In this paper, we present exact methods for solving the Time Dependent Minimum Duration Problem (TDMTDP) and the Time Dependent Delivery Man Problem (TD-DMP). Both methods are based on a Dynamic Discretization Discovery (DDD) approach for solving the Time Dependent Traveling Salesman Problem with Time Windows (TD-TSPTW). Unlike the TD-TSPTW, the problems we consider in this paper involve objective function values that depend in part on the time at which the vehicle departs the depot. As such, optimizing these problems adds a scheduling dimension to the problem. We present multiple enhancements to the DDD method, including enabling it to dynamically determine which waiting opportunities at the depot to model. With an extensive computational study we demonstrate that the resulting methods outperform all known methods for both the TD-MTDP and TD-DMP on instances taken from the literature