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In this paper, we present the first Assignment-Dial-A-Ride problem motivated by a real-life problem faced by medico-social institutions in France. Every day, disabled people use ride-sharing services to go to an appropriate institution where they receive personal care. These institutions have to manage their staff to meet the demands of the people they receive. They have to solve three interconnected problems: the routing for the ride-sharing services; the assignment of disabled people to institutions; and the staff size in the institutions. We formulate a general Assignment-Dial-A-Ride problem to solve all three at the same time. We first present a matheuristic that iteratively generates routes using a large neighborhood search in which these routes are selected with a mixed integer linear program. After being validated on two special cases in the literature, the matheuristic is applied to real instances in three different areas in France. Several managerial results are derived. In particular, it is found that the amount of cost reduction induced by the people assignment is equivalent to the amount of cost reduction induced by the sharing of vehicles between institutions., (© 2023. The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.)