  In this paper, we present formulations and an exact method to solve the Time
Dependent Traveling Salesman Problem with Time Window (TD-TSPTW) under a
generic travel cost function where waiting is allowed. A particular case in
which the travel cost is a non-decreasing function has been addressed recently.
With that assumption, because of both the First-In-First-Out property of the
travel time function and the non-decreasing property of the travel cost
function, we can ignore the possibility of waiting. However, for generic travel
cost functions, waiting after visiting some locations can be part of optimal
solutions. To handle the general case, we introduce new lower-bound
formulations that allow us to ensure the existence of optimal solutions. We
adapt the existing algorithm for TD-TSPTW with non-decreasing travel costs to
solve the TD-TSPTW with generic travel costs. In the experiment, we evaluate
the strength of the proposed lower bound formulations and algorithm by applying
them to solve the TD-TSPTW with the total travel time objective. The results
indicate that the proposed algorithm is competitive with and even outperforms
the state-of-art solver in various benchmark instances.
