## 6 Numerical Example
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Flow Vectors of** **(3;2,12)-MP\_{\text{200}}} & \multirow{2}{*}{\begin{tabular}{c} **State Vector** \\ **Candidates of** \\ **(3;2,12)-MP\_{\text{200}}** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Capacity of** \\ **Edge**\(\leq\) **2** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Verify by** \\ **Theorem 4** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Real State** \\ **Vectors of** \\ **(3;2,12)-MP\_{\text{200}}** \\ \end{tabular} } \\ \hline (2,1,0) & (3,0,2,0,1,0,2,1) & N & & & \\ (2,0,1) & (2,1,2,1,0,0,2,1) & Y & N & Y \\ (1,2,0) & (3,0,1,0,2,0,1,2) & N & & & \\ (1,0,2) & (1,2,1,2,0,0,1,2) & Y & N & Y \\ (1,1,1) & (2,1,1,1,1,0,1,2) & Y & N & Y \\ (0,2,1) & (2,1,0,1,2,0,0,3) & N & & & \\ (0,1,2) & (1,2,0,2,1,0,0,3) & N & & & \\ \hline \hline \end{tabular}  
* Use the information about the real state vectors of (3,2,12)-MP\_{200} obtained in Table 6 and of the distribution of the capacity listed in Table 2 to calculate the system reliability of the transfer of three units of goods driving on \(\text{Rd}_{\text{h}}\) from Taipei to Kaohsiung, denoted as \(\text{R}_{(3;2,12)-\text{MP}\_{200}}\) in the multi-flow network of Figure 6. \(\text{R}_{(3;2,12)-\text{MP}\_{200}}\) was obtained by the inclusion–exclusion principle shown in Equation (21).  
\[\begin{array}{rl}\text{R}_{(3;2,12)-\text{MP}\_{200}}&=\sum\limits_{i=1}^{3} P(x_{i})-\sum\limits_{j=2}^{3}\sum\limits_{i=1}^{j-1}P(x_{i}\cap x_{j})+(-1)^{3+1}P(x_{i} \cap x_{j}\cap x_{k})\\ &=1.665198-1.411024+0.4210704\\ &=0.675244\end{array} \tag{21}\]  
The reliability equals 67.5244% for the distribution of three units of goods from Taipei in northern Taiwan, i.e., the source node s, to Kaohsiung in the south, i.e., the sink node t, as shown in the MFN for driving on the highway (\(\text{Rd}_{\text{h}}\)) in Figure 6.  
### Drive on \(\text{Rd}_{g}\)  
Secondly, the choice of truck to drive on the slow roads (\(\text{Rd}_{\text{g}}\)) is limited to the vehicle loading capacity and cannot exceed three units of weight. The total transport lead time is less than 12 h, and the calculation process of the MFN reliability is as follows:  
* Find all MPs in Figure 6. A total of four MPs were found as below. & & & \\ \(\text{P}_{1}=\{\text{e}_{1},\text{e}_{3},\text{e}_{7}\}\), & \(\text{P}_{2}=\{\text{e}_{1},\text{e}_{5},\text{e}_{8}\}\), & \(\text{P}_{3}=\{\text{e}_{2},\text{e}_{4},\text{e}_{8}\}\), & \(\text{P}_{4}=\{\text{e}_{2},\text{e}_{6},\text{e}_{7}\}\) & \\ Step 2. & The length between nodes of each MP is calculated using the coordinates in Figure 5 and Equation (2), as shown in Table 3. & & & \\ \end{tabular}
* The above information is used to calculate the single transport lead time using Equation (3) and is listed in Equations (22)-(25) for \(\text{P}_{1}\) through \(\text{P}_{4}\), respectively. The relevant calculation results are shown in Table 7. The total transport lead time of each MP is the number of turns multiplied by the single transport lead time, as shown in the sixth column of Table 7.  
\end{table} 
Table 7: Transport lead time on \(\text{Rd}_{\text{g}}\). The reliability equals 85.2753% to distribute three units of goods from Taipei in northern Taiwan, i.e., the source node s, to Kaohsiung in the south, i.e., sink node t, as the MFN, as shown in Figure 6, for driving on the slow roads (\(\text{Rd}_{\text{g}}\)).