A BI-OBJECTIVE PROGRAMMING APPROACH TO SOLVE MATRIX GAMES WITH PAYOFFS OF ATANASSOV’S TRIANGULAR INTUITIONISTIC FUZZY NUMBERS

Document Type: Research Paper

Authors

1 School of Management, Fuzhou University, No. 2, Xueyuan Road, Daxue New District, Fuzhou 350108, Fujian, China

2 School of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China

Abstract

The intuitionistic fuzzy set has been applied to game theory very
rarely since it was introduced by Atanassov in 1983. The aim of this paper is
to develop an effective methodology for solving matrix games with payoffs of
Atanassov’s triangular intuitionistic fuzzy numbers (TIFNs). In this methodology,
the concepts and ranking order relations of Atanassov’s TIFNs are defined.
A pair of bi-objective linear programming models for matrix games with
payoffs of Atanassov’s TIFNs is derived from two auxiliary Atanassov’s intuitionistic
fuzzy programming models based on the ranking order relations of
Atanassov’s TIFNs defined in this paper. An effective methodology based on
the weighted average method is developed to determine optimal strategies for
two players. The proposed method in this paper is illustrated with a numerical
example of the market share competition problem.

Keywords


[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.
[2] K. T. Atanassov, Intuitionistic fuzzy sets, Springer-Verlag, Heidelberg, Germany, 1999.
[3] K. T. Atanassov and G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and
Systems, 31 (1989), 343-349.
[4] K. T. Atanassov, Answer to D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk and H. Prade’s
paper Terminological difficulties in fuzzy set theory - the case of ”intuitionistic fuzzy sets”,
Fuzzy Sets and Systems, 156 (2005), 496-499.
[5] C. R. Bector and S. Chandra, Fuzzy mathematical programming and fuzzy matrix games,
Springer Verlag, Berlin, Germany, 2005.
[6] C. R. Bector, S. Chandra and V. Vijay, Matrix games with fuzzy goals and fuzzy linear
programming duality, Fuzzy Optimization and Decision Making, 3 (2004), 255-269.
[7] C. R. Bector, S. Chandra and V. Vijay, Duality in linear programming with fuzzy parameters
and matrix games with fuzzy pay-offs, Fuzzy Sets and Systems, 46(2) (2004), 253-269.
[8] R. A. Borzooei and Y. B. Jun, Intuitionistic fuzzy hyper bck-ideals of hyper bck-al gebras,
Iranian Journal of Fuzzy Systems, 1(1) (2004), 65-78.
[9] P. Burillo and H. Bustince, Vague sets are intuitionistic fuzzy sets, Fuzzy Sets and Systems,
79 (1996), 403-405.
[10] L. Campos, Fuzzy linear programming models to solve fuzzy matrix games, Fuzzy Sets and
Systems, 32 (1989), 275-289.
[11] L. Campos and A. Gonzalez, Fuzzy matrix games considering the criteria of the players,
Kybernetes, 20 (1991), 17-23.
[12] L. Campos, A. Gonzalez and M. A. Vila, On the use of the ranking function approach to
solve fuzzy matrix games in a direct way, Fuzzy Sets and Systems, 49 (1992), 193-203.
[13] S. K. De, R. Biswas and A. R. Roy, An application of intuitionistic fuzzy sets in medical
diagnosis, Fuzzy Sets and Systems, 117 (2001), 209-213.
[14] G. Deschrijver, Arithmetic operators in interval-valued fuzzy set theory, Information Sciences,
177(14) (2007), 2906-2924.
[15] G. Deschrijver and E. E. Kerre, On the relationship between some extensions of fuzzy set
theory, Fuzzy Sets and Systems, 133 (2003), 227-235.
[16] G. Deschrijver and E. E. Kerre, On the position of intuitionistic fuzzy set theory in the
framework of theories modeling imprecision, Information Sciences, 177 (2007), 1860-1866.
[17] D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk and H. Prade, Terminological difficulties in
fuzzy set theory-the case of ”Intuitionistic Fuzzy Sets”, Fuzzy Sets and Systems, 156 (3)
(2005), 485-491.
[18] D. Dubois and H. Prade, fuzzy sets and systems: theory and applications, Mathematics in
Science and Engineering, Academic Press, Berlin, Germany, 144 (1980).
[19] J. G. Garc and S. E. Rodabaugh, Order-theoretic, topological, categorical redundancies of
interval-valued sets, grey sets, vague sets, interval-valued ”intuitionistic” sets, ”intuitionistic”
fuzzy sets and topologies, Fuzzy Sets and Systems, 156(3) (2005), 445-484.

[20] W. L. Gau and D. J. Buehrer, Vague sets, IEEE Transaction on Systems, Man, and Cybernetics,
23 (1993), 610-614.
[21] J. Goguen, L-fuzzy sets, Journal of Mathematical Analysis and Applications, 18 (1967),
145-174.
[22] E. E. Kerre and J. N. Mordeson, A historical overview of fuzzy mathematicas, New Mathematics
and Natural Computation, 1(1) (2005), 1-26.
[23] A. Khan, Y. B. Jun and M. Shabir, Ordered semigroups characterized by their intuitionistic
fuzzy bi-ideals, Iranian Journal of Fuzzy Systems, 7(2) (2010), 55-69.
[24] D. F. Li, Fuzzy constrained matrix games with fuzzy payoffs, The Journal of Fuzzy Mathematics,
7(4) (1999), 873-880.
[25] D. F. Li, A fuzzy multiobjective programming approach to solve fuzzy matrix games, The
Journal of Fuzzy Mathematics, 7(4) (1999), 907-912.
[26] D. F. Li, Multiattribute decision making models and methods using intuitionistic fuzzy sets,
Journal of Computer and System Sciences, 70 (2005), 73-85.
[27] D. F. Li, A note on using intuitionistic fuzzy sets for fault-tree analysis on printed circuit
board assembly, Microelectronics Reliability, 48 (2008), 1741.
[28] D. F Li, A ratio ranking method of triangular intuitionistic fuzzy numbers and its application
to MADM problems, Computers and Mathematics with Applications, 60(6) (2010), 1557-
1570.
[29] D. F. Li, Representation of level sets and extension principles for Atanassov’s intuitionistic
fuzzy sets and algebraic operations, Critical View, 4 (2010), 63-74.
[30] D. F. Li and C. T. Cheng, New similarity measures of intuitionistic fuzzy sets and application
to pattern recognitions, Pattern Recognition Letters, 23 (2002), 221-225.
[31] 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, 17(4) (2009), 585-607.
[32] S. T. Liu and C. Kao, Solution of fuzzy matrix games: an application of the extension
principle, International Journal of Intelligent Systems, 22 (2007), 891-903.
[33] A. Maturo, On some structures of fuzzy numbers, Iranian Journal of Fuzzy Systems, 6(4)
(2009), 49-59.
[34] P. K. Nayak and M. Pal, Bi-matrix games with intuitionistic fuzzy goals, Iranian Journal of
Fuzzy Systems, 7(1) (2010), 65-79.
[35] I. Nishizaki and M. Sakawa, Equilibrium solutions in multiobjective bimatrix games with
fuzzy payoffs and fuzzy goals, Fuzzy Sets and Systems, 111(1) (2000), 99-116.
[36] I. Nishizaki and M. Sakawa, Solutions based on fuzzy goals in fuzzy linear programming
games, Fuzzy Sets and Systems, 115(1) (2000), 105-119.
[37] I. Nishizaki and M. Sakawa, Fuzzy and multiobjective games for conflict resolution, Physica-
Verlag, Springer Verlag Company, Berlin, Germany, 2001.
[38] E. Pasha, A. Saiedifar and B. Asady, The percentiles of fuzzy numbers and their applications,
Iranian Journal of Fuzzy Systems, 6(1) (2009), 27-44.
[39] M. Sakawa and I. Nishizaki, A lexicographical solution concept in an n-person cooperative
fuzzy game, Fuzzy Sets and Systems, 61 (1994), 265-275.
[40] M. Sakawa and I. Nishizaki, Max-min solutions for fuzzy multiobjective matrix games, Fuzzy
Sets and Systems, 67 (1994), 53-69.
[41] E. Savas, (A)-Double sequence spaces of fuzzy numbers via orlicz function, Iranian Journal
of Fuzzy Systems, 8(2) (2011), 91-103.
[42] E. Szmidt and F. Baldwin, Intuitionistic fuzzy set functions, mass assignment theory, possibility
theory and histograms, 2006 IEEE International Conference on Fuzzy Systems Sheraton
Vancouver Wall Centre Hotel, Vancouver, BC, Canada, July 16-21, (2006), 35-41.
[43] G. Takeuti and S. Titani, Intuitionistic fuzzy logic and intuitionistic fuzzy set theory, The
Journal of Symbolic Logic, 49(3) (1984), 851-866.
[44] V. Vijay, S. Chandra and C. R. Bector, Matrix games with fuzzy goals and fuzzy payoffs,
Omega, 33 (2005), 425-429.

[45] Z. S. Xu, Models for multiple attribute decision making with intuitionistic fuzzy information,
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 15 (2007),
285-297.
[46] Z. S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems,
15 (2007), 1179-1187.
[47] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-356.
[48] L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision
processes interval-valued fuzzy sets, IEEE Transactions on Systems, Man, and Cybernetics,
3 (1973), 28-44.