truck and drone routing problem with soft time windows

Due to narrow streets, dense traffic, and transportation restrictions, city logistics operations are under increasing pressure and need innovative approaches. recently coordination of trucks and drones has been used as a new solution, which can improve the efficiency of city logistics operations. this paper also focuses on a truck and drone delivery system. as the major contributions, this paper develops a new mix integer programming model to formulate the hybrid truck and drone routing problem with soft time windows and proposes an effective two-phased metaheuristic algorithm. to evaluate the performance of the proposed metaheuristic, we carried out numerous computational experiments, where the results show the efficiency of the proposed metaheuristic. finally; a detailed sensitivity analysis is performed.