Solution of Matrix Game with Triangular Intuitionistic Fuzzy Pay-Off Using Score Function

Affiliation(s)

Bankura Christian College, Bankura, India.

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India.

Bankura Christian College, Bankura, India.

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India.

ABSTRACT

Using score function in a matrix game is very rare. In the proposed paper we have considered a matrix game with pay-off as triangular intuitionistic fuzzy number and a new ranking order has been proposed using value judgement index, available definitions and operations. A new concept of score function has been developed to defuzzify the pay-off matrix and solution of the matrix game has been obtained. A numerical example has been given in support of the proposed method.

Cite this paper

S. Bandyopadhyay, P. Kumar Nayak and M. Pal, "Solution of Matrix Game with Triangular Intuitionistic Fuzzy Pay-Off Using Score Function,"*Open Journal of Optimization*, Vol. 2 No. 1, 2013, pp. 9-15. doi: 10.4236/ojop.2013.21002.

S. Bandyopadhyay, P. Kumar Nayak and M. Pal, "Solution of Matrix Game with Triangular Intuitionistic Fuzzy Pay-Off Using Score Function,"

References

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[5] P. K. Nayak and M. Pal, “Linear Programming Technique to Solve Two Person Matrix Games with Interval Pay- Offs,” Asia-Pacific Journal of Operational Research, Vol. 26, No. 2, 2009, pp. 285-305. doi:10.1142/S0217595909002201

[6] A. L. Narayan, A. R. Meenakshi and A. M. S. Ramasamy, “Fuzzy Games,” The Journal of Fuzzy Mathematics, Vol. 10, 2002, pp. 817-829.

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[12] M. R. Seikh, M. Pal and P. K. Nayak, “Application of Triangular Intuitionistic Fuzzy Numbers in Bi-Matrix Games,” International Journal of Pure and Applied Ma- thematics, Vol. 79, No. 2, 2012, pp. 235-247.

[13] J.-X. Nan, D.-F. Li and M.-J. Zhang, “A Lexicographic Method for Matrix Games with Payoffs of Triangular Intuitionistic Fuzzy Numbers,” International Journal of Computational Intelligence Systems, Vol. 3, No. 3, 2010, pp. 280-289. doi:10.2991/ijcis.2010.3.3.4

[14] K. Atanassov, “Intuitionistic Fuzzy Sets: Theory and Applications,” Physica-Verlag, Berlin, 1999. doi:10.1007/978-3-7908-1870-3

[15] M. R. Seikh, M. Pal and P. K. Nayak, “Notes on Triangular Intuitionistic Fuzzy Numbers,” International Journal Mathematics in Operation Research, 2012.

[16] G. S. Mahapatra and T. K. Roy, “Reliability Evaluation Using Triangular Intuitionistic Fuzzy Numbers Arithmetic Operations,” International Journal of Computational and Mathematical Sciences, Vol. 3, No. 5, 2009, pp. 225- 231.

[17] H. B. Mitchell, “Ranking Intuitionistic Fuzzy Numbers,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 12, No. 3, 2004, pp. 377-386. doi:10.1142/S0218488504002886

[18] S.-M. Chen and J.-M. Tan, “Handling Multicriteria Fuzzy Decision Making Problems Based on Vague Set Theory,” Fuzzy Sets and Systems, Vol. 67, No. 2, 1994, pp. 163- 172. doi:10.1016/0165-0114(94)90084-1

[19] D. H. Hong and C.-H. Choi, “Multicriteria Fuzzy Decision Making Problems Based on Vague Set Theory,” Fuzzy Sets and Systems, Vol. 144, No. 1, 2000, pp. 103- 113.

[20] H. W. Liu, “Vague Set Methods of Multicriteria Fuzzy Decision Making,” System Engineering, Theory and Practice, Vol. 5, No. 5, 2004, pp. 214-220.

[21] J. V. Neumann and O. Morgenstern, “Theory of Games and Economic Behaviour,” Princeton University Press, Princeton, 1947.

[1] I. Nishizaki and M. Sakawa, “Equilibrium Solutions for Multiobjective Bimatrix Games Incorporating Fuzzy Goals,” Journal of Optimization Theory and Applications, Vol. 86, No. 2, 1995, pp. 433-457. doi:10.1007/BF02192089

[2] I. Nishizaki and M. Sakawa, “Max-Min Solution for Fuzzy Multiobjective Matrix Games,” Fuzzy Sets and Systems, Vol. 61, No. 1, 1994, pp. 265-275.

[3] P. K. Nayak and M. Pal, “Solution of Rectangular Fuzzy Games,” OPSEARCH, Vol. 44, No. 3, 2009, pp. 211-226.

[4] P. K. Nayak and M. Pal, “Solution of Interval Games Using Graphical Method,” Tamsui Oxford Journal of Mathematical Sciences, Vol. 22, No. 1, 2006, pp. 95-115.

[5] P. K. Nayak and M. Pal, “Linear Programming Technique to Solve Two Person Matrix Games with Interval Pay- Offs,” Asia-Pacific Journal of Operational Research, Vol. 26, No. 2, 2009, pp. 285-305. doi:10.1142/S0217595909002201

[6] A. L. Narayan, A. R. Meenakshi and A. M. S. Ramasamy, “Fuzzy Games,” The Journal of Fuzzy Mathematics, Vol. 10, 2002, pp. 817-829.

[7] R. E. Moore, “Method and Application of Interval Analysis,” Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1979. doi:10.1137/1.9781611970906

[8] H.-J. Zimmermann, “Fuzzy Mathematical Programming,” Computer Operational Research, Vol. 10, No. 4, 1983, pp. 291-298. doi:10.1016/0305-0548(83)90004-7

[9] K. Atanassov, “Intuitionistic Fuzzy Sets,” Fuzzy Sets and Systems, Vol. 20, No. 1, 1986, pp. 87-96. doi:10.1016/S0165-0114(86)80034-3

[10] K. T. Atanassov, “Ideas for Intuitionistic Fuzzy Equations, Inequalities and Optimization,” Notes on Intuitionistic Fuzzy Sets, Vol. 1, No. 1, 1995, pp. 17-24.

[11] D. F. Li and J. X. Nan, “A Nonlinear Programming Approach to Matrix Games with Payoffs of Atanassov’s Intuitionistic Fuzzy Sets,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 17, No. 4, 2009, pp. 585-607.

[12] M. R. Seikh, M. Pal and P. K. Nayak, “Application of Triangular Intuitionistic Fuzzy Numbers in Bi-Matrix Games,” International Journal of Pure and Applied Ma- thematics, Vol. 79, No. 2, 2012, pp. 235-247.

[13] J.-X. Nan, D.-F. Li and M.-J. Zhang, “A Lexicographic Method for Matrix Games with Payoffs of Triangular Intuitionistic Fuzzy Numbers,” International Journal of Computational Intelligence Systems, Vol. 3, No. 3, 2010, pp. 280-289. doi:10.2991/ijcis.2010.3.3.4

[14] K. Atanassov, “Intuitionistic Fuzzy Sets: Theory and Applications,” Physica-Verlag, Berlin, 1999. doi:10.1007/978-3-7908-1870-3

[15] M. R. Seikh, M. Pal and P. K. Nayak, “Notes on Triangular Intuitionistic Fuzzy Numbers,” International Journal Mathematics in Operation Research, 2012.

[16] G. S. Mahapatra and T. K. Roy, “Reliability Evaluation Using Triangular Intuitionistic Fuzzy Numbers Arithmetic Operations,” International Journal of Computational and Mathematical Sciences, Vol. 3, No. 5, 2009, pp. 225- 231.

[17] H. B. Mitchell, “Ranking Intuitionistic Fuzzy Numbers,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 12, No. 3, 2004, pp. 377-386. doi:10.1142/S0218488504002886

[18] S.-M. Chen and J.-M. Tan, “Handling Multicriteria Fuzzy Decision Making Problems Based on Vague Set Theory,” Fuzzy Sets and Systems, Vol. 67, No. 2, 1994, pp. 163- 172. doi:10.1016/0165-0114(94)90084-1

[19] D. H. Hong and C.-H. Choi, “Multicriteria Fuzzy Decision Making Problems Based on Vague Set Theory,” Fuzzy Sets and Systems, Vol. 144, No. 1, 2000, pp. 103- 113.

[20] H. W. Liu, “Vague Set Methods of Multicriteria Fuzzy Decision Making,” System Engineering, Theory and Practice, Vol. 5, No. 5, 2004, pp. 214-220.

[21] J. V. Neumann and O. Morgenstern, “Theory of Games and Economic Behaviour,” Princeton University Press, Princeton, 1947.