We call an optimization problem an optimization problem with controllable uncertainty if a) it contains uncertain input data and b) prior to deciding the optimization variables, the optimizer can for a certain cost reduce the scenario set of this uncertain data. In particular, we are interested in situations where each uncertain parameter is a priori known to lie in a closed, bounded interval and the optimizer has a fractional choice to shrink each interval. We study this setting for optimization problems with uncertain non-negative linear cost functions and non-negative decision variables. Moreover, we assume the scenario set to be restricted by so-called budgeted uncertainty, i.e. the sum over all intervals of relative deviations from the lower boundary cannot exceed a given budget. In particular, we show that the resulting three-stage problem can be solved efficiently as a single linear program, if the underlying optimization problem is a linear program given by a TDI description.