The impact of a financial constraint on the spatial structure of public transport services

Abstract

Using a single line model, it has been shown recently that the presence of a stringent financial constraint induces a less than optimal bus frequency and larger than optimal bus size. This occurs because the constraint induces a reduction of the importance of users’ costs (their time); in the extreme, users’ costs disappear from the design problem. In this paper we show that such a constraint also has an impact on the spatial structure of transit lines. This is done departing from the single line model using an illustrative urban network that could be served either with direct services (no transfers) or with corridors (transfers are needed). First, the optimal structure of lines is investigated along with frequencies and vehicle sizes when the full costs for users and operators are minimized (unconstrained case); the optimal lines structure is shown to depend upon the patronage level, the values of time and the cost of providing bus capacity. Then the same problem is solved for the extreme case of a stringent financial constraint, in which case users’ costs have relatively little or no effect in determining the solution; in this case the preferred outcome would be direct services under all circumstances, with lower frequencies and larger bus sizes. The impact of the financial constraint on the spatial structure of transit lines is shown to be caused by the reduction in cycle time under direct services; the introduction of users’ costs in the objective function makes waiting times reverse this result under some circumstances.

Appendices

Appendix 1. Numerical comparison of total cost from Eq. (19)

See Tables 2 and 3

Table 2 Values of the parameters used in the numerical evaluation

Table 3 Numerical comparison of total cost without and with approximation

Total cost is equal for direct and corridor structures for Y = 6,536 passengers per hour. The third term of Eq. (19) is equal for both structures so it cancels out. The first term is never larger than 3.1 % of the second term for both structures. When only this latter is used, the difference in cost becomes nil for Y = 7,439 pax/h, 13.8 % larger than the exact value.

Appendix 2. Optimal waiting and in-vehicle times

Replacing optimal frequency from Eq. (15) into the expressions for the waiting time (10) and in-vehicle time (14) yields.

$$ t_{w}^{ * } = \frac{{\varepsilon \left( {N + 2 + 4\tau } \right)\sqrt {c_{0} T_{0} } }}{{\sqrt {t\left[ {3c_{1} \left( {1 + \tau } \right) + P_{v} \left( {\frac{9}{8} + \frac{\tau }{2}(2N - 1)(1 - 2\tau )} \right)} \right] + \frac{{P_{w} \varepsilon }}{Y}\left( {N + 2 + 4\tau } \right)} }} $$

(A2.1)

$$ t_{v}^{ * } = \frac{3}{4}T_{0} Y + \frac{{tY\left[ {\frac{9}{8} + \frac{\tau }{2}\left( {2N - 1} \right)\left( {1 - 2\tau } \right)} \right]\sqrt {c_{0} T_{0} } }}{{\sqrt {t\left[ {3c_{1} \left( {1 + \tau } \right) + P_{v} \left( {\frac{9}{8} + \frac{\tau }{2}\left( {2N - 1} \right)\left( {1 - 2\tau } \right)} \right)} \right] + \frac{{P_{w} \varepsilon }}{Y}\left[ {N + 2 + 4\tau )} \right]} }} $$

(A2.2)

The waiting times for each structure are obtained replacing (N, τ) by (2, 1/4) for corridors and (4, 0) for direct lines. This yield that total waiting time for corridors is lower than for direct lines when

$$ 75\,tc_{1} + 28.125\,tP_{v} + 150\frac{{P_{w} \varepsilon }}{Y} < 135\,tc_{1} + 47.25\,tP_{v} + 180\frac{{P_{w} \varepsilon }}{Y} $$

(A2.3)

which is always true. Analogously, in-vehicle time for corridors is larger than for direct lines when (A2.4) is valid, which is always true.

$$ 147\,tc_{1} + 55.125\,tP_{v} + 294\frac{{P_{w} \varepsilon }}{Y} > 135\,tc_{1} + 47.25\,tP_{v} + 180\frac{{P_{w} \varepsilon }}{Y} $$

(A2.4)

Appendix 3

See Table 4

Table 4 Differences in total cost components as a function of Y