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

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s11067-007-9058-z

- Cite this article as:
- Li, ZC., Lam, W.H.K. & Wong, S.C. Netw Spat Econ (2009) 9: 191. doi:10.1007/s11067-007-9058-z

## 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.

### Keywords

Transit fare Market regime Network equilibrium Reliability Monopoly Social optimum Oligopolistic competition### Notation

*G*modified transit network with

*G*= (*N*,*S*)*N*set of nodes representing centroids and transit stops, in which passengers can board, alight or change vehicles

*S*set of links in the transit network

*G*;*S*=*S*_{1}∪*S*_{2}*S*_{1}set of transit links which connect two transit stops

*S*_{2}set of walking links including the access links from origin to transit stops or the egress links from alighting points to destination

*W*set of network origin-destination (OD) pairs,

*wεW**R*_{w}set of paths connecting OD pair

*wεW*in the transit network- \(P^{w}_{r} \)
probability that path

*r*is chosen for a trip between OD pair*wεW**θ*parameter representing the perception variation of passengers on travel disutility

*u*_{r}expected travel disutility on path

*r**u*_{s}expected travel disutility on link

*s**δ*_{sr}indicator variable; it equals to 1 if link

*s*is on path*r*, and 0 otherwise*h*_{r}passenger flow on path

*r**v*_{s}passenger flow on link

*s**S*_{w}expected minimum disutility between OD pair

*wεW**g*_{w}total resultant passenger demand between OD pair

*wεW*;*g*_{w}=*G*_{w}(*S*_{w})- \(g^{0}_{w} \)
potential (or latent) passenger demand between OD pair

*wεW**π*_{w}parameter of demand sensitivity to travel disutility between OD pair

*wεW**L*set of transit lines in the transit network

- \(p^{l}_{s} \)
fare of line

*l*on link*s**N*_{l}number of vehicles or fleet size on line

*l*- \(C^{0}_{l} \)
fixed operating cost of line

*l*- \(C^{1}_{l} \)
operating cost per vehicle-hour on line

*l**K*set of transit operators in the transit network

*L*_{k}set of transit lines operated by operator

*k**Ф*_{k}profit of transit operator

*k**A*_{s}set of attractive lines on link

*s**T*_{s}actual travel time on link

*s*; a random variable with mean*t*_{s}[i.e.*t*_{s}=*E*(*T*_{s})] and standard deviation σ_{s}*T*_{s1}actual in-vehicle travel time on link

*s*; a random variable with mean*t*_{s1}[i.e.*t*_{s1}=*E*(*T*_{s1})] and standard deviation σ_{s1}*T*_{s2}actual waiting time on link

*s*; a random variable with mean*t*_{s2}[i.e.*t*_{s2}=*E*(*T*_{s2})] and standard deviation σ_{s2}*g*_{s}in-vehicle crowding discomfort cost on transit link

*s**f*(*σ*_{s})unreliability cost of transit services on link

*s*; a function of the standard deviation σ_{s}of the travel time on link*s**τ*_{1},*τ*_{2}parameters for converting the different quantities to the same unit

*ρ*_{s},*σ*_{s}parameters for measuring the relationship between mean and variance of travel time

- \(t^{0}_{s} \)
free-flow travel time on link

*s*- \(t^{l}_{s} \)
mean in-vehicle travel time of line

*l*passing through link*s*- \(x^{l}_{s} \)
probability of passengers on link

*s*choosing line*l**f*_{l}frequency of line

*l*; a random variable with mean*E*(*f*_{l}) and standard deviation σ(*f*_{l})*f*_{s}total frequency on link

*s*; a random variable with mean*E*(*f*_{s}) and standard deviation σ(*f*_{s}); \(f_{s} = {\sum\nolimits_{l \in A_{s} } {f_{l} } }\)- \(g^{l}_{s} \)
in-vehicle discomfort cost of line

*l*passing through link*s*- \(g^{{l0}}_{s} \)
baseline discomfort level or riding quality of line

*l*passing through link*s*- \(v^{l}_{s} \)
passenger flow of line

*l*passing through link*s*- \(\overline{v} ^{l}_{s} \)
passenger flow competing with \(v^{l}_{s} \) for the same common capacity of line

*l*on link*s*- \(\overline{v} _{s} \)
passenger flow competing with

*v*_{s}for the same common capacity on link*s*- κ
_{l} capacity of transit vehicle on line

*l**K*_{l}capacity of line

*l*;*K*_{l}= κ_{l}*f*_{l}*K*_{s}total vehicle capacity on link

*s*; \(K_{s} = {\sum\nolimits_{l \in A_{s} } {K_{l} } }\)*i*(*s*)tail node of link

*s*- \(A^{{l + }}_{{i{\left( s \right)}}} \)
set of links going out from node

*i*(*s*) on which line*l*is included as an attractive line but link*s*is excluded- \(\overline{A} ^{l}_{{i{\left( s \right)}}} \)
set of links on which line

*l*is included as an attractive line, with origin node before*i*(*s*) and end node after*i*(*s*)- λ
_{1} parameter for measuring the degree of unreliability of transit services

- \(t_{s^{ + \left( - \right)} } \)
walking time in direction + (−) on walkway

*s*with bi-directional flows*C*_{s}capacity of physical walkway

*s*under unidirectional flow conditions*Γ*_{l}()**v**cycle journey time of a transit vehicle on line

*lεL*; a random variable with mean*E*(Γ_{l}(**v**)) and standard deviation σ(Γ_{l}(**v**))- \(dt_l^n \left( v \right)\)
dwelling time for the transit vehicle at node

*n*on line*l*; a random variable with mean \(\overline d t_l^n \left( v \right)\) and standard deviation \(\sigma \left( {dt_l^n \left( v \right)} \right)\)**h**vector of path passenger flow; \(h = \left( {h_r ,r \in R_w ,w \in W} \right)\)

**v**vector of link passenger flow; \(v = \left( {v_s ,s \in S} \right)\)

**p**vector of transit fare; \(p = \left( {p_s^l ,s \in S_1 ,l \in L} \right)\)

**g**vector of OD demand; \(g = \left( {g_w ,w \in W} \right)\)