Branching approaches for integrated vehicle and crew scheduling

Abstract

The integrated multi-depot vehicle and crew scheduling problem simultaneously builds vehicle blocks and crew duties. We present an integer mathematical formulation that combines a multi-commodity flow model with a mixed set partitioning/covering model. We propose solution approaches that start by solving the linear programming relaxation of the model. Whenever the resulting linear programming solution is not integer, three branching alternative strategies can be applied: a branch-and-bound algorithm and two branch-and-price schemes. The branch-andbound algorithm performs branching over the set of feasible crew duties generated while solving the linear relaxation. In the first branch-and-price scheme the linear programming relaxation is solved approximately, while in the second one it is solved exactly. Computational experience is reported over two different types of problems: randomly generated data publicly available for benchmarking in the Internet and data from a bus company operating in Lisbon.

Computing strong lower and upper bounds for the integrated multiple-depot vehicle and crew scheduling problem with branch-and-price

“…That is, a feasible solution for a given approach may not be feasible for another one, and vice-versa. As we mentioned in Section 2.1, our assumptions comply with those of Huisman et al (2005); Steinzen et al (2010), however, they differ from the assumptions of Mesquita and Paias (2008); Mesquita et al (2009).…”

“…In (Huisman, 2004;Huisman et al, 2005) the authors propose their randomly generated instances which are widely used in the literature (Borndörfer et al, 2008;Mesquita and Paias, 2008;Mesquita et al, 2009;Steinzen, 2007;Steinzen et al, 2010) and in this paper as well.…”

“…The pricing problems are modeled as resource constrained shortest path problems and are solved by a dynamic programming algorithm. Mesquita et al (2009) propose exact and non-exact branch-and-price procedures for the same problem definition and formulation as in (Mesquita and Paias, 2008). Steinzen (2007) and Steinzen et al (2010) use a similar modeling approach for the multipledepot case as in (Huisman, 2004), however, their mathematical formulation is based on timespace networks.…”

“…To our best knowledge, the only paper proposing an exact method for the integrated multipledepot vehicle and crew scheduling problem is that of Mesquita et al (2009), where a variant of the problem is studied in which some of the common assumptions we and other authors make on feasible crew schedules are neglected. Their MIP formulation, unlike ours, models vehicle and crew schedules separately and contains additional linking constraints to join the two parts.…”

A GRASP with efficient neighborhood search for the integrated maintenance and bus scheduling problem

“…Various papers give overviews of the models and solution approaches that are adopted (see Bodin and Golden (1981), Daduna and Paixão (1995), Bunte and Kliewer (2009) and Wren et al (2003)). Further considerations such as robustness and the integration of vehicle and crew scheduling are addressed by papers such as Békési et al (2009) and Mesquita et al (2009). Various practical extensions are also considered, such as multiple vehicles types (see Hassold and Ceder (2014)), which is NP-hard even for the single-depot case (see Lenstra and Rinnooy Kan (1981)), variable arrival and departure times (which is NP-hard even for one vehicle and one depot; see Savelsbergh (1985)), and route constraints (see Bodin and Golden (1981)).…”