This is a bit general, but is there a name to this type of problem? It looks like a directed graph traversal problem, but you have multiple paths going on, and timing may be important.
You operate a public library system with multiple branches. Every day, people check out and return materials. Library users can return materials at any branch. Additionally, library users can place a hold on an item and ask to pick it up at any branch. Essentially, on any given day you have items that need to be delivered between library branches.
You have hired a set number of drivers to move materials between locations. Assume that the transit time between branches is known and time-invariant, the carrying capacity of the vehicles is unlimited, drivers are on for a set amount of time each day, and the drivers can start and end at any library branch. Given a list of what must be moved, is there an optimal set of delivery routes that minimizes driving time?