- Email: [email protected]
Dynamics of modal choice of heterogeneous travelers with responsive transit services
Transportation Research Part C 68 (2016) 333–349
Contents lists available at ScienceDirect
Transportation Research Part C journal homepage: www.elsevier.com/locate/trc
Dynamics of modal choice of heterogeneous travelers with responsive transit services Xinwei Li ⇑, Hai Yang Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, PR China
a r t i c l e
i n f o
Article history: Received 9 January 2016 Received in revised form 18 April 2016 Accepted 18 April 2016
Keywords: Dynamic modal choice Responsive transit strategy Equilibrium Stability Profit
a b s t r a c t In this paper, we investigate travelers’ day-to-day modal choice in a bi-modal transportation system with responsive transit services under various economic objectives. A group of travelers with heterogeneous preferences adjust their modal choice each day based on their perceived travel cost of each mode, aiming to minimize their travel cost. Meanwhile, the transit operator sets frequency each period according to the realized transit demand and previous frequency, trying to achieve different profit targets. For a given profit target, the fixed point of equilibrium may not be unique. We establish the condition for existence of multiple fixed points and examine the stability of the fixed points in each case. Furthermore, in view of a socially desirable mode choice, we also investigate the impacts of total travel demand and bus size on the convergence of the system to various fixed points associated with different targeted mode split. Finally, we use several numerical examples to illustrate the theoretical results and their practical implications for the transit operator to design appropriate transit schemes in a dynamic transportation environment. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction A large pool of literature relies on the steady-state equilibrium paradigm to provide theoretical basis for the transport planners through the deterministic user equilibrium (DUE) or stochastic user equilibrium (SUE) approach. However, the steady-state analysis is limited for it only shows an end result rather than the choice adjustment process. The dynamic transportation approach can characterize the evolution of traffic states over time and its value of modeling the traffic system is well acknowledged. There are two major categories of dynamic approaches. One is within-day dynamic (e.g. Friesz et al., 1993; Lam and Huang, 1995; Ran et al., 1996), which focuses on dynamic traffic assignment (DTA) research and assumes the system is unchanging over days. The other one is day-to-day dynamics approach with dynamics between-day scale and constant within-day scale. Day-to-day dynamics approach can date back to the works of Horowitz (1984), Smith (1984) and Cascetta (1987, 1989); the present paper also lies in this research area. A bi-modal system is often as an example to analyze the interaction between private cars and public transits in the literature. For instance, Arnott and Yan (2000) and Kraus (2003) indicated the underpricing of auto travel is a source of market distortion and a resource waste in transportation system with private and public transport modes. Li et al. (2012) investigated the intermodal equilibrium with two alternative modes. David and Foucart (2014) provided a game-theoretical model
⇑ Corresponding author. E-mail address: [email protected] (X. Li). http://dx.doi.org/10.1016/j.trc.2016.04.014 0968-090X/Ó 2016 Elsevier Ltd. All rights reserved.
334
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
in which travelers have heterogeneous preferences choose rationally between using the car or public transportation. Zhang et al. (2014) studied the conditions for occurrence of the Downs-Thomson Paradox under different economic objectives. However, these studies are concerned with a steady state analysis; neither travelers’ day-to-day mode choice nor transit operator’s periodic operating strategy are considered. It is known that more (fewer) transit passengers lead to more (less) frequent service, which leads to even more (fewer) passengers. This so-called ‘‘Mohring effect”, named for the wellknown economist Herbert Mohring, can be better analyzed in a dynamic environment. A few recent studies are carried out on the dynamic bi-modal problem in the literature. Cantarella et al. (2013) conducted an analysis of day-to-day dynamic mode choice in a transportation system with homogeneous travelers; bus operating strategies are demand-responsive but with a fixed fleet size. Bar-Yosef et al. (2013) examined the phenomena of a vicious cycle of a bus service line; nevertheless, mode choice of non-captive travelers is based only on the willingness-to-wait for the bus rather than the total trip cost. Besides, neither of the two studies take account of the operation cost and specific transit operating schemes. Three recent dynamic modeling approaches are relevant but purely in the context of congestion pricing. Tan et al. (2015) proposed a discrete-time tolled scheme by using the implicit Runge–Kutta method with heterogeneous users and investigated the equilibrium properties of the dynamic system. Farokhi and Johansson (2015) considered a piecewise-constant congestion taxing policy for repeated routing games of fixed demand. Ye et al. (2015) investigated the convergence of the trial-and-error tolling procedure for achieving system optimum with day-to-day flow dynamics when the observed link flow pattern is in disequilibrium. In the latter two studies, the link toll is constant in each period and calculated based on marginal cost pricing with the flow observed at the beginning of the tolling period. In the same spirit, we assume that the transit operator adjusts the transit frequency from period to period and keeps the frequency unchanged throughout each period. Transit frequency is reset at the beginning of each period based on the realized transit demand at the end of the previous period. This paper is intended to fill a gap in the literature of bi-modal problems by examining the double dynamics of heterogeneous travelers’ day-to-day modal choice and transit operator’s periodic operating policy. Transit operator adjusts service frequency to drive the dynamic system towards a desired equilibrium while achieving a profit target. For various combinations of economic objectives and operating schemes, the interactions between travelers and transit operator are investigated in a dynamic environment. Existence, uniqueness and stability of the equilibrium fixed point of the double dynamic system are established. Particularly, occurrence of the vicious cycle considered by Bar-Yosef et al. (2013) is analyzed under different levels of travel demand and bus capacity. The rest of the paper is organized as follows. Section 2 introduces the bi-modal transportation system with heterogeneous travelers. Section 3 analyzes the bi-modal system in a period with given transit frequency, and shows the existence, uniqueness and stability of user equilibrium. Inter-period dynamic frequency adjustment of a responsive transit operator is considered under different economic objectives in Section 4. The impacts of travel demand and bus capacity on the system performance and the occurrence of the vicious cycle are examined in Section 5. Section 5 provides a numerical example and Section 6 draws the conclusions and highlights avenue for future research. 2. Model formulation We consider one origin–destination (OD) pair connected by a congested highway running in parallel to an exclusive transit line. On a typical day, the total travel demand for this OD pair is fixed at d. Travelers have to make a discrete choice between using a private transport (auto) and public transit (bus). For simplicity, the occupancy of each private car is assumed to be 1. Travelers are heterogeneous as they have different intrinsic preferences for the use of private transport relative to public transit. The share of auto users is represented by z, then the share of public transport users is 1 z. Transit users have to pay for the transit fare, spend time in waiting and riding. Following the model in David and Foucart (2014), the travel cost of commuters by each mode consists of the travel time cost and the monetary cost and the disutility of a traveler i is specified as follows:
pti ¼ wðf Þ þ s þ
ei 2
and
pci ¼ tðzÞ
ei 2
;
where pti stands for public transit cost and pci for private car cost, f and s denote the transit frequency and uniform ticket price (transit fare), respectively. The function wðf Þ represents the travel time cost of public transport as a function of service frequency. It is assumed to be strictly decreasing and differentiable with respect to transit frequency f, i.e., w0 ¼ w0 ðf Þ < 0. The function tðzÞ represents the travel cost of private car, including both travel time cost and monetary cost, it is assumed to be strictly increasing and differentiable with respect to auto share z, i.e., t 0 ¼ t0 ðzÞ > 0 and tð0Þ > 0 at free flow. We also assume that w0 is bounded for all f > 0 and t 0 is bounded for any z 2 ½0; 1. The commuter-specific parameter, ei , reflects individual preference between private car and public transit, it is continuously distributed with a cumulative distribution function e GðeÞ, where G is assumed to be strictly increasing, continuous and differentiable over its support ð1; þ1Þ. The support implies that some individuals prefer the private transport so
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
335
much and will never choose public transport (ei ! þ1), while others will always choose public transit (ei ! 1). The density function of ei is represented by g ðeÞ and G0 ðeÞ ¼ g ðeÞ > 0. The transit operation cost is represented by function kðf Þ, where kðf Þ is strictly increasing and differentiable with respect 0 0 0 to transit frequency f, i.e., k ¼ k ðf Þ > 0. k is bounded for any finite frequency. Let p denote the transit operation profit, which is equal to the operation revenue minus operation cost, i.e., p ¼ sð1 zÞd kðf Þ. In this paper, for analytical tractability, we assume that the responsive transit operator changes transit frequency from period to period only in responses to the change in transit demand, the transit ticket price s is fixed throughout all the operating periods. The major reason for this assumption is that the strategic combination of ðs; f Þ is not unique for achieving a given economic objective such as a certain level of transit profit. 3. Intra-period day-to-day modal choice with given transit frequency and fare 3.1. Static analysis We assume each rational traveler chooses the transport mode to maximize his utility given his expectation on the cost of each travel mode. Hence, traveler i commutes by car if pci < pti , i.e., ei > tðzÞ wðf Þ s. Let Dðz; f Þ ¼ tðzÞ wðf Þ s represent the additional cost (either positive or negative) faced by auto users in comparison with the cost faced by public transport users. Thus, the condition for traveler i to choose private transport is
ei > Dðz; f Þ:
ð1Þ
And the user equilibrium condition is
Z z¼
þ1
Dðz;f Þ
g ðeÞde:
ð2Þ
The existence of a user equilibrium for any given f is guaranteed in the following proposition. Proposition 1. For any given f, there exists a unique user equilibrium.
R þ1 Proof. For any given f, at equilibrium, z is pinned down by hðzÞ ¼ 0, where hðzÞ ¼ z Dðz;f Þ g ðeÞde. Since R þ1 0 monotonically increases with z. Also by hð0Þ ¼ Dð0;f Þ g ðeÞde < 0 and h ðzÞ ¼ 1 þ g ðDðz; f ÞÞt0 > 0; hðzÞ R þ1 R Dð1;f Þ hð1Þ ¼ 1 Dð1;f Þ g ðeÞde ¼ 1 g ðeÞde > 0, we can see that z is unique, so is the equilibrium. h 3.2. Intra-period day-to-day modal choice n
Suppose the transit frequency is fixed in each period and let f represent the frequency in the nth period. Here we analyze travelers’ day-to-day modal choice in a representative period. The general form of our day-to-day dynamic modal choice model in continuous time is
z_ ¼ d
Z
þ1
Dðz;f n Þ
!
g ðeÞde z ;
ð3Þ
R þ1 Dðz;f n Þ g ðeÞde z provides a flow changing direction. Dynamic system (3) means that, on any day, the share of auto users tends to move from the current auto share R þ1 R þ1 to the ‘‘target” auto share Dðz;f n Þ g ðeÞde, based on the current day situation. The target share Dðz;f n Þ g ðeÞde describes travelers’ n n cost-minimization behaviors. Specifically, as mentioned before, travelers with ei > D z; f will choose car while ei < D z; f choose transit to minimize their perceived travel cost when the current auto share is z; in a word, at any day-to-day time travelers tend to switch to the current cost-saving travel mode. This form of day-to-day dynamic is somehow consistent with the link-based day-to-day network model proposed by He et al. (2010) and their model is also investigated by Guo et al. (2013) and Guo et al. (2015) in both discrete and continuous time. Let zn denote the stationary point of dynamic system (3), which is given by the solution of equation z_ ¼ 0. Then where d is a positive constant parameter determining the flow changing rate, and
Z
zn ¼
þ1
Dðzn ; f n Þ
g ðeÞde:
ð4Þ
Clearly, zn coincides with the static equilibrium solution of Eq. (2) and it is unique as proved in Proposition 1. We are now ready to prove that the stationary point of the dynamic system (3) is asymptotically stable by employing Lyapunov’s stability theorem, which is stated below.
336
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
Lemma 1. [Khalil, 2002, Theorem 4.1] Let z be a stationary point of the system z_ ¼ hðzÞ and Z 2 Rn be a domain containing z . Let a Lyapunov function V : Z ! R be continuously differentiable such that
V ðz Þ ¼ 0 and VðzÞ > 0; _ VðzÞ 6 0;
8z 2 Z n fz g;
ð5Þ
8z 2 Z:
ð6Þ
Then z is stable. Moreover, if
_ VðzÞ < 0;
8z 2 Z n fz g;
ð7Þ
Then z is asymptotically stable. Based on Lemma 1, the following proposition holds.
Proposition 2. The unique stationary point zn of the dynamic system (3) is asymptotically stable. Proof. The proof is given in Appendix A.1. h The following Example 1 is provided to illustrate Propositions 1 and 2 on the uniqueness and stability of the stationary point of the dynamic system (3). Exampleh 1. Consider the two-mode network described in Section 2. The travel cost function of the private mode is i tðzÞ ¼ 10 1 þ 0:15ð5zÞ4 þ 20. Since the transit frequency and fare are fixed, the travel cost of public transport is also fixed and assumed to be wðf Þ þ s ¼ 35 þ 3 ¼ 38. All the cost units are in HK$. e follows the bimodal exponential power jej distribution BEF ð0; 1; 1; 1Þ, i.e. g ðeÞ ¼ j2eCjeð2Þ . n From Eq. (4), the stationary point is computed to be zn ¼ 0:3640 and the stationary cost difference is D zn ; f ¼ 8:4580. The following Fig. 1 depicts the evolution of the auto share day by day obtained by the dynamic system (3) with different initial values. The flow changing rate parameter d is set to be 0.05. From Fig. 1, we can see that, when the length of the period is long enough, such as 30 days, the dynamic system (3) will converge to the stationary point whatever the initial auto share is, which means the stationary point is asymptotically stable. 4. Inter-period transit operating strategy with intra-period day-to-day modal choice In this section, the transit operator adjusts the transit frequency on a periodic basis based on the transit demand observed at the end of the previous period, aiming to achieve various target profits. Then the frequency can be regarded as a function of the share of auto users z, i.e. f ¼ f ðzÞ and DðzÞ ¼ Dðz; f ðzÞÞ ¼ tðzÞ wðf ðzÞÞ s. 4.1. Static analysis with responsive transit operator When facing a decreasing demand, a transit operator will tend to reduce service to cut costs while when facing increasing demand, the transit operator will tend to increase supply to earn more profit. Thus, it is reasonable to make the following assumption. Assumption 1. The transit frequency is continuous and decreasing with respect to the share of auto users, i.e.,
df 6 0: dz
ð8Þ
Suppose the transit frequency has a realistic minimal value f min . The following proposition shows the existence of a user equilibrium and provides the condition for the existence of multiple equilibriums. Proposition 3. There exists at least one user equilibrium. Furthermore, multiple user equilibriums exist if and only if there exists at least one solution zk such that
R þ1 8 zk ¼ Dðzk Þ g ðeÞde > > > < R þ1 d g ðeÞde DðzÞ > >1 > dz > :
ð9Þ
z¼zk
Proof. The proof is given in Appendix A.2. h 0
Here we do not consider the extreme case in which there exists a user equilibrium zk such that h ðzk Þ ¼ 0, because it 0 implies that hðzk Þ ¼ 0 and h ðzk Þ ¼ 0 occur simultaneously, which is unlikely to happen in practice.
337
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349 1
Auto share
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
Days Fig. 1. The evolution auto share with d ¼ 0:05.
4.2. Inter-period transit operating strategies for a target profit When the length of the period is limited, the bi-modal system may not reach the equilibrium state at the end of each period. The transit operator adjusts the frequency at the beginning of a period based on the transit share observed at the end of the previous period. In other words, the transit operator adjusts frequency from period to period based on a still evolving disequilibrium modal split pattern. Suppose the transit operator resets its frequency from period to period in an attempt to achieve a given target profit p0 (for a revenue-neutral operator, p0 ¼ 0), the target profit is less than or equal to the maximum achievable profit. Let zn be the share of auto users at the end of the nth period and suppose that the transit operator’s response is characterized by Eq. (10), where bn 2 ð0; 1 for all n 2 N þ . bn ¼ 1 means that the transit operator changes the transit frequency strictly following the n rule according to the share of auto users at the end of the previous period to get the target profit p0 . Given zn and f , the transit frequency in the ðn þ 1Þth period is
f where
nþ1
¼ bn f ðzn Þ þ ð1 bn Þf
n
h i 1 f ðzn Þ ¼ max k ðsdð1 zn Þ p0 Þ; f min
ð10Þ
ð11Þ
As shown in (10) and (11), the new transit frequency is determined once the realized mode share is observed at the end of the previous period. To establish the convergence and stability of the two symbiotic processes of modal choice dynamics and frequency adjustments, we need to know how the auto share z changes from period to period. Nonetheless, it is difficult to determine znþ1 explicitly owing to its implicit dependence on zn and parameter d as well as the period length. However, the following Proposition 4 shows that, when the day-to-day dynamic modal choice follows Eq. (3), the resulting znþ1 is always between zn and the equilibrium auto share for the ðn þ 1Þth period. Proposition 4. When the day-to-day dynamic modal choice follows the dynamic system (3), znþ1 is always between zn and the equilibrium auto share for the ðn þ 1Þth period, that is
znþ1 ¼ an zðnþ1Þ þ ð1 an Þzn where
zðnþ1Þ ¼
Z
þ1
Dðzðnþ1Þ ;f nþ1 Þ
g ðeÞde
ð12Þ
ð13Þ
and an 2 ð0; 1 for all n 2 N þ : Proof. The proof is given in Appendix A.3. h Proposition 4 is clearly displayed in Fig. 1. It is seen from Fig. 1 that the share of auto users on any day in a period is always between its initial value and the equilibrium value for that period, regardless of the period length. Furthermore, we note in passing that an ¼ 1 Eq. (12) implies that the ðn þ 1Þth period is long enough for the dynamic system to settle down to its equilibrium. Given the periodic transit operating strategy specified by Eq. (10) and the end period auto share given by (12) resulting from the day-to-day modal choice dynamics (3), we now move to investigate the properties of the double dynamic system. First we introduce the following matrix.
338
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
0
@f nþ1 n @ @f
Dn ¼
1
@f nþ1 @zn
@znþ1 @f n
A¼
@znþ1 @zn
1 bn an ð1 bn ÞBn
where
( An ¼
n
k0 ðfsðdzn ÞÞ ; 0;
for f – f min n
for f ¼ f min
;
bn An an bn An Bn þ 1 an
ð14Þ
nþ1 nþ1 w0 f g D zðnþ1Þ ; f : Bn ¼ nþ1 nþ1 t0 z f 1 þ g D zðnþ1Þ ; f
n nþ1 . Then the determinant of Dn is jDn j ¼ ð1 an Þð1 bn Þ 2 ½0; 1Þ for all n 2 N þ , which is independent of zn ; f or znþ1 ; f
nþ1
1
From Proposition 1, zðnþ1Þ is unique when f is fixed. Since kðf Þ is a monotonic function, k ðsdð1 zn Þ p0 Þ and f ðzn Þ n nþ1 from Eqs. (10) and (12). Let ~z ; ~f are also unique when zn is fixed. Hence, given zn ; f , we can obtain a unique znþ1 ; f represent the fixed point of system (10) and (12), then,
~z ¼
Z
þ1
Dð~z ;~f Þ
g ðeÞde
ð15Þ
h i ~f ¼ max k1 ðsdð1 ~z Þ p0 Þ; f min :
ð16Þ
From Eq. (11), we know that the transit frequency is decreasing with respect to the share of auto users, which satisfies Assumption 1. As proved in Proposition 3, there exists at least one equilibrium fixed point ~z ; ~f for given target profit
p0 (hereinafter, calledtarget profit equilibrium fixed point). Multiple fixed points exist if and only if there exists at least one fixed point ~z ; ~f
such that
R þ1 d Dðz;f ðzÞÞ g ðeÞde dz
z¼~z
0 1 s d ¼ g D ~z ; ~f @t0 ð~z Þ þ w0 ~f A > 1: 0 k ~f
ð17Þ
Furthermore, we can analyze the stability of each fixed point. First we provide the following lemmas. Lemma 2. Let A and B be two constants, and denote k1 ; k2 the two eigenvalues of
J¼
1a
aA
ð1 aÞbB abAB þ 1 b
;
where a; b 2 ð0; 1, then the moduli of k1 and k2 are both less than 1, i.e. jkj < 1, if and only if ð2 aÞð2 bÞ=ab < AB < 1. Proof. The proof is given in Appendix A.4. h Definition 1. For any set D of matrices, we take an arbitrary point x0 2 R as initial point and define the following recurrence
xtþ1 2 fDit xt : Dit 2 Dg:
ð18Þ
A series ðx0 ; x1 ; . . . ; xk ; . . .Þ satisfying the inclusion (18), xtþ1 ¼ Dit xt , for some ðit Þ sequence, is a trajectory in the Rn space. Let DLIðDÞ denote the discrete linear inclusion set fxk : k P 0g, consisting of all possible sequences of vectors in Rn generated using the relation (18). Lemma 3. [Theys, 2005, Proposition 3.2] For any bounded set D of matrices, the joint spectral radius of the set D is defined as n o
1=k qðDÞ ¼ limk!1 max Di1 Dik : Di 2 D . Then qðDÞ < 1 () DLIðDÞ is asymptotic stability.
Based on Lemmas 2 and 3, we can investigate the stability of system (10) and (12) at each fixed point. The stable condition is given in Proposition 5. 0 t0 ð~z Þ þ sd=k ~f w0 ~f < 1, the corresponding interior fixed point is (locally) Proposition 5. When g D ~z ; ~f asymptotically stable and the boundary fixed point is always (locally) asymptotically stable.
Proof. The proof is given in Appendix A.5. h
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
339
From Proposition 5 we can see that the convergence of the dynamic system and the stability of the fixed points over periods are independent of the values of updating parameters. Now we can further investigate the stability of all fixed points in the following Proposition 6. Of particular interest is the fixed point with a minimal share of auto users, since the auto volume is highly related to road congestion. Proposition 6. For system (10) and (12), when there is only one target profit equilibrium fixed point, it must be asymptotically stable. When there are multiple target profit equilibrium fixed points, at least one fixed point is unstable and the fixed point with the lowest and the highest share of auto users is asymptotically stable. Proof. The proof is given in Appendix A.6. h 4.3. Inter-period transit operating strategies for a periodic maximum profit In this subsection, the transit is operated by a periodic profit-maximizing authority; the operator chooses the frequency in each period to maximize its profit only based on the share of transit users at the end of the previous period, without taking into account the impact of frequency change on travelers’ modal choice in the period. We assume that the marginal cost of carrying one additional traveler is less than the marginal profit. This means that the transit operator will try to make every transit bus loaded as full as possible for profit maximization. Let m denote the predetermined desirable number of passengers per bus (or equivalently, predetermined desirable load factor). The value of m evidently depends on the maximal capacity of a bus, but will be somewhat lower than that to avoid overcrowded buses and higher than the number of passengers necessary to merely cover operation costs (Bar-Yosef et al., 2013; Meignan et al., 2007). Then the transit operator’s response in terms of frequency setting from period to period is described by
f
nþ1
n
¼ bn f ðzn Þ þ ð1 bn Þf ;
where
f ðzn Þ ¼ max
ð19Þ
ð1 zn Þd ; f min : m
ð20Þ
In Eq. (19), bn 2 ð0; 1 and bn ¼ 1 means that the transit operator only follows the periodic profit-maximizing rule to change transit frequency according to the share of transit users at the end of the previous period. Here we should mention that transit operator’s choice of frequency in the current period is based on the realized demand at the end of the previous period, but, travelers’ response of modal choice or auto share in the current period is given by Eq. (12), which depends on the chosen frequency. Let z ; f represent the fixed points of system (12) and (19), then
z ¼
Z
þ1
Dðz ;f Þ
g ðeÞde
ð21Þ
f ¼ max ð1 z Þd ; f min : m
ð22Þ
Thus df =dz ¼ d=m < 0, which also satisfies Assumption 1. As proved in Proposition 3, there exists at least one periodic profit-maximizing equilibrium fixed point z ; f . Multiple periodic profit-maximizing fixed points exist if and only if there exists at least one periodic profit-maximizing equilibrium fixed point z ; f such that
R þ1 d Dðz;f ðzÞÞ g ðeÞde dz
¼ g D z ; f t 0 ðz Þ þ w0 f d=m > 1:
ð23Þ
z¼z
For the periodical dynamic system (12) and (19), we define
0
En ¼
@f nþ1 n @ @f @znþ1 n @f
@f nþ1 @zn @znþ1
1
A¼
@zn
1 bn
an ð1 bn ÞBn an bn An Bn þ 1 an
where
( An ¼
n
d=m for f – f min n
0 for f – f min
bn An
;
;
nþ1 nþ1 w0 f g D zðnþ1Þ ; f Bn ¼ : nþ1 t 0 ðzðnþ1Þ Þ 1 þ g D zðnþ1Þ ; f
ð24Þ
340
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
0 Similar to the proof in subsection 4.2, by changing An ¼ sd=k ~f in subsection 4.2 to An ¼ d=m and following the same processes, we have the following corresponding Propositions 7 and 8. Proposition 7. When g D z ; f t 0 ðz Þ þ ðd=mÞw0 f < 1, the corresponding interior fixed point is (locally) asymptotically stable and the boundary fixed point is always (locally) asymptotically stable. Proposition 8. For system (12) and (19), when there is only one periodic profit-maximizing equilibrium fixed point, it must be globally stable. When there are multiple periodic profit-maximizing equilibrium fixed points, at least one fixed point is unstable and the fixed points with the lowest and highest shares of auto users are locally stable. We conclude this section by pointing out that, as long as the dynamic system takes the form of Eqs. (10) and (12), where zðnþ1Þ is determined by Eq. (13), and f ðzn Þ satisfies df ðzn Þ=dzn 6 0, the stability condition of the fixed points is irrelevant to the value of the updating parameters an and bn . 5. The impacts of travel demand and bus capacity When the transit operator is responsive to the share of transit users, we always have df =dz < 0, no matter whether the operator is to achieve a target profit or to maximize profit periodically. Thus, when facing with decreasing transit demand, a transit operator tends to reduce frequency to cut expenses. Furthermore, based on Eqs. (12) and (13), we have dz=df < 0, which implies that reducing the transit frequency may induce some of the travelers to change to auto, resulting in fewer transit users. The decrease in the share of transit users may cause a further decrease in transit frequency, then a vicious cycle continues until only travelers with very strong preference for transit (very small ei ) stay on transit. However, a large share of auto users means more vehicles (auto and bus) on roads, which may contribute more to roadside pollution. Therefore, we further investigate the impact of travel demand (dÞ and bus capacity (mÞ and understand when vicious cycle of a transit line can be avoided. 5.1. The impact of travel demand Let us consider the travel demand as a model parameter and define
Z
þ1
hðz; dÞ ¼ z
Dðzd;f ðdÞÞ
g ðeÞde;
ð25Þ
where Dðzd; f ðdÞÞ ¼ t ðzdÞ wðf ðdÞÞ s and v ¼ zd is defined to denote the volume of auto users. Thus, dtðzdÞ=dd ¼ zt 0 ðv Þ > 0. 0 When the transit operator is revenue-neutral, we have df =dd ¼ sð1 zÞ=k ðf Þ > 0; when the transit operator is periodic profit-maximizing, we have df =dd ¼ ð1 zÞ=m > 0. Thus, regardless of the transit strategy, we have
@hðz; dÞ df > 0: ¼ g ðDðz; f ðdÞÞÞ zt 0 ðv Þ w0 ðf Þ dd @d
ð26Þ
Therefore, hðz; dÞ is strictly increasing with respect to travel demand. At equilibrium, z is pinned down by hðz; dÞ ¼ 0. Since hð0; dÞ < 0 and hð1; dÞ > 0 as proved before, the graph of hðz; dÞ with different travel demands can be shown in the following Fig. 2 with d1 < d2 < d3 . From Fig. 2, we can conclude that when travel demand is low (such as d ¼ d1 Þ, there is only one equilibrium z . This equilibrium point is stable and with relatively large share of auto users, thus a vicious cycle will occur for the transit line. As the travel demand increases, the system can turn into a situation with two stable equilibriums (such as d ¼ d2 Þ or even one (such as d ¼ d3 Þ. Thus, when the travel demand is high, the vicious cycle for the transit will be unlikely to occur and the system will converge to the equilibrium point with a relatively small share of auto users. 5.2. The impact of bus capacity For the revenue-neutral transit operator, we do not consider the bus capacity, since the frequency is determined by the transit demand at the end of the previous period. In this subsection, we only consider periodic profit-maximizing transit operator. The bus capacity is reflected by parameter m, which is convenient for analyzing the impact of bus capacity. Define
Z hðz; mÞ ¼ z
þ1
Dðz;f ðmÞÞ
g ðeÞde:
ð27Þ
Then
@hðz; mÞ ð1 zÞd ¼ g ðDðz; f ðmÞÞÞw0 ðf Þ < 0: @m m2
ð28Þ
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
341
Fig. 2. Evolution of model equilibriums with different travel demands (d1 < d2 < d3 Þ.
Similar to the proof in subsection 5.1 and from the following Fig. 3, we can conclude that when the bus capacity is large, there is only one stable equilibrium with relatively large share of auto users and the vicious cycle will occur for the transit line. Then by reducing the transit capacity, the system will move to a situation with one or two stable equilibriums. Thus, when the bus size is small, the vicious cycle for the transit will be unlikely to occur and the system will converge to the equilibrium point with relatively small share of auto users. This is because, when the total travel demand is fixed, using smaller buses will increase bus frequency and thus decrease transit cost, which in turn brings up an increase in transit demand. When the capacity is fixed, the transit profit is given by
pðzÞ ¼ sð1 zÞd kðf ðzÞÞ ¼ sð1 zÞd kðð1 zÞd=mÞ:
ð29Þ
0
Profitability of bus services requires that sm > k ðf Þ, which means that the maximal profit can be earned when the marginal revenue for running one additional bus is larger than its marginal cost. Thus
p0 ðzÞ ¼ sd þ k0 ðf Þ d=m < 0;
ð30Þ
which means that a transit operator can make a maximal profit at the stable fixed point with a minimal auto share. This underscores the importance of investigation on how to avoid vicious cycle and find the attraction domain for the dynamic system to converge to the stable fixed point with the lowest auto share of travelers. From Fig. 3 and our previous analysis, when the total demand is fixed, smaller buses can increase the transit frequency then decrease the transit travel cost and attract more travelers to choose transit, and finally increase the transit income. However, increasing transit frequency will also increase the transit operation cost. Thus, the choice of bus capacity by a periodic profit-maximizing operator depends on the specific forms of the operation cost function. In practice, the operation cost depends on both bus frequency and bus capacity. Thus, we define the operation cost function as kðf ; mÞ ¼ kðð1 zÞd=m; mÞ and assume @kðf ; mÞ=@m > 0. From Fig. 3, we can conclude that dz=dm > 0 in the attraction domain where the system converges to the fixed point with the lowest share of auto users. If
dkðf ; mÞ @kðf ; mÞ ð1 zÞd @kðf ; mÞ ¼ > 0; þ dm @f m2 @m which is equivalent to
1 @kðf ; mÞ 1 @kðf ; mÞ > ; f @m m @f
ð31Þ
the operation cost is increasing with bus capacity and
dpðmÞ dz dkðf ; mÞ ¼ sd < 0; dm dm dm
ð32Þ
which means that the transit profit is decreasing with bus capacity. Thus, smaller buses are preferred by the transit operator to earn more profit. Eq. (31) means that the marginal cost of transit system capacity by increasing bus size is larger than the corresponding marginal cost by increasing transit frequency. If Eq. (31) does not hold, such as when the operation cost is not very sensitive to bus size, extremely when @kðf ; mÞ=@m ¼ 0, it is no longer guaranteed that smaller buses are better for the periodic profit-maximizing authority. 5.3. A numerical example This example is presented to show the two symbiotic evolution processes of the day-to-day modal choice (3) and periodic frequency adjustments (10). It demonstrates the existence of multiple equilibrium fixed points, and illustrates the impact of the total demand and bus capacity.
342
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
Fig. 3. Evolution of model equilibriums with different desirable number of passengers per bus (m1 < m2 < m3 Þ.
Table 1 Function specifications. Function
Public transport travel time cost function
Specification 4 tðzÞ ¼ VOT 8 1 þ 0:15 zd þc H wðf Þ ¼ VOT 2f1 þ 10
Transit operation cost function
kðf Þ ¼ kf
Highway travel cost function
Consider the two-mode network described in Section 2 with function specifications given in Table 1 and parameter values in Table 2. Note that all the propositions only require that e is continuously distributed with a cumulative distribution function e GðeÞ, where G is assumed to be strictly increasing, continuous and differentiable over its support ð1; þ1Þ. Here we jej
consider a bimodal exponential power distribution BEF ð0; 1; 1; 3Þ of e, which means g ðeÞ ¼ 32jeCje 2 . In each period, the day-toð3Þ day dynamic system follows Eq. (3) with d ¼ 0:2. For periodic system dynamics, bn ¼ 0:8 for all periods. The length of each period is 30 (days). When transit operator follows the break-even strategy (i.e. p0 ¼ 0Þ, from Eqs. (15) and (16) we can obtain three different ~z1 ; ~f 1 ¼ ð0:5442; 0:0570Þ; ~z2 ; ~f 2 ¼ ð0:8504; 0:0187Þ and break-even equilibrium fixed points, i.e. ~ ~z3 ; f 3 ¼ ð0:9596; 0:0125Þ. Fig. 4 plots the trajectories of the auto share and the bus frequency starting from different initial states. It can be seen from Fig. 4 that the entire set is divided into two attraction domains, A and B. Initial states in domain A 0 converge to ~z3 ; ~f 3 and initial states in domain B converge to ~z1 ; ~f 1 . Since g D ~z1 ; ~f 1 t0 ~z1 þ sd=k ~f 1 w0 ~f 1 <1 ~ ~ ~ and ~z3 ; f 3 is the boundary fixed point, Fig. 4 shows that ~z1 ; f 1 and ~z3 ; f 3 are stable fixed points, this validates Propo sition 5. The boundary of domain A and B is formed by the trajectories towards the unstable fixed point ~z2 ; ~f 2 . This observation is consistent with the finding in Bie and Lo (2010). In this example, the double dynamic system is much more likely to converge to ~z ; ~f due to the much larger area of domain B, which means the vicious cycle is unlikely to occur. Only when 1
1
the initial auto share is very high (such as z ¼ 0:9Þ and the initial transit frequency is very low (such as f ¼ 1=50Þ the vicious cycle will occur. However, in practice, the initial auto share is usually below 0.8, thus the double dynamic system will likely move into a virtuous cycle with the functions and parameters given in Tables 1 and 2. This example also points to the design of policies for government intervention to avoid the occurrence of a vicious cycle. When the initial dynamic system is in domain A, the system will converge to the boundary fixed point. The level of transit service has to be subsidized in order to keep the minimal frequency because the profit is negative at the boundary fixed point. However, with increasing car ownership, people may have higher preference for auto, and thus, a vicious cycle will become inevitable if no sufficient subsidies can be introduced. For a transit line operating near the level (the red dotted line in Fig. 4) that separates the stable equilibriums with low and high share of auto users, the system may enter either a vicious or a virtuous cycle. Subsidizing the transit system and increasing the bus frequency may lead to a virtuous cycle, thereby driving the system to converge to a desirable equilibrium. When the initial dynamic system is already near the stable equilibrium with low auto share, there is no need to increase transit subsidies, the system will self-enter a virtuous cycle. Fig. 5 shows the change of auto share across the periods from different initial values with the same initial frequency of f ¼ 1=40. As indicated in Fig. 5, different stable fixed-points are reached from different starting states. More periods are needed to reach the target equilibrium auto share if the initial auto share is far from the equilibrium. The change of the difference between the actual auto share and the target equilibrium auto share is shown in Fig. 6 with the length of each period being 15 days (the blue1 line) and 30 days (the red line). For each scenario, the initial auto share is 0.7. It is clearly shown that 1
For interpretation of color in Fig. 6, the reader is referred to the web version of this article.
343
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
the difference approaches zero after 50 periods when the length of each period is 15 days and after 33 periods when the length of each period is 30 days. Fig. 6 indicates that, more (fewer) periods are needed to reach the target equilibrium with shorter (longer) period length. It is thus interesting to choose a proper length of each period to minimize the whole duration needed to get the target equilibrium. Fig. 7 shows the impact of the total demand. When the total demand is scaled from 0:98d to 1:04d, three fixed points emerge. When travel demand is small (such as d ¼ 0:9dÞ, there is only one equilibrium with relatively large share of auto
Table 2 Value of parameter. Parameter
Value
Base total demand Desirable number of passengers per bus Highway capacity Coefficients in operation cost function Value of time (VOT) Monetary cost by auto Transit fare Minimum transit frequency
d ¼ 1200 (person/h) m ¼ 150 (person) H ¼ 500 (veh/h) k ¼ 9600 (HK$/run) VOT ¼ 1 (HK$/min) c ¼ 8 (HK$) s ¼ 1 (HK$) f min ¼ 1=80 (run/min)
1 0.9
A
Initial Equilibrium
0.8
Auto share
0.7 0.6 0.5 0.4 0.3
B
0.2 0.1 0 0.0125 0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
Transit frequency Fig. 4. Attraction domains of the dynamic bi-modal system under break-even strategy.
1 0.9 0.8
Auto share
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
5
10
15
20
25
Period number Fig. 5. The evolution of the auto share.
30
35
344
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
Gap between the target equilibrium
0.16 Period length = 15 days Period length = 30 days
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
5
10
15
20
25
30
35
40
45
50
Period number Fig. 6. The difference between the actual auto share and the target equilibrium with different length of each period.
1
Auto share
0.8 0.6 0.4 0.2 0.9
0.92
0.94
0.96
0.98
1
1.02
1.04
1.06
1.08
1.1
Scale of demand Fig. 7. Fixed points of the dynamic system (10) and (12) with black (white) dots representing stable (unstable) fixed points.
1
A
Initial Equilibrium
0.9 0.8
Auto share
0.7 0.6 0.5 0.4
B
0.3 0.2 0.1 0 0.0125 0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
Transit frequency Fig. 8. The trajectories of the dynamic system under periodic profit-maximizing strategy.
users, thus vicious cycle occurs for the transit line. When the travel demand is large (such as d ¼ 1:1dÞ, the vicious cycle for the transit is unlikely to occur and the system converges to the equilibrium point with relatively small share of auto users. When the transit operator follows the periodic profit-maximizing strategy, from Eqs. (21) and (22) we can obtain three z1 ; f 1 ¼ ð0:5001; 0:0667Þ; z2 ; f 2 ¼ ð0:8693; 0:0174Þ and different break-even equilibrium fixed points i.e. z3 ; f 3 ¼ ð0:9596; 0:0125Þ. We track the trajectories of the auto share and the transit frequency starting from different initial states and obtain the results as shown in Fig. 8. Fig. 8 shows that the fixed points z ; f and z ; f are stable while z ; f is 1
unstable.
1
3
3
2
2
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
345
1 0.9 0.8
Auto share
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
60
70
80
90
100 110 120 130 140 150 160 170 180 190 200
Desirable number of passengers per bus Fig. 9. Fixed points of the dynamic system (12) and (19) with black (white) dots representing stable (unstable) fixed points.
Fig. 9 illustrates the impact of bus capacity. When the desirable number of passengers per bus is between 70 to 180, there are three fixed points. When the bus capacity is large (such as m ¼ 200Þ, there is only one equilibrium with relatively large share of auto users, thus vicious cycle occurs for the transit line. When the bus capacity is small (such as m ¼ 60Þ, the vicious cycle for the transit is unlikely to occur and the system converges to the equilibrium point with relatively small share of auto users. We conclude this section by pointing out that similar results are obtained under a normal distribution of e, and thus not presented here.
6. Conclusions In this paper, we investigated the dynamics of modal choice of heterogeneous travelers with responsive transit service under different economic objectives. The problem of interest is modeled as a double dynamic system where travelers adjust their modal choice from day to day while transit operator sets transit service frequency from period to period in response to changing demand. We analyzed travelers’ modal choice in each period based on the travelers’ rational behavior that travelers tend to change to more attractive mode. The dynamic system of modal choice in each period has a unique fixed point which is identical with the user equilibrium auto share. The stability of this unique fixed point can be established by Lyapunov stability theorem. Over periods, the transit frequency is responsive to the transit demand under different economic objectives, which may result in multiple stable fixed points. In the presence of multiple stable fixed points, the fixed point with the lowest or highest auto share is stable and the stable condition is independent of the updating parameters. We also investigated the impacts of total travel demand and bus capacity. The analysis and results are useful for setting operating strategies to drive the dynamic system towards a desirable equilibrium fixed point. Further research is expected to extend the double dynamic model to a general network, to seek the optimal length of each period for fast system convergence. Intelligent transportation system design (Cantarella, 2013) for the network is also an interesting extension. Other factors pertaining to modal choice behavior, such as willingness-to-wait for buses (Peterson et al., 2006), bounded-rationality (Wu et al., 2013) could be incorporated into the current modeling framework. The transit fare is assumed to be fixed through this study; this assumption should be relaxed for better representing realistic transit operating strategies of both fare choice and frequency setting. In addition, the continuous day-to-day traffic dynamic model developed in this study suffers from the limitations as pointed out in Watling and Hazelton (2003). A discrete day-to-day dynamic traffic model shall add realism but in this case the auto share may fluctuate with discrete time modal choice, giving rise to the failure of Proposition 4 and the non-convergence of the double dynamic system. It is a challenging extension about how to establish the double dynamic system embodied with a discrete-time dynamic modal choice. In this regard, we should point out that, as long as the auto share evolves closer to the equilibrium share of that period, namely, nþ1 z zðnþ1Þ < zn zðnþ1Þ , then the properties of the periodic dynamic model established so far are still valid (see proof in Appendix A.7). In this case, the proposed model can still help the transit operators to reach the target equilibriums even the system does not reach a stable state at the end of each period.
Acknowledgements The authors would like to express our sincere thanks to three enthusiastic anonymous reviewers for their constructive and helpful comments. The work described in this paper was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKUST16205715) and the National Natural Science Foundation of China (71371020).
346
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
Appendix A A.1. Proof for Proposition 2 Define the following Lyapunov function
Z VðzÞ ¼
þ1
Dðz;f n Þ
!2 g ðeÞde z
:
ð33Þ
On one hand, since GðeÞ is assumed to be strictly increasing, continuous and differentiable over its support ð1; þ1Þ; VðzÞ is also continuously differentiable with respect to z 2 ½0; 1, also, V ðzn Þ ¼ 0 and
VðzÞ > 0; 8z 2 ½0; 1 n zn :
ð34Þ n
On the other hand, for any z 2 ½0; 1 n z , we have
_ VðzÞ ¼ 2d
Z
þ1
Dðz;f n Þ
!2 g ðeÞde z
n g D z; f t0 ðzÞ 1 < 0;
ð35Þ
and V_ ðzn Þ ¼ 0. By Lemma 1, the stationary point zn is asymptotically stable.
A.2. Proof for Proposition 3 Similar to the proof in Proposition 1, define hðzÞ ¼ z
R þ1 DðzÞ
g ðeÞde. Since GðeÞ is continuous and differentiable over its sup-
port ð1; þ1Þ , hðzÞ is also continuous and differentiable for all z 2 ½0; 1. Then R þ1 Dð0Þ < tð0Þ s < þ1; Dð1Þ > tð1Þ wðf min Þ s > 1. Hence hð0Þ ¼ Dð0Þ g ðeÞde < 0 and R þ1 R Dð1Þ hð1Þ ¼ 1 Dð1Þ g ðeÞde ¼ 1 g ðeÞde > 0. Since hðzÞ is continuous with respect to z, based on the intermediate value theorem (Clarke, 1971), there exists at least one zk 2 ð0; 1Þ such that hðzk Þ ¼ 0, which means that there exists at least one user equilibrium. Here, we also show that condition (9) is a sufficient and necessary condition for the existence of multiple equilibriums. Note that condition (9) is equivalent to the following condition (36).
hðzk Þ ¼ 0
ð36Þ
0
h ð zk Þ < 0
(1) If there exists a zk satisfying the above conditions, then for any arbitrarily small g > 0, we have hðzk gÞ > 0; hðzk Þ ¼ 0 and hðzk þ gÞ < 0. Also by hð0Þ < 0 and hð1Þ > 0, based on the intermediate value theorem, hðzÞ must have at least three zero points. Therefore, there exists at least three equilibriums and condition (9) is a sufficient condition for the existence of multiple equilibriums. 0 (2) Assume that, at any equilibrium zk ; h ðzk Þ > 0. Then for any g > 0, we have hðzk gÞ < 0; hðzk Þ ¼ 0 and hðzk þ gÞ > 0. Hence, for any 0 6 z0 < zk < z00 6 1, we have hðz0 Þ < 0 and hðz00 Þ > 0. This implies that hðzÞ has only one zero point, therefore, condition (9) is a necessary condition for the existence of multiple equilibriums. From (1) and (2), condition (9) is, indeed, a necessary and sufficient condition. A.3. Proof for Proposition 4 In the ðn þ 1Þth period, the initial auto share is zn and the transit frequency is f we can obtain
€z ¼ d g D z; f nþ1 t 0 ðzÞ 1 < 0
nþ1
. Based on the dynamic system Eq. (3),
ð37Þ
for all z 2 ½0; 1. Thus,
R þ1 8 z_ > 0 if z < Dðzðnþ1Þ ;f nþ1 Þ g ðeÞde > > > < R þ1 z_ ¼ 0 if z ¼ Dðzðnþ1Þ ;f nþ1 Þ g ðeÞde > > > : z_ < 0 if z > R þ1 nþ1 g ðeÞde Dðzðnþ1Þ ;f Þ
ð38Þ
R þ1 R þ1 g ðeÞde z_ is always positive until z ¼ Dðzðnþ1Þ ;f nþ1 Þ g ðeÞde and the auto share does not change any Dðzðnþ1Þ ;f nþ1 Þ more. Otherwise, there exists z1 2 ½0; 1 such that z_ 1 < 0 at time t 1 in the ðn þ 1Þth period. From (38) we can get Therefore, if zn <
347
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
R þ1 g ðeÞde. Since z is continuous with respect to time, based on the intermediate value theorem, there exists a Dðzðnþ1Þ ;f nþ1 Þ stationary point z2 ¼ zðnþ1Þ at time t2 in the ðn þ 1Þth period with 0 < t2 < t1 . This contradicts the fact that the stationary R þ1 point is stable as proved in Proposition 2. In the same vein, if zn > Dðzðnþ1Þ ;f nþ1 Þ g ðeÞde; z_ is always negative until R þ1 z ¼ Dðzðnþ1Þ ;f nþ1 Þ g ðeÞde and the auto share does not change any more. Thus znþ1 is always between zn and the equilibrium auto share of the ðn þ 1Þth period, as described in Eq. (12).
z1 >
A.4. Proof for Lemma 2 k1 and k2 are the solutions of the quadratic equation k2 k trJ þ det J ¼ 0 where trJ ¼ k1 þ k2 ¼ abAB þ ð1 aÞ þ ð1 bÞ and det J ¼ k1 k2 ¼ ð1 aÞð1 bÞ 2 ½0; 1Þ. Based on the sign of tr2 J 4 det J, we have the following three different cases. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1) If tr2 J 4 det J P 0 and trJ P 0, both k1 and k2 are real and k1 ; k2 ¼ trJ tr2 J 4 det J =2 P 0. Thus jkj < 1 requires
8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi trJþ tr2 J4 det J > > <1 < 2
)
tr J 4 det J P 0 > > : trJ P 0 2
1 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi2 1 a 1 b 6 AB < 1: ab
(2) If tr2 J 4 det J P 0 and trJ < 0, both k1 and k2 are real and k1 ; k2 ¼
8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi trJ tr2 J4 det J > > > 1 < 2 tr J 4 det J P 0 > > : trJ < 0 2
)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
trJ
tr2 J4 det J 2
6 0. Thus jkj < 1 requires
1 1 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi2 ð2 aÞð2 bÞ < AB 6 1aþ 1b : ab ab
1=2 (3) If tr2 J 4 det J < 0; k1 and k2 are of a complex conjugate pair, with jk1 j ¼ jk2 j ¼ det J < 1. Thus jkj < 1 requires
tr2 J det J < 0 )
1 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi2 1 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi2 1 a þ 1 b < AB < 1a 1b : ab ab
Combining the above three cases we can conclude that the moduli of k1 and k2 are both less than 1, if and only if ð2 aÞð2 bÞ=ab < AB < 1. A.5. Proof for Proposition 5 The dynamic system over periods is ruled by a set of matrices D ¼ fD1 ; D2 ; . . . ; Dn ; . . .g. Since h i 1 z 2 ½0; 1; f 2 f min ; k ðsd p0 Þ , all the functions are continuous and bounded, then D is a bounded set of matrices. Based on Lemma 3, the fixed point is asymptotically stable if the joint spectral radius of the set D ¼ fD1 ; D2 ; . . . ; Dn ; . . .g is less than n o 1=k 1 at the fixed point. The joint spectral radius of the set D is defined as qðDÞ ¼ limk!1 max Di Di : Di 2 D . 1
Then
k
1=k n o qðDÞ ¼ lim max Di1 Dik 1=k : Di 2 D ¼ lim max Dki : Di 2 D k!1
k!1
1=k ¼ max lim Dki : Di 2 D ¼ max fqðDi Þ : Di 2 Dg: k!1
The last equation is based on Gelfand’s Formula. According to Eq. (14), at the interior fixed point,
An ¼ Thus,
sd
0 k ~f
;
w0 ~f g D ~z ; ~f : Bn ¼ 1 þ g D ~z ; ~f t 0 ð~z Þ
0 ~ ~ sd w f g D ~z;f An Bn ¼ 0
k ~f 1 þ g D ~z ; ~f
t 0 ð~z Þ
> 0;
which means that term ð2 an Þð2 bn Þ=an bn in Lemma 2 is unbounded. Therefore, for all n 2 N þ , when An Bn < 1, the moduli of the two eigenvalues of Dn are both less than 1, which means when
348
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
0 ~ ~ sd w f g D ~z;f 0
k ~f 1 þ g D ~z ; ~f
t 0 ð~z Þ
< 1;
qðDÞ < 1
and the interior fixed point is asymptotically stable. Since
k ~f 1 þ g D ~z ; ~f
0 1 sd 0 ~ A ~ 0 ~ @ ~ t ðz Þ þ 0 < 1; w f < 1 () g D z ; f t 0 ð~z Þ k ~f
0 ~ ~ sd w f g D ~z;f 0
0 t 0 ð~z Þ þ sd=k ~f w0 ~f < 1, the system (10) and (12) is (locally) asymptotwe can conclude that, when g D ~z ; ~f ~ ically stable at the corresponding interior fixed point ~z ; f . When the fixed point is at the boundary, based on Eq. (14) An ¼ 0, thus ð2 an Þð2 bn Þ=an bn < An Bn ¼ 0 < 1 for all n 2 N þ . Similar to the above before, we can conclude that the system (10) and (12) is always (locally) asymptotically stable at the boundary fixed point. A.6. Proof for Proposition 6 When there is only one target profit equilibrium fixed point. If the fixed point is a boundary fixed point, as indicated in Proposition 5, it must be asymptotically stable. If the fixed point is an interior fixed point, according to Proposition 3, at this fixed point ~z ; ~f ,
R þ1 d Dðz;f ðzÞÞ g ðeÞde dz
z¼~z
0 1 s d ¼ g D ~z ; ~f @t0 ð~z Þ þ 0 w0 ~f A < 1: k ~f
Also by Proposition 5, we can conclude that the system is asymptotically stable at this target profit equilibrium fixed point. When there are multiple target profit equilibrium fixed points, according to Proposition 3, there exists at least one target profit equilibrium fixed point ~z ; ~f such that k
R þ1 d Dðz;f ðzÞÞ g ðeÞde dz
z¼~z
k
0 1 s d 0 0 ~ ~ @ t ð~z Þ þ 0 w f A > 1: ¼ g D ~z ; f k ~f
Thus, the system is unstable at this fixed point.
Like the proof of Proposition 3, hð0Þ < 0 and hð1Þ > 0. Define ~zl ; ~f l and ~zh ; ~f h as the fixed points with the lowest and
highest shares of auto users, respectively. Then we have hð~zl Þ ¼ hð~zh Þ ¼ 0; hðzÞ < 0 for any z 2 ð0; ~zl Þ, and hðzÞ > 0 for any 0 0 z 2 ð~zh ; 1Þ. Also for any arbitrarily small g > 0, we have hð~zl þ gÞ > 0; hð~zh gÞ < 0. Thus h ð~zl Þ > 0 and h ð~zh Þ > 0. That is to say,
0 1 s d 0 0 ~ ~ g D ~zl ; f l @t ð~zl Þ þ 0 w f l A < 1 k ~f l and if ~zh ; ~f h is an interior fixed point,
0 1 s d 0 0 ~ ~ @ g D ~zh ; f h t ð~zh Þ þ 0 w f h A < 1: k ~f h If ~zh ; ~f h is a boundary fixed point, it must be asymptotically stable as proved in Proposition 5. Therefore, the target profit fixed points with the lowest and highest shares of auto users is locally stable. A.7. For each period, if the auto share evolves closer to the equilibrium auto share of that period, i.e., znþ1 zðnþ1Þ < zn zðnþ1Þ , then, the properties of the periodic dynamic model is still valid. Proof. The condition znþ1 zðnþ1Þ 6 zn zðnþ1Þ is equivalent to znþ1 ¼ cn zðnþ1Þ þ ð1 cn Þzn , where cn 2 ð0; 2Þ. Now we assume an 2 ð0; 2Þ in Eq. (12), and show it would not affect the following propositions by proving that Lemma 2 still holds when a 2 ð0; 2Þ.
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
349
When a 2 ð1; 2Þ; det J ¼ k1 k2 ¼ ð1 aÞð1 bÞ 2 ð1; 0Þ and tr2 J 4 det J > 0. Thus both k1 and k2 are real and jkj < 1 requires
ffi 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < trJþ tr2 J4 det J < 1 ð2 aÞð2 bÞ 2 < AB < 1: ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : trJ tr2 J4 det J ab > 1 2
Combining the above case with Lemma 2, we can conclude that when a 2 ð0; 2Þ, the condition for the moduli of k1 and k2 being less than 1, remains ð2 aÞð2 bÞ=ab < AB < 1. Following the proof of Proposition 5, since an 2 ð0; 2Þ, the term ð2 an Þð2 bn Þ=an bn is still unbounded. Therefore, when an 2 ð0; 2Þ, Propositions 5–8 all remain valid. References Arnott, R., Yan, A., 2000. The two-mode problem: second-best pricing and capacity. Rev. Urban Reg. Dev. Stud. 12 (3), 170–199. Bar-Yosef, A., Martens, K., Benenson, I., 2013. A model of the vicious cycle of a bus line. Transport. Res. B: Methodol. 54, 37–50. Bie, J., Lo, H., 2010. Stability and attraction domains of traffic equilibria in a day-to-day dynamical system formulation. Transport. Res. B: Methodol. 44 (1), 90–107. Cantarella, G., 2013. Day-to-day dynamic models for intelligent transportation systems design and appraisal. Transport. Res. C: Emerging Technol. 29, 117– 130. Cantarella, G., Velonà, P., Watling, D., 2013. Day-to-day dynamics & equilibrium stability in a two-mode transport system with responsive bus operator strategies. Netw. Spatial Econ. 15 (3), 485–506. Cascetta, E., 1987. Static and Dynamic Models of Stochastic Assignment to Transportation Networks. Flow Control of Congested Networks. Springer, Berlin, Heidelberg, pp. 91–111. Cascetta, E., 1989. A stochastic process approach to the analysis of temporal dynamics in transportation networks. Transport. Res. B: Methodol. 23 (1), 1–17. Clarke, D., 1971. Foundations of Analysis. Appleton-Century-Crofts, 284. David, Q., Foucart, R., 2014. Modal choice and optimal congestion. Reg. Sci. Urban Econ. 48, 12–20. Farokhi, F., Johansson, K., 2015. A piecewise-constant congestion taxing policy for repeated routing games. Transport. Res. B: Methodol. 78, 123–143. Friesz, T., Bernstein, D., Smith, T., Tobin, R., Wei, B., 1993. A variational inequality formulation of the dynamic network user equilibrium problem. Oper. Res. 41, 179–191. Guo, R., Yang, H., Huang, H., 2013. A discrete rational adjustment process of link flows in traffic networks. Transport. Res. C: Emerging Technol. 34, 121–137. Guo, R., Yang, H., Huang, H., Tan, Z., 2015. Link-based day-to-day network traffic dynamics and equilibria. Transport. Res. B: Methodol. 71, 248–260. He, X., Guo, X., Liu, H., 2010. A link-based day-to-day traffic assignment model. Transport. Res. Part B: Methodol. 44 (4), 597–608. Khalil, H., 2002. Nonlinear Systems, third ed. Prentice Hall, Upper Saddle River, New Jersey. Horowitz, J., 1984. The stability of stochastic equilibrium in a two-link transportation network. Transport. Res. B: Methodol. 18 (1), 13–28. Kraus, M., 2003. A new look at the two-mode problem. J. Urban Econ. 54 (3), 511–530. Lam, W., Huang, H., 1995. Dynamic user optimal traffic assignment model for many to one travel demand. Transport. Res. B: Methodol. 29 (4), 243–259. Li, Z., Lam, W., Wong, S., 2012. Modeling intermodal equilibrium for bimodal transportation system design problems in a linear monocentric city. Transport. Res. B: Methodol. 46 (1), 30–49. Meignan, D., Simonin, O., Koukam, A., 2007. Simulation and evaluation of urban bus-networks using a multiagent approach. Simul. Modell. Pract. Theory 15 (6), 659–671. Peterson, D., Hough, J., Ulmer, D., Urban, S., Center, R.T., 2006. Express Bus Transit Study: A Case Study (No. MPC Report No. 06-178). Mountain Plains Consortium. Ran, B., Hall, R., Boyce, D., 1996. A link-based variational inequality model for dynamic departure time/route choice. Transport. Res. B: Methodol. 30 (1), 31– 46. Smith, M., 1984. The stability of a dynamic model of traffic assignment: an application of a method of Lyapunov. Transport. Sci. 18 (3), 245–252. Tan, Z., Yang, H., Guo, R., 2015. Dynamic congestion pricing with day-to-day flow evolution and user heterogeneity. Transport. Res. C: Emerging Technol. 61, 87–105. Theys, J., 2005. Joint Spectral Radius: theory and approximations. Doctoral dissertation, UCL. Watling, D., Hazelton, M., 2003. The dynamics and equilibria of day-to-day assignment models. Netw. Spatial Econ. 3 (3), 349–370. Wu, J., Sun, H., Wang, D., Zhong, M., Han, L., Gao, Z., 2013. Bounded-rationality based day-to-day evolution model for travel behavior analysis of urban railway network. Transport. Res. C: Emerging Technol. 31, 73–82. Ye, H., Yang, H., Tan, Z., 2015. Learning marginal-cost pricing via a trial-and-error procedure with day-to-day flow dynamics. Transport. Res. B: Methodol. 81, 794–807. Zhang, F., Yang, H., Liu, W., 2014. The Downs–Thomson Paradox with responsive transit service. Transport. Res. A: Policy Pract. 70, 244–263.
Contents lists available at ScienceDirect
Transportation Research Part C journal homepage: www.elsevier.com/locate/trc
Dynamics of modal choice of heterogeneous travelers with responsive transit services Xinwei Li ⇑, Hai Yang Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, PR China
a r t i c l e
i n f o
Article history: Received 9 January 2016 Received in revised form 18 April 2016 Accepted 18 April 2016
Keywords: Dynamic modal choice Responsive transit strategy Equilibrium Stability Profit
a b s t r a c t In this paper, we investigate travelers’ day-to-day modal choice in a bi-modal transportation system with responsive transit services under various economic objectives. A group of travelers with heterogeneous preferences adjust their modal choice each day based on their perceived travel cost of each mode, aiming to minimize their travel cost. Meanwhile, the transit operator sets frequency each period according to the realized transit demand and previous frequency, trying to achieve different profit targets. For a given profit target, the fixed point of equilibrium may not be unique. We establish the condition for existence of multiple fixed points and examine the stability of the fixed points in each case. Furthermore, in view of a socially desirable mode choice, we also investigate the impacts of total travel demand and bus size on the convergence of the system to various fixed points associated with different targeted mode split. Finally, we use several numerical examples to illustrate the theoretical results and their practical implications for the transit operator to design appropriate transit schemes in a dynamic transportation environment. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction A large pool of literature relies on the steady-state equilibrium paradigm to provide theoretical basis for the transport planners through the deterministic user equilibrium (DUE) or stochastic user equilibrium (SUE) approach. However, the steady-state analysis is limited for it only shows an end result rather than the choice adjustment process. The dynamic transportation approach can characterize the evolution of traffic states over time and its value of modeling the traffic system is well acknowledged. There are two major categories of dynamic approaches. One is within-day dynamic (e.g. Friesz et al., 1993; Lam and Huang, 1995; Ran et al., 1996), which focuses on dynamic traffic assignment (DTA) research and assumes the system is unchanging over days. The other one is day-to-day dynamics approach with dynamics between-day scale and constant within-day scale. Day-to-day dynamics approach can date back to the works of Horowitz (1984), Smith (1984) and Cascetta (1987, 1989); the present paper also lies in this research area. A bi-modal system is often as an example to analyze the interaction between private cars and public transits in the literature. For instance, Arnott and Yan (2000) and Kraus (2003) indicated the underpricing of auto travel is a source of market distortion and a resource waste in transportation system with private and public transport modes. Li et al. (2012) investigated the intermodal equilibrium with two alternative modes. David and Foucart (2014) provided a game-theoretical model
⇑ Corresponding author. E-mail address: [email protected] (X. Li). http://dx.doi.org/10.1016/j.trc.2016.04.014 0968-090X/Ó 2016 Elsevier Ltd. All rights reserved.
334
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
in which travelers have heterogeneous preferences choose rationally between using the car or public transportation. Zhang et al. (2014) studied the conditions for occurrence of the Downs-Thomson Paradox under different economic objectives. However, these studies are concerned with a steady state analysis; neither travelers’ day-to-day mode choice nor transit operator’s periodic operating strategy are considered. It is known that more (fewer) transit passengers lead to more (less) frequent service, which leads to even more (fewer) passengers. This so-called ‘‘Mohring effect”, named for the wellknown economist Herbert Mohring, can be better analyzed in a dynamic environment. A few recent studies are carried out on the dynamic bi-modal problem in the literature. Cantarella et al. (2013) conducted an analysis of day-to-day dynamic mode choice in a transportation system with homogeneous travelers; bus operating strategies are demand-responsive but with a fixed fleet size. Bar-Yosef et al. (2013) examined the phenomena of a vicious cycle of a bus service line; nevertheless, mode choice of non-captive travelers is based only on the willingness-to-wait for the bus rather than the total trip cost. Besides, neither of the two studies take account of the operation cost and specific transit operating schemes. Three recent dynamic modeling approaches are relevant but purely in the context of congestion pricing. Tan et al. (2015) proposed a discrete-time tolled scheme by using the implicit Runge–Kutta method with heterogeneous users and investigated the equilibrium properties of the dynamic system. Farokhi and Johansson (2015) considered a piecewise-constant congestion taxing policy for repeated routing games of fixed demand. Ye et al. (2015) investigated the convergence of the trial-and-error tolling procedure for achieving system optimum with day-to-day flow dynamics when the observed link flow pattern is in disequilibrium. In the latter two studies, the link toll is constant in each period and calculated based on marginal cost pricing with the flow observed at the beginning of the tolling period. In the same spirit, we assume that the transit operator adjusts the transit frequency from period to period and keeps the frequency unchanged throughout each period. Transit frequency is reset at the beginning of each period based on the realized transit demand at the end of the previous period. This paper is intended to fill a gap in the literature of bi-modal problems by examining the double dynamics of heterogeneous travelers’ day-to-day modal choice and transit operator’s periodic operating policy. Transit operator adjusts service frequency to drive the dynamic system towards a desired equilibrium while achieving a profit target. For various combinations of economic objectives and operating schemes, the interactions between travelers and transit operator are investigated in a dynamic environment. Existence, uniqueness and stability of the equilibrium fixed point of the double dynamic system are established. Particularly, occurrence of the vicious cycle considered by Bar-Yosef et al. (2013) is analyzed under different levels of travel demand and bus capacity. The rest of the paper is organized as follows. Section 2 introduces the bi-modal transportation system with heterogeneous travelers. Section 3 analyzes the bi-modal system in a period with given transit frequency, and shows the existence, uniqueness and stability of user equilibrium. Inter-period dynamic frequency adjustment of a responsive transit operator is considered under different economic objectives in Section 4. The impacts of travel demand and bus capacity on the system performance and the occurrence of the vicious cycle are examined in Section 5. Section 5 provides a numerical example and Section 6 draws the conclusions and highlights avenue for future research. 2. Model formulation We consider one origin–destination (OD) pair connected by a congested highway running in parallel to an exclusive transit line. On a typical day, the total travel demand for this OD pair is fixed at d. Travelers have to make a discrete choice between using a private transport (auto) and public transit (bus). For simplicity, the occupancy of each private car is assumed to be 1. Travelers are heterogeneous as they have different intrinsic preferences for the use of private transport relative to public transit. The share of auto users is represented by z, then the share of public transport users is 1 z. Transit users have to pay for the transit fare, spend time in waiting and riding. Following the model in David and Foucart (2014), the travel cost of commuters by each mode consists of the travel time cost and the monetary cost and the disutility of a traveler i is specified as follows:
pti ¼ wðf Þ þ s þ
ei 2
and
pci ¼ tðzÞ
ei 2
;
where pti stands for public transit cost and pci for private car cost, f and s denote the transit frequency and uniform ticket price (transit fare), respectively. The function wðf Þ represents the travel time cost of public transport as a function of service frequency. It is assumed to be strictly decreasing and differentiable with respect to transit frequency f, i.e., w0 ¼ w0 ðf Þ < 0. The function tðzÞ represents the travel cost of private car, including both travel time cost and monetary cost, it is assumed to be strictly increasing and differentiable with respect to auto share z, i.e., t 0 ¼ t0 ðzÞ > 0 and tð0Þ > 0 at free flow. We also assume that w0 is bounded for all f > 0 and t 0 is bounded for any z 2 ½0; 1. The commuter-specific parameter, ei , reflects individual preference between private car and public transit, it is continuously distributed with a cumulative distribution function e GðeÞ, where G is assumed to be strictly increasing, continuous and differentiable over its support ð1; þ1Þ. The support implies that some individuals prefer the private transport so
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
335
much and will never choose public transport (ei ! þ1), while others will always choose public transit (ei ! 1). The density function of ei is represented by g ðeÞ and G0 ðeÞ ¼ g ðeÞ > 0. The transit operation cost is represented by function kðf Þ, where kðf Þ is strictly increasing and differentiable with respect 0 0 0 to transit frequency f, i.e., k ¼ k ðf Þ > 0. k is bounded for any finite frequency. Let p denote the transit operation profit, which is equal to the operation revenue minus operation cost, i.e., p ¼ sð1 zÞd kðf Þ. In this paper, for analytical tractability, we assume that the responsive transit operator changes transit frequency from period to period only in responses to the change in transit demand, the transit ticket price s is fixed throughout all the operating periods. The major reason for this assumption is that the strategic combination of ðs; f Þ is not unique for achieving a given economic objective such as a certain level of transit profit. 3. Intra-period day-to-day modal choice with given transit frequency and fare 3.1. Static analysis We assume each rational traveler chooses the transport mode to maximize his utility given his expectation on the cost of each travel mode. Hence, traveler i commutes by car if pci < pti , i.e., ei > tðzÞ wðf Þ s. Let Dðz; f Þ ¼ tðzÞ wðf Þ s represent the additional cost (either positive or negative) faced by auto users in comparison with the cost faced by public transport users. Thus, the condition for traveler i to choose private transport is
ei > Dðz; f Þ:
ð1Þ
And the user equilibrium condition is
Z z¼
þ1
Dðz;f Þ
g ðeÞde:
ð2Þ
The existence of a user equilibrium for any given f is guaranteed in the following proposition. Proposition 1. For any given f, there exists a unique user equilibrium.
R þ1 Proof. For any given f, at equilibrium, z is pinned down by hðzÞ ¼ 0, where hðzÞ ¼ z Dðz;f Þ g ðeÞde. Since R þ1 0 monotonically increases with z. Also by hð0Þ ¼ Dð0;f Þ g ðeÞde < 0 and h ðzÞ ¼ 1 þ g ðDðz; f ÞÞt0 > 0; hðzÞ R þ1 R Dð1;f Þ hð1Þ ¼ 1 Dð1;f Þ g ðeÞde ¼ 1 g ðeÞde > 0, we can see that z is unique, so is the equilibrium. h 3.2. Intra-period day-to-day modal choice n
Suppose the transit frequency is fixed in each period and let f represent the frequency in the nth period. Here we analyze travelers’ day-to-day modal choice in a representative period. The general form of our day-to-day dynamic modal choice model in continuous time is
z_ ¼ d
Z
þ1
Dðz;f n Þ
!
g ðeÞde z ;
ð3Þ
R þ1 Dðz;f n Þ g ðeÞde z provides a flow changing direction. Dynamic system (3) means that, on any day, the share of auto users tends to move from the current auto share R þ1 R þ1 to the ‘‘target” auto share Dðz;f n Þ g ðeÞde, based on the current day situation. The target share Dðz;f n Þ g ðeÞde describes travelers’ n n cost-minimization behaviors. Specifically, as mentioned before, travelers with ei > D z; f will choose car while ei < D z; f choose transit to minimize their perceived travel cost when the current auto share is z; in a word, at any day-to-day time travelers tend to switch to the current cost-saving travel mode. This form of day-to-day dynamic is somehow consistent with the link-based day-to-day network model proposed by He et al. (2010) and their model is also investigated by Guo et al. (2013) and Guo et al. (2015) in both discrete and continuous time. Let zn denote the stationary point of dynamic system (3), which is given by the solution of equation z_ ¼ 0. Then where d is a positive constant parameter determining the flow changing rate, and
Z
zn ¼
þ1
Dðzn ; f n Þ
g ðeÞde:
ð4Þ
Clearly, zn coincides with the static equilibrium solution of Eq. (2) and it is unique as proved in Proposition 1. We are now ready to prove that the stationary point of the dynamic system (3) is asymptotically stable by employing Lyapunov’s stability theorem, which is stated below.
336
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
Lemma 1. [Khalil, 2002, Theorem 4.1] Let z be a stationary point of the system z_ ¼ hðzÞ and Z 2 Rn be a domain containing z . Let a Lyapunov function V : Z ! R be continuously differentiable such that
V ðz Þ ¼ 0 and VðzÞ > 0; _ VðzÞ 6 0;
8z 2 Z n fz g;
ð5Þ
8z 2 Z:
ð6Þ
Then z is stable. Moreover, if
_ VðzÞ < 0;
8z 2 Z n fz g;
ð7Þ
Then z is asymptotically stable. Based on Lemma 1, the following proposition holds.
Proposition 2. The unique stationary point zn of the dynamic system (3) is asymptotically stable. Proof. The proof is given in Appendix A.1. h The following Example 1 is provided to illustrate Propositions 1 and 2 on the uniqueness and stability of the stationary point of the dynamic system (3). Exampleh 1. Consider the two-mode network described in Section 2. The travel cost function of the private mode is i tðzÞ ¼ 10 1 þ 0:15ð5zÞ4 þ 20. Since the transit frequency and fare are fixed, the travel cost of public transport is also fixed and assumed to be wðf Þ þ s ¼ 35 þ 3 ¼ 38. All the cost units are in HK$. e follows the bimodal exponential power jej distribution BEF ð0; 1; 1; 1Þ, i.e. g ðeÞ ¼ j2eCjeð2Þ . n From Eq. (4), the stationary point is computed to be zn ¼ 0:3640 and the stationary cost difference is D zn ; f ¼ 8:4580. The following Fig. 1 depicts the evolution of the auto share day by day obtained by the dynamic system (3) with different initial values. The flow changing rate parameter d is set to be 0.05. From Fig. 1, we can see that, when the length of the period is long enough, such as 30 days, the dynamic system (3) will converge to the stationary point whatever the initial auto share is, which means the stationary point is asymptotically stable. 4. Inter-period transit operating strategy with intra-period day-to-day modal choice In this section, the transit operator adjusts the transit frequency on a periodic basis based on the transit demand observed at the end of the previous period, aiming to achieve various target profits. Then the frequency can be regarded as a function of the share of auto users z, i.e. f ¼ f ðzÞ and DðzÞ ¼ Dðz; f ðzÞÞ ¼ tðzÞ wðf ðzÞÞ s. 4.1. Static analysis with responsive transit operator When facing a decreasing demand, a transit operator will tend to reduce service to cut costs while when facing increasing demand, the transit operator will tend to increase supply to earn more profit. Thus, it is reasonable to make the following assumption. Assumption 1. The transit frequency is continuous and decreasing with respect to the share of auto users, i.e.,
df 6 0: dz
ð8Þ
Suppose the transit frequency has a realistic minimal value f min . The following proposition shows the existence of a user equilibrium and provides the condition for the existence of multiple equilibriums. Proposition 3. There exists at least one user equilibrium. Furthermore, multiple user equilibriums exist if and only if there exists at least one solution zk such that
R þ1 8 zk ¼ Dðzk Þ g ðeÞde > > > < R þ1 d g ðeÞde DðzÞ > >1 > dz > :
ð9Þ
z¼zk
Proof. The proof is given in Appendix A.2. h 0
Here we do not consider the extreme case in which there exists a user equilibrium zk such that h ðzk Þ ¼ 0, because it 0 implies that hðzk Þ ¼ 0 and h ðzk Þ ¼ 0 occur simultaneously, which is unlikely to happen in practice.
337
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349 1
Auto share
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
Days Fig. 1. The evolution auto share with d ¼ 0:05.
4.2. Inter-period transit operating strategies for a target profit When the length of the period is limited, the bi-modal system may not reach the equilibrium state at the end of each period. The transit operator adjusts the frequency at the beginning of a period based on the transit share observed at the end of the previous period. In other words, the transit operator adjusts frequency from period to period based on a still evolving disequilibrium modal split pattern. Suppose the transit operator resets its frequency from period to period in an attempt to achieve a given target profit p0 (for a revenue-neutral operator, p0 ¼ 0), the target profit is less than or equal to the maximum achievable profit. Let zn be the share of auto users at the end of the nth period and suppose that the transit operator’s response is characterized by Eq. (10), where bn 2 ð0; 1 for all n 2 N þ . bn ¼ 1 means that the transit operator changes the transit frequency strictly following the n rule according to the share of auto users at the end of the previous period to get the target profit p0 . Given zn and f , the transit frequency in the ðn þ 1Þth period is
f where
nþ1
¼ bn f ðzn Þ þ ð1 bn Þf
n
h i 1 f ðzn Þ ¼ max k ðsdð1 zn Þ p0 Þ; f min
ð10Þ
ð11Þ
As shown in (10) and (11), the new transit frequency is determined once the realized mode share is observed at the end of the previous period. To establish the convergence and stability of the two symbiotic processes of modal choice dynamics and frequency adjustments, we need to know how the auto share z changes from period to period. Nonetheless, it is difficult to determine znþ1 explicitly owing to its implicit dependence on zn and parameter d as well as the period length. However, the following Proposition 4 shows that, when the day-to-day dynamic modal choice follows Eq. (3), the resulting znþ1 is always between zn and the equilibrium auto share for the ðn þ 1Þth period. Proposition 4. When the day-to-day dynamic modal choice follows the dynamic system (3), znþ1 is always between zn and the equilibrium auto share for the ðn þ 1Þth period, that is
znþ1 ¼ an zðnþ1Þ þ ð1 an Þzn where
zðnþ1Þ ¼
Z
þ1
Dðzðnþ1Þ ;f nþ1 Þ
g ðeÞde
ð12Þ
ð13Þ
and an 2 ð0; 1 for all n 2 N þ : Proof. The proof is given in Appendix A.3. h Proposition 4 is clearly displayed in Fig. 1. It is seen from Fig. 1 that the share of auto users on any day in a period is always between its initial value and the equilibrium value for that period, regardless of the period length. Furthermore, we note in passing that an ¼ 1 Eq. (12) implies that the ðn þ 1Þth period is long enough for the dynamic system to settle down to its equilibrium. Given the periodic transit operating strategy specified by Eq. (10) and the end period auto share given by (12) resulting from the day-to-day modal choice dynamics (3), we now move to investigate the properties of the double dynamic system. First we introduce the following matrix.
338
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
0
@f nþ1 n @ @f
Dn ¼
1
@f nþ1 @zn
@znþ1 @f n
A¼
@znþ1 @zn
1 bn an ð1 bn ÞBn
where
( An ¼
n
k0 ðfsðdzn ÞÞ ; 0;
for f – f min n
for f ¼ f min
;
bn An an bn An Bn þ 1 an
ð14Þ
nþ1 nþ1 w0 f g D zðnþ1Þ ; f : Bn ¼ nþ1 nþ1 t0 z f 1 þ g D zðnþ1Þ ; f
n nþ1 . Then the determinant of Dn is jDn j ¼ ð1 an Þð1 bn Þ 2 ½0; 1Þ for all n 2 N þ , which is independent of zn ; f or znþ1 ; f
nþ1
1
From Proposition 1, zðnþ1Þ is unique when f is fixed. Since kðf Þ is a monotonic function, k ðsdð1 zn Þ p0 Þ and f ðzn Þ n nþ1 from Eqs. (10) and (12). Let ~z ; ~f are also unique when zn is fixed. Hence, given zn ; f , we can obtain a unique znþ1 ; f represent the fixed point of system (10) and (12), then,
~z ¼
Z
þ1
Dð~z ;~f Þ
g ðeÞde
ð15Þ
h i ~f ¼ max k1 ðsdð1 ~z Þ p0 Þ; f min :
ð16Þ
From Eq. (11), we know that the transit frequency is decreasing with respect to the share of auto users, which satisfies Assumption 1. As proved in Proposition 3, there exists at least one equilibrium fixed point ~z ; ~f for given target profit
p0 (hereinafter, calledtarget profit equilibrium fixed point). Multiple fixed points exist if and only if there exists at least one fixed point ~z ; ~f
such that
R þ1 d Dðz;f ðzÞÞ g ðeÞde dz
z¼~z
0 1 s d ¼ g D ~z ; ~f @t0 ð~z Þ þ w0 ~f A > 1: 0 k ~f
ð17Þ
Furthermore, we can analyze the stability of each fixed point. First we provide the following lemmas. Lemma 2. Let A and B be two constants, and denote k1 ; k2 the two eigenvalues of
J¼
1a
aA
ð1 aÞbB abAB þ 1 b
;
where a; b 2 ð0; 1, then the moduli of k1 and k2 are both less than 1, i.e. jkj < 1, if and only if ð2 aÞð2 bÞ=ab < AB < 1. Proof. The proof is given in Appendix A.4. h Definition 1. For any set D of matrices, we take an arbitrary point x0 2 R as initial point and define the following recurrence
xtþ1 2 fDit xt : Dit 2 Dg:
ð18Þ
A series ðx0 ; x1 ; . . . ; xk ; . . .Þ satisfying the inclusion (18), xtþ1 ¼ Dit xt , for some ðit Þ sequence, is a trajectory in the Rn space. Let DLIðDÞ denote the discrete linear inclusion set fxk : k P 0g, consisting of all possible sequences of vectors in Rn generated using the relation (18). Lemma 3. [Theys, 2005, Proposition 3.2] For any bounded set D of matrices, the joint spectral radius of the set D is defined as n o
1=k qðDÞ ¼ limk!1 max Di1 Dik : Di 2 D . Then qðDÞ < 1 () DLIðDÞ is asymptotic stability.
Based on Lemmas 2 and 3, we can investigate the stability of system (10) and (12) at each fixed point. The stable condition is given in Proposition 5. 0 t0 ð~z Þ þ sd=k ~f w0 ~f < 1, the corresponding interior fixed point is (locally) Proposition 5. When g D ~z ; ~f asymptotically stable and the boundary fixed point is always (locally) asymptotically stable.
Proof. The proof is given in Appendix A.5. h
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
339
From Proposition 5 we can see that the convergence of the dynamic system and the stability of the fixed points over periods are independent of the values of updating parameters. Now we can further investigate the stability of all fixed points in the following Proposition 6. Of particular interest is the fixed point with a minimal share of auto users, since the auto volume is highly related to road congestion. Proposition 6. For system (10) and (12), when there is only one target profit equilibrium fixed point, it must be asymptotically stable. When there are multiple target profit equilibrium fixed points, at least one fixed point is unstable and the fixed point with the lowest and the highest share of auto users is asymptotically stable. Proof. The proof is given in Appendix A.6. h 4.3. Inter-period transit operating strategies for a periodic maximum profit In this subsection, the transit is operated by a periodic profit-maximizing authority; the operator chooses the frequency in each period to maximize its profit only based on the share of transit users at the end of the previous period, without taking into account the impact of frequency change on travelers’ modal choice in the period. We assume that the marginal cost of carrying one additional traveler is less than the marginal profit. This means that the transit operator will try to make every transit bus loaded as full as possible for profit maximization. Let m denote the predetermined desirable number of passengers per bus (or equivalently, predetermined desirable load factor). The value of m evidently depends on the maximal capacity of a bus, but will be somewhat lower than that to avoid overcrowded buses and higher than the number of passengers necessary to merely cover operation costs (Bar-Yosef et al., 2013; Meignan et al., 2007). Then the transit operator’s response in terms of frequency setting from period to period is described by
f
nþ1
n
¼ bn f ðzn Þ þ ð1 bn Þf ;
where
f ðzn Þ ¼ max
ð19Þ
ð1 zn Þd ; f min : m
ð20Þ
In Eq. (19), bn 2 ð0; 1 and bn ¼ 1 means that the transit operator only follows the periodic profit-maximizing rule to change transit frequency according to the share of transit users at the end of the previous period. Here we should mention that transit operator’s choice of frequency in the current period is based on the realized demand at the end of the previous period, but, travelers’ response of modal choice or auto share in the current period is given by Eq. (12), which depends on the chosen frequency. Let z ; f represent the fixed points of system (12) and (19), then
z ¼
Z
þ1
Dðz ;f Þ
g ðeÞde
ð21Þ
f ¼ max ð1 z Þd ; f min : m
ð22Þ
Thus df =dz ¼ d=m < 0, which also satisfies Assumption 1. As proved in Proposition 3, there exists at least one periodic profit-maximizing equilibrium fixed point z ; f . Multiple periodic profit-maximizing fixed points exist if and only if there exists at least one periodic profit-maximizing equilibrium fixed point z ; f such that
R þ1 d Dðz;f ðzÞÞ g ðeÞde dz
¼ g D z ; f t 0 ðz Þ þ w0 f d=m > 1:
ð23Þ
z¼z
For the periodical dynamic system (12) and (19), we define
0
En ¼
@f nþ1 n @ @f @znþ1 n @f
@f nþ1 @zn @znþ1
1
A¼
@zn
1 bn
an ð1 bn ÞBn an bn An Bn þ 1 an
where
( An ¼
n
d=m for f – f min n
0 for f – f min
bn An
;
;
nþ1 nþ1 w0 f g D zðnþ1Þ ; f Bn ¼ : nþ1 t 0 ðzðnþ1Þ Þ 1 þ g D zðnþ1Þ ; f
ð24Þ
340
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
0 Similar to the proof in subsection 4.2, by changing An ¼ sd=k ~f in subsection 4.2 to An ¼ d=m and following the same processes, we have the following corresponding Propositions 7 and 8. Proposition 7. When g D z ; f t 0 ðz Þ þ ðd=mÞw0 f < 1, the corresponding interior fixed point is (locally) asymptotically stable and the boundary fixed point is always (locally) asymptotically stable. Proposition 8. For system (12) and (19), when there is only one periodic profit-maximizing equilibrium fixed point, it must be globally stable. When there are multiple periodic profit-maximizing equilibrium fixed points, at least one fixed point is unstable and the fixed points with the lowest and highest shares of auto users are locally stable. We conclude this section by pointing out that, as long as the dynamic system takes the form of Eqs. (10) and (12), where zðnþ1Þ is determined by Eq. (13), and f ðzn Þ satisfies df ðzn Þ=dzn 6 0, the stability condition of the fixed points is irrelevant to the value of the updating parameters an and bn . 5. The impacts of travel demand and bus capacity When the transit operator is responsive to the share of transit users, we always have df =dz < 0, no matter whether the operator is to achieve a target profit or to maximize profit periodically. Thus, when facing with decreasing transit demand, a transit operator tends to reduce frequency to cut expenses. Furthermore, based on Eqs. (12) and (13), we have dz=df < 0, which implies that reducing the transit frequency may induce some of the travelers to change to auto, resulting in fewer transit users. The decrease in the share of transit users may cause a further decrease in transit frequency, then a vicious cycle continues until only travelers with very strong preference for transit (very small ei ) stay on transit. However, a large share of auto users means more vehicles (auto and bus) on roads, which may contribute more to roadside pollution. Therefore, we further investigate the impact of travel demand (dÞ and bus capacity (mÞ and understand when vicious cycle of a transit line can be avoided. 5.1. The impact of travel demand Let us consider the travel demand as a model parameter and define
Z
þ1
hðz; dÞ ¼ z
Dðzd;f ðdÞÞ
g ðeÞde;
ð25Þ
where Dðzd; f ðdÞÞ ¼ t ðzdÞ wðf ðdÞÞ s and v ¼ zd is defined to denote the volume of auto users. Thus, dtðzdÞ=dd ¼ zt 0 ðv Þ > 0. 0 When the transit operator is revenue-neutral, we have df =dd ¼ sð1 zÞ=k ðf Þ > 0; when the transit operator is periodic profit-maximizing, we have df =dd ¼ ð1 zÞ=m > 0. Thus, regardless of the transit strategy, we have
@hðz; dÞ df > 0: ¼ g ðDðz; f ðdÞÞÞ zt 0 ðv Þ w0 ðf Þ dd @d
ð26Þ
Therefore, hðz; dÞ is strictly increasing with respect to travel demand. At equilibrium, z is pinned down by hðz; dÞ ¼ 0. Since hð0; dÞ < 0 and hð1; dÞ > 0 as proved before, the graph of hðz; dÞ with different travel demands can be shown in the following Fig. 2 with d1 < d2 < d3 . From Fig. 2, we can conclude that when travel demand is low (such as d ¼ d1 Þ, there is only one equilibrium z . This equilibrium point is stable and with relatively large share of auto users, thus a vicious cycle will occur for the transit line. As the travel demand increases, the system can turn into a situation with two stable equilibriums (such as d ¼ d2 Þ or even one (such as d ¼ d3 Þ. Thus, when the travel demand is high, the vicious cycle for the transit will be unlikely to occur and the system will converge to the equilibrium point with a relatively small share of auto users. 5.2. The impact of bus capacity For the revenue-neutral transit operator, we do not consider the bus capacity, since the frequency is determined by the transit demand at the end of the previous period. In this subsection, we only consider periodic profit-maximizing transit operator. The bus capacity is reflected by parameter m, which is convenient for analyzing the impact of bus capacity. Define
Z hðz; mÞ ¼ z
þ1
Dðz;f ðmÞÞ
g ðeÞde:
ð27Þ
Then
@hðz; mÞ ð1 zÞd ¼ g ðDðz; f ðmÞÞÞw0 ðf Þ < 0: @m m2
ð28Þ
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
341
Fig. 2. Evolution of model equilibriums with different travel demands (d1 < d2 < d3 Þ.
Similar to the proof in subsection 5.1 and from the following Fig. 3, we can conclude that when the bus capacity is large, there is only one stable equilibrium with relatively large share of auto users and the vicious cycle will occur for the transit line. Then by reducing the transit capacity, the system will move to a situation with one or two stable equilibriums. Thus, when the bus size is small, the vicious cycle for the transit will be unlikely to occur and the system will converge to the equilibrium point with relatively small share of auto users. This is because, when the total travel demand is fixed, using smaller buses will increase bus frequency and thus decrease transit cost, which in turn brings up an increase in transit demand. When the capacity is fixed, the transit profit is given by
pðzÞ ¼ sð1 zÞd kðf ðzÞÞ ¼ sð1 zÞd kðð1 zÞd=mÞ:
ð29Þ
0
Profitability of bus services requires that sm > k ðf Þ, which means that the maximal profit can be earned when the marginal revenue for running one additional bus is larger than its marginal cost. Thus
p0 ðzÞ ¼ sd þ k0 ðf Þ d=m < 0;
ð30Þ
which means that a transit operator can make a maximal profit at the stable fixed point with a minimal auto share. This underscores the importance of investigation on how to avoid vicious cycle and find the attraction domain for the dynamic system to converge to the stable fixed point with the lowest auto share of travelers. From Fig. 3 and our previous analysis, when the total demand is fixed, smaller buses can increase the transit frequency then decrease the transit travel cost and attract more travelers to choose transit, and finally increase the transit income. However, increasing transit frequency will also increase the transit operation cost. Thus, the choice of bus capacity by a periodic profit-maximizing operator depends on the specific forms of the operation cost function. In practice, the operation cost depends on both bus frequency and bus capacity. Thus, we define the operation cost function as kðf ; mÞ ¼ kðð1 zÞd=m; mÞ and assume @kðf ; mÞ=@m > 0. From Fig. 3, we can conclude that dz=dm > 0 in the attraction domain where the system converges to the fixed point with the lowest share of auto users. If
dkðf ; mÞ @kðf ; mÞ ð1 zÞd @kðf ; mÞ ¼ > 0; þ dm @f m2 @m which is equivalent to
1 @kðf ; mÞ 1 @kðf ; mÞ > ; f @m m @f
ð31Þ
the operation cost is increasing with bus capacity and
dpðmÞ dz dkðf ; mÞ ¼ sd < 0; dm dm dm
ð32Þ
which means that the transit profit is decreasing with bus capacity. Thus, smaller buses are preferred by the transit operator to earn more profit. Eq. (31) means that the marginal cost of transit system capacity by increasing bus size is larger than the corresponding marginal cost by increasing transit frequency. If Eq. (31) does not hold, such as when the operation cost is not very sensitive to bus size, extremely when @kðf ; mÞ=@m ¼ 0, it is no longer guaranteed that smaller buses are better for the periodic profit-maximizing authority. 5.3. A numerical example This example is presented to show the two symbiotic evolution processes of the day-to-day modal choice (3) and periodic frequency adjustments (10). It demonstrates the existence of multiple equilibrium fixed points, and illustrates the impact of the total demand and bus capacity.
342
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
Fig. 3. Evolution of model equilibriums with different desirable number of passengers per bus (m1 < m2 < m3 Þ.
Table 1 Function specifications. Function
Public transport travel time cost function
Specification 4 tðzÞ ¼ VOT 8 1 þ 0:15 zd þc H wðf Þ ¼ VOT 2f1 þ 10
Transit operation cost function
kðf Þ ¼ kf
Highway travel cost function
Consider the two-mode network described in Section 2 with function specifications given in Table 1 and parameter values in Table 2. Note that all the propositions only require that e is continuously distributed with a cumulative distribution function e GðeÞ, where G is assumed to be strictly increasing, continuous and differentiable over its support ð1; þ1Þ. Here we jej
consider a bimodal exponential power distribution BEF ð0; 1; 1; 3Þ of e, which means g ðeÞ ¼ 32jeCje 2 . In each period, the day-toð3Þ day dynamic system follows Eq. (3) with d ¼ 0:2. For periodic system dynamics, bn ¼ 0:8 for all periods. The length of each period is 30 (days). When transit operator follows the break-even strategy (i.e. p0 ¼ 0Þ, from Eqs. (15) and (16) we can obtain three different ~z1 ; ~f 1 ¼ ð0:5442; 0:0570Þ; ~z2 ; ~f 2 ¼ ð0:8504; 0:0187Þ and break-even equilibrium fixed points, i.e. ~ ~z3 ; f 3 ¼ ð0:9596; 0:0125Þ. Fig. 4 plots the trajectories of the auto share and the bus frequency starting from different initial states. It can be seen from Fig. 4 that the entire set is divided into two attraction domains, A and B. Initial states in domain A 0 converge to ~z3 ; ~f 3 and initial states in domain B converge to ~z1 ; ~f 1 . Since g D ~z1 ; ~f 1 t0 ~z1 þ sd=k ~f 1 w0 ~f 1 <1 ~ ~ ~ and ~z3 ; f 3 is the boundary fixed point, Fig. 4 shows that ~z1 ; f 1 and ~z3 ; f 3 are stable fixed points, this validates Propo sition 5. The boundary of domain A and B is formed by the trajectories towards the unstable fixed point ~z2 ; ~f 2 . This observation is consistent with the finding in Bie and Lo (2010). In this example, the double dynamic system is much more likely to converge to ~z ; ~f due to the much larger area of domain B, which means the vicious cycle is unlikely to occur. Only when 1
1
the initial auto share is very high (such as z ¼ 0:9Þ and the initial transit frequency is very low (such as f ¼ 1=50Þ the vicious cycle will occur. However, in practice, the initial auto share is usually below 0.8, thus the double dynamic system will likely move into a virtuous cycle with the functions and parameters given in Tables 1 and 2. This example also points to the design of policies for government intervention to avoid the occurrence of a vicious cycle. When the initial dynamic system is in domain A, the system will converge to the boundary fixed point. The level of transit service has to be subsidized in order to keep the minimal frequency because the profit is negative at the boundary fixed point. However, with increasing car ownership, people may have higher preference for auto, and thus, a vicious cycle will become inevitable if no sufficient subsidies can be introduced. For a transit line operating near the level (the red dotted line in Fig. 4) that separates the stable equilibriums with low and high share of auto users, the system may enter either a vicious or a virtuous cycle. Subsidizing the transit system and increasing the bus frequency may lead to a virtuous cycle, thereby driving the system to converge to a desirable equilibrium. When the initial dynamic system is already near the stable equilibrium with low auto share, there is no need to increase transit subsidies, the system will self-enter a virtuous cycle. Fig. 5 shows the change of auto share across the periods from different initial values with the same initial frequency of f ¼ 1=40. As indicated in Fig. 5, different stable fixed-points are reached from different starting states. More periods are needed to reach the target equilibrium auto share if the initial auto share is far from the equilibrium. The change of the difference between the actual auto share and the target equilibrium auto share is shown in Fig. 6 with the length of each period being 15 days (the blue1 line) and 30 days (the red line). For each scenario, the initial auto share is 0.7. It is clearly shown that 1
For interpretation of color in Fig. 6, the reader is referred to the web version of this article.
343
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
the difference approaches zero after 50 periods when the length of each period is 15 days and after 33 periods when the length of each period is 30 days. Fig. 6 indicates that, more (fewer) periods are needed to reach the target equilibrium with shorter (longer) period length. It is thus interesting to choose a proper length of each period to minimize the whole duration needed to get the target equilibrium. Fig. 7 shows the impact of the total demand. When the total demand is scaled from 0:98d to 1:04d, three fixed points emerge. When travel demand is small (such as d ¼ 0:9dÞ, there is only one equilibrium with relatively large share of auto
Table 2 Value of parameter. Parameter
Value
Base total demand Desirable number of passengers per bus Highway capacity Coefficients in operation cost function Value of time (VOT) Monetary cost by auto Transit fare Minimum transit frequency
d ¼ 1200 (person/h) m ¼ 150 (person) H ¼ 500 (veh/h) k ¼ 9600 (HK$/run) VOT ¼ 1 (HK$/min) c ¼ 8 (HK$) s ¼ 1 (HK$) f min ¼ 1=80 (run/min)
1 0.9
A
Initial Equilibrium
0.8
Auto share
0.7 0.6 0.5 0.4 0.3
B
0.2 0.1 0 0.0125 0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
Transit frequency Fig. 4. Attraction domains of the dynamic bi-modal system under break-even strategy.
1 0.9 0.8
Auto share
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
5
10
15
20
25
Period number Fig. 5. The evolution of the auto share.
30
35
344
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
Gap between the target equilibrium
0.16 Period length = 15 days Period length = 30 days
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
5
10
15
20
25
30
35
40
45
50
Period number Fig. 6. The difference between the actual auto share and the target equilibrium with different length of each period.
1
Auto share
0.8 0.6 0.4 0.2 0.9
0.92
0.94
0.96
0.98
1
1.02
1.04
1.06
1.08
1.1
Scale of demand Fig. 7. Fixed points of the dynamic system (10) and (12) with black (white) dots representing stable (unstable) fixed points.
1
A
Initial Equilibrium
0.9 0.8
Auto share
0.7 0.6 0.5 0.4
B
0.3 0.2 0.1 0 0.0125 0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
Transit frequency Fig. 8. The trajectories of the dynamic system under periodic profit-maximizing strategy.
users, thus vicious cycle occurs for the transit line. When the travel demand is large (such as d ¼ 1:1dÞ, the vicious cycle for the transit is unlikely to occur and the system converges to the equilibrium point with relatively small share of auto users. When the transit operator follows the periodic profit-maximizing strategy, from Eqs. (21) and (22) we can obtain three z1 ; f 1 ¼ ð0:5001; 0:0667Þ; z2 ; f 2 ¼ ð0:8693; 0:0174Þ and different break-even equilibrium fixed points i.e. z3 ; f 3 ¼ ð0:9596; 0:0125Þ. We track the trajectories of the auto share and the transit frequency starting from different initial states and obtain the results as shown in Fig. 8. Fig. 8 shows that the fixed points z ; f and z ; f are stable while z ; f is 1
unstable.
1
3
3
2
2
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
345
1 0.9 0.8
Auto share
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
60
70
80
90
100 110 120 130 140 150 160 170 180 190 200
Desirable number of passengers per bus Fig. 9. Fixed points of the dynamic system (12) and (19) with black (white) dots representing stable (unstable) fixed points.
Fig. 9 illustrates the impact of bus capacity. When the desirable number of passengers per bus is between 70 to 180, there are three fixed points. When the bus capacity is large (such as m ¼ 200Þ, there is only one equilibrium with relatively large share of auto users, thus vicious cycle occurs for the transit line. When the bus capacity is small (such as m ¼ 60Þ, the vicious cycle for the transit is unlikely to occur and the system converges to the equilibrium point with relatively small share of auto users. We conclude this section by pointing out that similar results are obtained under a normal distribution of e, and thus not presented here.
6. Conclusions In this paper, we investigated the dynamics of modal choice of heterogeneous travelers with responsive transit service under different economic objectives. The problem of interest is modeled as a double dynamic system where travelers adjust their modal choice from day to day while transit operator sets transit service frequency from period to period in response to changing demand. We analyzed travelers’ modal choice in each period based on the travelers’ rational behavior that travelers tend to change to more attractive mode. The dynamic system of modal choice in each period has a unique fixed point which is identical with the user equilibrium auto share. The stability of this unique fixed point can be established by Lyapunov stability theorem. Over periods, the transit frequency is responsive to the transit demand under different economic objectives, which may result in multiple stable fixed points. In the presence of multiple stable fixed points, the fixed point with the lowest or highest auto share is stable and the stable condition is independent of the updating parameters. We also investigated the impacts of total travel demand and bus capacity. The analysis and results are useful for setting operating strategies to drive the dynamic system towards a desirable equilibrium fixed point. Further research is expected to extend the double dynamic model to a general network, to seek the optimal length of each period for fast system convergence. Intelligent transportation system design (Cantarella, 2013) for the network is also an interesting extension. Other factors pertaining to modal choice behavior, such as willingness-to-wait for buses (Peterson et al., 2006), bounded-rationality (Wu et al., 2013) could be incorporated into the current modeling framework. The transit fare is assumed to be fixed through this study; this assumption should be relaxed for better representing realistic transit operating strategies of both fare choice and frequency setting. In addition, the continuous day-to-day traffic dynamic model developed in this study suffers from the limitations as pointed out in Watling and Hazelton (2003). A discrete day-to-day dynamic traffic model shall add realism but in this case the auto share may fluctuate with discrete time modal choice, giving rise to the failure of Proposition 4 and the non-convergence of the double dynamic system. It is a challenging extension about how to establish the double dynamic system embodied with a discrete-time dynamic modal choice. In this regard, we should point out that, as long as the auto share evolves closer to the equilibrium share of that period, namely, nþ1 z zðnþ1Þ < zn zðnþ1Þ , then the properties of the periodic dynamic model established so far are still valid (see proof in Appendix A.7). In this case, the proposed model can still help the transit operators to reach the target equilibriums even the system does not reach a stable state at the end of each period.
Acknowledgements The authors would like to express our sincere thanks to three enthusiastic anonymous reviewers for their constructive and helpful comments. The work described in this paper was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKUST16205715) and the National Natural Science Foundation of China (71371020).
346
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
Appendix A A.1. Proof for Proposition 2 Define the following Lyapunov function
Z VðzÞ ¼
þ1
Dðz;f n Þ
!2 g ðeÞde z
:
ð33Þ
On one hand, since GðeÞ is assumed to be strictly increasing, continuous and differentiable over its support ð1; þ1Þ; VðzÞ is also continuously differentiable with respect to z 2 ½0; 1, also, V ðzn Þ ¼ 0 and
VðzÞ > 0; 8z 2 ½0; 1 n zn :
ð34Þ n
On the other hand, for any z 2 ½0; 1 n z , we have
_ VðzÞ ¼ 2d
Z
þ1
Dðz;f n Þ
!2 g ðeÞde z
n g D z; f t0 ðzÞ 1 < 0;
ð35Þ
and V_ ðzn Þ ¼ 0. By Lemma 1, the stationary point zn is asymptotically stable.
A.2. Proof for Proposition 3 Similar to the proof in Proposition 1, define hðzÞ ¼ z
R þ1 DðzÞ
g ðeÞde. Since GðeÞ is continuous and differentiable over its sup-
port ð1; þ1Þ , hðzÞ is also continuous and differentiable for all z 2 ½0; 1. Then R þ1 Dð0Þ < tð0Þ s < þ1; Dð1Þ > tð1Þ wðf min Þ s > 1. Hence hð0Þ ¼ Dð0Þ g ðeÞde < 0 and R þ1 R Dð1Þ hð1Þ ¼ 1 Dð1Þ g ðeÞde ¼ 1 g ðeÞde > 0. Since hðzÞ is continuous with respect to z, based on the intermediate value theorem (Clarke, 1971), there exists at least one zk 2 ð0; 1Þ such that hðzk Þ ¼ 0, which means that there exists at least one user equilibrium. Here, we also show that condition (9) is a sufficient and necessary condition for the existence of multiple equilibriums. Note that condition (9) is equivalent to the following condition (36).
hðzk Þ ¼ 0
ð36Þ
0
h ð zk Þ < 0
(1) If there exists a zk satisfying the above conditions, then for any arbitrarily small g > 0, we have hðzk gÞ > 0; hðzk Þ ¼ 0 and hðzk þ gÞ < 0. Also by hð0Þ < 0 and hð1Þ > 0, based on the intermediate value theorem, hðzÞ must have at least three zero points. Therefore, there exists at least three equilibriums and condition (9) is a sufficient condition for the existence of multiple equilibriums. 0 (2) Assume that, at any equilibrium zk ; h ðzk Þ > 0. Then for any g > 0, we have hðzk gÞ < 0; hðzk Þ ¼ 0 and hðzk þ gÞ > 0. Hence, for any 0 6 z0 < zk < z00 6 1, we have hðz0 Þ < 0 and hðz00 Þ > 0. This implies that hðzÞ has only one zero point, therefore, condition (9) is a necessary condition for the existence of multiple equilibriums. From (1) and (2), condition (9) is, indeed, a necessary and sufficient condition. A.3. Proof for Proposition 4 In the ðn þ 1Þth period, the initial auto share is zn and the transit frequency is f we can obtain
€z ¼ d g D z; f nþ1 t 0 ðzÞ 1 < 0
nþ1
. Based on the dynamic system Eq. (3),
ð37Þ
for all z 2 ½0; 1. Thus,
R þ1 8 z_ > 0 if z < Dðzðnþ1Þ ;f nþ1 Þ g ðeÞde > > > < R þ1 z_ ¼ 0 if z ¼ Dðzðnþ1Þ ;f nþ1 Þ g ðeÞde > > > : z_ < 0 if z > R þ1 nþ1 g ðeÞde Dðzðnþ1Þ ;f Þ
ð38Þ
R þ1 R þ1 g ðeÞde z_ is always positive until z ¼ Dðzðnþ1Þ ;f nþ1 Þ g ðeÞde and the auto share does not change any Dðzðnþ1Þ ;f nþ1 Þ more. Otherwise, there exists z1 2 ½0; 1 such that z_ 1 < 0 at time t 1 in the ðn þ 1Þth period. From (38) we can get Therefore, if zn <
347
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
R þ1 g ðeÞde. Since z is continuous with respect to time, based on the intermediate value theorem, there exists a Dðzðnþ1Þ ;f nþ1 Þ stationary point z2 ¼ zðnþ1Þ at time t2 in the ðn þ 1Þth period with 0 < t2 < t1 . This contradicts the fact that the stationary R þ1 point is stable as proved in Proposition 2. In the same vein, if zn > Dðzðnþ1Þ ;f nþ1 Þ g ðeÞde; z_ is always negative until R þ1 z ¼ Dðzðnþ1Þ ;f nþ1 Þ g ðeÞde and the auto share does not change any more. Thus znþ1 is always between zn and the equilibrium auto share of the ðn þ 1Þth period, as described in Eq. (12).
z1 >
A.4. Proof for Lemma 2 k1 and k2 are the solutions of the quadratic equation k2 k trJ þ det J ¼ 0 where trJ ¼ k1 þ k2 ¼ abAB þ ð1 aÞ þ ð1 bÞ and det J ¼ k1 k2 ¼ ð1 aÞð1 bÞ 2 ½0; 1Þ. Based on the sign of tr2 J 4 det J, we have the following three different cases. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1) If tr2 J 4 det J P 0 and trJ P 0, both k1 and k2 are real and k1 ; k2 ¼ trJ tr2 J 4 det J =2 P 0. Thus jkj < 1 requires
8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi trJþ tr2 J4 det J > > <1 < 2
)
tr J 4 det J P 0 > > : trJ P 0 2
1 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi2 1 a 1 b 6 AB < 1: ab
(2) If tr2 J 4 det J P 0 and trJ < 0, both k1 and k2 are real and k1 ; k2 ¼
8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi trJ tr2 J4 det J > > > 1 < 2 tr J 4 det J P 0 > > : trJ < 0 2
)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
trJ
tr2 J4 det J 2
6 0. Thus jkj < 1 requires
1 1 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi2 ð2 aÞð2 bÞ < AB 6 1aþ 1b : ab ab
1=2 (3) If tr2 J 4 det J < 0; k1 and k2 are of a complex conjugate pair, with jk1 j ¼ jk2 j ¼ det J < 1. Thus jkj < 1 requires
tr2 J det J < 0 )
1 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi2 1 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi2 1 a þ 1 b < AB < 1a 1b : ab ab
Combining the above three cases we can conclude that the moduli of k1 and k2 are both less than 1, if and only if ð2 aÞð2 bÞ=ab < AB < 1. A.5. Proof for Proposition 5 The dynamic system over periods is ruled by a set of matrices D ¼ fD1 ; D2 ; . . . ; Dn ; . . .g. Since h i 1 z 2 ½0; 1; f 2 f min ; k ðsd p0 Þ , all the functions are continuous and bounded, then D is a bounded set of matrices. Based on Lemma 3, the fixed point is asymptotically stable if the joint spectral radius of the set D ¼ fD1 ; D2 ; . . . ; Dn ; . . .g is less than n o 1=k 1 at the fixed point. The joint spectral radius of the set D is defined as qðDÞ ¼ limk!1 max Di Di : Di 2 D . 1
Then
k
1=k n o qðDÞ ¼ lim max Di1 Dik 1=k : Di 2 D ¼ lim max Dki : Di 2 D k!1
k!1
1=k ¼ max lim Dki : Di 2 D ¼ max fqðDi Þ : Di 2 Dg: k!1
The last equation is based on Gelfand’s Formula. According to Eq. (14), at the interior fixed point,
An ¼ Thus,
sd
0 k ~f
;
w0 ~f g D ~z ; ~f : Bn ¼ 1 þ g D ~z ; ~f t 0 ð~z Þ
0 ~ ~ sd w f g D ~z;f An Bn ¼ 0
k ~f 1 þ g D ~z ; ~f
t 0 ð~z Þ
> 0;
which means that term ð2 an Þð2 bn Þ=an bn in Lemma 2 is unbounded. Therefore, for all n 2 N þ , when An Bn < 1, the moduli of the two eigenvalues of Dn are both less than 1, which means when
348
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
0 ~ ~ sd w f g D ~z;f 0
k ~f 1 þ g D ~z ; ~f
t 0 ð~z Þ
< 1;
qðDÞ < 1
and the interior fixed point is asymptotically stable. Since
k ~f 1 þ g D ~z ; ~f
0 1 sd 0 ~ A ~ 0 ~ @ ~ t ðz Þ þ 0 < 1; w f < 1 () g D z ; f t 0 ð~z Þ k ~f
0 ~ ~ sd w f g D ~z;f 0
0 t 0 ð~z Þ þ sd=k ~f w0 ~f < 1, the system (10) and (12) is (locally) asymptotwe can conclude that, when g D ~z ; ~f ~ ically stable at the corresponding interior fixed point ~z ; f . When the fixed point is at the boundary, based on Eq. (14) An ¼ 0, thus ð2 an Þð2 bn Þ=an bn < An Bn ¼ 0 < 1 for all n 2 N þ . Similar to the above before, we can conclude that the system (10) and (12) is always (locally) asymptotically stable at the boundary fixed point. A.6. Proof for Proposition 6 When there is only one target profit equilibrium fixed point. If the fixed point is a boundary fixed point, as indicated in Proposition 5, it must be asymptotically stable. If the fixed point is an interior fixed point, according to Proposition 3, at this fixed point ~z ; ~f ,
R þ1 d Dðz;f ðzÞÞ g ðeÞde dz
z¼~z
0 1 s d ¼ g D ~z ; ~f @t0 ð~z Þ þ 0 w0 ~f A < 1: k ~f
Also by Proposition 5, we can conclude that the system is asymptotically stable at this target profit equilibrium fixed point. When there are multiple target profit equilibrium fixed points, according to Proposition 3, there exists at least one target profit equilibrium fixed point ~z ; ~f such that k
R þ1 d Dðz;f ðzÞÞ g ðeÞde dz
z¼~z
k
0 1 s d 0 0 ~ ~ @ t ð~z Þ þ 0 w f A > 1: ¼ g D ~z ; f k ~f
Thus, the system is unstable at this fixed point.
Like the proof of Proposition 3, hð0Þ < 0 and hð1Þ > 0. Define ~zl ; ~f l and ~zh ; ~f h as the fixed points with the lowest and
highest shares of auto users, respectively. Then we have hð~zl Þ ¼ hð~zh Þ ¼ 0; hðzÞ < 0 for any z 2 ð0; ~zl Þ, and hðzÞ > 0 for any 0 0 z 2 ð~zh ; 1Þ. Also for any arbitrarily small g > 0, we have hð~zl þ gÞ > 0; hð~zh gÞ < 0. Thus h ð~zl Þ > 0 and h ð~zh Þ > 0. That is to say,
0 1 s d 0 0 ~ ~ g D ~zl ; f l @t ð~zl Þ þ 0 w f l A < 1 k ~f l and if ~zh ; ~f h is an interior fixed point,
0 1 s d 0 0 ~ ~ @ g D ~zh ; f h t ð~zh Þ þ 0 w f h A < 1: k ~f h If ~zh ; ~f h is a boundary fixed point, it must be asymptotically stable as proved in Proposition 5. Therefore, the target profit fixed points with the lowest and highest shares of auto users is locally stable. A.7. For each period, if the auto share evolves closer to the equilibrium auto share of that period, i.e., znþ1 zðnþ1Þ < zn zðnþ1Þ , then, the properties of the periodic dynamic model is still valid. Proof. The condition znþ1 zðnþ1Þ 6 zn zðnþ1Þ is equivalent to znþ1 ¼ cn zðnþ1Þ þ ð1 cn Þzn , where cn 2 ð0; 2Þ. Now we assume an 2 ð0; 2Þ in Eq. (12), and show it would not affect the following propositions by proving that Lemma 2 still holds when a 2 ð0; 2Þ.
X. Li, H. Yang / Transportation Research Part C 68 (2016) 333–349
349
When a 2 ð1; 2Þ; det J ¼ k1 k2 ¼ ð1 aÞð1 bÞ 2 ð1; 0Þ and tr2 J 4 det J > 0. Thus both k1 and k2 are real and jkj < 1 requires
ffi 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < trJþ tr2 J4 det J < 1 ð2 aÞð2 bÞ 2 < AB < 1: ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : trJ tr2 J4 det J ab > 1 2
Combining the above case with Lemma 2, we can conclude that when a 2 ð0; 2Þ, the condition for the moduli of k1 and k2 being less than 1, remains ð2 aÞð2 bÞ=ab < AB < 1. Following the proof of Proposition 5, since an 2 ð0; 2Þ, the term ð2 an Þð2 bn Þ=an bn is still unbounded. Therefore, when an 2 ð0; 2Þ, Propositions 5–8 all remain valid. References Arnott, R., Yan, A., 2000. The two-mode problem: second-best pricing and capacity. Rev. Urban Reg. Dev. Stud. 12 (3), 170–199. Bar-Yosef, A., Martens, K., Benenson, I., 2013. A model of the vicious cycle of a bus line. Transport. Res. B: Methodol. 54, 37–50. Bie, J., Lo, H., 2010. Stability and attraction domains of traffic equilibria in a day-to-day dynamical system formulation. Transport. Res. B: Methodol. 44 (1), 90–107. Cantarella, G., 2013. Day-to-day dynamic models for intelligent transportation systems design and appraisal. Transport. Res. C: Emerging Technol. 29, 117– 130. Cantarella, G., Velonà, P., Watling, D., 2013. Day-to-day dynamics & equilibrium stability in a two-mode transport system with responsive bus operator strategies. Netw. Spatial Econ. 15 (3), 485–506. Cascetta, E., 1987. Static and Dynamic Models of Stochastic Assignment to Transportation Networks. Flow Control of Congested Networks. Springer, Berlin, Heidelberg, pp. 91–111. Cascetta, E., 1989. A stochastic process approach to the analysis of temporal dynamics in transportation networks. Transport. Res. B: Methodol. 23 (1), 1–17. Clarke, D., 1971. Foundations of Analysis. Appleton-Century-Crofts, 284. David, Q., Foucart, R., 2014. Modal choice and optimal congestion. Reg. Sci. Urban Econ. 48, 12–20. Farokhi, F., Johansson, K., 2015. A piecewise-constant congestion taxing policy for repeated routing games. Transport. Res. B: Methodol. 78, 123–143. Friesz, T., Bernstein, D., Smith, T., Tobin, R., Wei, B., 1993. A variational inequality formulation of the dynamic network user equilibrium problem. Oper. Res. 41, 179–191. Guo, R., Yang, H., Huang, H., 2013. A discrete rational adjustment process of link flows in traffic networks. Transport. Res. C: Emerging Technol. 34, 121–137. Guo, R., Yang, H., Huang, H., Tan, Z., 2015. Link-based day-to-day network traffic dynamics and equilibria. Transport. Res. B: Methodol. 71, 248–260. He, X., Guo, X., Liu, H., 2010. A link-based day-to-day traffic assignment model. Transport. Res. Part B: Methodol. 44 (4), 597–608. Khalil, H., 2002. Nonlinear Systems, third ed. Prentice Hall, Upper Saddle River, New Jersey. Horowitz, J., 1984. The stability of stochastic equilibrium in a two-link transportation network. Transport. Res. B: Methodol. 18 (1), 13–28. Kraus, M., 2003. A new look at the two-mode problem. J. Urban Econ. 54 (3), 511–530. Lam, W., Huang, H., 1995. Dynamic user optimal traffic assignment model for many to one travel demand. Transport. Res. B: Methodol. 29 (4), 243–259. Li, Z., Lam, W., Wong, S., 2012. Modeling intermodal equilibrium for bimodal transportation system design problems in a linear monocentric city. Transport. Res. B: Methodol. 46 (1), 30–49. Meignan, D., Simonin, O., Koukam, A., 2007. Simulation and evaluation of urban bus-networks using a multiagent approach. Simul. Modell. Pract. Theory 15 (6), 659–671. Peterson, D., Hough, J., Ulmer, D., Urban, S., Center, R.T., 2006. Express Bus Transit Study: A Case Study (No. MPC Report No. 06-178). Mountain Plains Consortium. Ran, B., Hall, R., Boyce, D., 1996. A link-based variational inequality model for dynamic departure time/route choice. Transport. Res. B: Methodol. 30 (1), 31– 46. Smith, M., 1984. The stability of a dynamic model of traffic assignment: an application of a method of Lyapunov. Transport. Sci. 18 (3), 245–252. Tan, Z., Yang, H., Guo, R., 2015. Dynamic congestion pricing with day-to-day flow evolution and user heterogeneity. Transport. Res. C: Emerging Technol. 61, 87–105. Theys, J., 2005. Joint Spectral Radius: theory and approximations. Doctoral dissertation, UCL. Watling, D., Hazelton, M., 2003. The dynamics and equilibria of day-to-day assignment models. Netw. Spatial Econ. 3 (3), 349–370. Wu, J., Sun, H., Wang, D., Zhong, M., Han, L., Gao, Z., 2013. Bounded-rationality based day-to-day evolution model for travel behavior analysis of urban railway network. Transport. Res. C: Emerging Technol. 31, 73–82. Ye, H., Yang, H., Tan, Z., 2015. Learning marginal-cost pricing via a trial-and-error procedure with day-to-day flow dynamics. Transport. Res. B: Methodol. 81, 794–807. Zhang, F., Yang, H., Liu, W., 2014. The Downs–Thomson Paradox with responsive transit service. Transport. Res. A: Policy Pract. 70, 244–263.