In recent decades, scientific interest has grown in tackling the vehicle routing problem with a sustainable approach (Green VRP). There are numerous studies in the literature addressing environmental problems from the point of view of efficient planning that allows visualizing the benefits associated with the use of the new technologies in electric vehicles. This paper focuses on the electric vehicle routing problem and considers the batteries’ state of charge (SoC). The problem considers a set of customers, where each one has a specific demand and a time window. Deliveries are performed through a homogeneous fleet of electric vehicles with a fixed charging capacity and limited autonomy. In particular, when the vehicle is traveling, it consumes an amount of energy proportional to the distance it travels; therefore, it must visit battery recharging stations to continue and complete its route. The objective is to determine the performed routes with the minimum cost (time), while seeking to visit the recharging stations as many times as possible. In this way, overcharging and deep discharges are avoided by protecting the battery from degradation. In this paper, four models are proposed: the first model requires that the battery be fully charged in the stations; the second model allows partial recharging; the third formulation limits deep discharge; and a fourth formulation adheres to a limitation associated with overcharging and tries to keep the battery in its most comfortable place. The efficiency of the proposed formulations is tested in structured instances of different sizes. The results obtained show the efficiency of the formulations proposed for the electric vehicle routing problem when considering battery degradation.

Hub location problems (HLPs) support decision making on multimodal transport strategic planning. It is related to the location of hubs and the allocation of origin/destination (O/D) flow in a system. Classical formulations assume that these flows are predefined paths and direct delivery is not available. This applied research presents a mixed integer linear programming (MILP) model for a capacitated multimodal, multi-commodity HLP. Furthermore, an application on the export process in a Latin American country is detailed. The new proposed model, unlike the traditional HLP, allows direct shipment, and its O/D flows are part of the decision model. Situations with up to 100 nodes, six products, and two transport modes are used, working with initial and projected flows. All instances can be solved optimally using the commercial solver, Gurobi 7.5.0, in computational times less than a minute. Results indicate that only one hub is profitable for the case study, both for the initial and projected scenarios. The installation of a hub generates transport savings over 1% per year. Two factors affect the location decision: low concentration and distance between the hubs and destinations. Long distances involve an exhaustive use of trains instead of trucks, which leads to lower transport cost per unit.

This paper introduces a novel variant of the cumulative vehicle routing problem (CCVRP) that deals with home health care (HHC) logistics. It includes multiple nonfixed depots and emergency trips from patients to the closest depot. The aim is to minimize the system's delayed latency by satisfying mandatory visit times. Delayed latency corresponds to caregivers' total overtime hours worked while visiting patients. A new mixed-integer linear programming model is proposed to address this problem. Computational experiments, with more than 165 new benchmark instances, are carried out using the CPLEX and Gurobi MIP solvers. The results indicate that patients' geographical distribution directly impacts the complexity of the problem. An analysis of the model parameters proves that instances with more depots/vehicles or longer workdays are significantly easier to solve than are original cases. The results show that Gurobi outperforms CPLEX in 55% of the instances analyzed, while CPLEX performs better in only 16% of them. To the best of our knowledge, this is the first VRP that minimizes delayed latency and the first HHC routing study to use a cumulative objective function.