GRASP and VNS approaches for a vehicle routing problem with step cost functions

This article addresses a vehicle routing problem in which companies hire a third-party logistics company whose freight charges are calculated using discontinuous step cost functions with fixed costs for each distance range according to the type of vehicle being used. The focused problem also takes into account a diverse fleet of vehicle that can be chosen and time windows for every client and in the depot. This problem is known as the fleet size and mixed vehicle routing problem with time windows and step cost functions. Given its complexity and its relevance for real-word situations, efficient customized metaheuristics are proposed based on the trajectory metaheuristics, namely, variable neighborhood search and greedy randomized adaptive search procedure. A hybrid metaheuristic that aims to exploit the best characteristics of the basic methodologies is also proposed. A computational comparative analysis is conducted on benchmark instances from the literature, small-size instances with known optimal solution, and instances based on a real-world problem. Numerical experiments indicate that the proposed methods are suitable for solving practical instances of the problem and show good performance in different scenarios.