An interval-valued programming approach to matrix games with payoffs of triangular intuitionistic fuzzy numbers

Document Type : Research Paper


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


The purpose of this paper is to develop a methodology for solving a new type of matrix games in which payoffs are expressed with triangular intuitionistic fuzzy numbers (TIFNs). In this methodology, the concept of solutions for matrix games with payoffs of TIFNs is introduced. A pair of auxiliary intuitionistic fuzzy programming models for players are established to determine optimal strategies and the value of the matrix game with payoffs of TIFNs. Based on the cut sets and ranking order relations between TIFNs, the intuitionistic fuzzy programming models are transformed into linear programming models, which are solved using the existing simplex method. Validity and applicability of the proposed methodology are illustrated with a numerical example of the market share problem.


A. Aggarwal, A. Mehra and S. Chandra, \emph{Application of linear programming with I-fuzzy sets to matrix games with I-fuzzy goals},
Fuzzy Optimization and Decision Making, \textbf{11} (2012), 465-480.
K. T. Atanassov, \emph{Intuitionistic fuzzy sets}, In: V. Sgurev, ed., VII ITKR's session, Sofia,
June 1983.
K. T. Atanassov, \emph{Intuitionistic fuzzy sets}, Fuzzy Sets and
Systems, \textbf{20} (1986), 87-96.
K. T. Atanassov, Intuitionistic Fuzzy Sets, Springer-Verlag,
Heidelberg, Germany, 1999.
C. R. Bector and S. Chandra, Fuzzy Mathematical Programming and
Fuzzy Matrix Games, Springer-Verlag, Berlin, Germany, 2005.
C. R. Bector, S. Chandra and V. Vijay, \emph{Matrix games with
fuzzy goals and fuzzy linear programming duality}, Fuzzy
Optimization and Decision Making, \textbf{3} (2004), 255-269.
C. R. Bector, S. Chandra and V. Vijay, \emph{Duality in linear
programming with fuzzy parameters and matrix games with fuzzy
pay-offs}, Fuzzy Sets and Systems, \textbf{46(2)} (2004),
L. Campos, \emph{Fuzzy linear programming models to solve fuzzy
matrix games}, Fuzzy Sets and Systems, \textbf{32} (1989),
L. Campos and A. Gonzalez, \emph{Fuzzy matrix games considering
the criteria of the players}, Kybernetes, \textbf{20} (1991),
L. Campos, A. Gonzalez and M. A. Vila, \emph{On the use of the
ranking function approach to solve fuzzy matrix games in a direct
way}, Fuzzy Sets and Systems, \textbf{49} (1992), 193-203.
S. K. De, R. Biswas and A. R. Roy, \emph{An application of
intuitionistic fuzzy sets in medical diagnosis}, Fuzzy Sets and
Systems, \textbf{117} (2001), 209-213.
D. Dubois and H. Prade, \emph{Fuzzy sets and systems: theory and
applications}, Academic
Press, Berlin, Germany, 1980.
L. Dymova and P. Sevastjanov, \emph{An interpretation of intuitionistic
 fuzzy sets in terms of evidence theory: decision making aspect}, Knowledge-Based Systems,
 \textbf{ 23(8)} (2010), 772-782.
H. Ishibuchi and H. Tanaka, \emph{Multiobjective programming in optimization of the interval objective function},European Journal of Operational Research,
\textbf{48} (1990), 219-225.
S. T. Liu and C. Kao, \emph{Solution of fuzzy matrix games: an
application of the extension principle}, International Journal of
Intelligent Systems, \textbf{22 }(2007), 891-903.
S. K. Mahato and A. K. Bhunia, \emph{Interval-arithmetic-oriented interval computing technique for global optimization}, Applied Mathematics Research Express, (2006),  1-19.
P. K. Nayak and M. Pal, \emph{Linear programming technique to solve two person matrix games with interval pay-offs}, Asia-Pacific Journal of Operational Research, \textbf{26(2)}
(2007), 1-10.
P. K. Nayak and M. Pal, \emph{Bi-matrix games with intuitionistic fuzzy payoffs}, Notes on Intuitionistic Fuzzy Sets, \textbf{13(3)}
(2009), 285-305.
I. Nishizaki and M. Sakawa, \emph{Solutions based on fuzzy goals
in fuzzy linear programming games}, Fuzzy Sets and
Systems,\textbf{ 115(1)} (2000), 105-119.
A. Sengupta and T. K. Pal, \emph{On comparing interval numbers}, European Journal of Operational Research,
 \textbf{127(1)} (2000), 28-43.
V. Vijay, S. Chandra and C. R. Bector, \emph{Matrix games with
fuzzy goals and fuzzy payoffs}, Omega, \textbf{33} (2005),
L. A. Zadeh, \emph{Fuzzy sets,} Information and Control,\textbf{
8} (1965), 338-353.