95

c1

c2

15.21

7.605

3

10

2

10

1

10

0

10 Gap

Category Unit

−1

10

−2

10

−3

10

−4

10

0

50

100

150

200

250

300

350

Number of Iterations

Fig. 6. Convergence curves of the multi-route example.

400

s1 2400

Time N/A s2 1600

t 8 a.m.

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It is worth noting that, although the ratio b=c is still constant in this example, the initial solution of the decomposition algorithm (equilibrium solution ignoring link interactions) no longer coincides with the analytical solution, due to the existence of another route. Consequently, hundreds of iterations are required to achieve a satisfactory convergence. 7. Concluding remarks This paper proposes a new approach for solving the dynamic equilibrium problem of a bottleneck model. This semi-analytical approach transforms the dynamic equilibrium problem of a bottleneck with heterogeneous users into a static traffic assignment problem. We show that the commute cost function is not separable and the link interaction is asymmetric. Since no equivalent optimization formulation exists for the equivalent traffic assignment problem, it is formulated and solved as a variational inequality problem. The variational inequality formulation provides a new perspective to examine the properties of the bottleneck model with heterogenous users. Importantly, alternative proofs for the solution existence and uniqueness are presented. Our numerical experiments suggest that the proposed methodology promises efficient solution of the bottleneck model with general user heterogeneity. Thus, it offers a useful numerical tool to perform policy analysis based on the bottleneck model. An example is the welfare analysis conducted in our first experiment, which reveals that users with intermediate value of time could suffer the greatest loss from a system optimal toll. The solution algorithm adopted in this study is a simple diagonalization algorithm. More sophisticated algorithms may provide better convergence performance than demonstrated herein, and hence warrant further investigation. The semi-analytical approach proposed in this paper can be extended to solve dynamic user equilibrium problems in more realistic and complicated contexts, such as considering extra dimensions of travel preferences (e.g., desired arrival time), or solving the equilibrium under step tolls (i.e., tolls invariant within a certain period). Some of these extensions are on-going research efforts. Performing more in-depth welfare analysis of congestion pricing using the proposed tool is another possible direction for further research. Such an analysis may consider, among other things, how the impact of route choice and the effectiveness of toll revenue redistribution schemes may be affected by the consideration of realistic user heterogeneity. Finally, an interesting and challenging question is how an infinite number of commuter groups (e.g., from a continuous joint distribution of travel preference) may be handled by the proposed methodology. Acknowledgement The authors wish to thank Professor Kenneth Small from the University of California at Irvine, Professor Robin Lindsey from the University of British Columbia, and two anonymous referees for their valuable comments on the earlier versions of the paper. This study is partially funded by a National Science Foundation project (the award number CMMI-1256021). The remaining errors are the authors’ alone. Appendix A. Proofs of Lemmas 2 and 3 Proof of Lemma 2. Note that

2

1 0

 0

32

a1 h1

6 1 1    0 76 0 76 6 76 A ¼ LU ¼ 6 76 . 6 .. .. . . 4. . . 0 54 .. 1 1  1 0

a2 h1



a2 ðh2  h1 Þ    .. . 0

an h1

3

an ðh2  h1 Þ 7 7

7 7 5 .    an ðhn  hn1 Þ ..

that is, there exists a lower triangular matrix L and an upper triangular matrix U both with positive diagonal elements such that A ¼ LU. Per Lemma 1, this is equivalent to all leading principal minors of A is are positive. It can be easily verified that any principal minor of A, i.e., AðMÞ where M is a subset of indices f1; 2; . . . ; ng, also belong to class A. As proved above, all leading principal minors of AðMÞ are positive, including AðMÞ itself. Thus, we conclude all the principal minors of A are positive. According to Definition 1, A is a P-matrix.  Proof of Lemma 3. According to Definition 1, to every vector x – 0, there exists an index k such that xk ðA1 xÞk > 0. Therefore,   x to every vector – 0, there exists an index k such that: w

xk



A1 0

0 A2



x w



k

¼ xk



A1 x A2 w



¼ xk ðA1 xÞk > 0

k

We invoke Definition 1 again, and conclude that J is a P-matrix. Proof is completed.



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