We consider vehicle routing and crew scheduling problems that involve a lexico- graphic bi-level objective function (for instance, minimizing first the number of vehi- cles and second the operational costs) and can be solved by column generation where the subproblem is a resource constrained shortest path problem. Such problems are often modeled using a single-level objective function with a large fixed cost (weight) for ensuring the minimization of the primary objective. In this paper, we study the impact on the solution time of the fixed cost value. First, we present computational results which show that the solution time increases as the fixed cost value gets larger. Then, we develop an exact dynamic fixed cost procedure compatible with column gen- eration that starts with a relatively small fixed cost value and increases it iteratively until optimality is reached. To prove optimality, a shortest path problem with resource constraints needs to be solved. Through a series of computational experiments on two types of problems, we show that this procedure can reduce solution times by up to 50% when compared to an approach relying on one very large fixed cost value.
Published March 2005 , 27 pages