An exact algorithm for a Vehicle-and-Driver Scheduling Problem.

We introduce the Vehicle-and-Driver Scheduling Problem (VDSP), a new task scheduling problem that can be modeled as a two-depot vehicle routing problem with two types of vehicles, each type performing different kinds of routes.We propose a mixed integer linear programming formulation to model the problem, and present several families of valid inequalities to strengthen its linear relaxation.We describe a branch-and-cut algorithm for solving the problem that is able to cope with instances with up to 50 nodes. This article introduces a combinatorial optimization problem that consists of assigning tasks to machines and operators, and sequencing the tasks assigned to each one. Two configurations exist. Machines alternate configurations, while the operators must start and finish the process in the same configuration. Moreover, machines and operator have limited capacities. The sequencing of the tasks must guarantee that each one is performed by a machine and an operator at the same time, and it is determined in order to minimize an overall cost function. Two critical aspects of the problem are the need of synchronizing the machine and the operator performing each task, and the need of minimizing the changeovers, which are pairs of tasks done consecutively by the same machine but by different operators. The problem is modeled as a vehicle routing problem with two types of vehicles and with two depots. We propose a mixed integer programming formulation, and introduce valid inequalities to strengthen its linear programming relaxation. We describe separation routines for these inequalities and design a branch-and-cut algorithm for the problem. The algorithm is tested on a set of benchmark instances showing that it is able to solve to optimality instances with up to 50 customers.