## 3 The Multi-Echelon Vehicle Routing Problem  
In this Section our Vehicle Routing Problem model is detailed. Our VRP model and test instance is based on our previous work [26]. Figure 1 illustrates the topology of our problem. Figure 1 illustrates an example of a VRP sketch on which our own VRP example is based. Figure 1 indicates that our VRP consists of not only customers, indicated with \(\mathit{CU}\ =\ \{\mathit{cu_{1},cu_{2},\ldots,cu_{nc}}\}\) and a depot, indicated with \(D\ =\ \{\mathit{d_{1}}\}\), but also satellites. However, the satellites are also different, in levels. The product is shipped from a central depot to the first level satellite \(\mathit{SL_{1}}\ =\ \{\mathit{sl_{1,1},sl_{2},\ldots,sl_{1,ns_{1}}}\}\), than to the second level satellite \(\mathit{SL_{2}}\ =\ \{\mathit{sl_{2,1},sl_{2},\ldots,sl_{2,ns_{2}}}\}\), etc., and then to the customers. The system contains a single type of product, indicated with \(\mathit{PR}\ =\ \{\mathit{pr_{1}}\}\). We can also define the set of locations, with the union of the depot, satellites and customers: \(\mathit{LO}\ =\ \mathit{D}\cup\mathit{SL_{1}}\cup\mathit{SL_{2}}\ldots\cup \mathit{CUI}\ =\ \{\mathit{l_{0}},\ldots,\mathit{l_{0}}\}\). The system also contains several vehicles, indicated with \(\mathit{VE}\ =\ \{\mathit{ve_{1},\ldots,ve_{nav}}\}\). Our system contains several components, which can be categorized in the following way: node components, vehicle components, time components, product components, cost components, and metrics (objective function components). To formalize the components, the following notations are introduced: \(i,j\) is the node index, \(k\) is the vehicle index. The product has not got any special index, because the system contains one type of product. In the following, we use a function-oriented description of the costs associated with the main components of the VRP system in order to provide a more general formalism. The arguments of the functions are equal to the main cost parameters of the VRP. The objective is the minimization of the length of the route \((\min z_{lr})\), fuel consumption \((\min z_{fc})\), vehicle rental fee \((\min z_{rf})\), route time \((\min z_{rt})\), unvisited customers \((\min z_{uc})\), loading time \((\min z_{lt})\), unloading time \((\min z_{lt})\), unloading time \((\min z_{uc})\), administration cost \((\min z_{ac})\), quality control cost \((\min z_{qcc})\) and the maximization of the route status \((\min z_{rs})\). For the objective function, it is necessary to normalize the individual components of the objective function. The objective function is a combination of these components, which can be written in the following way: \(Z\ =\ \{(w_{lr}\ \times\ z_{lr}^{norm})^{2}\)\(+(w_{fc}\times z_{fc}^{norm})^{2}\ +\ \left(w_{rf}\times z_{rf}^{norm}\right)^{2}\ +\ (w_{rt}\times z_{rt}^{norm})^{2}\ +\ (w_{uc}\times z_{uc}^{norm})^{2}\ +\ \left(w_{lt}\ \times z_{lt}^{norm} \right)^{2}\ +\ (w_{lt}\ \times z_{lt}^{norm})^{2}\ +\ (w_{lt}\ \times z_{lt}^{norm})^{2} +\ (w_{lt}\ \times\ z_{lt}^{norm})^{2}\ +\ (w_{uc}\ \times\ z_{ac}^{norm})^{2} +\ (w_{qcc}\ \times\ z_{qcc}^{norm})^{2}\ +\ (w_{rs}\ \times\ z_{rs}^{norm})^{2}\}^{1/2}\), where \(z^{norm}\) means the normalization of an objective function component, and \(w\) means the weights of each components. To the node component belongs the travel time between the nodes, which can be written with the following function: \(\mathit{TravelTime}\big{(}\mathit{l_{0}},\mathit{l_{0}},\mathit{v_{c}}\big{)}\), so the travel time varies by locations and vehicle types. The travel distance between the nodes can be written in the following way: \(\mathit{TravelDistance}\big{(}\mathit{l_{0}},\mathit{l_{0}},\mathit{v_{c}} \big{)}\), so the travel distance varies also by locations and vehicle types. Route status between the nodes is also defined: \(\mathit{RoutStatus}\big{(}\mathit{l_{0}},\mathit{l_{0}},\mathit{v_{c}}\big{)}\), it also varies by locations and vehicles.