Managing Fixed Costs in Vehicle Routing and Crew Scheduling Problems Solved by a Branch-and-Price Approach

In this paper, we address vehicle routing and crew scheduling problems that can be solved by a branch-and-price approach where the column generation subproblem corresponds to a shortest path problem with or without resource constraints. As surveyed in [1] and [2], branch-and-price approaches have been successfully used to solve a wide variety of vehicle routing and crew scheduling applications such as the vehicle routing problem with time windows, the multi-depot vehicle scheduling problem, and the aircrew pairing problem. More specifically, we are interested in applications that involve a two-level hierarchical objective function where the primary objective aims at minimizing the number of vehicles or crews used in the solution, while the secondary one consists of minimizing operational costs. It is well known that such a two-level hierarchical objective function can be converted into a single-level objective function that involves the operational costs and a large artificial fixed cost for each vehicle used (see [6]). It seems, however, that no efficient methodology has yet been proposed in the literature to determine the value of this fixed cost. Indeed, the only known algorithms for this purpose ([6] and [5]) can often produce values that are unusable in practice due to their huge magnitude. Consequently, this fixed cost is often set to an arbitrary large value that is believed to yield a minimal number of vehicles. We recently noticed that the value of the fixed cost f has an impact on the solution time of branch-and-price approaches. Indeed, the time required to solve a linear relaxation generally increases with f until it reaches a plateau for a relatively large value of f . This behavior was 1In order to lighten the text, we will mostly use the vehicle routing terminology (that is, vehicle and route) for the