**6. Conclusions**

This work presented an evaluation through evolving fuzzy neural network models to address auction fraud problems. The eFNN-SODA model proposed in this paper achieved more than 98% in correctly classifying fraudulent situations or not within a highly unbalanced data set. eFNN-SODA can be considered the most straightforward approach compared with the other models evaluated in the experiment, as it performed the fraud classification tasks with fewer fuzzy rules and greater assertiveness. The model also presented distinguishable fuzzy rules because their antecedents and consequents are dissimilar.

Another factor verified in the experiments is that the inconsistencies of the fuzzy rules during the experiment were minimal and promptly corrected by training eFNN-SODA. Regarding the coverage criterion, as they are Gaussian neurons formed in the fuzzification process, eFNN-SODA also meets the requirement. Finally, the model meets the criterion of completeness throughout the experiment. Therefore, it can be concluded that the model's fuzzy rules presented at the end of the training meet the criteria of consistency, coverage, and completeness.

The evaluation of problem features helped the classifier to correctly identify the classes by assigning to each fuzzy rule antecedent a weight corresponding to its relevance in correctly identifying the problem classes. Factors such as the expansion of knowledge through a new approach, the visualization of the interpretability of the problem, and the analysis of antecedents and consequents were also presented in this article. They corroborate the statement that the fuzzy rules generated by the eFNN-SODA model are efficient and interpretable in solving fraud identification problems in an auction.

The model proposed in this paper can generate technological products that evolve their knowledge as the behavior of fraud changes. This can happen by creating auxiliary expert systems to validate or investigate certain behaviors in the auctions. Fuzzy rules can be shown to system administrators and users, allowing corrective or analysis actions. With this type of system, both parties have clear and understandable logical relationships. Finally, creating an expert system to assist in data validation after the auction is also possible.

Possible future work can be conducted to develop new training techniques for model evolution and to use other neural structures to improve model accuracy. Another interesting topic to address is to evaluate or propose new methods for verifying the interpretability of models evolving neuro-fuzzy networks. The validation of the generated fuzzy rules can also be a potential focus of future research. Other (less unbalanced) data sets and auction fraud assessments are also encouraged to verify the model's adaptability in solving the problems of this behavior.

**Author Contributions:** Conceptualization, P.V.d.C.S. and E.L.; methodology, P.V.d.C.S. and H.R.B.; software, P.V.d.C.S.; validation, E.L.; formal analysis, E.L.; investigation, P.V.d.C.S. and A.J.G.; resources, P.V.d.C.S.; data curation, E.L.; writing—original draft preparation, P.V.d.C.S., H.R.B., and A.J.G.; writing—review and editing, E.L. and H.R.B.; visualization, P.V.d.C.S., A.J.G., and H.R.B.; supervision, E.L.; project administration, E.L.; funding acquisition, E.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** Open Access Funding by the Austrian Science Fund (FWF), contract number P32272-N38, acronym IL-EFS.

**Data Availability Statement:** Publicly available datasets were analyzed in this study. These data can be found here: https://archive.ics.uci.edu/ml/datasets/Shill+Bidding+Dataset (accessed on 12 August 2022).

**Conflicts of Interest:** The authors declare no conflict of interest.
