Hesitant fuzzy sets (HFSs), which allow the membership degree of an element to a set represented by several possible values, can be considered as a powerful tool to express uncertain information in the process of group decision making. We derive some correlation coefficient formulas for HFSs and apply them to clustering analysis under hesitant fuzzy environments. Two real world examples, i.e. software evaluation and classification as well as the assessment of business failure risk, are employed to illustrate the actual need of the clustering algorithm based on HFSs, which can incorporate the difference of evaluation information provided by different experts in clustering processes. In order to extend the application domain of the clustering algorithm in the framework of HFSs, we develop the interval-valued HFSs and the corresponding correlation coefficient formulas, and then demonstrate their application in clustering with interval-valued hesitant fuzzy information through a specific numerical example. © 2012 Elsevier Inc.
A complex matrix P∈Cn×n is said to be a generalized reflection if P=PH=P-1. Let P∈Cn×n and Q∈Cn×n be two generalized reflection matrices. A complex matrix A∈Cn×n is called a generalized centro-symmetric with respect to (P;Q), if A=PAQ. It is obvious that any n×n complex matrix is also a generalized centro-symmetric matrix with respect to (I;I). In this work, we consider the problem of finding a simple way to compute a generalized centro-symmetric solution pair of the generalized coupled Sylvester matrix equations (GCSY)∑i=1lAiXBi+∑i=1lCiYDi=M, ∑i=1lEiXFi+∑i=1lGiYHi=N, (including Sylvester and Lyapunov matrix equations as special cases) and to determine solvability of these matrix equations over generalized centro-symmetric matrices. By extending the idea of conjugate gradient (CG) method, we propose an iterative algorithm for solving the generalized coupled Sylvester matrix equations over generalized centro-symmetric matrices. With the iterative algorithm, the solvability of these matrix equations over generalized centro-symmetric matrices can be determined automatically. When the matrix equations are consistent over generalized centro-symmetric matrices, for any (special) initial generalized centro-symmetric matrix pair [X(1), Y(1)], a generalized centro-symmetric solution pair (the least Frobenius norm generalized centro-symmetric solution pair) can be obtained within finite number of iterations in the absence of roundoff errors. Also, the optimal approximation generalized centro-symmetric solution pair to a given generalized centro-symmetric matrix pair [X~, Y~] can be derived by finding the least Frobenius norm generalized centro-symmetric solution pair of new matrix equations. Moreover, the application of the proposed method to find a generalized centro-symmetric solution to the quadratic matrix equation Q(X)=AX2+BX+C=0 is highlighted. Finally, two numerical examples are presented to support the theoretical results of this paper. © 2011 Elsevier Inc.