New solutions of LR fuzzy linear systems using ranking functions and ABS algorithms

Abstract

We propose an approach for computing the general compromised solution of an LR fuzzy linear system by use of a ranking function when the coefficient matrix is a crisp m × n matrix. The solution is so that mean values of a compromised solution satisfies the corresponding crisp linear system. We show that if the corresponding crisp system is incompatible, then the fuzzy linear system lacks any solution. Otherwise, we solve a constrained least squares problem to compute a compromised solution. If the optimal value of the constrained least squares problem is zero, then we obtain the LR solution, namely the exact solution, of the system with respect to a ranking function. On the other hand, if the optimal value of the constrained least squares problem is nonzero, then no exact solution exists and thus we introduce and compute approximate (or weak) solutions of the system with respect to a ranking function. Also, when the ranking function is a member of a certain class of ranking functions, we propose a class of algorithms, based on ABS class of algorithms, to compute the general compromised solution.