# A q-rung orthopair basic probability assignment and its application in medical diagnosis

• Bulendra Limboo Department of Mathematics, Dibrugarh University, Assam, India
• Palash Dutta Department of Mathematics, Dibrugarh University, Assam, India
Keywords: Fuzzy soft set, q-rung belief function, Association coefficient, Medical diagnosis

### Abstract

Dempster-Shafer theory is widely used in decision-making and considered as one of the potential mathematical tools in order to fuse the evidence. However, existing studies in this theory show disadvantage due to conflicting nature of standard evidence set and the combination rule of evidence. In this paper, we have constructed the framework of q-rung evidence set to address the issue of conflicts based on the q-rung fuzzy number due to its more comprehensive range of advantage compared to the other fuzzy or discrete numbers. The proposed q-rung evidence set has the flexibility in assessing a parameter through the q-rung orthopair basic probability assignment consisting of membership and non-membership belief degree. Moreover, as the proposed q-rung orthopair basic probability assignment consists of pair of belief degrees, the possibility of conflicts cannot be ignored entirely. In this regard, a new association coefficient measure is introduced where each component of the belief degrees is modified through the weighted average mass technique. This paper uses various concept such as fuzzy soft sets, Deng entropy, association coefficient measure and score function for decision-making problem. Firstly, to obtain the initial q-rung belief function, we have implemented the Intuitionistic fuzzy soft set to assess the parameter of the alternatives and Deng entropy to find the uncertainty of the parameters. Secondly, the association coefficient measure is used to avoid the conflict through the modified form of evidence. Finally, we combined the evidence and found the score value of the Intuitionistic fuzzy numbers for the ranking of the alternatives based on the score values of alternatives. This study is validated with the case study in the medical diagnosis problem from the existing paper and compared the ranking of alternatives based on the score function of belief measures of the alternatives.

### References

Atanassov, K. (1986). Intuitionistic fuzzy sets. Fuzzy Sets & Systems, 20(1), 87-96. DOI: https://doi.org/10.1016/S0165-0114(86)80034-3

Basu, T.M., Mahapatra, N.K., & Mondal, S.K. (2012). A balanced solution of a fuzzy soft set based decision-making problem in medical science. Applied Soft Computing, 12, 3260-3275. DOI: https://doi.org/10.1016/j.asoc.2012.05.006

Çelik, Y., & Yamak, S. (2013). Fuzzy soft set theory applied to medical diagnosis using fuzzy arithmetic operations. Journal of Inequality and Applications, 2013, 82. DOI: https://doi.org/10.1186/1029-242X-2013-82

Chen, L., Diao, L., & Jun, S. (2019). A novel weighted evidence combination rule based on improved entropy function with a diagnosis application. International Journal of Distributed Sensor Networks, 15(1), 1-13.

Cheng, C., & Xiao, F. (2019). A new distance measure of belief function in evidence theory. IEEE Access, 7, 2169-3536.

Cuong, B.C. (2014). Picture fuzzy sets. Journal of Computer Science and Cybernetics, 30(4), 409-420. DOI: https://doi.org/10.15625/1813-9663/30/4/5032

Das, S., Kar, M.B. & Kar, S. (2013). Group multi-criteria decision making using intuitionistic multi-fuzzy sets. Journal of Uncertainty Analysis and Applications, 1, 1-16. DOI: https://doi.org/10.1186/2195-5468-1-10

Das, S., Kar, S. & Pal, T. (2017). Robust decision making using intuitionistic fuzzy numbers. Granular Computing, 2(1), 41–54. DOI: https://doi.org/10.1007/s41066-016-0024-3

Dempster, A. (1967). Upper and lower probabilities induced by a multi-valued mapping. Annals of Mathematical Statistics, 38(2), 325-339. DOI: https://doi.org/10.1214/aoms/1177698950

Deng, Y. (2015). Deng entropy: a generalized Shannon entropy to measure uncertainty, viXra, https://doi.org/10.5281/zenodo.32211.

Deng, Y. (2016). Deng entropy. Chaos, Solitons & Fractals, 91, 549-553. DOI: https://doi.org/10.1016/j.chaos.2016.07.014

Deng, Y., Shi, W., Zhu, Z., & Liu, Q. (2004). Combining belief functions based on distance of evidence. Decision Support System, 38(3), 489-493. DOI: https://doi.org/10.1016/j.dss.2004.04.015

Dubois, D., & Prade, H. (1986). A set theoretic view on belief functions: Logical operations and approximations by Fuzzy sets. International Journal of General Systems, 12(3), 193-226. DOI: https://doi.org/10.1080/03081078608934937

Dutta, P., & Limboo, B. (2017). Bell-shaped Fuzzy Soft Sets and Their Application in Medical Diagnosis, Fuzzy Information and Engineering, 9(1), 67-91. DOI: https://doi.org/10.1016/j.fiae.2017.03.004

Fei, L., & Deng, Y. (2018). A new divergence measure for basic probability assignment and its applications in extremely uncertain environments. International Journal of Intelligent System, 34(4), 584-600.

Gao, X., & Deng, Y. (2019). Quantum model of mass function. Internal Journal of Intelligent System, 35(2), 267-282.

Gorzałczany, M.B., (1987). A method of inference in approximate reasoning based on Interval-valued fuzzy sets. Fuzzy Sets and Systems, 21(1), 1-17. DOI: https://doi.org/10.1016/0165-0114(87)90148-5

Hussain, A., Muhammad, Ali., Mahmood, T., & Muhammad, M. (2020). q‐Rung orthopair fuzzy soft average aggregation operators and their application in multicriteria decision‐making. International Journal Intelligent System, 35(4), 1–29.

Inagaki, T. (1991). Interdependence between safety control policy and multiple sensor schemes via Dempster-Shafer theory. IEEE Transaction on Reliability, 40(2), 182-188. DOI: https://doi.org/10.1109/24.87125

Jiang, W. (2016). A correlation coefficient for belief functions. International Journal of Approximate Reasoning, 103, 94-106.

Jiang, W., Wei, B., Xie, C., & Zhou, D. (2016). An evidential sensor fusion method in fault diagnosis. Advances in Mechanical Engineering, 8(3), 1–7. DOI: https://doi.org/10.1177/1687814016641820

Jiang, W., Zhuang, M., Qin, X., & Tang, Y. (2016). Conflicting evidence combination based on uncertainty measure and distance of evidence. Springer Plus, 5(1), 1217. DOI: https://doi.org/10.1186/s40064-016-2863-4

Jousselme, A., Grenier, D., & Bosse, E. (2001). A new distance between two bodies of evidence. Information Fusion, 2(2), 91-101. DOI: https://doi.org/10.1016/S1566-2535(01)00026-4

Krishankumar, R., Ravichandran, K.S., Kar, S., Cavallaro, F., Zavadskas, E.K., & Mardani, A. (2019). Scientific Decision Framework for Evaluation of Renewable Energy Sources under Q-Rung Orthopair Fuzzy Set with Partially Known Weight Information, Sustainability, 11(15), 4202(1-21).

Li, Y., & Deng, Y. (2019). Intuitionistic Evidence Sets. IEEE Access, 7, 106417-106426.

Li, Z., Wen, G., & Xie, N. (2015). An approach to fuzzy soft sets in decision-making based on grey relational analysis and Dempster-Shafer theory of evidence: An application in medical diagnosis. Artificial Intelligence in Medicine, 64(3), 161–171. DOI: https://doi.org/10.1016/j.artmed.2015.05.002

Maji, P., Biswas, R., & Roy, A. (2001). Fuzzy soft sets. Journal of Fuzzy Mathematics, 9(3), 589‐602.

Maji, P., Biswas, R., & Roy, A. (2001). Intuitionistic fuzzy soft sets. Journal of Fuzzy Mathematics, 9(3), 677‐692.

Mishra, A.R., Rani P, Krishankumar, R., Ravichandran, K.S., & Kar, S. (2021). An extended fuzzy decision-making framework using hesitant fuzzy sets for the drug selection to treat the mild symptoms of Coronavirus Disease 2019 (COVID-19), Applied Soft Computing, 103, 107155, 1-18.

Molodtsov, D. (1999). Soft set theory‐first results. Computers and Mathematics with Applications, 37(4–5), 19‐ 31. DOI: https://doi.org/10.1016/S0898-1221(99)00056-5

Murphy, C. (2000). Combining belief functions when evidence conflicts. Decision Support Systems, 29(1), 1-9. DOI: https://doi.org/10.1016/S0167-9236(99)00084-6

Pan, L., & Deng, Y. (2019). An association coefficient of a belief function and its application in a target recognition system. International Journal of Intelligent System, 35(1), 85-104. DOI: https://doi.org/10.1002/int.22200

Peng, X, Dai, J., & Garg, H. (2018). Exponential operation and aggregation operator for q-rung orthopair fuzzy set and their decision-making method with a new score function. International Journal of Intelligent System, 33(11), 2255– 2282.

Peng, X., & Yang, Y. (2015). Some results for Pythagorean fuzzy sets. International Journal of Intelligent Systems, 30(11), 1133–1160. DOI: https://doi.org/10.1002/int.21738

Peng, X., Yang, Y., Song, J., & Jiang, Y. (2015). Intuitionistic fuzzy soft set and its application. Computer Engineering, 41(7), 224-229.

Peng, X., Yang, Y., Song, J., & Jiang, Y. (2015). Pythagorean fuzzy soft set and its application, Computer Engineering, 41(7), 2-24.

Shafer, G. (1976). A mathematical theory of evidence. Princeton: Princeton University Press.

Si, A., Das, S., & Kar, S. (2021). Picture fuzzy set-based decision-making approach using Dempster–Shafer theory of evidence and grey relation analysis and its application in COVID-19 medicine selection, Soft Computing, 1-15, doi: 10.1007/s00500-021-05909-9.

Song, Y., Wang, X., & Zhang, H. (2015). A distance measure between intuitionistic fuzzy belief functions. Knowledge-Based Systems, 86, 288-298. DOI: https://doi.org/10.1016/j.knosys.2015.06.011

Wang, J., Hu, Y., & Xiao, F. (2016). A novel method to use fuzzy soft sets in decision making based on ambiguity measure and Dempster-Shafer theory of evidence: an application in medical diagnosis. Artificial Intelligence Medicine, 69, 1–11. DOI: https://doi.org/10.1016/j.artmed.2016.04.004

Xiao, F. (2018). A hybrid fuzzy soft sets decision making method in medical diagnosis. IEEE Access, 6, 25300-25312. DOI: https://doi.org/10.1109/ACCESS.2018.2820099

Xiao, F. (2018). An improved method for combining conflicting evidences based on the similarity measure and belief function entropy. International Journal of Fuzzy System, 20(4), 1256-1266. DOI: https://doi.org/10.1007/s40815-017-0436-5

Xiao, F. (2019). Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy. Information Fusion, 46, 23-32. DOI: https://doi.org/10.1016/j.inffus.2018.04.003

Xiao, F. (2020). Generalization of Dempster–Shafer theory: A complex mass function. Applied Intelligence, 50(10), 3266-3275. DOI: https://doi.org/10.1007/s10489-019-01617-y

Xiao, F., & Qin, B. (2018). A weighted combination method for conflicting evidence in multi-sensor data fusion. Sensors, 18(5), 74-87. DOI: https://doi.org/10.3390/s18051487

Yager, R.R. (1987). On the Dempster-Shafer framework and new combination rules. Information Science, 41(2), 93-137. DOI: https://doi.org/10.1016/0020-0255(87)90007-7

Yager, R.R. (2001). Dempster-Shafer belief functions with interval valued focal weights. Internal Journal of Intelligent System, 16(4), 497–512. DOI: https://doi.org/10.1002/int.1020

Yager, R.R. (2013). Pythagorean fuzzy subsets. Proceedings of the Joint IFSA World Congress and NAFIPS Annual Meeting Edmonton Canada, 57‐61. DOI: https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375

Yager, R.R. (2014). An intuitionistic view of the Dempster–Shafer belief structure. Soft Computing, 18, 2091-2099. DOI: https://doi.org/10.1007/s00500-014-1320-y

Yager, R.R. (2016). Generalized orthopair fuzzy sets. IEEE Transactions Fuzzy Systems, 25(5), 1222‐1230. DOI: https://doi.org/10.1109/TFUZZ.2016.2604005

Zadeh, L. (1965). Fuzzy sets. Information Control, 8(3), 338‐356. DOI: https://doi.org/10.1016/S0019-9958(65)90241-X

Zhang, L. (1994). Representation, independence, and combination of evidence in the Dempster-Shafer theory. Advances in the Dempster-Shafer Theory of Evidence (pp. 51-69), New York: John Wiley & Sons Inc.

Zhou, Q., Mo, H., & Deng, Y. (2020). A new divergence measure of Pythagorean fuzzy sets based on belief function and its application in medical diagnosis. Mathematics, 8(1), 142.

Published
2022-05-20
How to Cite
Limboo, B., & Dutta, P. (2022). A q-rung orthopair basic probability assignment and its application in medical diagnosis. Decision Making: Applications in Management and Engineering, 5(1), 290-308. https://doi.org/10.31181/dmame191221060l
Section
Regular articles