Soft Computing Algorithm for Arithmetic Multiplication of Fuzzy - - PowerPoint PPT Presentation

Soft Computing Algorithm for Arithmetic Multiplication

• f Fuzzy Sets Based on

Universal Analytic Models

Yuriy Kondratenko, Volodymyr Kondratenko y_kondrat2002@yahoo.com

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

2

While performing the tasks of organizational control there always happen such situations when the initial conditions of decision making are not clearly defined and characterized by insufficient information for the human-operator who makes decisions, particularly in conflict situations or under extreme conditions. For mathematical formalization of processes and systems of this class there appeared a need to use a new mathematical approach - a theory of fuzzy sets and fuzzy logic developed by professor L.Zadeh and

• ther well-known scientists.

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

Introduction

3

Fuzzy set theory has a special notion of membership function that exists in the interval [0,1]. Each element of the fuzzy set, for example set , corresponds to a specific value of the membership function . Thus, fuzzy set that is specified on the basis

• f the universal set Е, is called the set of pairs

where  Е, .

x

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

A

 

[0,1]

A x

 

A

 

 

,

A

x

x 

( ) [0,1]

A x

 

x

4

Fuzzy sets and fuzzy logic are used for tasks of decision making and control in uncertainty, in particular for problems of routes planning and trajectory optimization, investments in uncertainty and so on. The solution of the such problems causes the necessity of fulfilling the operations of fuzzy arithmetic, in particular operations with fuzzy sets including addition, subtraction, multiplication and division.

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

5

Inverse models of resulting membership functions that are based on using

• cuts

do not always provide high performance of computing operations and often lead to comp- lications in solving control problems in real time. The development of generalized analytic models, based on the direct approach that allow to formalize fuzzy arithmetic operations and to improve their operating speed and accuracy, is an important direction of research that is associated with increased dependability of intelligent systems.

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine



6

One of the most difficult fuzzy arithmetic

• perations in terms of its mathematical for-

malization is an operation of multiplication.

Computational algorithms for the operations of multiplication on the basis of using -cuts of the relevant fuzzy sets (inverse approach) have high computational complexity, as it is performed in turn for all -levels ( ) with the step of discreteness , which value, taking into consideration that , significantly affects the accuracy and operating speed of the computational procedures.

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

 

 

0, 1 , 0,1,...,

i

i r   

 

1 i i

  

 

 

7

Therefore, -cuts of the fuzzy set is

• rdinary (in terms of conditions ) subset

that contains elements whose degree of membership to a set is not less than value , that is . Subsets and that determine the appropria- te -cuts of fuzzy sets and can be written as follows: , where , , and arithmetic opera- tion of multiplication can be written as , (1)

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine



A R 

( )

A x

  

x R 

A 

 

{ ( ) }, 0,1

A

A x x



     

A



B

 A

B

 

1 2

( ), ( ) A a a



  

 

1 2

( ), ( ) B b b



  

 

0, 1   , A B R 

       

1 2 1 2 1 1 2 2

( ), ( ) ( ), ( ) ( ) ( ), ( ) ( ) A B a a b b a b a b

 

           

8

The aim of this work is a synthesis of inverse and direct analytical models of resulting MF for fuzzy arithmetic operations, which will give the opportunity to significantly reduce the volume, complexity and accuracy, and to improve their operating speed. A detailed analysis of the properties of arithmetic operation of multiplication of fuzzy numbers for illustration of inverse and direct approaches will be given.

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

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The triangular fuzzy number is called fuz- zy number whose membership function is of triangular shape and mathematical presen- tation of triangular fuzzy number has the form , where .

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

A

 

A x



 

1 2

, , A a a a   

1

0;

A a

 

 

1;

A a

 

 

2 A a

 

• Fig. 1. Triangular Fuzzy Number A

R 

10

Generalized model , synthesized on the basis of inverse approach, and direct model in a form of a triangular membership function of triangular fuzzy number are determined by the appropriate relevant dependencies (2) and (3): , (2) (3)

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

A



 

A x

 A

       

1 2 1 1 2 2

, , A a a a a a a a a



                  

                 

1 2 1 1 1 2 2 2

0, / , / ,

A

x a x a x x a a a a x a a x a a a x a                    

11

Synthesis of analytical models in R+: inverse and direct approaches

We shall illustrate the method of forming direct generalized analytical model of resulting MF for the multiplication of fuzzy triangular numbers. Firstly we’ll form the inverse generalized model for a given triangular fuzzy number in the set of non-negative real numbers .

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

 

C x



   

1 2

, A a a



      

 

1 2

, , A a a a 

R

12

Let’s analyze the left branch

• f the triangular FN for -cut inverse model

, where , , as , since .

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

   

1 1 1

a a a a     

A         

1 2 1 1 2 2

, , A a a a a a a a a



                 

 

1 1 0

a a  

1

a a  

1

a a  A R 

13

Let’s introduce the designations: ; where and , regarding this the left component can be written as: .

1 1

K a a  

2 1,

K a 

1

K 

2

K 

 

1 2 1

a K K    

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

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Let’s similarly analyze the right branch of fuzzy number : Having marked , because ; , as , we shall receive model . However, as there is inequality then respectively.

A  

2 2 2

( ) a a a a     

3 2,

K a a  

4 2

K a 



3

K 

 

2 2

a a a  

4

K 



A R 

 

2 4 3

a K K    

2 2 ,

a a a  

4 3

K K 

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

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The -cut for the left and right branches of the triangular fuzzy number (inverse model) can be represented as follows , (4) (taking into account coefficients ; ; ; ; ).



A

   

 

1 2 2 1 4 3

, , A a a K K K K



           

, 1...4:

i

K i 

1

K 

2

K 

3

K 

4

K 

4 3

K K 

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

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For triangular fuzzy number corresponding cut has the form . Having marked we’ll receive the modified cut model , (5) (taking into account the coefficients ).

 

1 2

, , B b b b 

 

   

1 2

, B b b



      

1 1 2 1 3 2 4 2

, , , , S b b S b S b b S b      

 

B

   

 

1 2 2 1 4 3

, , B b b S S S S



           

, 1...4:

i

S i 

1 2 3 4 4 3

0; 0; 0; 0; S S S S S S     

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

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Based on (4) and (5) we shall receive an inverse model for cut of a resulting fuzzy set : : (6)

  ( ) C A B  

                     

1 2 1 2 1 1 2 2 1 2

( ) , , , , C A B a a b b a b a b C C

  

                                 

         

2 1 2 1 4 3 4 3 2 1 1 1 2 2 1 2 2 2 3 3 3 4 4 3 4 4

, , K K S S K K S S K S K S K S K S K S K S K S K S                             

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

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For the forming direct model we’ll consider in a more detailed way a constituent for cut (6) of the resulting fuzzy set , formed by the operation of multiplying : .

 

C x



 

1

C 

 

C

( ) C A B  

   

2 1 1 1 1 2 2 1 2 2

C K S K S K S K S       

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

19

The solution of the next equation will have the following roots . (7)

   

 

2 1 1 1 2 2 1 2 2 1

K S K S K S K S C        

 



1,2 1 2 2 1

K S K S     

   

2 1 2 2 1 1 1 1 1

4 / 2 K S K S K S C K S    

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

20

Let’s analyze the roots and accor- ding to (7), which can be written as , taking into consideration the signs To form a direct model of the resulting nonli- near membership function it is necessary to check the performance of the condition .

1



2



 

1,2 1 1 1 1 1 1 1

/ / / V Q W V W Q W       

   

1 1 2 2 1 1 1 1 2 1 1 2 2 1 1 1 1

, 2 , 4 . V K S K S W K S Q K S K S K S C       

 

c x



 

1,2

0;1  

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

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Let’s consider in detail the components : а) , taking into account correla- tions we have inequality and respectively inequality takes place; b) : as , because , then always ; c) , as then .

1 1 1

, , V Q W

1 1 2 2 1

V K S K S  

1 2 1 2

0, 0, 0, K K S S    

1

V 

1

V      

2 1 1 2 2 1 1 1 1

4 Q K S K S K S C    

 

1

C  

, A B R 

1 1

0, K S  

1

Q 

1 1 1

2 W K S 

1 1

0, 0, K S  

1

W 

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

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As , and the roots have requirements , then the root (that satisfies the given condition under , ) will be only , that is (8)

1 1

/ V W  

1,2



 

1,2

0;1  

1 1

0, Q V  

1

W 

 

1 1 1 1

/ V Q W    

     

 

2 1 2 2 1 1 2 2 1 1 1 1 1 1 1

4 2 K S K S K S K S K S C K S        

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

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The root does not satisfy the condition , as there is always a condition . The transition from inverse to direct approach shows that is a parameter of the function that is , where , and then (8) can be represented as (9)

 

2 1 1 1

/ V Q W    

 

2

0;1  

2

 

x  

 

1

, f C   

 

f x  

 

1

, x c c 

   

 

2 1 2 2 1 1 2 2 1 1 1 1 1 1

4 2 K S K S K S K S K S x K S       

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

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Let’s consider in detail the second component . The solution of the equation will be the roots

   

2 2 3 3 3 4 4 3 4 4

C K S K S K S K S       

   

 

2 3 3 3 4 4 3 4 4 2

K S K S K S K S C        

     

 

2 3 4 4 3 3 4 4 3 3 3 2 3,4 3 3

4 2 K S K S K S K S K S C K S        

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

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The analysis of roots and shows, that the root does not satisfy the condition , and therefore the only acceptable root will be root . The transition from inverse to direct approach allows to transform into , and thus (10)

3



4



3

1  

 

3

0;1  

4



 

 

2

f C   

 ,

f x  

 

2

, x c c 

   

 

2 3 4 4 3 3 4 4 3 3 3 4 3 3

4 2 K S K S K S K S K S x K S       

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

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The analysis of the roots and allows to make a conclusion that nonlinear dependence under intervals , is one-valued function. The investigations allow to form a direct analytical model for one-valued nonlinear resulting membership function of fuzzy set that is formed by multiplying triangular fuzzy num- bers and in :

1



4



 

f x  

 

1

, x c c 

 

2

, x c c 

 

C x



 

C A B  

A

B

R

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

27

                 

2 2 4 4 2 1 2 2 1 1 2 2 1 1 1 2 2 1 1 1 2 2 1 2 2 1 1 2 3 4 4 3 3 4 4 3 3 3 3 3 3 4 4 3 4 4 4 4 3 3 1 1 1 2 2 1 2 2 3 3

0, 4 , , 2 4 , , 2 1, =

C

x K S x K S K S K S K S K S K S x x K S K S K S K S K S K S x K S K S K S K S K S x x K S K S K S K S K S K S x c K S K S K S K S K S                                 

3 4 4 3 4 4

K S K S K S              10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

Direct analitical model of result MF

28

The given direct approach allows to form nonlinear resulting membership function

• n the basis of known coefficients
• f fuzzy numbers

and in . By substituting previously set marks we’ll get a direct model (Tabl.1) that is realized directly on the basis of parameters

• f triangular fuzzy numbers

and in .

 

C x



 

, 1..4

i i

K S i 

A

B

R

 

C x



1 2

, , , a a a

1 2

, , b b b A

B

R

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

29

Table 1: Analytical model : direct approach

 

C x



10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

30

The chart of the corresponding resulting membership function under the realizing the operation of multiplication of triangular fuzzy numbers and using a direct model, presented in Table 1, shown in

• Fig. 2.

 

C x



(5, 7, 12) A  (2, 9, 14) B 

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

Example 1

31

• Fig. 2. Implementation of the direct model  

C x



10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

32

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

Direct and inverse analytical models of fuzzy sets with bell-shape MFs

 

2

1/ 1

A

x b x c                  

 

1 2

1 1 ( ), ( ) 1, 1 A a a b c b c



               

33

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

Inverse analytical model for multiplication

• f fuzzy sets with bell-shape MF

         

1 1 2 2 1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 1 1 1 , 1 1 1 1 1 1 1 1 , = 1 1 1 1 p s p s C A B p s p s p p p s s p s s p p p s s p s s

  

                                                                                                       

   

1 2

= , . c c      

34

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

Direct analytical model for multiplication of fuzzy sets with bell-shape MF

 

1 2

0, : x p p  

 

   

2 2 2 2 2 1 3 3 3 2 1 3 2 2 1 3 3 3 2 2 2 2 1 3 1 3 2 3 2 2 2 2 1 3 1 3 1 3 2

(2 2 2 ) 4( 2 2 2 2 2 2 ) ; 2 2 2 2

C x

k k k k x k k k k k k k x k x k k k x k x k k k k x k k k x k x k                         

 

1 2,

: x p p   

 

   

2 2 2 2 2 1 3 3 3 2 1 3 2 2 1 3 3 3 2 2 2 2 1 3 1 3 2 3 2 2 2 2 1 3 1 3 1 3 2

(2 2 2 ) 4( 2 2 2 2 2 2 ) . 2 2 2 2

C x

k k k k x k k k k k k k x k x k k k x k x k k k k x k k k x k x k                          1 2 :

x p p 

 

1;

C x

 

35

Synthesis of analytical models in R: inverse and direct approaches

The task of synthesis of inverse and direct analytical models becomes complicated while per- forming the operation of multiplication of triangular fuzzy numbers (Fig. 3) and , that exist in the set of all real numbers .

A

B

R

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

36

• Fig. 3. Triangular Fuzzy Number in

We shall use the approach discussed above that is based on the analysis of the corresponding square roots for synthesis of inverse and direct models while implementing the operation of multi- plication in .

R

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

R

37

The algorithm of implementing the multipli- cation operation for fuzzy numbers and while using

• cuts

is based on the following inverse model:

 

1 2

, , A a a a 

 

1 2

, , B b b b 



   

1 2 1 1 2 2

( ), ( ) ( ) , ( ) , A a a a a a a a a



          

   

1 2 1 1 2 2

( ), ( ) ( ) , ( ) B b b b b b b b b



          

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

38

         



                       



1 2 1 1 2 1 1 2 2 2 1 1 2 1 1 2 2 2

( ), ( ) min , , , , max , , , C c c a b a b a b a b a b a b a b a b



                        

           

1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 2

( ) ( ) , ( ) ( ) , min , ( ) ( ) , ( ) ( ) a a a b b b a a a b b b a a a b b b a a a b b b                                       

           

1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 2

( ) ( ) , ( ) ( ) , max ( ) ( ) , ( ) ( ) a a a b b b a a a b b b a a a b b b a a a b b b                                     (11) where

   

1 1 1 2 1 1 2 2 2

min , , , , c a b a b a b a b 

   

2 1 1 2 1 1 2 2 2

max , , , ; c a b a b a b a b 

   

1 2 0 0

1 1 c c a b  

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

Inverse analytical model of result MF

39

The direct model will be formed in the following way: (12) where In the direct model (12) the roots of four square equations are used. These equations are formed while analyzing every

• f four components of the inverse model (11).

 

( ) ( )

C A B

x x  





       

 

 

 

1 2 * * 1 0 0 0 0 2 0 0

0, ( ) , , 1...8, , , , 1,

C i

x G x G x i x G a b x a b G x a b                        

 

1 1 1 2 1 1 2 2 2

min , , , , G a b a b a b a b 

 

2 1 1 2 1 1 2 2 2

max , , , G a b a b a b a b 

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

40

Let’s consider all different components of model (11): а) for component we form the equation , the roots of which are

1 1 1 1 1 1 2 1 1 1 1 1 1 1 1

[ ( ) ( )] [ ( )][ ( )] ( )( ) [( ( ) ( )] , a b a a a b b b a a b b a b b b a b a b                    

2 1 1 1 1 1 1 1 1

( )( ) [ ( ) ( ) ( ) a a b b a b b b a a a b x           

2 1 1 1 1 1 1 1 1 1 1 1 1 1,2 1 1

[ ( ) ( )] [ ( ) ( )] 4( )( )( ) ; 2( )( ) a b b b a a a b b b a a a a b b a b x a a b b                      

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41

b) for component the roots of the formed equation are     

2 1 2 2 1 1

( ) ( ) ( ) ( ) a b a a a b b b         

2 2 1 1 2 2 1 1 2

( )( ) [ ( ) ( ) ( ) b b a a a b b b a a a b x           

2 2 1 1 2 2 1 1 2 2 1 1 2 3,4 2 1

[ ( ) ( )] [ ( ) ( )] 4( )( )( ) ; 2( )( ) b a a a b b b a a a b b b b a a a b x b b a a                     

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

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c) for component the roots of the formed equation are

    

1 2 1 1 2 2

( ) ( ) ( ) ( ) a b a a a b b b         

2 2 1 2 1 1 2 2 1

( )( ) [ ( ) ( ) ( ) a a b b a b b b a a a b x           

2 2 1 1 2 2 1 1 2 2 1 2 1 5,6 2 1

[ ( ) ( )] [ ( ) ( )] 4( )( )( ) 2( )( ) a b b b a a a b b b a a a a b b a b x a a b b                      

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

43

d) for component the roots of the formed equation are

  

 

2 2 2 2 2 2

( ) ( ) ( ) ( ) a b a a a b b b         

2 2 1 1 2 2 1 1 2

( )( ) [ ( ) ( ) ( ) b b a a a b b b a a a b x           

2 2 2 2 2 2 2 2 2 2 2 2 2 7,8 2 2

[ ( ) ( )] [ ( ) ( )] 4( )( )( ) 2( )( ) a b b b a a a b b b a a a a b b a b x a a b b                      

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44

The algorithm according to which the value

• f is selected has the following interpre-
• tation. We define a subset of indices under

the condition . The subset of indices belongs to set of indices

• f all roots, defined above

. The investigations show that the presence of some roots that satisfy the condition the para- meter is defined as follows

*



1

I I 

 

 

1

0,1

i

I i i I    

 

1,2,3,4,5,6,7,8 I  , 1...8

i i

 

 

0,1

i

 

*



 

1

*

max .

i i I

 





10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

1

I

45

Functional dependences of the roots from the parameter while implementing the operation of multiplication of triangular fuzzy numbers and are given in the Fig. 4-8.

 ,

1...8

i i

f x i   

x

( 3,1, 8) A  

( 2, 2, 4) B  

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

Example 2

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

• Fig. 4. Roots

 

1 1

, f x  

 

2 2

f x  

 

1 1

f x  

 

2 2

f x  

Modelling results (1)

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

 

4 4

f x  

Fig 5. Roots

 

3 3

f x  

 

3 3

, f x  

 

4 4

f x  

Modelling results (2)

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

Fig 6. Roots

 

6 6

f x  

 

5 5

f x  

 

5 5

, f x  

 

6 6

f x  

Modelling results (3)

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

Fig 7. Roots

 

7 7

f x  

 

8 8

f x  

 

7 7

, f x  

 

8 8

f x  

Modelling results (4)

50

• Fig. 8. Functional dependences

 ,

1...8

i i

f x i   

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

 

7 7

f x  

 

4 4

f x  

 

6 6

f x  

 

1 1

f x  

 

8 8

f x  

 

2 2

f x  

 

3 3

f x  

 

5 5

f x  

Modelling results (5)

51

The chart of the resulting membership function for model (12) is presented in the Fig. 9. Fig.9 illustrates that during the process of chan- ging from -16 to +32, the function takes place according to the chain (points 1,2,3,4 in Fig. 9).

 

*

f x   5 3 1 8

      

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

x

52

• Fig. 9. Resulting fuzzy set

( ) : , C A B A R B R    

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

* 5

  

* 3

  

* 1

  

* 8

  

Modelling results (6)

53

Conclusions

• 1. So, this presentation deals with the investigations
• f increasing dependability of computing
• perations for fuzzy numbers with different MFs.
• 2. Special attention is paid to synthesis of analytic

models of the MFs for results of fuzzy arithmetic

• perations.
• 3. New analytical models of the result's MFs with

the description of synthesis procedures for the multiplication operation with fuzzy numbers in R+ and R are presented in the universal style.

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54

• 4. The general analytic models for determi-

nation of -cuts parameters and values

• f result MF (direct and inverse approaches) for

result fuzzy sets are given.

• 5. The usage of the developed analytical models

has significant advantage for accuracy of calculations, time of modeling and program implementation of the formed models in comparison with step-by-step models of fuzzy numbers multiplication based on the algorithms of sorting and max-min convolutions.



10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine

 

C x



55

• 6. The developed models allow increasing the

reliability of software/hardware realization of corresponding digital devices.

• 7. Modeling results confirms the efficiency of

the proposed models and fuzzy arithmetic algorithms for fuzzy information processing.

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56

Thank you for your attention!

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine