The vehicle routing problem with synchronization constraints and support vehicle-dependent service times

Many production processes require the cooperation of various resources. Especially when using expensive machines, their utilization plays a decisive role in efficient production. In agricultural production or civil construction processes, e.g., harvesting or road building, the machines are typically mobile, and synchronization of different machine types is required to perform operations. In addition, the productivity of one type often depends on the availability of another type. In this paper, we consider two types of vehicles, called primary and support vehicles. Primary vehicles perform operations and are assisted by at least one support vehicle, with more support vehicles resulting in faster service times for primary vehicles. We call this practical problem the vehicle routing and scheduling problem with support vehicle-dependent service times and introduce two mixed-integer linear programming models. The first represents each support vehicle individually with binary decision variables, while the second considers the cumulative flow of support vehicles with integer decision variables. Furthermore, the models are defined on a graph that allows easy transformation into multiple variants. These variants are based on allowing or prohibiting switching support vehicles between primary vehicles and splitting services among primary vehicles. We show in our extensive computational experiments that: i) the integer representation of support vehicles is superior to the binary representation, ii) the benefit of additional vehicles is subject to saturation effects and depends on the ratio of support and primary vehicles, and iii) switching and splitting lead to problems that are more difficult to solve, but also result in better solutions with higher primary vehicle utilization.