## 3 Presentation of the Proposed Model
### Sub-Modules of the Presented Model
#### 3.3.4 ANN Learning Module  
The developed system allows integration with the MatLab software to model ANN. MatLab provides many algorithms for ANN training. Applying different learning algorithms results in different  
Figure 2: Relationships between order-picking time and input parameters for individual analysis, source: own elaboration. outcomes [27], some of which are used in this publication. As a function of the objective, minimization of the mean square error (MSE) was assumed. The consequential floating-point results were mathematically rounded off to the integers corresponding to the class of product affiliation. Afterwards, statistical analyses were performed to determine which of the methods allows one to designate a category of products that would allow for shorter order-picking times. Among the available learning algorithms, the most popular ones were used, e.g., [28]:  
* Trainoss--One step secant backpropagation;
* Traingd--Gradient descent backpropagation;
* Trains--Sequential order incremental training with learning functions;
* Trainlm--Levenberg-Marquardt backpropagation. To select the best learning algorithm, the simulations for the same model with different learning algorithms were done. The learning algorithms were compared by the quality of the simulation result. The criterion for selecting this algorithm was the minimization of the order-picking time. The achieved results are presented in Figure 3.  
As can be seen in Figure 3, the most different results, with comparison to others, are obtained after applying the trains learning function--a sequential method of modifying weights with a learning function. By making a preliminary review of statistical data, it can be concluded that the _trainlm_ function can achieve the best result since it has the lowest average value and median, while maintaining the smallest standard deviation, scope and maximum duration of order-picking (Table 4). For a selected case study, the following relationship of training data sets was made:  
* 70% of the collection--training data;
* 15% of the collection--validation data;
* 15% of the collection--testing data. During data preparation, the following Matlab functions were applied: Mapminmax, Removeconstantrows. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Learning** & \multirow{2}{*}{**Average**} & \multirow{2}{*}{**Standard**} & \multirow{2}{*}{**Median**} & \multirow{2}{*}{**Coefficient of**} & \multirow{2}{*}{**Min.**} & \multirow{2}{*}{**Max.**} & \multirow{2}{*}{**Range**} \\
**Function** & & & & & & & \\ \hline traingd & **938.3** & **312.8** & **929.7** & 0.333 & 198.4 & **1926.4** & **1728.0** \\ \hline trainlm & 872.1 & 288.9 & 857.5 & 0.331 & **201.2** & 1771.2 & 1570.0 \\ \hline trainoss & 886.7 & 292.1 & 879.2 & 0.329 & 201.2 & 1958.0 & **1756.8** \\ \hline trains & **971.5** & **334.3** & **958.2** & **0.344** & 198.4 & 1894.0 & 1695.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistics of order-picking time depending on the ANN learning function, sec. Figure 3: Time of product picking depending on the ANN learning function used, sec. ; on **left—**results for each case as a point cloud, on **right—**results as box-and-whisker chart; source: own elaboration (Statgraphics Centurion 15.2.11.0). The _Mapminmax_ function is used to normalize the input data, which also reduces the necessary calculation time. Before training, the inputs and targets are scaled so that they always fall within a specified range. The function _mapminmax_ scales inputs and targets so that they fall in the range [-1,1]. The original network inputs and targets are given in the matrices \(p\) and \(t\). The normalized inputs and targets \(pn\) and \(tn\) contain the settings; in this case, the minimum and maximum values of the original inputs and targets. After the network has been trained, the ps settings should be used to transform any future inputs that are applied to the network. They effectively become a part of the network, just like the network weights and biases.