Abstract
One of the applications of linear programing is to get solutions for fully fuzzy linear system (FFLS) when the near-zero fuzzy number is considered. This usage could be applied to interpret the nature of FFLS solution according to the nature of FFLS solution in the work of Babbar et al. (Soft Comput. 17:1-12, 2012) and Kumar et al. (Advances in Fuzzy Systems 2011:1-8, 2011). This paper shows that the nature of FFLS solutions must not depend upon the nature of linear programming (LP) solutions, because LP is not enough to obtain all the exact solutions for FFLS which contradicts the claims of researchers. Counter examples are provided in order to falsify those claims. Numerically, we confirm that the nature of the possible way of solving FFLS is completely different from that of the linear system. For instance, FFLS may have two unique solutions which contradict the uniqueness that can be obtained through only one unique solution.
Highlights
Linear system of equations is the simplest framework and the most beneficial mathematical model for many problems considered by applied mathematics
We show that the nature of the solutions of the fully fuzzy linear system (FFLS) is completely different from the nature of the solutions of the linear programming (LP) and that it is insufficient to obtain all exact solutions for FFLS, using numerical examples
We provide the final result without considering the used methods because the aim of this paper is to show the weakness of the LP method to detect all exact feasible solutions, and we want to confirm that the nature of solution for LP which has been gained by optimal solution cannot explain the nature of the solution for FFLS
Summary
Linear system of equations is the simplest framework and the most beneficial mathematical model for many problems considered by applied mathematics. The LP technique cannot obtain all feasible solutions for FLS and FFLS Kumar and his colleagues illustrated examples of one solution while it has two unique fuzzy solutions or infinite number of solutions as will be illustrated through examples in this paper. The system has a unique solution, X~ 1⁄4 ðm1x; α1x; β1xÞ 1⁄4 ð0; 6; 3Þ: Verification for the solution: 8 < ð3; 1; 1Þ⊗ð0; 6; 3Þ 1⁄4 ð0; 24; 12Þ; : ð2; 1; 0Þ⊗ð0; 6; 3Þ 1⁄4 ð0; 12; 6Þ: The following non-square FFLS has many infinite solutions where the number of equations is less than the number of parameters, the result contrary to the result in Example 4. By solving the associated linear system, we provide the general form solution, which cannot be obtained by LP, and some particular solutions are produced: G′
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