## 4 Numerical Example
### Results of Dataset 1  
Table 3 lists the estimated parameter values for each model obtained from dataset 1. In this study, the proposed model has four parameters: \(\alpha\), \(\beta\), \(\gamma\), and \(N\), and the estimated values of dataset 1 are \(\hat{\alpha}=0.7342\), \(\hat{\beta}=0.001093\), \(\hat{\gamma}=0.001092\), and \(\hat{N}=2594.496\). Figure 2 shows the cumulative number of failures and estimates for the 15 models at each time point from Dataset 1. The black dots represent the actual cumulative number of failures in Dataset 1, and the red line represents the estimated value of the software reliability model proposed in this study. Compared with the estimates of other existing software reliability models, the estimates of the proposed model are closest to the actual cumulative number of failures. In addition, the thick black dashed and thin black dotted lines represent the 95% and 99% confidence intervals of the proposed software reliability model, and all the actual values are within the confidence intervals. Tables 4 and 5 present the results for the calculated criteria and multi-criteria based on Table 3. \(MSE\), \(R^{2}\), \(Adj\_R^{2}\), \(AIC\), \(BIC\), \(PRV\), \(RMSPE\), \(TS\), and \(PIC\) have the best results, with 273.018, 0.999, 0.999, 518.486, 526.515, 16.057, 16.058, 1.393, and 13,928.1, respectively, whereas \(PP\) and \(MAE\) have the second-best results, with 0.351 and 13.483, respectively. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**No.** & **Model** & \(\hat{\alpha}\) & \(\hat{b}\) & \(\hat{\alpha}\) & \(\hat{\beta}\) & \(\hat{N}\) & \(\hat{\gamma}\) & \(\hat{\varepsilon}\) & \(\hat{p}\) & \(\hat{q}\) & \(\hat{\lambda}\) \\ \hline \hline
1 & GO & 8874.738 & 0.004367 & & & & & & & \\ \hline
2 & DS & 2116.271 & 0.060638 & & & & & & & \\ \hline
3 & IS & 2006.643 & 0.070081 & & 4.556844 & & & & & \\ \hline
4 & YID1 & 8381.19 & 0.004642 & \(7.73\times 10^{-5}\) & & & & & & \\ \hline
5 & YID2 & 437.5306 & 0.087785 & 0.078345 & & & & & & \\ \hline
6 & YR & 2269.766 & & & 0.923509 & 0.001184 & 1.905442 & & & \\ \hline
7 & YE & 8911.425 & & & 0.619214 & \(1.94\times 10^{-5}\) & 362.1234 & & & \\ \hline
8 & PZ & 162,359.5 & 0.118437 & 0.000134 & 4.351755 & & 836.2331 & & \\ \hline
9 & PNZ & 839.1432 & 0.118376 & 0.025688 & 4.364091 & & & & \\ \hline
10 & IFD & 368.735 & 0.076678 & & & & & & & \\ \hline
11 & TC & 0.000221 & 1.475181 & 7509.944 & 6.142695 & 2155.745 & & & \\ \hline
12 & TP & 301.2389 & 0.182161 & 1.137592 & 0.847396 & & 3.508369 & 0.087644 & 0.005934 \\ \hline
13 & 3P & 0.315812 & 0.18365 & & 16.45598 & 3854.995 & 3.561905 & & \\ \hline
14 & Vrub & 2.109497 & 0.617114 & 0.148282 & 23.85692 & 3197.256 & & & \\ \hline
15 & NEW & & & 0.734214 & 0.001093 & 2594.496 & 0.001092 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter estimation of model from Dataset 1. The proposed model has the third-best result in \(MCDM\) with 0.3864 and the best results in \(MCDM\) and \(MCDM\) with 0.0062 and 0.1731, respectively. TC had the best result for \(MCDM\) but the fifth-best result for \(MCDM\) and \(MCDM\). This demonstrates the importance of assigning weights for a comprehensive evaluation. Figure 3 is a graphical representation of the three-dimensional plots of the \(MCDM\), \(MCDM\), and \(MCDM\) for the top five of the 15 models listed in Table 1 for Dataset 1.