# A Network-Based Model for Optimization of Container Location Assignment at Inland Terminals
## 4 Mathematical Model
### Consistent Approximation Based on an Interview  
For each inconsistency stream \(\mathcal{F}\), at least one consistent approximation stream \(\mathcal{F}_{0}\) could be obtained that satisfied Equation (3). A deflection of approximation \(\mu(\mathcal{F}_{0})\) is the ratio  
\[\mu(\mathcal{F}_{0})=\frac{\parallel\mathcal{F}_{0}-\mathcal{F}\parallel}{ \parallel\mathcal{F}\parallel}, \tag{7}\]  
where the square of the absolute flow value, \(\parallel\mathcal{F}\parallel^{2}\), is given by  
\[\parallel\mathcal{F}\parallel^{2}=\frac{1}{2}\cdot\sum_{i,j\in V}(\mathcal{F} (i\gets j))^{2}. \tag{8}\]  
Analogously, it follows that  
\[\parallel\mathcal{F}-\mathcal{F}_{0}\parallel^{2}=\frac{1}{2}\cdot\sum_{i,j \in V}(\mathcal{F}_{0}(i\gets j)-\mathcal{F}(i\gets j))^{2}. \tag{9}\]  
Application of Equation (8) leads to \(\parallel\mathcal{F}\parallel^{2}=16+1+16+4+1+1+4+1+0+4=48\). To create the stream \(\mathcal{F}_{0}\) with minimal \(\mu(\mathcal{F}_{0})\), we respect the maximal potentials together with maximal preferences of \(\mathcal{F}\). The maximal potential of \(\mathcal{F}\), given in Table 3, is \(X(a)=11\). Thus, the new potential \(X_{0}(a)=11\) appears in \(\mathcal{F}_{0}\). Maximal preferences incident to criterion \(a\) are \(\mathcal{F}(a\gets b)=\mathcal{F}(a\gets d)=4\). Then the new potentials of consistent approximation \(\mathcal{F}_{0}\) are \(X_{0}(d)=X_{0}(b)=7\) and criterion \(d\), \(b\), and \(a\) have new potentials. The maximum preferences of replacements incident to the criteria with new potentials are \(\mathcal{F}(a\gets e)=\mathcal{F}(b\gets c)=\mathcal{F}(e\gets d)=2\). Because \(X_{0}(a)=11\geq 7=X_{0}(d)=X_{0}(b)\), then the new potential \(X_{0}(e)=9\). By analogy, \(X_{0}(c)=9\). The undirected complete graph with new potentials is shown in Figure 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Criterion** & **Criterion Potential (X)** \\ \hline Container dimension \((a)\) & \(11\) \\ \hline Container occupancy \((b)\) & \(2\) \\ \hline Cargo type \((c)\) & \(0\) \\ \hline Container owner \((d)\) & \(-5\) \\ \hline Distance to the container’s final destination \((e)\) & \(-8\) \\ \hline
**Total** & \(\mathbf{0}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Potential of inconsistent flow. The new replacement preferences are calculated by considering the new potentials as a tool. The consistent stream \(\mathcal{F}_{0}\) is obtained and presented in Figure 3.  
Plugging the relevant values into Equations (9) and (7) leads to \(\parallel\boldsymbol{\mathcal{F}}-\boldsymbol{\mathcal{F}_{0}}\parallel^{2}= \boldsymbol{0}+\boldsymbol{1}+\boldsymbol{0}+\boldsymbol{0}+\boldsymbol{1}+ \boldsymbol{1}+\boldsymbol{0}+\boldsymbol{1}+\boldsymbol{4}+\boldsymbol{0}= \boldsymbol{8}\) and \(\boldsymbol{\mu}(\boldsymbol{\mathcal{F}_{0}})=\boldsymbol{0.41}\). The potentials calculated by consistent approximation preferences created in Figure 3 change the ordinal scale at the lowest preferred potentials. The criteria were ranked according to Equation (6), where \(\alpha=1.15\) was defined arbitrarily in order to obtain a suitable range of criterion ranks [54]. Potentials and ranks are shown in Table 4. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Criterion Potential** & **Criterion Rank** \\ \hline Criterion & \(X\) & \(w\) \\ \hline Container dimension \((a)\) & \(12\) & \(0.618\) \\ \hline Distance to the container’s final destination \((e)\) & \(2\) & \(0.153\) \\ \hline \(\text{Cargo type }(c)\) & \(2\) & \(0.153\) \\ \hline Container owner \((d)\) & \(-8\) & \(0.038\) \\ \hline Container occupancy \((b)\) & \(-8\) & \(0.038\) \\ \hline
**Total** & \(\boldsymbol{0}\) & \(\boldsymbol{1.000}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Criterion potentials and ranks based on the consistent approximation preference stream. Figure 3: Consistent approximation preference stream. Figure 2: An unoriented complete graph with new potentials.