Minimal time to ride all ski slopes 
Suppose we want to know what the minimum time is to ride all ski slopes on a mountain. We know the time it takes to ride a slope, and we know the time it takes to take a ski lift to get from one ski station to another, given that we have to end up where we started. 

This screems Minimum-Cost-Flow-Problem to me. Thus my first idea was to construct a directed graph, whose vertices represent the ski stations and the edges represent the slopes and lifts. Let the cost of an edge equal the time it takes to ride that slope/lift. Now I want to find the minimal time it takes to ride every slope (note: not necessarily every lift), but I can't seem to find a way to ensure that we have to ride every slope atleast once.
Is there a chance to do it this way or am I completely wrong with my attempt?

As far as I can tell, your problem is a lightly disguised version of Traveling Salesman (note that since you have to ski all the slopes, for algorithmic purposes the skiing time is irrelevant, all that matters is the time to travel between slopes.)
