Numerical technique for solving fully fuzzy systems of linear equations


Аuthors

Panteleyev A. V.*, Luneva S. Y.**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: PanteleevAV@mai.ru
**e-mail: LunevaSY@mai.ru

Abstract

The article considers the problem of numerical solution of a linear system of equations with a fuzzy rectangular matrix and a fuzzy right-hand side. An uncertainty of the parameters, described by the intervals of possible values, presents, as a rule, in the practice of engineering and economic calculations. Besides, the level of confidence, set in the fuzzy set theory by the so-called membership functions, may be assigned to the numerical value from the interval. One of the possible types of membership functions are triangular ones, which set triangular fuzzy numbers. The authors suggest employing the triangular numbers description in the form of an average value and deviations from the average value. A technique for fully fuzzy system solution, employing the more accurate formula for the fuzzy numbers product, correct in the absence of the assumption of scatter smallness around the average value, as well as the apparatus for obtaining pseudo-solutions of the systems of linear algebraic equations, was obtained in the article. The class of problems to be solved is limited to finding positive solutions of systems of linear equations, provided that the fuzzy numbers included in the matrix of the system and the right-hand side are also positive. Quite stringent condition were obtained, under which the solution of the system is positive. The article presents four examples, illustrating the proposed method application for systems with matrices of various sizes and ratios of the rows and columns number, as well as comparison of the obtained results with solutions obtained using well-known formulas.

Keywords:

fuzzy numbers, completely fuzzy linear system of equations, triangular numbers, pseudo-inverse matrix, pseudo-solution

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