### Mathematical Model
It is possible to use three transport modes between transportation points: road, railway and waterway transport. In case of railway and inland waterway transportation, it is necessary use \"last mile\" solutions (Figure 3) [27]. Analysis of the modes of transport between Klaipeda port and Kaunas showed that the power of every truck engine is about 300-350 kW (total used engine power is about 15,000 kW), and train locomotive engine power, on average, is about 3000 kW. It is worth noting that, on average, one freight train can replace about 50 trucks [8; 20; 36; 38]. The \"Last  
Figure 3: Road, railway and inland waterway transport system between Klaipeda port and Kaunas free economic zone ([[https://www.vle.lt/straipsnis/lietuvos-transportas/](https://www.vle.lt/straipsnis/lietuvos-transportas/)]([https://www.vle.lt/straipsnis/lietuvos-transportas/](https://www.vle.lt/straipsnis/lietuvos-transportas/)), accessed on 3 September 2022) [27]. mile\" destination should be organized between the railway station and the destination point. Finally, it is also possible to use an inland waterway transport option between Klaipeda port and the Kaunas area using the Nemunas river, and in this case, the same cargo quantities transportation request one inland waterway ship, which uses an engine with power of about 400 kW [27]. The graph model for the selected transport systems in general between Klaipeda port and Kaunas free economic zone (FEZ) for customers' transportation situation (distances) can be explained as follows (Figure 4) [41]:  
In Figure 4, the distances are as follows: 330 km by railway from Klaipeda container terminal to Kaunas intermodal terminal; 7 km by road transport from Kaunas intermodal terminal to Kaunas free economic zone; 230 km by road transport from Klaipeda container terminal to Kaunas FEZ free economic zone; 270 km by inland waterway from Klaipeda container terminal to inland waterways loading/unloading place in the Kaunas area; and 9 km by road transport from the inland waterway barges loading place in Kaunas area to Kaunas FEZ free economic zone. A typical transportation case in Lithuania, based on data received during analysis, considers a train, includes a propelled barge, which can transport about 80 TEU (Figure 5). For some 20\({}^{\prime}\) (TEU--Twenty Foot Equivalent Unit) and 40\({}^{\prime}\) foot or 2 TEU containers, it will be necessary to have approximately 50 road transport trucks. Propelled barge engine power is about 400 kW, and the average speed is about 15 km/h (barges work on Lithuania inland water ways). The engine power of a train locomotive, which can transport 80 TEU, is about 4000 kW, and the average speed is about 35 km/h (in this case, according to our investigations, the average power of a train locomotive is about 300 kW). The engine power of one road truck is about 300 kW, and the average speed is about 60 km/h [5; 7; 8; 23]. The abovementioned transport modes parameters should be taken as a basis for the calculation of transportation time, costs and energy demand using the methodology described in Section 3 of this paper. It is also necessary to take distribution procedures at intermodal terminal into account in cases in which railway transport and road transport on \"last mile\" is to be used. The latter shall also apply in cases in which waterway transport and road transport on the \"last mile\" are used. Figure 4: Case study graph: \(v_{1}\)—Klaipeda container terminal; \(v_{2}\) —Kauras intermodal terminal; \(v_{3}\) —Kauras free economic zone; \(v_{4}\) —inland waterway loading place. Figure 5: About 80 TEU can be transported by inland waterway transport mode (barge). The adjacency matrix for the transportation distances at Figure 4 could be made as follows:  
\[A_{D}= \begin{bmatrix}0&330&230&270\\ 330&0&7&0\\ 230&7&0&9\\ 270&0&9&0\end{bmatrix} \]  
Inland waterway barges carry a lot of cargo with a relatively low-power engine, and may keep a constant speed in case of the absence of any locks or other obstacles on the waterway.