Game-Theoretic Applications of Pentagonal Fuzzy Numbers Through New Representation and Defuzzification Schemes

Authors

  • Ganesh Kumar * Department of Mathematics, University of Rajasthan, Jaipur-302004, India.
  • Vinod Jangid Department of Mathematics, University of Rajasthan, Jaipur-302004, India.
  • Govind Shay Sharma Department of Mathematics, Acharya Narendra Dev College University of Delhi, India.

https://doi.org/10.22105/opt.vi.96

Abstract

This research paper presents novel insights into the representation, ranking, and defuzzification of pentagonal fuzzy numbers. The fundamental foundation of this study lies in the development of a robust pentagonal fuzzy number, which is explored through diverse representations that harness the principles of continuity within the membership function. To facilitate practical implementation, a variety of defuzzification methods are meticulously applied, resulting in the transformation of fuzzy data into crisp. The significance of pentagonal fuzzy numbers is further illuminated through their application in the context of game theory, specifically within the domain of matrix games. This strategic analysis explains the pragmatic relevance of pentagonal fuzzy numbers in deciphering complex real-world scenarios and optimizing decision-making processes. Theoretical constructs are bolstered by numerical examples, empirically showcasing the practical applicability of the developed theory.

Keywords:

Pentagonal fuzzy number, Defuzzification, Two person zero sum game

References

  1. [1] Abbasbandy, S., & Hajjari, T. (2009). A new approach for ranking of trapezoidal
  2. [2] fuzzy numbers. Computers & mathematics with applications, 57(3), 413-419. https://doi.org/10.1016/j.camwa.2008.10.090
  3. [3] Asady, B., & Zendehnam, A. (2007). Ranking fuzzy numbers by distance mini-
  4. [4] mization. Applied mathematical modelling, 31(11), 2589-2598.https://doi.org/10.1016/j.apm.2006.10.018
  5. [5] Atanassov, K. T. (1999). Intuitionistic fuzzy sets. In Intuitionistic fuzzy sets: theory
  6. [6] and applications (pp. 1-137). Heidelberg: Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-1870-3_1
  7. [7] Atanassov, K. T. (1986) ‘Intuitionistic fuzzy sets’, Fuzzy Sets Syst, 20(1), 87–96. [5] Buckley, J. J. (1988). Possibilistic linear programming with triangular fuzzy
  8. [8] numbers. Fuzzy sets and Systems, 26(1), 135-138. https://doi.org/10.1016/0165-0114 (88)90013-9
  9. [9] Chakraborty, A., Mondal, S. P., Alam, S., Ahmadian, A., Senu, N., De, D., & Salahshour, S. (2019). The pentagonal fuzzy number: its different representations, properties, ranking, defuzzification and application in game problems. Symmetry, 11(2), 248.
  10. [10] Chen, S. M. (2011). Analyzing fuzzy risk based on a new fuzzy ranking method between generalized fuzzy numbers. Expert Systems with Applications, 38(3), 2163-2171. https://doi.org/10.1016/j.eswa.2010.08.002
  11. [11] Cheng, C. H. (1998). A new approach for ranking fuzzy numbers by distance method. Fuzzy sets and systems, 95(3), 307-317. https://doi.org/10.1016/S0165-0114
  12. [12] (96)00272-2
  13. [13] Chou, C. C. (2003). The canonical representation of multiplication operation on
  14. [14] triangular fuzzy numbers. Computers & Mathematics with Applications, 45(10-11),
  15. [15] -1610. https://doi.org/10.1016/S0898-1221(03)00139-1
  16. [16] Chutia, R., & Chutia, B. (2017). A new method of ranking parametric form of
  17. [17] fuzzy numbers using value and ambiguity. Applied soft computing, 52, 1154-
  18. [18] https://doi.org/10.1016/j.asoc.2016.09.013
  19. [19] Dubois, D., & Prade, H. (1978). Operations on fuzzy numbers. International jour-
  20. [20] nal of systems science, 9(6), 613-626. https://doi.org/10.1080/00207727808941724
  21. [21] Dubois, D., & Prade, H. (Eds.). (2012). Fundamentals of fuzzy sets (Vol. 7).
  22. [22] Springer Science & Business Media. https://link.springer.com/book/10.1007/978-1-4615
  23. [23] -4429-6
  24. [24] Garg, H. (2018). Some arithmetic operations on the generalized sigmoidal fuzzy
  25. [25] numbers and its application. Granular computing, 3(1), 9-25.. https://doi.org/10.1007/s41066-017-0052-7
  26. [26] Heilpern, S. (1997). Representation and application of fuzzy numbers. Fuzzy sets and
  27. [27] systems, 91(2), 259-268. https://doi.org/10.1016/S0165-0114(97)00146-2
  28. [28] Kaur, A., & Kumar, A. (2012). A new approach for solving fuzzy transportation prob-
  29. [29] lems using generalized trapezoidal fuzzy numbers. Applied soft computing, 12(3), 1201-1213. https://doi.org/10.1016/j.asoc.2011.10.014
  30. [30] Kosiń´ski, W., Prokopowicz, P., & Ślęzak, D. (2003). Ordered fuzzy numbers.
  31. [31] bulletin of the polish academy of sciences, 51(3), 327-338.
  32. [32] http://users.pja.edu.pl/~wkos/11-KOSIN.PDF
  33. [33] Massanet, S., Riera, J. V., Torrens, J., & Herrera-Viedma, E. (2014). A new linguistic computational model based on discrete fuzzy numbers for computing with words. Information sciences, 258, 277-290. https://doi.org/10.1016/j.ins.2013.06.055
  34. [34] Nayagam, V. L. G., Jeevaraj, S., & Sivaraman, G. (2016). Complete ranking of
  35. [35] intuitionistic fuzzy numbers. Fuzzy Information and engineering, 8(2), 237-254. https://doi.org/10.1016/j.fiae.2016.06.007
  36. [36] Nehi, H. M. (2010). A new ranking method for intuitionistic fuzzy numbers.
  37. [37] International journal of fuzzy systems, 12(1).https://www.researchgate.net/publica-
  38. [38] tion/279900044_A_New_Ranking_Method_for_Intuitionistic_Fuzzy_Numbers
  39. [39] Panda, A., & Pal, M. (2015). A study on pentagonal fuzzy number and its corresponding matrices. Pacific science review B: humanities and social sciences, 1(3), 131-139. https://doi.org/10.1016/j.psrb.2016.08.001
  40. [40] Qiu, D., & Zhang, W. (2013). Symmetric fuzzy numbers and additive equivalence
  41. [41] of fuzzy numbers. Soft computing, 17(8), 1471-1477.https://doi.org/10.1007/s00500-
  42. [42] -1000-3
  43. [43] Rouhparvar, H., & Panahi, A. (2015). A new definition for defuzzification of generalized fuzzy numbers and its application. Applied Soft Computing, 30, 577-584. https://doi.org/10.1016/j.asoc.2015.01.053
  44. [44] Seresht, N. G., & Fayek, A. R. (2019). Computational method for fuzzy arith-
  45. [45] metiac operations on triangular fuzzy numbers by extension principle.
  46. [46] International journal of approximate reasoning, 106, 172-193. https://doi.org/10.1016/j.ijar.2019.01.005
  47. [47] Smarandache, F. (1999). A unifying field in logics neutrosophy: neutrosophic
  48. [48] probability. Set and Logic.
  49. [49] Wang, Y. M., Yang, J. B., Xu, D. L., & Chin, K. S. (2006). On the centroids of
  50. [50] fuzzy numbers. Fuzzy sets and systems, 157(7), 919-926. https://doi.org/10.1016/j.fss.2005.11.006
  51. [51] Xu, Z., Shang, S., Qian, W., & Shu, W. (2010). A method for fuzzy risk analysis based on the new similarity of trapezoidal fuzzy numbers. Expert Systems with Applications, 37(3), 1920-1927. https://doi.org/10.1016/j.eswa.2009.07.015
  52. [52] Yager, R. R. (1986). On the theory of bags. International journal of general
  53. [53] system, 13(1), 23-37. https://doi.org/10.1080/03081078608934952
  54. [54] Zad´eh, L.A. (1965) ‘Fuzzy sets’, Information and control, 8(3), 338–353.
  55. [55] Zadeh, L. A. (1975). The concept of a linguistic variable and its application to ap-
  56. [56] proximate reasoning—I. Information sciences, 8(3), 199-249. https://doi.org/10.1016/0020-0255(75)90036-5

Downloads

Published

2026-06-19

Issue

Vol. 3 No. 2 (2026): Optimality

Section

Articles

Categories

How to Cite

Kumar, G., Jangid, V., & Shay Sharma, G. (2026). Game-Theoretic Applications of Pentagonal Fuzzy Numbers Through New Representation and Defuzzification Schemes. Optimality, 3(2), 157-185. https://doi.org/10.22105/opt.vi.96