A simulating annealing algorithm to solve the green vehicle routing & scheduling problem with hierarchical objectives and weighted tardiness

We present a green vehicle routing and scheduling problem (GVRSP) considering general time-dependent traffic conditions with the primary objective of minimizing CO₂ emissions and weighted tardiness. A new mathematical formulation is proposed to describe the GVRSP with hierarchical objectives and weighted tardiness. The proposed formulation is an alternative formulation of the GVRSP in the way that a vehicle is allowed to travel an arc in multiple time periods. The schedule of a vehicle is determined based on the actual distance that the vehicle travels each arc in each time period instead of the time point when the vehicle departs from each node. Thereby, more general time dependent traffic patterns can be considered in the model. The proposed formulation is studied using various objectives functions, such as minimizing the total CO₂ emissions, the total travel distance, and the total travel time. Computational results show that up to 50% reduction in CO₂ emissions can be achieved with average reductions of 12% and 28% compared to distance-oriented solutions and travel-time-oriented solutions, respectively. In addition, a simulated annealing (SA) algorithm is introduced to solve large-sized problem instances. To reduce the search space, the SA algorithm searches only for vehicle routes and rough schedules, and a straightforward heuristic procedure is used to determine near-optimal detailed schedules for a given set of routes. The performance of the SA algorithm is tested on large-sized problems with up to 100 nodes and 10 time periods.