The solver of my Network Analyst extension is able to properly consider Historical Traffic Speeds Data (that I gave as an input in the N.D) when using the "ROUTE" option of the extension. So a route has different travel time depending on the time of the day;
The problem is that when I use the " VEHICLE ROUTING PROBLEM" option these Historical Speeds data are not considered and the network dataset is the same. A set of route during the night has the same travel time of a set during the day. I could not find the reason.
VRP solver does not use historical traffic to optimize the route, instead, it always uses the time-neutral time information for optimization and reports the time-neutral output in its NA classes.
Once VRP solve finishes, you can click on "Directions" button to get the travel time with historical traffic. It will feed the order assignments and route start time from VRP results to the Route solver and populate the travel time information based on historical traffic.