The uncertain time dependent vehicle routing problem with soft time windows

This paper studies an uncertain time-dependent vehicle routing problem with soft time windows. A novel mathematical model which considers both transportation costs (total traveling distance and number of vehicles) and service costs (early and late arrivals) is developed, and the equations for calculating the expected total service costs are deduced under uncertainty and time-dependency. A variation of artificial bee colony algorithm is designed to solve the problem; computational results show the feasibility of this approach.