In this study, the discrete p-center problem with the presence of multilevel capacities and fixed (opening) cost of a facility under a limited budget is investigated. A mathematical model of the problem is produced, where we seek the location of open facilities, their corresponding capacities, and the allocation of the customers to the open facilities in order to minimize the maximum distance between customers and their as- signed facilities. Two matheuristic approaches are also proposed to deal with larger instances. The first approach is a hybridisation of a clustering-based technique, an ex- act method, while the second one is based on Variable Neighborhood Search (VNS). Computational experiments show that the proposed methods produce interesting and competitive results on newly and randomly generated datasets.