On the power sequence of a fuzzy interval matrix with max-min operation

An $$n \times n $$n×n interval matrix $${\mathcal {A}}= [\underline{A},\overline{A}]$$A=[A̲,A¯] is called to be a fuzzy interval matrix if $$0 \le \underline{A}_{ij} \le \overline{A}_{ij}\le 1$$0≤A̲ij≤A¯ij≤1 for all $$1 \le i, j \le n$$1≤i,j≤n. In this paper, we proposed the notion of max-min algebra of fuzzy interval matrices. We show that the max-min powers of a fuzzy interval matrix either converge or oscillate with a finite period. Conditions for limiting behavior of powers of a fuzzy interval matrix are established. Some properties of fuzzy interval matrices in max-min algebra are derived. Necessary and sufficient conditions for the powers of a fuzzy interval matrices in max-min algebra to be nilpotent are proposed as well.

[1]  Günter Mayer,et al.  On the convergence of powers of interval matrices , 1984 .

[2]  Chenyi Hu,et al.  Studying interval valued matrix games with fuzzy logic , 2008, Soft Comput..

[3]  H. Hashimoto,et al.  Transitivity of generalized fuzzy matrices , 1985 .

[4]  Tofigh Allahviranloo,et al.  A new approach to obtain algebraic solution of interval linear systems , 2012, Soft Comput..

[5]  G. Alefeld,et al.  Introduction to Interval Computation , 1983 .

[6]  Yung-Yih Lur,et al.  On nilpotent fuzzy matrices , 2004, Fuzzy Sets Syst..

[7]  H. Hashimoto,et al.  Convergence of powers of a fuzzy transitive matrix , 1983 .

[8]  M. Thomason.  Convergence of powers of a fuzzy matrix , 1977 .

[9]  J. Plávka,et al.  On the weak robustness of interval fuzzy matrices , 2015 .

[10]  Yin-Feng Xu,et al.  Consistency issues of interval pairwise comparison matrices , 2014, Soft Computing.

[11]  Yung-Yih Lur,et al.  On the Stability of Takagi–Sugeno Fuzzy Systems With Time-Varying Uncertainties , 2008, IEEE Transactions on Fuzzy Systems.

[12]  Hiroshi Hashimoto,et al.  Reduction of a nilpotent fuzzy matrix , 1982, Inf. Sci..

[13]  Yung-Yih Lur,et al.  On simultaneously nilpotent fuzzy matrices , 2003 .

[14]  Martin Gavalec,et al.  Interval Eigenproblem in Max-Min Algebra , 2014 .

[15]  Yung-Yih Lur,et al.  A new proof of Mayer's theorem , 2002 .

[16]  Emrah Akyar,et al.  Brown–Robinson method for interval matrix games , 2011, Soft Comput..

[17]  Mau-Hsiang Shih,et al.  An inequality for the spectral radius of an interval matrix , 1998 .

[18]  Lazim Abdullah,et al.  A new preference scale mcdm method based on interval-valued intuitionistic fuzzy sets and the analytic hierarchy process , 2016, Soft Comput..