Dual hesitant fuzzy matrix games: based on new similarity measure

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The dual hesitant fuzzy set is an effective mathematical approach to deal with the data which are imprecise, uncertain or incomplete information. Dual hesitant fuzzy set is an extension of hesitant fuzzy set which encloses fuzzy set, intuitionistic fuzzy set and hesitant fuzzy set as a special one. In this paper, the axiomatic definition of similarity measure between the dual hesitant fuzzy set is presented. A new similarity measure by considering membership and non-membership functions of dual hesitant fuzzy set is introduced. It is shown that the corresponding distance measure can be obtained from the proposed similarity measure. To check the utility, the proposed similarity measure is applied in a bidirectional approximate reasoning system into matrix game. Matrix game with precise data is hardly applicable in real-life decision-making problem. In view of more realistic sense, we choose the elements as dual hesitant fuzzy into the payoff of the matrix game, which is treated as dual hesitant fuzzy matrix game. Mathematical formulation of dual hesitant fuzzy matrix game with on restriction (DHFMGR) is described. Four algorithms are emerged on the proposed similarity measure, which are provoked to find the optimal value of the DHFMGR. A numerical example is incorporated to illustrate the applicability and feasibility of the proposed measure in dual hesitant fuzzy matrix game. The paper ends with the conclusions including an outlook for future study in this direction.

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  1. Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96

  2. Bector CR, Chandra S (2005) Fuzzy mathematical programming and fuzzy matrix games. Springer, Berlin

  3. Bhaumik A, Roy SK, Li DF (2017) Analysis of triangular intuitionistic fuzzy matrix games using robust ranking. J Intell Fuzzy Syst 33(1):327–336

  4. Campos L (1989) Fuzzy linear programming models to solve fuzzy matrix games. Fuzzy Sets Syst 32(3):275–289

  5. Chen SM, Tan JM (1994) Handling multi-criteria fuzzy decision making problems based on vague sets. Fuzzy Sets Syst 67(2):163–172

  6. Chen SM, Hsiao WH, Jong WT (1997) Bidirectional approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets Syst 91(3):339–353

  7. Hung WL, Yang MS (2007) Similarity measures of intuitionistic fuzzy sets based on \(L_p\) metric. Int J Approx Reason 46(1):120–136

  8. Li YH, Olson DL, Qin Z (2007) Similarity measures between intuitionistic fuzzy (vague) sets: a comparative analysis. Pattern Recognit Lett 28(2):278–285

  9. Liang ZZ, Shi PF (2003) Similarity measures on intuitionistic fuzzy sets. Pattern Recognit Lett 24(15):2687–2693

  10. Mula P, Roy SK, Li DF (2015) Birough programming approach for solving bi-matrix games with birough payoff elements. J Intell Fuzzy Syst 29(2):863–875

  11. Neumann J, Morgenstern O (1944) Theory of games and economic behaviour. Princeton University Press, Princeton

  12. Ning Y, Chen X, Wang Z, Li X (2017) An uncertain multi-objective programming model for machine scheduling problem. Int J Mach Learn Cybern 8(5):1493–1500

  13. Owen G (1982) Game theory. Academic press, New York

  14. Roy SK (2007) Fuzzy programming approach to two-person multicriteria bimatrix games. J Fuzzy Math 15(1):141–153

  15. Roy SK (2010) Game theory under MCDM and fuzzy set theory. VDM (Verlag Dr.Muller), Germany

  16. Roy SK, Bhaumik A (2018) Intelligent water management: a triangular type-2 intuitionistic fuzzy matrix games approach. Water Resour Manag 32(3):949–968

  17. Roy SK, Mondal SN (2016) An approach to solve fuzzy interval valued matrix game. Int J Oper Res 26(3):253–267

  18. Roy SK, Mula P (2013) Bimatrix game in bifuzzy rough environment. J Uncertain Anal Appl 1:11.

  19. Roy SK, Mula P (2015) Rough set approach to bimatrix game. Int J Oper Res 23(2):229–244

  20. Roy SK, Mula P (2016) Solving matrix game with rough payoffs using genetic algorithm. Oper Res Int J 16(1):117–130

  21. Roy SK, Mula P, Mondal SN (2011) A new solution concept in credibilistic game. CIIT Int J Fuzzy Syst 3(3):115–120

  22. Singh P (2014) A new method for solving dual hesitant fuzzy assignment problems with restriction based on similarity measure. Appl Soft Comput 24:559–571

  23. Szmidt E, Kacprzyk J (2000) Distance between intutionistic fuzzy sets. Fuzzy Sets Syst 114(3):505–518

  24. Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25:529–539

  25. Torra V, Narukawa Y (2009) On hesitant fuzzy sets and decision. In: The 18th IEEE international conference on fuzzy syst, Jeju Island, Korea, pp 1378–1382

  26. Wu D, Mendel JM (2009) A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets. Inf Sci 179(8):1169–1192

  27. Xu FS (2009) A new method on measures of similarity between vague sets. World Congress Comput Sci Inf Eng 4:297–300

  28. Xu Z (2010) A method based on distance measure for interval-valued intuitionistic fuzzy group decision making. Inf Sci 180(1):181–190

  29. Xu Z, Xia M (2011) Distance and similarity measures for hesitant fuzzy sets. Inf Sci 181(1):2128–2138

  30. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–356

  31. Zhu B, Xu Z, Xia M (2012) Dual hesitant fuzzy sets. J Appl Math.

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The author, Jishu Jana, would like to thank to Council of Scientific and Industrial Research (CSIR) for supporting the financial support to continue this research work under JRF scheme with sanctioned Grant No. 09/599(0067)/2016-EMR-I dated 20/10/2016. The research of Sankar Kumar Roy is partially supported by the Portuguese Foundation for Science and Technology (“ FCT-Fundação para a Ciência e a Tecnologia”), through the CIDMA—Center for Research and Development in Mathematics and Applications, University of Aveiro, Portugal, within project UID/MAT/04106/2013. The authors are very much thankful to the Managing Editor, Prof. Raffaele Cerulli and anonymous reviewer for their precious comments that help us so much to rigorously improve the quality of the paper.

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Correspondence to Sankar Kumar Roy.

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Jana, J., Roy, S.K. Dual hesitant fuzzy matrix games: based on new similarity measure. Soft Comput 23, 8873–8886 (2019) doi:10.1007/s00500-018-3486-1

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  • Matrix games
  • Hesitant fuzzy set
  • Dual hesitant fuzzy set
  • Similarity measure
  • Bidirectional approximate reasoning system