### Mathematical Model
Finally, the transport mode comparative index (\(K_{i}\)), could be expressed as follows:  
\[K_{i}=\frac{1}{\eta_{k}}(k_{P}\cdot\frac{P_{i}}{P_{0}}+k_{T}\cdot\frac{T_{i}} {T_{0}}+k_{E}\cdot\frac{E_{i}}{E_{0}}+k_{i}\frac{A_{i}}{A_{0}}) \tag{8}\]  
where: \(\eta_{K}\)--correlation coefficient, for the transport corridors, in case of similar factors; for example, the \"last mile\" situation for the railway and inland waterway transport modes and so on, could be 0.97-0.99, using more factors could improve the identification of correlation coefficients by using a matrix system [18]; \(P_{i},T_{i},E_{i}\)--costs, time and energy demand for the analyzing transport corridor; \(P_{0},T_{0},E_{0}\)--costs, time and energy demand for the etalon, which can be selected by expert method, transport corridor; \(k_{P}\)--weight coefficient of the transport corridor's costs. Depending on the type of cargo, it could be taken between 0.3-0.4; \(k_{T}\)--weight coefficient of the transport corridor's transportation time. This depends on the type of cargo and could be taken between 0.2-0.3;\(k_{E}\) is the weight coefficient of the transport corridor's energy (fuel) demand. It depends on the transportation conditions and areas and could be taken between 0.2-0.3; \(k_{i}\), \(A_{i}\), \(A_{0}\)--other possible factors and weight coefficients; for example, emissions, cargo weights and other factors, which are important for the transportation cases. The sum of the weighting factors must be equal to one. Equation (8) is based on multi-criteria methods and can be used for many tasks; for example, for passenger and cargo transportation optimization, including environmental impact evaluation in case only a few transport corridors could be used, as well for the logistics chains evaluations. Using the mentioned equation is necessary to in order to pay attention to weight coefficients, which can be calculated via the matrix in the existing wide data base or could be used expert methods in case of limited real data. For the optimal transportation way and energy saving, it is necessary to find the best transport network, and this could be achieved using the graph theory method. The application of graph theory method is used, in which the model is built in such a way that incorporates a set of vertices, which represent possible transport roads or corridors, and a set of edges, which represents the main connecting points, such as terminals, railway stations (intermodal terminals) and final destination points. This could be modeled as a graph tree and expressed as follows [41; 42; 43; 44; 45]:  
\[G=(V,E) \tag{9}\]  
where: \(V\)--the set of vertices; \(E\)--the set of edges. Such a graph model can be expressed as shown in Figure 2.  
The departure point could be Seaport (\(v_{1}\)) or another departure point, and the destination point could be the same as cargo consignee (\(v_{4}\)). Depending on the transport mode, it could be a direct delivery by road transport mode (from door to door), which means that from (\(v_{1}\)) to (\(v_{4}\)), a railway mode could be used between points \(v_{1}\) and \(v_{2}\), from point \(v_{2}\) to point \(v_{3}\), and from point \(v_{3}\) to point \(v_{4}\), road transport could be used as the \"last mile destination\". Similarly, an inland water way transport mode could be used from seaport point \(v_{1}\) up to inland waterway ships loading/unloading place \(v_{5}\), and road transport could be used from inland waterway transport loading/unloading place \(v_{6}\) to final destination \(v_{4}\) as the \"last mile destination\".