# Reliability Analysis of Military Vehicles Based on Censored Failures Data
## 2 Methodology
### Neural Modelling  
Artificial neural networks are an advanced tool for modelling, classification and prediction of processes, phenomena and objects based on a set of data [48]. Multilayer Perceptron (MLP) is a neural network based on three types of neuron layers: input, hidden (or several hidden) and output [27; 49; 50]. The input layer neurons are responsible for sending input signals to the hidden layer neurons. For predictive models, the input signals are the normalised values of the explanatory variables. Hidden neurons aggregate the received input signals multiplied by the weights of the synaptic connections. The aggregated signal is the argument of the activation function, whose value is passed to the neurons of the next hidden layer or to the neurons of the output layer. The output layer neurons perform the same actions as the hidden neurons, except that the value obtained is equated to the value of the explanatory variable in the model [51; 52; 53].  
The use of a simple neural network to approximate functions on the basis of estimated values obtained from empirical studies makes it possible to develop models that fit real data with a very high accuracy. Additionally, a regression network with an uncomplicated structure allows the approximating function to be written in analytical form.  
Figure 3 shows a flowchart illustrating the operation of the proposed neural network used to approximate the reliability function. The input signal is the mileage between failures, for which the value of the reliability function has been estimated. The output signal is the value of the reliability function. In the proposed neural model, the activation function of the hidden layer is a hyperbolic tangent, which ensures a fast learning process of the neural network [54; 55]. For the output layer, on the other hand, activation occurs using a linear function.  
If the variables \(x\) and \(y\) represent the normalised values of the variables \(t\) and \(R(t)\) according to the following relations:  
\[x=\frac{2(t-t_{\min})}{t_{\max}-t_{\min}}-1, \tag{14}\]  
\[y=\frac{2\Big{(}\widehat{R}(t)-\widehat{R}(t)_{\min}\Big{)}}{\widehat{R}(t)_{ \max}-\widehat{R}(t)_{\min}}-1. \tag{15}\]  
Then the analytical form of the neural model is represented by the relation:  
\[y=\left[\sum_{n=1}^{h}\left(\left(\frac{2}{1+e^{-2(xw_{1n}+w_{0n})}}-1\right) \cdot v_{n1}\right)\right]+v_{01}, \tag{16}\]  
where \(h\) is the number of hidden neurons, \(w\) represents weights of synaptic connections between the input layer and the hidden layer, and \(v\) are weights of synaptic connections between the hidden layer and the output layer.  
During the training process, the neural network, using the Levenberg-Marquardt algorithm, seeks to minimise the value of its error function, which is calculated according to the following formula:  
\[E(n)=\frac{1}{2}\sum_{i=1}^{n}\Big{(}\widehat{R}(t_{i})-R_{out}(t_{i})\Big{)}^ {2}, \tag{17}\]  
where \(\widehat{R}(t_{i})\) denotes the estimated value of the reliability function at point \(t_{i}\), while \(R_{out}(t_{i})\) is the value obtained at the output of the neural network.