## 4 Experiments and Results
### \(P_{mtv}\) of Different Algorithms  
This paper randomly selects 20 networks from all testable networks and numbers them. Then, this paper calculates their probabilities of shorted circuit by Equation (2). The probabilities are arranged into a matrix in the form of Formula (1), which is shown as Table 2.  
After the probability of shorted circuit is obtained, according to the description in Section 3.4; this paper used the improved genetic algorithm to determine the parameters \(\beta_{k}\) of the heuristic function in GA-AO* approach. This paper set \(c_{end}\) to 5000. After iteration, the final value of \(\beta_{k}\) is shown in Table 3.  
Figure 7: (**a**) The module under test; (**b**) Boundary scan test system.  
The estimated \(P_{\text{MTV}}\) and actual \(P_{\text{MTV}}\) under each order of sub-matrix is shown in Figure 8, with an average relative error of 6.9%. The results show that our heuristic function has good estimation ability for \(P_{\text{MTV}}\).  
This paper used these three algorithms to generate MTVs according to the probability of shorted circuit in Table 2. The codes of these algorithms were implemented on MATLAB 2020. The CPU type of the computer running codes was \"Intel (R) Core (TM) i7-10750H [EMAIL_ADDRESS] GHz\". All algorithms generated MTVs for the same networks under optimal compactness, which means this paper set the number of columns of MTVs as the rounding up of \(log_{2}(n+2)\) (n is the number of networks under test). This paper set the \(P_{\text{MTV}}\) of MTVs which is generated by PSO as the initial value of \(P_{min}\). Finally, this paper set \(\theta\) to 80% and \(c_{s}\) to 100. PSO and Walk-1 are implemented according to the ways and parameters provided in reference [16] and reference [17], respectively. Under these conditions, this paper observed the changes of \(P_{\text{MTV}}\) of the three algorithms by changing the number of networks under the boundary scan test and evaluated the \(P_{\text{MTV}}\). When the number of networks is 5, 10, 15, and 20, respectively, the \(P_{\text{MTV}}\) of the MTVs generated by three algorithms are shown in Table 4.  
The \(P_{\text{MTV}}\) of the MTVs generated by the three algorithms varies with the number of networks as shown as Figure 9.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c} \hline \hline
**Net** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8**  
As can be seen from Table 4 and Figure 9, all three algorithms can generate fault-free MTVs when the number of networks is small. With the increase in network size, the \(P_{\rm MTV}\) of MTV generated by Walk-1 algorithm increases significantly because of fixed compactness index, while the MTVs generated by PSO and by our method can still maintain a low \(P_{\rm MTV}\). At the same time, because we take the \(P_{\rm MTV}\) of the MTVs generated by the PSO as \(P_{min}\) of our method, the \(P_{\rm MTV}\) of the MTVs generated by our method is always less than or equal to the \(P_{\rm MTV}\) of the MTVs generated by the PSO, so our method realizes the optimization. When the number of networks is 20, the running time of the three algorithms is 0.074 s, 248 s, and 1756 s. For the actual test, the generation of MTV for the same type of circuit is one-time and offline, so the generation time of MTV will not affect the test efficiency basically.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of Networks** & **Walk-1** & **PSO** & **Our Method** \\ \hline
5 & 0 & \(3.47\times 10^{-5}\) & 0 \\
10 & \(6.05\times 10^{-4}\) & \(4.10\times 10^{-4}\) & \(1.85\times 10^{-4}\) \\
15 & 0.0024 & 0.0016 & \(3.16\times 10^{-4}\) \\
20 & 0.0068 & 0.0030 & 0.0021 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The \(P_{\rm MTV}\) of each algorithm under different number of networks under test.  
Figure 8: The estimated \(P_{\rm MTV}\) and actual \(P_{\rm MTV}\) under each order of sub-matrix.  
Figure 9: The \(P_{\rm MTV}\) of MTVs generated by the three algorithms with the change in number of networks.