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

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine Soft Computing Algorithm for Arithmetic Multiplication of 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 Introduction 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 other well-known scientists. 2

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine Fuzzy set theory has a special notion of membership function that exists in the interval [0,1]. x Each element of the fuzzy set, for example set , corresponds to a specific value of the A     membership function . A x [0,1] A Thus, fuzzy set that is specified on the basis of the universal set Е , is called the set of pairs      x x , A   where  Е , . x ( ) [0,1] A x 3

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine 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. 4

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine 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. 5

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine One of the most difficult fuzzy arithmetic operations 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       0, 1 , i 0,1,..., r for all -levels ( ) with the i   step of discreteness , which value, taking into consideration that ,        i 1 i significantly affects the accuracy and operating speed of the computational procedures. 6

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine   A R Therefore, -cuts of the fuzzy set is    ordinary (in terms of conditions ) subset A x ( )  x R that contains elements whose degree of  A membership to a set is not less than value ,         A { x ( ) x }, 0,1 that is .  A A B  Subsets and that determine the appropria-   B A te -cuts of fuzzy sets and can be written as           follows: , A a ( ), a ( ) ( ), ( ) B b b   1 2 1 2   R     where , , and arithmetic opera- 0, 1 A B , tion of multiplication can be written as                A B a ( ), a ( ) b ( ), b ( ) , (1)   1 2 1 2        a ( ) ( ), b a ( ) b ( ) 7 1 1 2 2

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine 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. 8

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine The triangular fuzzy number is called fuz-    A A x zy number whose membership function is of triangular shape and mathematical presen- tation of triangular fuzzy number has the form            A a a a , , A a 0; A a 1; , where 1 0 2 1 0     . 0 A a 2 R   Fig. 1. Triangular Fuzzy Number A 9

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine A Generalized model , synthesized on the  basis of inverse approach, and direct model in a    form of a triangular membership function of A x A triangular fuzzy number are determined by the appropriate relevant dependencies (2) and (3):           A  a , a   1 2 , (2)              a a a , a a a   1 0 1 2 2 0         0, x a x a 1 2                  x x a / a a , a x a A 1 0 1 1 0 (3)             a x / a a , a x a  2 2 0 0 2 10

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine 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 C x numbers. Firstly we’ll form the inverse generalized model          A a , a    1 2 for a given triangular fuzzy number    A a a a , , 1 0 2 R  in the set of non-negative real number s . 11

Let’s analyze the left branch          a a a a 1 1 0 1  of the triangular FN for -cut inverse model A                       , A  a , a   a a a , a a a   1 2 1 0 1 2 2 0        a a where , , as , since a a 1 0 0 a a 0 0 1 1 0 1 R   A . 0 10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine 12

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine Let’s introduce the designations :   K a a ; 1 0 1  1 , K a 2 K  K  0 where and , 0 2 1 regarding this the left component can be written as:       a K K . 1 2 1 13

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine Let’s similarly analyze the right branch of A fuzzy number :        a a ( a a ) 2 2 0 2 Having marked    K a a 2 , K a 3 0 4 2       R   K  K  , because ; , as , 0 a a a A 0 0 2 2 0 0 3 4       we shall receive model . a K K 2 4 3   However, as there is inequality then 2 , a a a 2 0  K K respectively. 4 3 14

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine  The -cut for the left and right branches of the triangular fuzzy number ( inverse model ) A can be represented as follows                 , , A  a a  K K K K , (4)  1 2 2 1 4 3 K i  (taking into account coefficients , 1...4: i  K  K  K  K  K K 0 ; ; ; ; ). 0 0 0 4 3 1 4 2 3 15

10th Intern. Conf. ICTERI 2014 June 18-21, 2014, Kherson – Ukraine    B b b b , , For triangular fuzzy number 1 0 2   corresponding cut has the form          . , B  b b   1 2 Having marked       S b b S , b S , b b , S b , 1 0 1 2 1 3 0 2 4 2   we’ll receive the modified cut model B                  B b , b S S , S S   , (5)  1 2 2 1 4 3 S i  (taking into account the coefficients , 1...4: i      S 0; S 0; S 0; S 0; S S ). 1 2 3 4 4 3 16