Facility and covering location models are key elements in many decision aid tools in logistics, supply chain design, telecommunications, public infrastructure planning, and many other industrial and public sectors. In many applications, it is likely that customers are not dichotomously covered by facilities, but gradually covered according to, e.g., the distance to the open facilities. Moreover, customers are not served by a single facility, but by a collection of them, which jointly serve them. In this paper we study the recently introduced multiple gradual cover location problem (MGCLP). The MGCLP addresses both of the issues described above. We provide four different mixed-integer programming formulations for the MGCLP, all of them exploiting the submodularity of the objective function and developed a branch-and-cut framework based one these formulations. The framework is further enhanced by starting and primal heuristics and initialization procedures. The computational results show that our approach allows to effectively address different sets of instances. We provide optimal solution values for 13 instances from literature, where the optimal solution was not known, and additionally provide improved solution values for seven instances. Many of these instances can be solved within a minute. We also analyze the dependence of the solution-structure on instance-characteristics.