Document Type : Research Article
Authors
1 Department of Mathematics, Ganesh Dutt College, Begusarai, India.
2 Department of Mathematics, Lalit Narayan Mithila University, Darbhanga.
10.30473/coam.2025.73505.1287
Abstract
The incorporation of Pythagorean fuzzy sets into credit risk assessment represents a relatively innovative approach for predicting loan defaults, offering a more precise and adaptable tool for financial institutions. Key customer information—such as credit history, credit mix, credit utilization, duration of credit history, income level, and employment stability—is obtained as linguistic variables. These linguistic assessments are then transformed into Pythagorean fuzzy numbers. The combined Pythagorean fuzzy information is subsequently processed using the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). This approach employs a modified accuracy function to determine the Pythagorean fuzzy positive ideal solution and the Pythagorean fuzzy negative ideal solution. For distance calculations within the TOPSIS framework, spherical distance measurements are utilized. Alternatives are ranked based on the relative closeness coefficient and an adjusted index, collectively facilitating decision-making. The practical applicability of the proposed model is demonstrated through an illustrative numerical example.
Highlights
- Presents a novel approach that integrates Pythagorean fuzzy sets (PFS) into credit risk assessment to enhance loan default predictions.
- Converts linguistic variables—such as credit history, income, and employment stability—into Pythagorean fuzzy numbers to better manage uncertainty.
- Utilizes the TOPSIS method with spherical distance metrics to rank borrowers according to their credit risk.
- Demonstrates improved predictive accuracy and robustness compared to traditional credit scoring techniques.
- Includes a practical numerical example demonstrating the effectiveness of the proposed PFS-based framework.
- Outlines potential directions for future research, including model refinement, application across various financial sectors, and hybrid methods combining PFS with machine learning.
Keywords
- Pythagorean fuzzy sets
- Credit risk assessment
- Spherical distance measurement
- TOPSIS
- Revised index method
Main Subjects
[1] Adak, A.K., Kumar, D. (2023). “Spherical distance measurement method for solving MCDM problems under Pythagorean fuzzy environment”, Journal of Fuzzy Extension and Applications, 4(1), doi:10.22105/ jfea.2022.3516 77.1224.
[2] Adak, A.K., Kumar, G., Bhowmik, M. (2023). “Pythagorean fuzzy semi-prime ideals of ordered semi-groups”, International Journal of Computer Applications, 185(5), doi:10.5120/ijca2023922661.
[3] Astuti, I.F., Faizah, L., Khairina, D.M., Cahyadi, D. (2021). “A fuzzy Mamdani approach on community business loan feasibility assessment”, In 3rd 2021 East Indonesia Conference on Computer and Information Technology, EIConCIT 2021 (pp. 438–442). Institute of Electrical and Electronics Engineers Inc., doi:10.1109/EIConCIT50028.2021.9431899.
[4] Atanassov, K.T. (1989). “More on intuitionistic fuzzy sets”, Fuzzy Sets and Systems, 33(1), doi: 10.1016/0165-0114(89)90215-7.
[5] Bazmara, A., Donighi, S.S. (2014). “Bank customer credit scoring by using fuzzy expert system”, International Journal of Intelligent Systems and Applications in Engineering, 11, 29-35, doi: 10.5815/ijisa.2014.11.04.
[6] Bazmara, A., Sarkar, B. (2019). “Pythagorean fuzzy TOPSIS for multicriteria group decision-making with unknow weight information through entropy measure”, International Journal of Intelligent Systems, 34, 1108-1128, doi:10.1002/int.22088.
[7] Biryukov, A., Murzagalina, G., Kagarmanova, A., Kochetkova, S. (2022). “Concepts of improving fuzzy and neural network methods for simulating bankruptcy in risk management by the bank’s loan portfolio”, In Lecture Notes in Networks and Systems (vol. 381 LNNS, 513–524). Springer Science and Business Media Deutschland GmbH., doi:10.1007/978-3-030-93677-8_45
[8] Carvalho, J.P., Tome, J.A.B. (2009). “Rule based fuzzy cognitive map in socio-economic system”, In Proceedings of the 2009 International Fuzzy Systems Association World Congress and 2009 European Society for Fuzzy Logic and Technology Conference, Lisbon, Portugal, 1821-1826.
[9] Chen, Z.S., Zhou, J., Zhu, C.Y., Wang, Z.J., Xiong, S.H., Rodríguez, R.M., Skibniewski, M.J. (2023). “Prioritizing real estate enterprises based on credit risk assessment: An integrated multi-criteria group decision support framework”, Financial Innovation, 9(1), doi:10.1186/s40854-023-00517-y.
[10] Chourmouziadis, K., Chatzoglou, P.D. (2016). “An intelligent short term stock trading fuzzy system for assisting investors in portfolio management”, Expert Systems with Applications, 43, 298-311, doi:10.1016/j.eswa.2015.07.063.
[11] Garg, H. (2016). “A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem”, Journal of Intelligent & Fuzzy Systems, 31(1), 529-540, doi:10.3233/IFS-162165.
[12] He, X.X. , Li, Y.F., Qin, K.Y., Meng, D. (2019). “Distance measures on intuitionistic fuzzy based on intuitionistic fuzzy dissimilarity functions”, Soft Computing. 24(1), 523-541, doi:10.1007/ s00500-019-03932-5.
[13] Hesamian, G. (2020). “A fuzzy distance measure for fuzzy numbers”, Control and Optimization in Applied Mathematics, 5(1), 29-39, doi:10.30473/coam.2021.44290.1105.
[11] Garg, H. (2016). “A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem”, Journal of Intelligent & Fuzzy Systems, 31(1), 529-540, doi:10.3233/IFS-162165.
[12] He, X.X. , Li, Y.F., Qin, K.Y., Meng, D. (2019). “Distance measures on intuitionistic fuzzy based on intuitionistic fuzzy dissimilarity functions”, Soft Computing. 24(1), 523-541, doi:10.1007/ s00500-019-03932-5.
[13] Hesamian, G. (2020). “A fuzzy distance measure for fuzzy numbers”, Control and Optimization in Applied Mathematics, 5(1), 29-39, doi:10.30473/coam.2021.44290.1105.
[14] Hwang, C.L., Yoon, K. (1981). “Multiple attribute decision methods and applications: A state of the art survey”, Springer Verlag, New York, doi:10.1007/978-3-642-48318-9.
[15] Karimi, S.S. , Sohrabi, T., Bayat Tork, A. (2023). “Blockchain technology in optimizing logistics information security in business process technology transfer management”, Control and Optimization in Applied Mathematics, 8(2), 63-84, doi:10.30473/coam.2023.66693.1224.
[16] Krasavtseva, A. (2022). “Logical-linguistic method for assessing the risk of specialized lending (on the example of project financing)”, Economics and the Mathematical Methods, 58(4), 83, doi: 10.31857/s042473880020295-3.
[17] Lai, K.K., Yu, L., Wang, S., Zhou, L. (2006). “Neural Network Metalearning for Credit Scoring”, In: Huang, DS., Li, K., Irwin, G.W. (eds) Intelligent Computing. ICIC 2006. Lecture Notes in Computer Science, 4113. Springer, Berlin, Heidelberg, doi:10.1007/11816157_47.
[18] Li, D.Q., Zeng, W.Y. (2018). “Distance measure of Pythagorean fuzzy sets”, International Journal of Intelligent Systems, 33, 348-361, doi:10.1002/int.21934.
[19] Malhotra, R., Malhotra, D.K. (2002). “ Differentiating between good credits and bad credits using neuro-Fuzzy systems”, European Journal of Operational Research, 136, 190-211, doi:10.1016/ S0377-2217(01)00052-2.
[20] Makhazhanova, U., Kerimkhulle, S., Mukhanova, A., Bayegizova, A., Aitkozha, Z., Mukhiyadin, A., Azieva, G. (2022). “The evaluation of creditworthiness of trade and enterprises of service using the method based on fuzzy logic”, Applied Sciences (Switzerland), 12(22), doi:10.3390/app122211515.
[21] Nosratabadi, H.E., Nadali, A., Pourdarab, S. (2012). “Credit assessment of bank customers by a fuzzy expert system. based on rules extracted from association rules”, International Journal of Machine Learning and Computing, 2, 662-666, doi:10.7763/IJMLC.2012.V2.210.
[22] Peng, X.D., Li, W.Q. (2019). “Algorithms for interval-valued Pythagorean fuzzy sets in emergency decision making based on multi-parametric similarity measures and WSBA”, IEEE Access, 7, 7419-7441, doi:10.1109/ACCESS.2018.2890097.
[23] Peng, X., Yang, Y. (2015). “Some results for Pythagorean fuzzy sets”, International Journal of Intelligent Systems, 30, 1133-1160, doi:10.1002/int.21738.
[24] Polishchuk, V., Kelemen, M., Povkhan, I., Kelemen, M., Liakh, I. (2021). “Fuzzy model for assessing the creditworthiness of Ukrainian coal industry enterprises”, Acta Montanistica Slovaca, 26(3), 444-454, doi:10.46544/AMS.v26i3.05.
[25] Roy, P.K., Shaw, K. (2023). “An integrated fuzzy credit rating model using fuzzy-BWM and new fuzzy-TOPSIS-Sort-C”, Complex and Intelligent Systems, 9(4), 3581-3600, doi:10.1007/s40747-022-00823-5.
[26] Sartova, R., Mussina, A., Uakhitova, A. (2023). “Fuzzy logic application for credit risk assessment”. In AIP conference proceedings (vol. 2948), American Institute of Physics Inc., doi: 10.1063/5.0165250.
[27] Setiawan, T.H., Prihatini, L. (2023). “Tsukamoto fuzzy in optimizing the creditworthiness assessment process at savings and loan cooperatives”, Barekeng: Jurnal Ilmu Matematika Dan Terapan, 17(2), 0775-0786, doi:10.30598/barekengvol17iss2pp0775-0786.
[28] Wang, H.D., He, S.F., Pan, X. H. (2018). “A new bi-directional projection model based on Pythagorean uncertain linguistic variable”, Information, 9, 23-34, doi:10.3390/info9050104.
[29] Yager, R.R. (2016). “Properties and applications of Pythagorean fuzzy sets. In: Angelov, P., Sotirov, S. (eds) Imprecision and uncertainty in information representation and processing”, Studies in Fuzziness and Soft Computing, 332, Springer, Cham, doi:10.1007/978-3-319-26302-1_9.
[30] Yang, H., Yang, S. (2023). “Empirical research on credit system management for Chinese vocational college students based on personal mobile terminals”, International Journal of Intelligent Systems and Applications in Engineering, 12(6s), 386-400.
[31] Yu, L.P., Zeng, S.Z., Merigo, J.M., Zhang, C.H. (2019). “A new distance measure based on the weighted induced method and its application to Pythagorean fuzzy multiple attribute group decision making”, International Journal of Intelligent Systems, 34, 1440-1454, doi:10.1002/int.22102.
[32] Zadeh, L.A. (1965). “Fuzzy sets”, Information and Control, 8, 338-353.
[33] Zeng, W.Y., Li, D.Q., Yin, Q. (2018). “Distance and similarity measures of Pythagorean fuzzy sets and their applications to multiple criteria group decision making”, International Journal of Intelligent Systems, 33, 2236-2254, doi:10.1002/int.22027.
[34] Zhang, X.L., Xu, Z.S. (2014). “Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets”, International Journal of Intelligent Systems 29, 1016-1078, doi:10.1002/int.21676.
[35] Zhang, J., Guo, L., Lyu, T. (2021). “An enhanced personal credit identification coin-day destruction model based on blockchain technology fuzzy sets for region of China pearl river delta”, Journal of Intelligent and Fuzzy Systems, 41(3), 4519-4525, doi:10.3233/JIFS-189712.
)