## 4 Examples  
The three classical PN models shown in Figure 3, Figures 7 and 8 are used to evaluate and check our proposed IMFFP-2 in this section, based on two famous analysis software PN-Tools [53] and INA [54]. The first model, shown in Figure 3, has two idle places (i.e., \(p_{1}\) and \(p_{8}\)), six activity places (i.e., \(p_{2}\)-\(p_{7}\)), and three resource places (i.e., \(p_{9}\)-\(p_{11}\)). Additionally, there are 44 reachable markings (i.e., \(M_{1}\)-\(M_{44}\)) in total, including 8 FBMs (i.e., \(M_{9}\), \(M_{11}\), \(M_{17}\), \(M_{18}\), \(M_{23}\), \(M_{25}\), \(M_{30}\), and \(M_{37}\)) and 36 legal markings in its reachability graph, respectively. Its reachability graph is also shown in Figure 4.  
According to our proposed algorithm in Section 3, \(T_{PIT}\) and \(P_{PIP}\) must be identified first. The computational efficiency of the conventional MFFP can then be further enhanced, based on Definitions 5 and 6, and the \(T_{PIT}=\{t_{4},\ t_{8}\}\) and \(P_{PIP}=\{p_{4},\ p_{7}\}\) in Example 1. Next, following the vector covering approach, one can obtain \(\mathcal{M}_{L}^{\star}=\{M_{36},M_{38},M_{43},M_{44}\}=\{(2p_{2}+p_{3}+p_{4}), (p_{2}+p_{5}+p_{6}+p_{7}),\ (p_{2}+p_{3}+p_{4}+p_{7}),\ (p_{5}+p_{6}+2p_{7})\}\) and \(\mathcal{M}_{FBM}^{\star}=\{M_{9},M_{11},M_{18}\}=\{(2p_{2}+p_{5}),\ (p_{3}+p_{5}),\ (2p_{2}+p_{6})\}\), respectively. Please note that all the above markings present partial information. The detailed information is shown in Tables 2 and 3. Figure 3: Another simple PN model of FMS. Further, the IMFFP-2 for this model can then be obtained as follows. The first iteration in our IMFFP-2: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Marking** & \multicolumn{2}{c}{**Marking Information**} \\
**No.** & **Classification** & \(\left[\textbf{P}_{1}\textbf{P}_{2}\textbf{P}_{3}\textbf{P}_{4}\textbf{P}_{5} \textbf{P}_{6}\textbf{P}_{7}\textbf{P}_{8}\textbf{P}_{9}\textbf{P}_{10}\textbf{P }_{11}\right]^{\textbf{T}}\) \\ \hline \(M_{9}\) & \(M_{Q}\)/\(\mathcal{M}_{FBM}^{\star}\) & \(\left[2,\,2,\,0,\,0,\,1,\,0,\,0,\,3,\,0,\,1,\,0\right]^{\textbf{T}}\) \\ \(M_{11}\) & \(M_{Q}\)/\(\mathcal{M}_{FBM}^{\star}\) & \(\left[3,\,0,\,1,\,0,\,1,\,0,\,0,\,3,\,2,\,0,\,0\right]^{\textbf{T}}\) \\ \(M_{17}\) & \(M_{Q}\)/\(\mathcal{M}_{FBM}\) & \(\left[2,\,1,\,1,\,0,\,1,\,0,\,0,\,3,\,1,\,0,\,0\right]^{\textbf{T}}\) \\ \(M_{18}\) & \(M_{Q}\)/\(\mathcal{M}_{FBM}^{\star}\) & \(\left[2,\,2,\,0,\,0,\,0,\,1,\,0,\,3,\,0,\,0,\,1\right]^{\textbf{T}}\) \\ \(M_{23}\) & \(M_{D}\)/\(\mathcal{M}_{FBM}\) & \(\left[1,\,2,\,1,\,0,\,1,\,0,\,0,\,3,\,0,\,0,\,0\right]^{\textbf{T}}\) \\ \(M_{25}\) & \(M_{D}\)/\(\mathcal{M}_{FBM}\) & \(\left[2,\,2,\,0,\,0,\,1,\,1,\,0,\,2,\,0,\,0,\,0\right]^{\textbf{T}}\) \\ \(M_{30}\) & \(M_{Q}\)/\(\mathcal{M}_{FBM}\) & \(\left[3,\,0,\,1,\,0,\,1,\,0,\,1,\,2,\,1,\,0,\,0\right]^{\textbf{T}}\) \\ \(M_{37}\) & \(M_{Q}\)/\(\mathcal{M}_{FBM}\) & \(\left[2,\,1,\,1,\,0,\,1,\,0,\,1,\,2,\,0,\,0,\,0\right]^{\textbf{T}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of eight First Bad Markings. Figure 4: The initial and full reachability graph of Figure 3.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Marking** & \multicolumn{2}{c}{**Marking Information**} \\
**No. ** & **Classification** & \(\left[\textbf{P}_{1}\textbf{P}_{2}\textbf{P}_{3}\textbf{P}_{4}\textbf{P}_{5} \textbf{P}_{6}\textbf{P}_{7}\textbf{P}_{8}\textbf{P}_{9}\textbf{P}_{10} \textbf{P}_{11}\right]^{\textbf{  
\(max\)\(f^{*}=f_{1}+f_{2}+f_{3}\) S. t.  
\(2l_{2}+l_{3}\leq\beta\)  
\(l_{2}+l_{5}+l_{6}\leq\beta\)  
\(l_{2}+l_{3}\leq\beta\)  
\(l_{5}+l_{6}\leq\beta\)  
\(2l_{2}+l_{5}>\beta-Q(1-f_{1})\)  
\(l_{3}+l_{5}>\beta-Q(1-f_{2})\)  
\(2l_{2}+l_{6}>\beta-Q(1-f_{3})\)  
\(l_{i}\in\mathbb{N}\), \(\forall i\in\mathbb{N}_{A}\). \(\beta\in\mathbb{N}\). \(f_{k}\in\{0,\ 1\}\). In the following, solving the ILPP by using our proposed IMFFP-2, one can obtain two solutions.