The Optimal Transit Fare Structure under Different Market Regimes with Uncertainty in the Network

Abstract

This paper proposes a network-based model for investigating the optimal transit fare structure under monopoly and oligopoly market regimes with uncertainty in the network. The proposed model treats the interaction between transit operators and transit passengers in the market as a two-level hierarchical problem with the transit operator sub-model at the upper-level and the transit passenger sub-model at the lower-level. The upper-level problem is to determine the fare structure so as to optimize the objective function of the transit operators, whereas the lower-level problem represents the path choice equilibrium of the transit passengers. In order to consider the uncertainty effects on transit network, the proposed model incorporates the unreliability component of transit services into the passenger disutility function, which is mainly due to variations of the in-vehicle travel time and the dwelling time of transit vehicles at stops. With the use of the proposed model, a numerical example is given to assess the impacts of the market regimes and the unreliability of the transit services on the optimal transit fare structure.

Appendix

1.1 Appendix A. Link travel disutility

As stated in Section 2.1, there are two types of links in the transit network G = (N, S); namely, transit links and walking links. Let S ₁ and S ₂ represent the set of transit links and the set of walking links, respectively, and thus $S = S_1 \cup S_2$. In the following, we define in turn the travel disutility functions for the two types of links.

1.2 Transit link travel disutility

The travel disutility u _s on transit link s, measured in generalized time units, consists of the following four generalized travel cost components: travel time, transit fare, in-vehicle crowding discomfort, and perceived cost of the unreliability of transit services, i.e.,

$$u_s = E\left( {T_s } \right) + \frac{1}{{\alpha _1 }}p_s + \frac{{\beta _1 }}{{\alpha _1 }}g_s + \frac{{\beta _2 }}{{\alpha _2 }}f\left( {\sigma _s } \right),\quad \forall s \in S_1 $$

(17)

where T _s is the random variable of (actual) travel time on transit link s. E(T _s ) and σ _s are the mean and the standard deviation of T _s , respectively. p _s and g _s are the transit fare and the in-vehicle crowding discomfort cost on transit link s, respectively. The parameters α ₁, β ₁ and β ₂ are, respectively, the value of time, the value of discomfort, and the value of reliability of passengers, which are all measured in monetary value per unit time. The function $f\left( \cdot \right)$ measures the cost of service unreliability, which is a function of the standard deviation, σ _s , of the travel time on transit link s (Noland et al. 1998).The mean travel time E(T _s ) on transit link s comprises the mean in-vehicle travel time E(T _{s 1}) and the mean waiting time E(T _{s 2}) on link s, i.e.,

$$E\left( {T_s } \right) = \tau _1 E\left( {T_{s1} } \right) + \tau _2 E\left( {T_{s2} } \right),\quad \forall s \in S_1 $$

(18)

where T _{s 1} and T _{s 2} are the random variables of the actual in-vehicle travel time and the actual waiting time on link s, respectively. The parameters () are the reciprocal substitution factors for converting the different time components to the same unit. For the purpose of presentation, let t _s  = E(T _s ), t _{s 1} = E(T _{s 1}), and t _{s 2} = E(T _{s 2}), then Eq. (18) can be rewritten as

$$t_s = \tau _1 t_{s1} + \tau _2 t_{s2} ,\quad \forall s \in S_1 $$

(19)

Next, we define the mean in-vehicle link travel time t _{s 1}, the mean waiting time t _{s 2}, the crowding discomfort cost g _s , and the unreliability cost f(σ _s ) of transit services.

1.2.1 Mean in-vehicle link travel time

According to A4 in Section 2.2, the actual in-vehicle link travel time is an independent normally distributed random variable with mean t _{s 1} and standard deviation σ _{s 1}. The mean in-vehicle travel time t _{s 1} on link s can be estimated in terms of the mean in-vehicle travel times of all attractive lines on link s (De Cea and Fernandez 1993; Uchida et al. 2005), i.e.,

$$t_{s1} = \sum\limits_{l \in A_s } {t_s^l x_s^l ,\quad \forall s \in S_1 } $$

(20)

where $x_s^l$ is the probability of passengers on link s choosing line l. $t_s^l$ is the mean in-vehicle travel time of passengers using line l on link s, which is assumed to be a constant that is dependent on the length of line l.

We now derive the probability $x_s^l$ of passengers on link s choosing line l for traveling. Note that the transit service frequency f _s on link s can be formulated as the sum of the service frequencies of all attractive lines on link s, i.e.,

$$f_s = \sum\limits_{l \in A_s } {f_l } ,\quad \forall s \in S_1 $$

(21)

As the transit line frequency f _l is a random variable that is dependent on the level of the reliability of transit services, the link frequency f _s is also a random variable with

$$E\left( {f_s } \right) = \sum\limits_{l \in A_s } {E\left( {f_l } \right)} ,\quad \forall s \in S_1 $$

(22)

where E(f _l ) and E(f _s ) are the mean service frequencies of line l and link s, respectively.

Consequently, according to A3 in Section 2.2, the probability $x_s^l$ of passengers on transit link s choosing line l can be approximated by the proportion of the mean frequency of line l to the mean frequency of link s, i.e.,

$$x_s^l = \frac{{E\left( {f_l } \right)}}{{E\left( {f_s } \right)}} = \frac{{E\left( {f_l } \right)}}{{\sum\limits_{l \in A_s } {E\left( {f_l } \right)} }},\quad \forall l \in A_s ,s \in S_1 $$

(23)

where E(f _l ) can be determined according to Appendix B.

1.2.2 In-vehicle crowding discomfort cost

In general, passenger discomfort is affected by the degree of crowding in transit vehicles. Similar to (20), the crowding discomfort cost on link s can be estimated in terms of the mean discomfort costs of all attractive lines on link s, i.e.,

$$g_s = \sum\limits_{l \in A_s } {g_s^l x_s^l } ,\quad \forall s \in S_1 $$

(24)

where $g_s^l$ is the in-vehicle discomfort cost of line l on link s.

According to Spiess and Florian (1989); Wu et al. (1994), and Lo et al. (2003), the in-vehicle crowding discomfort cost, $g_s^l$, which is measured in terms of generalized time units, on line l passing through link s can be expressed in the form of the Bureau of Public Roads (BPR) type function with regard to the mean in-vehicle travel time, passenger volume, and vehicle capacity on the line, i.e.,

$$g_s^l = t_s^l \left( {g_s^{l0} + \gamma _1 \left( {\frac{{v_s^l + \bar v_s^l }}{{K_l }}} \right)^{n_1 } } \right),\quad \forall l \in A_s ,s \in S_1 $$

(25)

where $g_s^{l0}$ is the baseline discomfort level or riding quality of line l passing through link s, and ₁ and n ₁ are the positive calibrated parameters of the in-vehicle discomfort function. $v_s^l$ is the passenger flow using line l on link s, and can be estimated by

$$v_s^l = v_s x_s^l ,\quad \forall l \in A_s ,s \in S_1 $$

(26)

where v _s is the passenger flow on link s.

In Eq. (25), the capacity K _l of transit line l can be calculated by

$$K_l = \kappa _l f_l ,\quad \forall l \in L$$

(27)

where L is the set of transit lines and κ _l is the vehicle capacity on line l.

$\bar v_s^l$ is the passenger flow that competes with $v_s^l$ for the same common capacity of line l passing through link s. It consists of two components: (1) the number of passengers boarding at node i(s) (i.e., the tail node of link s), all other links that include line l as an attractive line, i.e., the first term on the right-hand side of Eq. (28) below; and (ii) the passenger volume boarding line l at a node before i(s) and alighting after i(s), i.e., the second term on the right-hand side of (28) below. $\bar v_s^l$ can be represented as

$$\bar v_s^l = \sum\limits_{e \in A_{i\left( s \right)}^{l + } } {v_e^l } + \sum\limits_{e \in \bar A_{i\left( s \right)}^l } {v_e^l ,} \quad \forall l \in A_s ,s \in S_1 $$

(28)

where $A_{i\left( s \right)}^{l }$ is the set of links going out from node i(s) on which line l is included as an attractive line but link s is excluded, and $\bar A_{i\left( s \right)}^l$ is the set of links on which line l is included as an attractive line, with an origin node before i(s) and an end node after i(s).

1.2.3 Mean waiting time

The waiting time that is experienced by a transit passenger includes the waiting time for the arrival of the transit vehicle and the overload delay at stops due to the insufficient capacity of the arriving vehicle. The former depends on the arrival distribution of passengers and the average arrival frequency of the vehicles on the wait link, and the latter depends on the passenger volumes boarding the same link and those already in the arriving vehicles. Similar to De Cea and Fernandez (1993) and Lo et al. (2003, 2004), the average waiting time t _{s 2}(v _s ) on link s can be described as the following volume-delay function,

$$t_{s2} \left( {v_s } \right) = E\left( {\frac{{\alpha _2 }}{{f_s }}} \right) + \gamma _2 \left( {\frac{{v_s + \bar v_s }}{{K_s }}} \right)^{n_2 } ,\quad \forall s \in S_1 $$

(29)

where α ₂, ₂, and n ₂ are positive calibrated parameters. The value of α ₂ is dependent on the distributions of transit vehicle headways and passenger arrival times. The typical value of α ₂ adopted in the literature is 0.5 with assumptions of a uniform random arrival distribution of the passengers and of a constant transit vehicle headway (Lam and Morrall 1982). The first term on the right-hand side of Eq. (29) represents the expected waiting time of passenger for the next arriving vehicle, while the second term captures the boarding congestion effect at the transit stops.

In Eq. (29), K _s is the total capacity of the transit vehicles on link s, and

$$K_s = \sum\limits_{l \in A_s } {K_l } ,\quad \forall s \in S_1 $$

(30)

where the capacity K _l of line l can be given by Eq. (27).

v _s is the passenger volume waiting to get on link s, and $\bar v_s$ is the passenger volume competing with v _s for the same common capacity on link s, which can be calculated by

$$\bar v_s = \sum\limits_{l \in A_s } {\bar v_s^l } ,\quad \forall s \in S_1 $$

(31)

where the passenger flow $\bar v_s^l$ can be determined by Eq. (28).

Similar to the derivation of the expected line frequency in Appendix B, the mean $E\left( {\frac{{\alpha _2 }}{{f_s }}} \right)$, which is frequency-dependent and used in Eq. (29), can be calculated by

$$E\left( {\frac{{\alpha _2 }}{{f_s }}} \right) = \alpha _2 E\left( {\frac{1}{{f_s }}} \right) = \frac{{\alpha _2 }}{{E\left( {f_s } \right)}}\left( {1 + \frac{{\left( {\sigma \left( {f_s } \right)} \right)^2 }}{{\left( {E\left( {f_s } \right)} \right)^2 }}} \right),\quad \forall s \in S_1 $$

(32)

where the mean E(f _s ) and variance $\left( {\sigma \left( {f_s } \right)} \right)^2$ of f _s can be given by, respectively,

$$E\left( {f_s } \right) = \sum\limits_{l \in A_s } {E\left( {f_l } \right)} ,\quad \forall s \in S_1 $$

(33)

$$\left( {\sigma \left( {f_s } \right)} \right)^2 = \sum\limits_{l \in A_s } {\left( {\sigma \left( {f_l } \right)} \right)} ^2 ,\quad \forall s \in S_1 $$

(34)

where E(f _l ) and σ(f _l ) can be determined according to Appendix B.

1.2.4 Measure of the unreliability of transit services

As stated in Eq. (17), the unreliability cost of transit services can be measured by the function f(σ _s ) of the standard deviation σ _s . In this paper, for simplicity, we define

$$f\left( {\sigma _s } \right) = \lambda _1 \sigma _s ,\quad \forall s \in S_1 $$

(35)

where ₁ measures the degree of the unreliability of transit services. The larger the value of ₁, the less reliable the transit services, and vice versa.

Similar to Eqs. (18) or (19), the variance $\sigma _s^2$ of the travel time on transit link s is the sum of the variances of the in-vehicle link travel time and waiting time on link s, i.e.,

$$\sigma _s^2 = \left( {\tau _1 \sigma _{s1} } \right)^2 + \left( {\tau _2 \sigma _{s2} } \right)^2 ,\quad \forall s \in S_1 $$

(36)

where σ _{s 1} and σ _{s 2} are the standard deviations of the in-vehicle travel time and waiting time on link s, respectively. They can be determined according to the relationship between mean and variance, as shown in A4 in Section 2.2.

1.3 Walking disutility function

The walking times for access to or egress from transit stops are often assumed to be flow independent in the previous literature (Wu et al. 1994). However, the empirical study of Lam et al. (2003) showed that on a bi-directional walkway with heavy opposing pedestrian flows, both the capacity of the walkway and the pedestrian walking speeds can be reduced significantly, particularly in the minor flow direction.

On the basis of their empirical studies, Lam et al. (2003) proposed a generalized walking time function to account for the bi-directional flow effects on the walkways under different flow conditions, ranging from free-flow to congested situations. Following Lam et al. (2003), the (expected) generalized walking time function that is used in the proposed model is

$$t_{s^{ + \left( - \right)} } \left( {v_{s^ + } ,v_{s^ - } } \right) = t_s^0 + B\left( {{{v_{s^{ + \left( - \right)} } } \mathord{\left/ {\vphantom {{v_{s^{ + \left( - \right)} } } {\left( {v_{s^ + } + v_{s^ - } } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {v_{s^ + } + v_{s^ - } } \right)}}} \right)^x \left( {{{\left( {v_{s^ + } + v_{s^ - } } \right)} \mathord{\left/ {\vphantom {{\left( {v_{s^ + } + v_{s^ - } } \right)} {C_s }}} \right. \kern-\nulldelimiterspace} {C_s }}} \right)^y ,{\text{ }}\forall s \in S_2 $$

(37)

where s ⁺ and s ⁻ are two walking links that represent the physical walkway s, $t_{s^{ + \left( - \right)} }$ is a unit of the walking time in direction + (−) on walkway s with bi-directional flows, $t_s^0$ is the free-flow walking time on walkway s, and C _s is the capacity of the physical walkway s under unidirectional flow conditions. B, x, and y are the parameters to be calibrated with observed data.

Similar to Eq. (35), we can define the uncertainty cost that is caused by the fluctuation of walking time on the congested walkways as below.

$$f\left( {\sigma _s } \right) = \lambda _2 \sigma _s ,\quad \forall s \in S_2 $$

(38)

where ₂ measures the degree of the uncertainty of the walking time. Again, the standard deviation σ _s of the walking time on walking link s can be determined according to the relationship between mean and variance, as shown in A4 in Section 2.2.

Consequently, the walking disutility for access to or egress from transit stops can be formulated as the sum of the walking time and the uncertainty cost that is caused by the fluctuation of the walking time, i.e.,

$$u_s = t_s + \frac{{\beta _2 }}{{\alpha _1 }}f\left( {\sigma _s } \right),\quad \forall s \in S_2 $$

(39)

where the walking time t _s on link s can be calculated by Eq. (37).

1.4 Appendix B. Random transit line frequency

As stated in A2 in Section 2.2, the variations of the in-vehicle travel time and the dwelling time of transit vehicles at stops can cause variability of the transit service frequency. Hence, the line frequency f _l is a random variable and its mean and variance are derived as follows.

Let N _l be the number of vehicles on line lεL, and Γ _l (v) be the cycle journey time of a transit vehicle on line lεL, and then the line frequency f _l can be obtained by

$$f_l = \frac{{N_l }}{{\Gamma _l \left( {\mathbf{v}} \right)}},\quad \forall l \in L$$

(40)

where v is the vector of passenger flows in the transit network.

The cycle journey time Γ _l (v) of a transit vehicle on line lεL is composed of the line-haul travel time, terminal time, and dwelling delays at transit stops (Fernandez and Marcotte 1992; Lam et al. 2002). The uncertainties of the line-haul travel time and the dwelling time of transit vehicles at stops would lead to the variation of the cycle journey time. Hence, Γ _l (v) is a random variable, and its mean E(Γ _l (v)) and variance (σ(Γ _l (v)))² can be calculated by, respectively,

$$E\left( {\Gamma _l \left( {\mathbf{v}} \right)} \right) = \zeta t_l^0 + \sum\limits_{m \in l} {t_l^m } + \sum\limits_{n \in l} {\bar d\bar t_l^n \left( {\mathbf{v}} \right)} ,\quad \forall l \in L$$

(41)

$$\left( {\sigma \left( {\Gamma _l \left( {\mathbf{v}} \right)} \right)} \right)^2 = \sum\limits_{m \in l} {\left( {\sigma _l^m } \right)} ^2 + \sum\limits_{n \in l} {\left( {\sigma \left( {dt_l^n \left( {\mathbf{v}} \right)} \right)} \right)} ^2 ,\quad \forall l \in L$$

(42)

where $t_l^0$ is the constant terminal time on line l and ζ is the number of terminal times on the circular line. mεl and nεl imply that line segment m and transfer node n lie on transit line l, respectively. $t_l^m$ and $\sigma _l^m$ are the mean and standard deviation of the travel time on line segment m on transit line l, respectively. $\bar d\bar t_l^n \left( {\mathbf{v}} \right)$ and $\sigma \left( {dt_l^n \left( {\mathbf{v}} \right)} \right)$ are the mean and standard deviation of the dwelling time $dt_l^n \left( {\mathbf{v}} \right)$ at node n on line l, respectively.

According to Lam et al. (1998), the transit vehicle dwelling time at a transit stop is governed by the number of boarding and alighting passengers, i.e., the total interchanging passenger volumes. The expected dwelling time can be expressed as a function with regard to the boarding and alighting volumes (Yin et al. 2004),

$$\bar d\bar t_l^n \left( {\mathbf{v}} \right) = \max \left( {\bar d\bar t_{l0}^n ,\eta _0 + \eta _1 Bo_l^n + \eta _2 Al_l^n } \right),{\text{ }}\forall l \in L$$

(43)

where $\bar d\bar t_{l0}^n$ is the minimal (scheduled) dwelling time of line l at stop n. $Bo_l^n$ and $Al_l^n$ are, respectively, the number of passengers boarding and the number of passengers alighting line l at stop n, and they can be determined by the method that is outlined in the study of Lam et al. (2002). The coefficients (η) are the positive parameters, which can be calibrated by the observed data (Lam et al. 1998, 2002).

From (40), for a given value of N _l , the mean E(f _l ) and variance (σ(f _l ))² of the line frequency f _l can be given by, respectively,

$$E\left( {f_l } \right) = E\left( {\frac{{N_l }}{{\Gamma _l \left( {\mathbf{v}} \right)}}} \right) = N_l E\left( {\frac{1}{{\Gamma _l \left( {\mathbf{v}} \right)}}} \right),\quad \forall l \in L$$

(44)

$$\left( {\sigma \left( {f_l } \right)} \right)^2 = \left( {\sigma \left( {\frac{{N_l }}{{\Gamma _l \left( {\mathbf{v}} \right)}}} \right)} \right)^2 = N_l^2 \left( {\sigma \left( {\frac{1}{{\Gamma _l \left( {\mathbf{v}} \right)}}} \right)} \right)^2 {\text{,}}\quad \forall l \in L$$

(45)

Applying a quadratic Taylor series approximation to Eqs. (44) and (45), we then have

Proposition B.1

The mean and variance of the transit line frequency f _l can be calculated by, respectively,

$$E\left( {f_l } \right) = \frac{{N_l }}{{E\left( {\Gamma _l \left( {\mathbf{v}} \right)} \right)}}\left( {1 + \frac{{\left( {\sigma \left( {\Gamma _l \left( {\mathbf{v}} \right)} \right)} \right)^2 }}{{\left( {E\left( {\Gamma _l \left( {\mathbf{v}} \right)} \right)} \right)^2 }}} \right),\quad \forall l \in L$$

(46)

$$\left( {\sigma \left( {f_l } \right)} \right)^2 = N_l^2 \frac{{\left( {\sigma \left( {\Gamma _l \left( {\mathbf{v}} \right)} \right)} \right)^2 }}{{\left( {E\left( {\Gamma _l \left( {\mathbf{v}} \right)} \right)} \right)^4 }},\quad \forall l \in L$$

(47)

where E(Γ _l (v)) and σ(Γ_l(v)) can be calculated by Eqs. (41) and (42), respectively.

Proof

For the purpose of presentation, let $\frac{1}{X} = \frac{1}{{\Gamma _l \left( {\mathbf{v}} \right)}}$, and then a quadratic Taylor series approximation of $\frac{1}{X}$ around X ₀ = E(X) can be represented as

$$\frac{1}{X} = \frac{1}{{X_0 }} - \frac{1}{{1!X_0^2 }}\left( {X - X_0 } \right) + \frac{2}{{2!X_0^3 }}\left( {X - X_0 } \right)^2 + \cdots$$

(48)

Taking the expectation on both sides of Eq. (48) and ignoring higher order terms yield

$$E\left( {\frac{1}{X}} \right) = \frac{1}{{X_0 }} - \frac{1}{{X_0^2 }}E\left( {X - X_0 } \right) + \frac{1}{{X_0^3 }}E\left( {\left( {X - X_0 } \right)^2 } \right)$$

(49)

Because $E\left( {X - X_0 } \right) = 0$ and $E\left( {\left( {X - X_0 } \right)^2 } \right) = \left( {\sigma \left( X \right)} \right)^2$, Eq. (49) can be written as

$$E\left( {\frac{1}{X}} \right) = \frac{1}{{E\left( X \right)}} + \frac{{\left( {\sigma \left( X \right)} \right)^2 }}{{E\left( X \right)^3 }} = \frac{1}{{E\left( X \right)}}\left( {1 + \frac{{\left( {\sigma \left( X \right)} \right)^2 }}{{\left( {E\left( X \right)} \right)^2 }}} \right)$$

(50)

Taking the variance on both sides of Eq. (48) and ignoring higher order terms, we obtain

$$\left( {\sigma \left( {\frac{1}{X}} \right)} \right)^2 = \frac{{\left( {\sigma \left( X \right)} \right)^2 }}{{\left( {E\left( X \right)} \right)^4 }}$$

(51)

Substituting $\frac{1}{X} = \frac{1}{{\Gamma _l \left( v \right)}}$ into Eqs. (50) and (51), Eqs. (44) and (45) then become (46) and (47), respectively. This completes the proof of Proposition B.1.