Time-dependent transit fare optimization with elastic and spatially distributed demand

Document Type

Journal Article

Publication Date


Subject Area

ridership - demand, ridership - elasticity, policy - fares, operations - capacity, infrastructure - fleet management, economics - pricing, economics - subsidy


Public transit pricingTime-dependent fareSocial welfare maximizationSpatial and temporal distribution of demand


Motivated by the lack of microeconomic models that optimize time-dependent transit fares based on realistic demand formulations, this paper presents a microeconomic model for the design of a time-dependent transit pricing scheme considering elastic and spatiotemporally distributed demand. To model the spatial distribution of demand, a transit line with multiple origin–destination pairs is considered. To model the cyclical demand fluctuations, transit operations in one day are divided into multiple time periods. In the proposed model we optimize fares, headway, vehicle capacity, and maximum fleet size, with the objective of maximizing social welfare, subject to fleet size and vehicle capacity constraints. We find time-dependent pricing could avoid cross-subsidization among travelers in different time periods. Under both pricing schemes, the time-dependent headways satisfy the same optimality condition: the total rider waiting cost equals the total fixed cost on the supplier side. We also demonstrate that both resource constraints (vehicle capacity and fleet size) can be binding in multiple time periods, unlike the usual assumption in the literature that resource constraints are binding only in the period with the highest demand. Two extensions (considering a financial constraint and a variable roundtrip time) are also investigated. The developed models can be used to facilitate the design of time-dependent pricing schemes for practical applications.


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


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