Traveling Salesman Tardiness

How fragile is the routing time window of delivery systems against spatial distributional uncertainty?

We study the tardiness risk of Traveling Salesman Problem (TSP) solutions with respect to a service deadline (target) over the routing time.

Using the robust satisficing model, we introduce the TSP tardiness index to quantify the target's fragility under distributional uncertainty in customer locations.

Assuming there are m potential customer locations from historical samples on a service region D (of area |D|), we prove that the TSP tardiness index is {\Theta}(n * sqrt(|D|m) / {\tau}) for n realized locations with respect to the routing time target {\tau}, under non-boundary conditions.

This result establishes a new scaling law that extends beyond the existing deterministic and probabilistic TSP bounds.

We further extend it to a multi-vehicle case and derive simple partition rules for managing delivery systems.

Our numerical experiments using synthetic and real-world routing data validate the value of the TSP tardiness index in characterizing and managing the overtime risk of routing systems.