This work proposes exact solution approaches for the m-machine robust permutation flow shop problem, where operation processing times are uncertain and vary in a given interval. Following the concept of budgeted uncertainty, the objective is to obtain a robust scheduling that minimizes the total weighted completion time of the restricted worst-case scenario, in which only a subset of operation processing times will deviate to worst-case values. We develop several robust counterpart formulations, which can be used to derive optimal solutions for medium-sized problem instances by using a Column-and-Constraint Generation algorithm. The efficacy of the solution methods is validated through experiments on three sets of randomly-generated instances. Finally, a case study for the maintenance schedule of Brazilian oil platforms is presented.