Multi-period profitable tour problem with electric vehicles and mandatory stops

The paper deals with a new variant of the profitable tour problem with electric vehicles. In order to avoid one of the main disadvantages encountered in this type of problem, i.e. the idle time imposed by recharging the battery, we propose a version where this operation is synchronized with lunch breaks. Such a policy requires multi-period agreements with clusters of restaurants. It leads to the multi-period profitable tour problem with electric vehicles and mandatory stops. This problem arises in scenarios of tourist trip design, blood mobile collection, or street marketing activities planning when the fleet is composed of electric vehicles. A mathematical model and a Branch-and-Price algorithm are proposed. In the solution method, the subproblem is solved following two different approaches, one exact approach using elementary paths and a relaxed approach allowing ng-paths. The numerical tests on instances with up to 100 clients and 3 periods show that both approaches produce excellent quality solutions in a very reasonable amount of time.