Optimizing fleet size for on-demand taxi and ridehailing services: application to the case of Madrid

Digitization and new business models are transforming urban mobility, particularly within the on-demand passenger transport sector, which includes taxi and ridehailing services. While these innovations benefit consumers, planners still face the challenge of determining the appropriate number, spatial distribution, and temporal deployment of vehicles to ensure service quality while mitigating negative externalities. Thhis study develops and employs a methodology for determining the optimal fleet size for taxi and ridehailing services, considering factors such as maximum passenger waiting times, vehicle depreciation costs, passengers' value of time, drivers' fees, and dynamic demand fluctuations. It also introduces a generalized, optimization-based framework for service-level-constrained fleet sizing and repositioning, incorporating an Erlang-C waiting-time term and inter-hour vehicle-flow continuity. This approach was applied to the case of Madrid (Spain) using two datasets from 2023: (i) zone-to-zone flows to capture inter-zonal dynamics; and (ii) trip request data to characterize temporal demand patterns. The analysis shows that 46 % of the taxi and ridehailing demand remains unmet in Madrid, ranging from 30 % on Mondays, Tuesdays, and Wednesdays, to 45 % on Thursdays and Fridays, and up to 60 % on Saturdays and Sundays. Increasing the number of vehicles per hour by 25–30 % would cover 87–89 % of the demand, while a 50–75 % increase would ensure near-complete coverage, thereby reducing almost all unmet demand, particularly during peak hours. These adjustments would enhance user satisfaction while ensuring operators’ profits. A detailed case study of ten origin–destination routes from metropolitan neighborhoods to Central Madrid during the late-night peak (23:00–01:00) on a typical Saturday reveals a pronounced midnight (00:00) peak in vehicles required per hour across all routes. The ≤10-min expected-wait target can be met with modest, largely symmetric repositioning (≈1–2 vehicles per route and per hour) to smooth short-term imbalances. Cost decomposition shows that operating and wage/depreciation costs dominate, while passenger waiting costs remain low. Sensitivity analysis further highlights that tightening service targets increases peak staffed vehicles per hour and total costs, whereas relaxing targets off-peak lowers costs with limited impact on reliability. Improvements in service rate and delivery speed provide comparable or better service at a lower cost than expanding vehicle fleets. Finally, scaling late-night demand by −20 %, +20 %, and +40 % confirms near-proportional hourly capacity needs for modest changes and steep nonlinear cost growth under larger increases, supporting temporary, time-bound vehicle-per-hour authorizations and preplanned outbound staging during predictable peaks.