## 4 Results
### A Model Based on Neural Networks for Predicting Customers Loyalty  
On the basis of our conclusions, 13 variables were used to project loyalty: (1) age of the respondents (K2); (2) the origin of the vehicle--new or used (S1) and the variables measuring the compatibility with the statements claiming that: (3) the materials concerning the provision of the services by the auto repair shop (leaflets, folders, website) are visually attractive (P4); (4) operation materials and spare parts that are used by the auto repair shop, which are available in the course of the provision of a service in case of need (P5); (5) the auto repair shop is clean and tidy (P6); (6) documentation connected with the provided services is correct (P10); (7) the services that are offered by the auto repair shop are in line with customers' expectations (P11); (8) the services are individualized and adjusted to the needs of the customers (P14); (9) the auto repair shop is prepared to provide a given service every time that it is needed (P16); (10) the workers employed in the auto repair shop are kind and polite (P17); (11) the worker of the auto repair shop is helpful in selecting operation materials and spare parts (P19); (12) the auto repair shop guarantees an alternative vehicle to be used during the repair (P20); and (13) the services that are offered by the auto repair shop are provided in the comprehensive way (P21). The remaining variables were not considered due to the correlation with the variables listed above or due to the marginal impact on the projected variable. In Table 1 we present 16 different variants of models which were tested.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Network** & **Learning** & **Number** & **Activation** & **ASE for** & **ASE for** & **MR for** \\
**Code** & **Algorithm** & **of Layers** & **Function** & **L Measure** & **SOW Measure** & **NPS Measure** \\ \hline Mod1 & Backprop & 3 & logistic & 0.288 & 0.166 & 0.752 \\ Mod2 & rprop\(+\) & 3 & logistic & 0.302 & 0.168 & 0.761 \\ Mod3 & rprop\(-\) & 3 & logistic & 0.328 & 0.171 & 0.759 \\ Mod4 & Sag & 3 & logistic & 0.314 & 0.173 & 0.760 \\ Mod5 & Backprop & 5 & logistic & 0.291 & 1.681 & 0.783 \\ Mod6 & rprop\(+\) & 5 & logistic & 0.293 & 0.172 & 0.785 \\ Mod7 & rprop\(-\) & 5 & logistic & 0.305 & 0.175 & 0.783 \\ Mod8 & Sag & 5 & logistic & 0.314 & 0.177 & 0.780 \\ Mod9 & Backprop & 3 & tanh & 0.301 & 0.201 & 0.754 \\ Mod10 & rprop\(+\) & 3 & tanh & 0.331 & 0.204 & 0.760 \\ \hline \end{tabular}
\end{table}
Table 1: Characteristics of analyzed neural networks.  
Figure 4: Conceptual model presenting the projection of the loyalty of the customers of the auto repair shops. Source: authors’ study.  
As can be seen in the last three columns in Table 1, from among analyzed neural networks, the model based on the backpropagation error with three hidden layers (in Table 1 marked as Mod1) proved to be the best. Average Squared Error for the loyalty expressed by the frequency of the visits (L) is 0.288 and for the loyalty in the value-oriented perspective (SOW) it is even lower and equals to 0.166. These low values translate into extremely high efficiency of the network (projection accuracy). Whereas, Misclassification Rate for the NPS measure turns out to be relatively high, with the amount of 0.75, while the projection with the use of this network gives much better results than the random classification.  
The forms of the obtained neural networks are presented in Figures 5-7 for the loyalty expressed by L, SOW, and NPS, respectively.  
In connection with the above, it may be stated that the created neural networks project the loyalty of the customers of the auto repair shops in appropriate manner.  
Figure 6: Neural network for the projection of the loyalty of customers by Share of Wallet (SOW) measure. Source: authors’ study.  
Figure 7: Neural network for the projection of the loyalty of customers by Net Promoter Score (NPS) measure. Source: authors’ study.