07-10-2016, 10:58 AM
1458149369-maincopyieee.docx (Size: 48.72 KB / Downloads: 7)
ABSTRACT
The field of decision making is most useful application of fuzzy set theory. In this paper, procedures are presented for fuzzy decision model. In real life situation uncertain information gathered for decision making. In this paper hexagonal fuzzy number matrix is applied to the decision making problems
INTRODUCTION:
In daily days we are facing so many problems to select the best alternative with multiple criteria in fuzzy environment. Fuzzy set theory is most useful to solving the such type of problems. Decision making procedure is used to find the best alternatives. In fuzzy environment fuzzy number is very important in decision making process. The idea of fuzzy set was introduced by Zadeh in 1965. Bellmann and Zadeh developed the concept of decision making in fuzzy environment. Fuzzy set theory is applied in many field such as medical, engineering , etc. In fuzzy set, membership element described in the interval [0,1]. It can be used to find the solution where the information is incomplete and imprecise.
The paper organised as follows. In section 2, the basic definition of fuzzy number. In section3, hexagonal fuzzy numbers are defined and based on the function principle are arithmetic operations such as addition, subtraction, scalar multiplication and maximum operation of hexagonal fuzzy number. In section 5, an application of hexagonal fuzzy number matrix in decision making problem is given with numerical example.
2. PRE-REQUISITES:
Definition 2.1 Fuzzy number:
A fuzzy set Â, defined on the universal set of real number R, is said to be a fuzzy number of its membership function has the following characteristics.
 is convex.
i.e., μ (λx1+(1-λ)x2) ≥ min(μÂ(x1),μÂ(x2)) for all x1,x2 ϵ R, λ ϵ [0,1].
 is normal is there exists x0 ϵ R that μÂ(x0) = 1.
μ is piecewise continuous.
Definition 2.2 Triangular fuzzy number:
It is a fuzzy number represented with three points as follows: Â = (a1,a2,a3)
This representation is interpreted as membership functions and holds the following conditions,
a1 to a2 is increasing function.
a2 to a3 is decreasing function.
a1 ≤ a2 ≤ a3 .
Definition 2.3 Trapezoidal fuzzy number:
A fuzzy number  is a trapezoidal fuzzy number denoted by (a1,a2,a3,a4) where a1,a2, a3 and a4 are real numbers and its membership function μÂ(x) is given below:
μ_A (x)={█( 0, x≤a_1@(x-a_1)/(a_2-a_1 ),& a_1≤x ≤a_█(2)@(a_4-x)/(a_4-a_3 ), a_3≤x ≤a_█(4)@ 0, x≥a_4 )┤
Definition 2.4 Pentagonal fuzzy number:
A pentagonal fuzzy number of a fuzzy set P ̃ is defined as P ̃ = (a1, a2, a3, a4, a5; k, ω), and its membership is given by,
μ_P ̃ (x)={█( 0 for X ≤ a_1@(■((x-a_1)/(a_2-a_1 ) )) for a_1 ≤X ≤ a_2 @(■((x-a_2)/(a_3-a_2 ) )) for a_2 ≤X ≤ a_3@ 1 for X = a_3@(■((a_4-x)/(a_4-a_3 ) )) for a_3 ≤X ≤ a_4@(■((a_5-x)/(a_5-a_4 ) )) for a_4 ≤X ≤ a_5@ 0 for X ≥a_5 )┤
3 Hexagonal fuzzy number:
Definition 3.1 Hexagonal fuzzy number:
A fuzzy number ÂH is a hexagonal fuzzy number denoted by, ÂH = (a1, a2, a3, a4, a5, a6) where a1, a2, a3, a4, a5, a6 are real numbers and its membership function μÂ(X) is given below.
〖μÃ〗_H (x)={█( 0 for X ≤ a_1@1/2 (■((x-a_1)/(a_2-a_1 ) )) for a_1 ≤X ≤ a_2@1/2+1/2 (■((x-a_2)/(a_3-a_2 ) )) for a_2 ≤X ≤ a_3@ 1 for a_3≤X≤ a_4@1-1/2 (■((x-a_4)/(a_5-a_4 ) )) for a_4 ≤X ≤ a_5@1/2 (■((a_6-x)/(a_6-a_5 ) )) for a_5 ≤X ≤ a_6@ 0 for X ≥a_6 )┤