# A Cost-Optimized Model for Dynamic Port Hinterland
## 3 Methodology
### Methodological Proposal
#### 3.2.2 Port-to-Warehouse Transport Cost (_Pwtc_) and Transport Distribution Cost (_Tdc_)  
Port-to-Warehouse Cost (_PWTC_) and Transport Distribution Cost (_TDC_) represent critical components in modern logistics, covering transportation from port to intermediate warehouse and final delivery, respectively [1]. While traditionally dominated by road and rail transport, these operations now incorporate diverse options, including inland waterways and short-sea shipping, with mode selection influenced by environmental factors, reliability, and cargo requirements beyond pure cost considerations [60; 72].  
The determination of _PWTC_ and _TDC_ values is predicated on identifying the most cost-effective transport option or combination of modes. This optimization process can be mathematically represented as follows:\[PTWC=min\left\{\begin{array}{cc}\sigma\cdot d_{PTCW}&(Road)\\ Trc+HC_{o}+HC_{d}&(Train)\\ Any other medium or combination\\ \end{array}\right., \tag{6}\]  
\[TDC=min\left\{\begin{array}{cc}\sigma\cdot d_{ldc}&(Road)\\ Trc+HC_{o}+HC_{d}&(Train)\\ Any other medium or combination\\ \end{array}\right., \tag{7}\]  
where  
* \(\sigma\) represents the cost per kilometer (km) by truck (EUR/km);
* \(d_{PVTC}\) is the minimum road distance (km) from the destination port to the intermediate warehouse;
* \(d_{ldc}\) is the minimum road distance (km) from the intermediate warehouse to the destination point;
* \(Trc\) denotes the cost of transportation by train to the nearest station for both endpoints of the journey--that is, from the origin point to its nearest train station and from the destination's nearest train station to the final destination;
* \(HC_{o}\) represents the haulage cost in the origin station;
* \(HC_{d}\) represents the haulage cost in the destination station.  
These equations, while foundational, have been subject to refinement in recent literature. For instance, [PERSON] et al. propose an extended model that incorporates time-dependent factors and congestion costs [75], reflecting the dynamic nature of urban logistics environments. Similarly, [PERSON] et al. introduce a stochastic element to these calculations, accounting for the inherent uncertainties in transport times and costs, particularly in the context of multimodal transportation [80].  
The growing emphasis on sustainability in logistics has also led to the incorporation of environmental costs into these calculations. The authors of [59] suggest an augmented model that includes a carbon pricing component:  
\[PTWC=min\left\{\begin{array}{cc}(\sigma+\varepsilon_{c})\cdot d_{ptvc}&(Road )\\ Trc+HC_{o}+HC_{d}+\varepsilon_{r}&(Train)\\ Any other medium or combination\\ \end{array}\right., \tag{8}\]  
\[TDC=min\left\{\begin{array}{cc}(\sigma+\varepsilon_{c})\cdot d_{dtc}&(Road )\\ Trc+HC_{o}+HC_{d}+\varepsilon_{r}&(Train)\\ Any other medium or combination\\ \end{array}\right., \tag{9}\]  
where \(\varepsilon_{c}\) and \(\varepsilon_{r}\) represent the carbon emission costs for road and rail transport, respectively, typically expressed in EUR/ton-km of CO\({}_{2}\) equivalent.  
Emerging technologies are transforming inland transport calculations, with autonomous vehicles potentially restructuring costs [81] and predictive analytics enabling dynamic optimization strategies [66]. While fundamental _PWTC_ and _TDC_ structures remain valid, future models must adapt to incorporate modern supply chain complexities, including sustainability, technological advancement, and network resilience.