A note on the time travel approach for handling time windows in vehicle routing problems

Nagata et al. (2010) introduce a new, highly efficient approach to calculate time window violations, which is based on the concept of time travel. To put it short, if a time window violation occurs at a certain customer i, a penalty of the actual arrival time minus the latest feasible arrival time is assigned to the route. However, the calculation of violations for the following customers is executed as if a travel back in time to the latest feasible arrival time at customer i had taken place. In this way, none of the subsequent customer visits are affected by the actual delay at customer i. This approach has two important advantages compared to the traditional approach of propagating time window violations along the route. First, it allows to evaluate potential time window violations for inter-route moves in constant time O(1) for most of the classical neighborhood operators like Relocate, Exchange, Or-Opt and 2-opt*. This significantly speeds up the calculation compared to the traditional approach that needs O(n) time. Second, the approach penalizes time window violations only at customers where they originate and feasible parts of routes are no longer penalized by earlier time window violations. To keep this note short, we stick exactly to the notations and definitions used in the original paper and forgo unnecessary reintroductions. To compute time window violations, Proposition 1 (Nagata et al., 2010; p. 736) states the following: If a route r = 〈0, . . . , x, v, y, . . . , 0〉 is constructed by inserting a node v in between two partial paths 〈0, . . . , x〉 and 〈y, . . . , 0〉, the time window violation of the route is: TW(r) = TWx + TW ← y + max (ãx + sx + cxv − (z̃y − cvy − sv), 0). (1)