### The multiple vehicle balancing problem

Marco Casazza, Alberto Ceselli, Daniel Chemla, Frédéric Meunier, Roberto Wolfler Calvo
<span title="2018-03-30">2018</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/p4qgtym3grdnngfxilwqjjxqy4" style="color: black;">Networks</a> </i> &nbsp;
This paper deals with the Multiple Vehicle Balancing Problem (MVBP). Given a fleet of vehicles of limited capacity, a set of stations with initial and target inventory levels and a distribution network, the MVBP requires to design a set of routes and pickup and delivery operations along each route such that inventory is redistributed among the stations without exceeding the vehicle capacities and such that routing costs are minimized. The MVBP arises in bicycle sharing systems, where
more &raquo; ... of bicycles is needed between stations to match their expected demands. The MVBP is NP-hard, generalizing several problems in transportation like the Split Delivery Vehicle Routing Problem. Using theoretical properties of the problem, we propose an integer linear programming formulation. Lower bounds are computed by a column generation routine embedding an adhoc pricing algorithm; we also introduce strengthening valid inequalities. Upper bounds are obtained by a memetic algorithm that focuses on routing and considers the pickup and delivery operations in a post-processing phase. We combine lower and upper bounding routines in both exact and matheuristic algorithms, obtaining proven optimal solutions for MVBP instances with up to 25 stations and an unbounded number of vehicles, or up to 20 stations and 5 vehicles.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/net.21822">doi:10.1002/net.21822</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nbgcg5vqynh6xcvaqwxqpcgxb4">fatcat:nbgcg5vqynh6xcvaqwxqpcgxb4</a> </span>
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