Alternative Determines Positivity of Hexagonal Fuzzy Numbers and Their Alternative Arithmetic

There are quite a lot of arithmetic operations for hexagonal fuzzy numbers, most of them only define positive fuzzy numbers and few are discussing negative fuzzy numbers. And rarely found inverse of a fuzzy hexagonal number. So, often the results obtained in a hexagonal fuzzy linear equation system are not compatible. In this paper, we will discuss arithmetic alternatives on fuzzy hexagonal numbers. In this paper will definitions of positive and negative fuzzy numbers based on the concept of wide area covered by hexagonal fuzzy numbers in quadrant I and in quadrant II (right and left segments called r). From the concept of positivity and negativity the hexagonal fuzzy numbers will be constructed arithmetic alternatives for hexagonal fuzzy numbers. At the final part be given an inverse for a hexagonal fuzzy number so that, so for any fuzzy number there is an inverse hexagonal fuzzy number and its multiplication produces an identity.


Introduction
Fuzzy logic is part of mathematics science introduced by L. A Zadeh in 1965 [8,9]. Fuzzy set theory permits the gradual assessment of the membership of elements in a set which is decribed in the interval [0, 1]. Fuzzy set theory is applied using the membership functions. Previous researchers defined fuzzy hexagonal with , , , , , and only defined positive fuzzy with results that are not compatible yet [10-13, 15, 16].
One that often appears in various forms of arithmetic for hexagonal fuzzy numbers is for any hexagonal fuzzy number not necessarily be valid =0 and also not necessarily there so that ⊗ like the arithmetic given by the study [10][11][12][13][15][16]. in [13] a positive fuzzy number is defined, if 0, for all 1, 2, 3, 4, 5, 6 and the opposite is said negative this condition does not answer for fuzzy hexagonal numbers contain 0. While the study [11] discusses in the α-cut form but the hexagon fuzzy number form does not convex and also the conditions mentioned above also do not apply. Other side [15] uses the concept of centroid, but still the above conditions are met and still do not solve the problem for fuzzy numbers that contain 0.
Based on the conditions above, the author feels the need to define the concept of positivity of a fuzzy hexagonal number by using the difference in the concept of wide area in quadrant I with quadrant II (the difference between the area to the right of the r axis and the area to the left of the r axis) and the arithmetic alternative will be constructed for fuzzy hexagonal numbers with convex. In particular the form will also be constructed for any fuzzy number so that ⊗ . While, for multiplication will be described in various cases, so fulfilled, positive fuzzy numbers multiplied by negative fuzzy numbers must be negative, negative fuzzy numbers multiplied by negative fuzzy numbers must be positive and form other cases. The author will define hexagonal fuzzy numbers in the form , , !, ", #, $ on condition % , β ≤ α, δ ≤γ with ! distance left from the center , " distance left from ! , # distance right from the center and $ distance right from & # . By determining alternative fuzzy arithmetic numbers offered. It is expected to solve all matrix problems whose elements are fuzzy hexagonal numbers.

Positive Fuzzy and Negative Fuzzy Numbers
In this section a new definition will be given to determine a fuzzy number which is said to be positive fuzzy numbers or negative fuzzy numbers wich will be use modify algebra in multiplying two fuzzy numbers.
Fuzzy numbers are said to be positive (negative) if area 2 , ≥ 0 2 , < 0 or ( ≥ D ( < D seen from the positive x and x negatives as follows: 1. If fuzzy numbers are only in the x-positive line area then it will be positive fuzzy number or ! 0 3. If fuzzy is in both x-positive and x-negative areas, it is divided into 3 cases, as follows: Case 1 Consider the following figure bellow From the figure above, it can be seen that some areas are at x-positive and others are at x-negative, so they can be divided into 6 (six) area sections, hence Then overall area will be obtained, as follows:  From the figure above, it can be seen that some areas are at x-positive and others are at x-negative, so they can be divided into 5 (five) area sections, hence Then overall area will be obtained, as follows: If From the figure above, it can be seen that some areas are at x-positive and others are at x-negative, so they can be divided into 5 (sive) area sections, hence Then overall area will be obtained, as follows: If  Then overall area will be obtained, as follows: If ≥ 0 and ≥ 0, said to be positive fuzzy number if ( − D ≥ 0 or can be written as + # − ! + $ − " ≥ 0, and to be said negative if ( − D ≤ 0 or can be written as + # − ! + $ − " ≤ 0.

New Arithmetic Hexagonal Fuzzy Number
After defining positive fuzzy numbers and negative fuzzy numbers and operation of fuzzy numbers then it will be applied to the multiplication of two fuzzy numbers will be explained in the following below: Arithmetic algebra operation will be for hexagonal fuzzy numbers. For = , , !, ", #, $ and = L, M, N, O, P, Q , then the parametric forms are as follows: Transforming back into the hexagonal form, from equation (1) (2) (3) and (4) Transforming back into the hexagonal form, from equation (9) (10) (11) and (12) Transforming back into the hexagonal form, from equation (17) (18) (19) and (20)

Conclusion
In this paper the author defines a new form of the hexagonal fuzzy number = , , !, ", #, $ . Furthermore, the authors define positive hexagonal fuzzy numbers and negative fuzzy numbers, so can determine the multiplication of two fuzzy numbers, which will produce compatible results. The next discussion can develop hexagonal fuzzy using the Cramer, Gauss-Jacobi and others methods.