PCVRP with prizes reduced over time
This has been called "Discounted Reward." In my brutish opinion, steer toward the CS guys and away from the OR guys. I just like the lit better.

See: "Approximation Algorithms for Orienteering and Discounted-Reward TSP." FOCS 2003 Which I just stole from my colleague's desk. 

They model reward as $\lambda^{-t}$, where $t$ is time, and $\lambda$ is constant, but otherwise it is the standard orienteering formulation: Maximize reward, subject to a budget, or minimize budget subject to a reward constraint.