## 4 Problem Definition and Mathematical Model  
A logistics company distributes new vehicles in Mexico, manufactured in another country. It carries out the delivery of thousands of vehicles, according to the demand of each of the dealerships responsible for the sale of vehicles. Currently, the logistics company uses an empirical allocation and routing method for the auto-carriers. The empirical method consists of the design of the route according to the experience of the operator of the auto-carrier, based on the vehicles that will be transported without the use of a heuristic or similar method for the optimization of the route. Similarly, the allocation corresponds to a simulation with fictitious vehicles of the load of the auto-carrier, positioning the vehicles in different levels of the auto-carrier, considering the dimensional restrictions. The process of the routing and loading of the auto-carriers, begins at the moment that the operators receive a list of the vehicles to be delivered to the different dealerships. In the NCSY, the operators confirm the vehicles to be transported with the manager of the NCSY, who to complete the loading process verifies that the vehicles in the auto-carrier correspond to the request of the dealerships. Empirical routing is inefficient because it does not consider restrictions as the time windows. The time windows are the hours in which an auto-carrier can perform the unloading of vehicles at a dealership, who defines an initial time and an end time to carry it out. Time windows are defined to not violate local traffic laws, thus avoiding monetary penalties. Avoiding various penalties and monetary losses for companies, are some reasons of importance for the VRPTW and its applications. An example of its application is in the cement industry, if the concrete mixer trucks do not arrive within the stipulated time window. It may be that part of the concrete dries, becoming unusable, and the work stops. In the case of the logistics company, if an auto-carrier arrives at the dealership at a time outside the time window, it causes a time penalty that is, the operator must wait at the dealership to unload vehicles. In addition to the time window restrictions, this case study includes a total of 44 dealerships, a demand with approximately 4000 vehicles of different dimensions (which add three restrictions to the allocation) and a variable number of auto-carriers of different capacity load (3, 6, 7, 10 and 11 vehicles). This paper describes the algorithm developed for a real-world problem of a logistics company; the problem can be summarized as follows:  
given a heterogeneous fleet of auto-carriers based at a NCSY and a set of dealerships each requiring a set of vehicles, the loading of the vehicles into the auto-carriers and route the auto-carriers through the road network to deliver all dealerships with minimum cost (total number of kilometers traveled) that start and ends in the NCSY, considering the restrictions of time windows, a LIFO policy for the loading/unloading of vehicles and maximizing the total use of the capacity of each auto-carrier  
The characteristics of the dealerships (time windows), the NCSY and the auto-carriers (capacity), as well as different operational restrictions on the routes, bring forth the VRPTW, several authors [14; 15; 16; 17; 18] have worked on this variant of the VRP. In this case, study the term vehicle denotes a transported item (e.g., a car, a van), the term auto-carrier denotes a truck transporting vehicles, and the term dealership denotes a delivery point (i.e., a customer requiring one or more vehicles). With the previously mentioned elements, the model can be described as follows [21]:  
* Network: Given a complete graph \(G=(V,E)\), where \(V=0,1,\ldots,n\) is the set of vertices and \(E\) the set of edges connecting each vertex pair. Vertex 0 corresponds to the NCSY, whereas vertices \(1,\ldots,n\) correspond to the \(n\) dealerships to be served.