A new and efficient method for elementary fuzzy arithmetic operations on pseudo-geometric fuzzy numbers

There are certain problems in the subtraction operator, division operator and obtaining the membership functions of operators and above all, dependence effect in the fuzzy arithmetic operations using the extension principle (in the domain of the membership function) or the interval arithmetics (in the domain of α cuts). In this regard, this paper provide a new method regarding the effective practical computation of elementary fuzzy arithmetic operations on pseudo-geometric fuzzy numbers. Therefore we eliminated such deficiency with the new proposed method and demonstrated that the new operators are more efficient. Finally, several illustrative examples were given to show the accomplishment and ability of the proposed method. The future prospect of this paper is a new attitude to fuzzy mathematics.


Introduction
In order to use fuzzy numbers and relations in any intelligent system one must be able to perform arithmetic operations, addition, subtraction, multiplication and division, employing these fuzzy quantities, the process of which is called fuzzy arithmetic.One of the most basic concepts of fuzzy set theory is the extension principle introduced by Zadeh, which was already implied in [10] in its rudimentary form and was finally presented in its fully-fledged form in [11,12,13].This principle provides a general method for extending crisp mathematical concepts to fuzzy quantities, that is, it allows the domain of functional mapping definition to be extended from crisp elements to fuzzy sets as the arguments of the function.The usual arithmetic operations on real numbers can be extended to the ones defined on fuzzy numbers by means of Zadeh's EP .In this context, direct implementation of this principle in fuzzy arithmetic is computationally expensive due to the requirement of solving a nonlinear programming problem [9].To overcome this deficiency, many researchers consider fuzzy numbers as a collection of α -levels, in which case, fuzzy arithmetics are performed using conventional interval arithmetic according to the α-cut representation.Interval arithmetic can verify as addition (+) and multiplication (×) operations on closed intervals with commutative and associative specifications, yet subtraction (-) and division (÷) are neither commutative nor associative.Therefore modified interval arithmetic operators operate under special circumstances or are only applied on special fuzzy numbers [7,8].Although with the revised definitions on subtraction and division, employing interval arithmetic for fuzzy operators has been permitted, because it always exists, yet, is not efficient since the results support is major agent(dependence effect).Additionally, we have not yet been able to calculate interval arithmetic in determining the membership function of operators based on Zadeh's EP.Hence, most researchers apply interval operators on triangular or trapezoidal fuzzy numbers; otherwise the results could not be easily obtained, as they would become dependent upon the max-min of non-linear functions.In this paper, we did not consider the fuzzy attitude on the fuzzy arithmetic based on interval arithmetic.Accordingly, with a completely fuzzy thought, we become able to define new fuzzy arithmetic operators based on TA (in the domain of the transmission, the average of support) and fix the problems associated with neutral members.We will further show that the results of the newly proposed operators are closer to reality than fuzzy arithmetic operations based on EP (in the domain of the membership function ) or the interval arithmetic (in the domain of the α-cuts).The future prospect of this paper is a new attitude to fuzzy mathematics.The paper is organized as follows.In section 2, we present the basic definitions and concepts related to the subject.In section 3, new fuzzy arithmetic operators based on TA are defined with the visual diagrams and its basic properties are investigated in section 4. Several numerical examples of pseudo-geometric and geometric fuzzy numbers to illustrate applying the method and comparing the results of the new method with the previous methods are given in section 5. Finally, conclusions and future research are drawn in section 6.

Preliminaries and notations
In this section, some notations and background about the concept are brought.Definition 2.1.[11,12,13]

be the Cartesian product of universes, and A
be fuzzy sets in each universe respectively.Also let Y be another universe and B ∈ Y be a fuzzy set such that B = f (A 1 × • • • × A n ), where f : X → Y is a monotonic mapping.Then Zadeh's EP is defined as follows: Definition 2.2.[11,12,13] (α -cut Representation) A fuzzy set, A can be represented (decomposed) by the union of all its α-cuts, i.e.
It is easy to check that the following holds:

[6] (Fuzzy number) A fuzzy set A in R is called a fuzzy number if it satisfies the following conditions:
(i) A is normal, (ii) A α is a closed interval for every α ∈ (0, 1], (iii) the support of A is bounded.
According to definition of fuzzy number mentioned above and the our emphasis on non-triangular and trapezoidal fuzzy numbers, we use define a pseudo-geometric fuzzy numbers in two case as follows: Definition 2.4.[4] (Pseudo-triangular and Pseudo-trapezoidal fuzzy numbers) A fuzzy number A is called a pseudotrapezoidal fuzzy number if its membership function µ A (x) is given by otherwise. (2.4) Where l A (x) and r A (x) are nondecreasing and non increasing functions, respectively.The pseudo-trapezoidal fuzzy number A is denoted by and the trapezoidal fuzzy number by (a, a 1 , a 2 , a, −, −), that, −, − means l A (x) and r A (x) are linear.
A pseudo-triangular fuzzy number is a particular pseudo-trapezoidal fuzzy number,when the a 1 = a 2 .The pseudotriangular fuzzy number A is denoted by and the triangular fuzzy number by (a, a, a, −, −).Definition 2.6.(a.c( Ã)) Let Ã be a pseudo-geometric fuzzy number.Then, we define from [4],

Definition 2.5. [14] ( Fuzzy arithmetic operations based on interval arithmetic(α-cut)) A popular way to carry out fuzzy arithmetic operations is by way of interval arithmetic. This is possible because any αcut of a fuzzy number is always an interval. Therefore, any fuzzy number may be represented as a series of intervals. Let us consider two interval numbers [a,b] and [c
3 The Description of the new method for elementary fuzzy arithmetic Theorem 3.1.
[5] (Mizumoto and Tanaka) For any non-degenerated fuzzy number Ã, there exist no inverse number Ã′ , Ã′′ under ⊕ and ⊗ respectively, such that Mizumoto and Tanaka's theorem exposes the uncertain as part of the essence of fuzzy systems, and we believe that the uncertainty is not related to the constraints on EP-based fuzzy arithmetic operations which are only presented for reverse operators in the fuzzy systems.Ergo, we provided the new method for elementary fuzzy arithmetic operations on pseudo-geometric fuzzy numbers, in a way that excludes all conditions or preconditions.Moreover, we proved that the newly proposed operations are more efficient than the operations based on interval arithmetic.with the following α-cut forms: In the following, we define fuzzy arithmetic operations based on TA for addition, subtraction, multiplication and division: ) ) (3.11) Remark 3.1.Division on pseudo-triangular fuzzy number 0 = (0, 0, 0, l 0(x), r 0(x)) is not able to define.

Graphical visualization
The proposed arithmetic operations on pseudo-trapezoidal fuzzy numbers are visualized in figures ( 4), ( 5) and ( 6), where the hypothetical ambiguities are shown in the diagram.

Theorems and Properties
In this section, some lemma and theorem about the concept are brought.

Lemma 4.1. (The membership functions of the fuzzy arithmetic operations TA for the triangular fuzzy numbers) Let
with the following α-cut forms:  Based on the definition of αcut A + B : Thus, from (4.24) we conclude that: Hence, considering (4.28): thus, we conclude from (4.27) : Hence, considering (4.32): Similar to the proof above, we have for the cases (ii), (iii), (iv).
Proof.We have the cases above, according to the definitions of the proposed arithmetic operations.Proof.We have the cases above, according to the definitions of the proposed arithmetic operations and definition (2.6).Then using the elementary fuzzy arithmetic operations based on the TA, we get: International Scientific Publications and Consulting Services ,d] where a ≤ b and c ≤ d.Then the following arithmetic operations proceed as shown below: (i) addition: [a , b ] + [ c , d ] = [ a+ c , b + d ] , (ii) subtraction: [ a , b ]-[ c , d ] = [ a-d , b-c ] , (iii) multiplication: [ a , b ] .[ c , d ] = [ m i n { a c , a d ,b c , b d } , m a x { a c , a d , b c , b d } ] , (iv) division: [ a ,b ]/ [ c , d ] = [ a , b ] .[ 1/d , 1/c ].

Definition 3 . 1 .
(The fuzzy arithmetic operations based on TA for pseudo-triangular fuzzy numbers) Consider two pseudo-triangular fuzzy number A = (a, a, a, l A (x), r A (x)), B = (b, b, b, l B (x), r B (x)),

Figure 1 :
Figure 1: The graph of addition TA operation on pseudo-triangular fuzzy numbers.

Figure 2 :
Figure 2: The graph of multiplication TA operation on pseudo-triangular fuzzy numbers.

Figure 3 :
Figure 3: The graph of addition and multiplication inverse TA operations on pseudo-triangular fuzzy number.

Figure 4 :
Figure 4: The graph of addition TA operation on pseudo-trapezoidal fuzzy numbers.

Figure 5 :
Figure 5: The graph of multiplication TA operation on pseudo-trapezoidal fuzzy numbers.

Figure 6 :
Figure 6: The graph of addition and multiplication inverse TA operations on pseudo-triangular fuzzy number.

Remark 4 . 1 .
In the case (iii) above, let a, b ≥ 0. Similar to above, we have the same results for (a, b ≤ 0) , (a ≥ 0, b ≤ 0) and (a ≤ 0, b ≥ 0).Lemma 4.2.(The membership functions of the fuzzy arithmetic operations TA for the trapezoidal fuzzy numbers) Let

Remark 4 . 3 .
Although subtraction division addition and multiplication are all defined, similar results were obtained of the lemma above for subtraction and division.

Remark 4 . 4 .
The membership functions of TA fuzzy arithmetic operations for pseudo-geometric fuzzy numbers cannot be obtained unless the Membership functions are known, in which case triangular and trapezoidal fuzzy numbers are dealt with.

Theorem 4 . 1 .
(Lack dependence effect) The results support of fuzzy arithmetic operations based on TA (in the domain of the transmission average of support) are smaller than fuzzy arithmetic operations based on the EP (in the domain of the membership function ) and the interval arithmetic (in the domain of the α-cuts).Proof.Let A = (a, a, a, l A (x), r A (x)), B = (b, b, b, l B (x), r B (x)).According to the fuzzy arithmetic operations based on the EP (α-cut) and the TA, we have the following steps.Based on the EP (α-cut):Ã + B = (a + b, a + b, a + b, l Ã+ B(x), r Ã+ B(x)),Ã * B = (min{a b, ab, ab, ab}, ab, max{a b, ab, ab, ab}, l Ã * B(x), r Ã * B(x)).(4.37)Based on the TA:A + B = ( (a + b) + (a + b) 2 , a + b, (a + b) + (a + b) 2 , l A+B (x),r A+B (x)), A * B = ( ab + ab 2 , ab, ab + ab 2 , l A * B (x), r A * B (x)).

Figure 8 :
Figure 8: The red graph is based on the extension principle (α-cut).The green graph is based on the transmission average.

Figure 9 :
Figure 9: The red graph is based on the extension principle (α-cut).The green graph is based on the transmission average.

Figure 10 :
Figure 10: The red graph is based on the extension principle (α-cut).The green graph is based on the transmission average.

Figure 11 :
Figure 11: The red graph is based on the extension principle (α-cut).The green graph is based on the transmission average.

Figure 12 :
Figure 12: The red graph is based on the extension principle (α-cut).The green graph is based on the transmission average.

Figure 13 :
Figure 13: The red graph is based on the extension principle (α-cut).The green graph is based on the transmission average.