# Theoretical Framework (Module) for Short-Sea Shipping System Evaluation
## 3 Theoretical Basis of the Evaluation of Short-Sea Shipping Processes
### Mathematical SSS System Model
In the case of SSS in different regions of the world [16,18] where transportation can be carried out mainly over relatively short distances, Formula (3) can be applied as follows:  
\[P_{S}=P_{OTHC}+P_{BAF}+P_{CAF}+P_{ISPS}+P_{CONG}+P_{PORT}+P_{THC}+P_{DOC}+P_{ IMP/EXP}+P_{RF}. \tag{4}\]  
Using the SSS system, shipping lines usually set a total (lump sum) price (\(P_{L}\)) for the transportation of cargo units and a system of possible discounts, into which direct terminal service and transportation vessel prices are entered (like ticket prices from entry into a departure port terminal and departure from an arrival port terminal); then, Formula (4) can be written as follows:  
\[P_{S}=P_{L}. \tag{5}\]  
The transportation services time function (\(T_{L}\)), using the SSS sytem or other transport modes for a specific quantity of cargo or units, including freight transport distances, vehicle speeds, and weight coefficients of transport modes, can be expressed as follows (Equation (6)):  
\[T_{L}=\frac{1}{\eta_{T}}\bigg{(}k_{ST}\frac{S_{S}}{v_{S}}N_{T}+k_{RT}\frac{S_ {R}}{v_{R}}N_{T}+k_{TT}\frac{S_{T}}{v_{T}}N_{T}+\cdots\bigg{)}, \tag{6}\]  
where \(\eta_{T}\) is the correlation coefficient in case of similarities between transport modes (this could be between 0.95 and 1.0 (in the case where one transport mode is used, this coefficient will be one)); \(k_{ST}\) is the time function coefficient for SSS transportation (which depends on weather conditions, waiting to enter the port, and so on; for liner shipping, this depends on the probability fluctuation from the planned time table); \(S_{S}\) is the sea distance between SSS ports; \(v_{S}\) is the ship's commercial speed; \(k_{RT}\) is the time function coefficient for road transportation, which depends on traffic and road conditions and mainly depends on the probability fluctuation from the planned time table; \(S_{R}\) is the distance by roads between ports or other locations; \(v_{R}\) is the average speed of road transport for the freight, including mandatory stops for drivers to refresh and so on; \(k_{TT}\) is the time function coefficient for railway transportation, which depends on the required stops during crossing countries' borders and so on and mainly depends on the probability fluctuation from the planned time table; \(S_{T}\) is the distance by railway between ports or other locations; \(v_{T}\) is the train's commercial speed; and \(+\cdots\) is other possible time elements such as the waiting time of the ship upon entering the port, as well as the time required for unloading and loading the ship or other means of transport, and any other possible additional time. The cost function (\(D_{L}\)) of storing, redirecting, and distributing goods in ports or other locations for a specific quantity of cargo units can be expressed, taking into account the periodicity of SSS transportation and the possibility of promptly redistributing cargo (similar to airports), as follows (Equation (7)):  
\[D_{L}=\frac{1}{\eta_{D}}\big{(}k_{SD}N_{T}^{\prime}+k_{RD}N_{T}^{\prime}+k_{TD }N_{T}^{\prime}+\cdots\big{)}, \tag{7}\]  
where \(\eta_{D}\) is the correlation coefficient in cases where there are similarities between cargo or goods distribution in different transport modes (this could be between 0.97 and 1.0 (in cases where one transport mode is used, this coefficient will be one)); \(k_{SD}\) is the cost function coefficient, which depends on the operations that should be carried out in ports or terminals and on SSS possibilities and transportation procedures; \(N_{T}^{\prime}\) is the quantities of the cargo or number of transport units for which additional operations in ports or terminals are required; \(k_{RD}\) is the cost function coefficient, which depends on the operations which should be carried out during transportation by roads (in the case of overweight freight, additional leasing, etc.