Step cost functions in a fleet size and mix vehicle routing problem with time windows

The vehicle routing problem is a traditional combinatorial problem with practical relevance for a wide range of industries. In the literature, several specificities have been tackled by dedicated methods in order to better reflect real-world situations. Following this trend, this article addresses the fleet size and mix vehicle routing problem with time windows in which companies hire a third-party logistics company. The shipping charges considered in this work are calculated using step cost functions, in which values are determined according to the type of vehicle and the total distance traveled, with fixed values for predefined distance ranges. A mixed integer linear programming model is introduced and two sequential insertion heuristics are proposed. The introduced methods are examined through a computational comparative analysis in small-sized instances with known optimal solution, 168 benchmark instances from the literature, and 3 instances based on a real-world problem from the civil construction industry. The numerical experiments show that the proposed methods are efficient and show good performance in different scenarios.