The Vehicle Routing Problem with Time Windows: Minimizing Route Duration

We investigate the implementation of edge-exchange improvement methods for the vehicle routing problem with time windows with minimization of route duration as the objective. The presence of time windows as well as the chosen objective cause verification of the feasibility and profitability of a single edge-exchange to require an amount of computing time that is linear in the number of vertices. We show how this effort can, on the average, be reduced to aconstant. I. Introduction Over the past ten years, operations researchers interested in vehicle routing and scheduling have emphasized the development of algorithms for real-life problems. The size of the problems solved has increased and practical side constraints are no longer ignored. One such constraint is the specification of time windows at customers, i.e., time intervals during which they must be served. These lead to mixed routing and scheduling problems. Obviously, the introduction of time windows at customers makes it harder to construct and maintain a feasible set of routes. Savelsbergh [1986] shows that in the special case of a single vehicle, Le., the traveling salesman problem with time windows, constructing a feasible route is already NP-hard. As a result, most of the research in this area has been directed towards controling the feasibility of a set of routes. However, the introduction oftime windows at customers also allows the specification of more realistic objective functions, compared to minimizing distance, such as minimizing waiting time, minimizing completion time. and minimizing route duration. Edge-exchange improvement methods form an important and popular class of algorithms in the context of vehicle routing problems. Recently a number of papers have been published that study efficient implementations of edge-exchange improvement methods for the vehicle routing problem with time windows, such as Savelsbergh [1986, 1990] and Solomon e.a. [1988]. However, they concentrate solely on the feasibility aspect, neglecting the fact that identifying profitable exchanges for realistic objective functions can be as difficult as identifying feasible exchanges. This paper studies efficient implementations of edge-exchange improvement methods when the objective is to minimize route duration and the departure time of a vehicle at the depot is not fixed, but has to fall within a time window, as is the case in many real-life situations. We will generalize and extend the techniques presented in Savelsbergh [1990], that enable incorporation of time window constraints in edge-exchange improvement methods without increasing their complexity. and show that minimizing route duration, with …