## 5 Simulation  
Monte Carlo simulation is widely employed in the literature to assess the performance of different models. For example, the authors of [22] employed it to estimate the reliability of logistics and supply chain networks. This section concentrates on a graphical demonstration of the performance under the applied reliability estimators for comparison and assessment purposes using Monte Carlo simulation. The simulation program was operated using the statistical software R, version 4.2.2. We conducted the simulation using a Windows 10 system with an 11 th Gen Intel(R) Core(TM) i5-1135G7 @ 2.42 GHz CPU. Different estimation methods were evaluated by comparing the mean squared error (MSE), which is defined as  
\[MSE\big{(}\hat{R}(t)\big{)}=\frac{1}{N}\sum_{i=1}^{N}\big{(}\hat{R}_{i}(t)-R( t)\big{)}^{2}, \tag{39}\]  
where \(N\) represents the number of iterations. Given that the \(MSE\) acts on a given value \(t\) for \(R(t)\), we then adopted the Mean Integrated Squared Error (\(MISE\)) to measure the performance of the discussed estimators for the reliability function globally. The \(MISE\) is defined as  
\[MISE\big{(}\hat{R}(t)\big{)}=\frac{1}{N}\sum_{i=1}^{N}\left(\frac{1}{T}\sum_{j=1 }^{T}\big{(}\hat{R}_{i}(t_{j})-R(t_{j})\big{)}^{2}\right)=\frac{1}{T}\sum_{j=1 }^{T}MSE\big{(}\hat{R}(t_{j})\big{)}, \tag{40}\]  
where \(t_{j}\) refers to the particular values of \(t\) selected from the reliability function \(R(t)\) and T is the number of the selected values denoted by \(t_{j}\). The simulation involved the following procedures:  
1. Before proceeding with the simulation, we initially determined the values of each known parameter in the model. The parameters \(\alpha\) and \(\beta\) appeared in the Burr XII distribution and were set as \(\alpha=2\) and \(\beta=0.5\). We employed sample sizes of \(n=10,n=30\), and \(n=100\). In the Bayesian estimators, the hyperparameters of the informative Gamma prior were \((2,1)\), and the values of \(m\) in the quasi-density prior were \(m=0.5,1\). As for the value \(p\), which occurred in the GEL function, it was set as \(p=0.5\) and \(p=2\). We considered \(q=0.5,2\) for the QEL function, as well as \(z=-0.5\), \(q=0.5\) and \(z=-1\), \(q=2\) for the WQEL function. For the choice of \(z=-0.5\), \(q=0.5\), we aimed to verify that the WQEL estimator for \(z=-0.5\), \(q=0.5\) was equivalent to the estimator under the SEL function. For the latter, it represented a choice of parameters for the WQEL estimator that exhibited the best performance, which we determined after many attempts. 2. We obtained \(n\) samples randomly selected from the Burr XII distribution using the inverse transforming sampling method with the given values of \(\alpha\) and \(\beta\). 3. We repeated step (2) \(N=1000\) times and calculated the \(MISE\) values. When \(N\) was large enough, the effects of random errors occurring each time we obtained the random sample in step (2) diminished, resulting in more informative simulation results. As observed in Figure 1, significant variations emerged with small values of \(t\) in the probability density plots. Thus, it should be noted that when computing the \(MISE\), we took the argument of the reliability function as \(t=(0.5,1,1.5,2,3,4)\). The simulation records are presented in Tables 1-5. The estimates were evaluated with varying values of \(\alpha\), \(n\), \(m\), \(p\), \(q\), and \(z\). From the Monte Carlo simulation results, we can conclude the following:  
1. By comparing Tables 2-5, it is evident the estimators under the WQEL function for the parameter \(q=2\), \(z=-1\) exceeded the estimators under other loss functions from a comprehensive point of view due to their smaller MISE values. This result is better visualized in Figure 5.
2. Based on the results in Tables 2 and 5, a Wilcoxon signed-rank test was conducted for the corresponding estimates under different priors in order to determine whether the estimates under the SEL and the WQEL with \(q=0.5\), \(z=-0.5\) were significantly different.