# Reliability Analysis of Military Vehicles Based on Censored Failures Data
## 3 Results and Discussion
### Weibull Model of Reliability  
The approximation of the reliability function \(R(t)\) with the Weibull model was performed using the nonlinear least-squares method, analogous to the exponential model. Selection of optimal values of parameters \(\alpha\) and \(\beta\) was carried out with minimisation of the objective function according to the relation [59; 60; 61]:  
\[S_{n}(\alpha,\beta)=\sum_{i=1}^{n}\left(\widehat{R}(t_{i})-f(t_{i},\alpha,\beta )\right)^{2}, \tag{24}\]  
where \(f(t_{i},\alpha,\beta)\) is the value of the Weibull model of the reliability function \(R(t)\) with parameters \(\alpha\) and \(\beta\) for the value of \(t_{i}\).  
\(S_{n}(\alpha,\beta)\) takes the minimum value for the critical point for which the following relations are valid:  
\[\frac{\partial S_{n}(\alpha,\beta)}{\partial\alpha}=0, \tag{25}\]  
\[\frac{\partial S_{n}(\alpha,\beta)}{\partial\beta}=0. \tag{26}\]  
Transforming the above relations, analogous to the case of the exponential model, the following two conditions are obtained:  
\[\sum_{i=1}^{n}\left[\widehat{R}(t_{i})-f(t_{i},\alpha,\beta)\right]\left(- \frac{\partial f(t_{i},\alpha,\beta)}{\partial\alpha}\right)=0, \tag{27}\]  
\[\sum_{i=1}^{n}\left[\widehat{R}(t_{i})-f(t_{i},\alpha,\beta)\right]\left(- \frac{\partial f(t_{i},\alpha,\beta)}{\partial\beta}\right)=0. \tag{28}\]  
After the approximation by the nonlinear least-squares method, using the Trust Region algorithm [62; 63], the two-parameter Weibull model was obtained, which is represented by the formula:  
\[R(t)=e^{-0.000574t^{0.889}}. \tag{29}\]  
Figure 6 shows the course of the approximation curve with the points corresponding to the values obtained from the estimation. The small distances between the curve and the points indicate a good fit of the Weibull model.  
Figure 5: Graph presenting exponential model fitting to estimated values of reliability function.  
The analysis of the \(MSE\) error values and the \(R\) correlation coefficient presented in Figures 7 and 8 show a slight improvement in the degree of fitting of the neural model to the estimated values of the reliability function. The graphs shown in Figures 9 and A1-A9 (Appendix A) illustrate the approximation of the reliability function by neural models with a number of neurons in the hidden layer ranging from 1 to 10. While the number of hidden neurons increases, the model tends to interpolate the function. For a developed model based on 9 and 10 neurons, the approximating function takes a non-monotonic form and therefore does not satisfy the condition of non-increasing reliability function. The uncomplicated form and good fit of the MLP 1-1-1 model led us to choose this neural network as the suitable one for further analyses and considerations.  
Figure 9 shows the course of the reliability function approximation curve in the MLP 1-1-1 neural model together with the estimated values and the difference between the estimated value and the modelled value. The degree of fitting in each dataset together  
Figure 8: Analysis of sensitivity of \(R\) correlation coefficient in neural model as a function of hidden neurons number.  
Figure 9: Reliability function approximated by MLP 1-1-1.  
with scatter plots are shown in Figure 10. The similar values of the \(R\) correlation coefficient indicate a valid approximation, without the occurrence of network overtraining and fitting only to the data from the training set.  
The low complexity of the MLP 1-1-1 neural model makes it easy to obtain its analytical form. For the analysed example, the MLP 1-1-1 model can be represented by relation (29):  
\[R(t)=\frac{32.925260+1.786689e^{0.000249t}}{1+34.674828e^{0.000249t}}. \tag{30}\]