Optimal sectional fare and frequency settings for transit networks with elastic demand

Document Type

Journal Article

Publication Date


Subject Area

economics - pricing, economics - profitability, place - asia, operations - frequency, planning - methods, planning - network design, ridership - elasticity


Fare optimization, Frequency setting, Transit assignment, Bilevel optimization, Transit network design, Sensitivity-based heuristic


Sectional fares have been used in transit services in practice but are rarely examined analytically and compared with flat and distance-based fares, especially under the considerations of path choice, elastic demand, service frequency, and profitability. This paper proposes a bilevel programming model to jointly determine the fare and frequency setting to maximize transit operator's profit. The preceding three fare structures can be incorporated into the bilevel model. To consider the path choice and elastic demand in the bilevel model, the existing approach-based stochastic user equilibrium transit assignment model for the fixed demand was extended to the elastic demand case and the resultant model was used in the lower level model. To solve the bilevel model, the sensitivity-based descent search method that takes into account the approach-based formulation for the elastic demand transit assignment is proposed, in which the approach-based formulation was solved by the cost-averaging self-regulated averaging method. Numerical studies and mathematical analyses were performed to examine the model properties and compare the three fare structures. The result of the Tin Shui Wai network instance is also provided to illustrate the performance of the solution method.

It is proven that when all passengers’ destinations are located at transit terminals, the sectional fare structure is always better than the other two fare structures in terms of profitability. For more general networks, the sectional fare structure is always better than the flat fare structure, but the choice between sectional and distance-based fare structures depends on the geometry of the network (e.g., the route structure and the distance between stops), the demand distribution, and the maximum allowable fares. It is also proven that the optimal profit (total vehicle mileage) is strictly monotonically decreasing (monotonically decreasing) with respect to the unit operating cost. Moreover, it is proven that the lower level approach-based assignment problem with elastic demand has exactly one solution. However, the bi-level problem can have multiple optimal solutions. Interestingly, it is found that from the operator's profitability point of view, providing better information to the passengers may not be good.


Permission to publish the abstract has been given by Elsevier, copyright remains with them.


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