## 3 Materials and Methods
### Implementation Methodology
#### 3.9.4 Neural Network Training
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Part No.** & **Nomenclature** & \begin{tabular}{l} **Quantity** \\ **Demanded** \\ \end{tabular} & \
\begin{tabular}{l} **Lumpiness Factor** \\ **Categorization** \\ \end{tabular} &
\begin{tabular}{l} **Demand** \\ **Categorization** \\ \end{tabular} \\ \hline
246-3904-000 & Tail rotor hub & 48 & 1.230240223 & Lumpy \\ \hline
246-3925-00 & Tail rotor blade & 44 & 1.391716506 & Lumpy \\ \hline LOPR-15875-2300-1 & Tail rotor chain & 41 & 1.543305508 & Lumpy \\ \hline KAY-115 AM & Hydraulic booster & 32 & 1.404591553 & Erratic \\ \hline HP-3 BM & Fuel control unit & 31 & 1.17969417 & Lumpy \\ \hline TB3-117 BM & Engine assembly & 30 & 2.228030758 & Lumpy \\ \hline AK50T-1 & Air compressor & 27 & 1.404358296 & Lumpy \\ \hline
8 AT-1250-00-02 & Vibration damper assembly & 24 & 1.334541546 & Lumpy \\ \hline
8 AT-2710-000 & Main rotor blade set of 5 & 24 & 1.723303051 & Lumpy \\ \hline
8-1930-00 & Main rotor hub & 22 & 1.423085356 & Lumpy \\ \hline Grand Total & 323 * & & \\ \hline \hline \end{tabular}  
* Selected entries are shown here from the full sample space of lumpy spares held at an aviation spare depot.  
\end{table} 
Table 2: Spare parts demand categorization. Summary July 2009–June 2015. Figure 6: Architecture of proposed ANN. The outcome of the trained model was simulated by the trained and validated ANN model [59]. After that, post-processing was carried out. The descaling of the outcome was performed by the inverse transformation equation [58]. \[x=x_{min}+\frac{(T-0.1)(x_{max}-x_{min})}{0.8} \tag{13}\]  
The mean absolute percentage error (MAPE) is a popular metric for measuring the accuracy of a predictive model. [PERSON] et al. [36] use MAPE as a criterium to evaluate the accuracy of different forecasting models. The MAPE is calculated as the average of the absolute percentage errors over all forecasted periods. The formula for calculating MAPE is as follows  
\[MAPE=\frac{1}{n}\sum_{t=1}^{n}|\frac{A_{t}-F_{t}}{A_{t}}| \tag{14}\]  
where \(F_{t}\) is the forecasted value at time period \(t\), \(A_{t}\) is the actual value at time period \(t\), \(n\) is the total number of time periods, and the absolute value and percentage are taken to ensure that the errors are positive and relative to the actual value. This criterion of MAPE is used to compare the forecasting performance of different models, including ARIMA, seasonal decomposition of time series, exponential smoothing, and neural networks, on various demand patterns for aircraft spare parts [36]. There are other metrics such as MAE, MASE and root mean square error (RMSE) that are also found to be used for similar purposes in the literature [60; 61; 62; 63]. MAE is a commonly used metric in time series forecasting to evaluate the performance of a forecasting model. MAE measures the average magnitude of errors between actual and predicted values. It measures the average absolute difference between the actual and predicted values of a variable. The formula for calculating MAE is  
\[MAE=\frac{1}{N}*\sum|y_{i}-\tilde{y}| \tag{15}\]  
where \(n\) is the total number of observations, \(y_{i}\) is the actual value of the \(i\)-\(th\) observation and \(\tilde{y}\) is the mean value of all actual values. In other words, it computes the absolute difference between each actual value and the mean value, sums them up, and then divides by the total number of observations. The MAE is a useful metric because it gives a sense of the magnitude of the errors in predictions, regardless of their direction. It is also relatively easy to interpret, as it is expressed in the same units as the variable being predicted. MASE is another common metric used to evaluate the accuracy of a time series forecasting model. It measures the forecast's accuracy relative to a naive forecast and allows for comparisons of different forecasting methods across different time series.