Solutions of Fuzzy System of Linear Equations

Solutions of Fuzzy System of Linear Equations

Laxminarayan Sahoo (Raniganj Girls' College, India)
DOI: 10.4018/978-1-7998-0190-0.ch002

Abstract

This chapter deals with solution methodology of fuzzy system of linear equations (FSLEs). In fuzzy set theory, finding solutions of FLSEs has long been a well-known problem to the researchers. In this chapter, the fuzzy number has been converted into interval number, and the authors have solved the interval system of linear equation for finding the fuzzy valued solution. Here, a fuzzy valued linear system has been introduced and a numerical example has been solved and presented for illustration of purpose.
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2 Some Mathematical Background

In this section some definitions and basic concepts related to fuzzy sets, fuzzy numbers and interval numbers used in this chapter are described briefly.

2.1 Preliminary Definitions About Fuzzy Set

A fuzzy set 978-1-7998-0190-0.ch002.m01 is defined by a membership function978-1-7998-0190-0.ch002.m02, which corresponds to each element x in X to a real number in the interval 0≤x≤1. The function 978-1-7998-0190-0.ch002.m03 represents the grade of membership of x in978-1-7998-0190-0.ch002.m04.

Definition 2.1 The α-cut (Zimmermann, 2001) of a fuzzy set 978-1-7998-0190-0.ch002.m05 is a crisp subset of X and is denoted by 978-1-7998-0190-0.ch002.m06, where 978-1-7998-0190-0.ch002.m07 is the membership function of 978-1-7998-0190-0.ch002.m08 and 0≤α≤1.

Definition 2.2 A fuzzy set 978-1-7998-0190-0.ch002.m09 is called a normal fuzzy set if there exists at least one xX such that 978-1-7998-0190-0.ch002.m10.

Definition 2.3 A fuzzy set 978-1-7998-0190-0.ch002.m11 is called convex iff for every pair of x1,x2X, the membership function of 978-1-7998-0190-0.ch002.m12 satisfies 978-1-7998-0190-0.ch002.m13, where 0≤λ≤1.

Definition 2.4 A fuzzy number 978-1-7998-0190-0.ch002.m14 is a fuzzy set which is both convex and normal.

Definition 2.5 The triangular fuzzy number (TFN) is a normal fuzzy number denoted as 978-1-7998-0190-0.ch002.m15=(a1,a2,a3) where a1a2a3 and its membership function 978-1-7998-0190-0.ch002.m16 is defined by

978-1-7998-0190-0.ch002.m17

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