Operations on Intuitionistic Trapezoidal Fuzzy Numbers using Interval Arithmetic

Transcription

1 Intern. J. Fuzzy Mathematical Archive Vol. 9, No. 1, 2015, ISSN: (P), (online) Published on 8 October International Journal of Operations on Intuitionistic Trapezoidal Fuzzy Numbers using Interval Arithmetic Thangaraj Beaula 1 and M. Priyadharsini 2 1 Department of Mathematics, T.B.M.L. College, Porayar Tamil Nadu edwinbeaula@yahoo.co.in 2 Bharathiyar College of Engineering and Technology, Tamil Nadu, Karaikal sethu_priya22@rediffmail.com Received 9 September 2015; accepted 1 October 2015 Abstract. In this paper operations based on, cut for intuitionistic trapezoidal fuzzy numbers are defined as addition, multiplication, scalar multiplication, subtraction etc., including exponentiation, extracting n th root, and taking logarithm using the degree of acceptance and rejection. Keywords: Intuitionistic trapezoidal fuzzy number,, cuts. AMS Mathematics Subject Classification (2010): 90C08 1 Introduction Transportation problem is a special kind of linear programming problem in which goods are transported from a set of sources to destinations with respect to the supply and demand respectively, such that the total cost of transportation is minimized. Intuitionistic fuzzy set (IFS) is one of the components of fuzzy set theory [10] and also introduced by Atanassov [1,2]. IFS is exaggerated by a degree of acceptance and a degree of rejection function so that the sum of both values is less than one [1]. Mahapatra and Roy [6] defined the triangular intuitionistic fuzzy number (TrIFN) and trapezoidal intuitionistic fuzzy number (TrIFN) and their arithmetic operations based on intuitionistic fuzzy extension principle using, cuts method. Li [4] developed the ratio ranking method for ranking trapezoidal intuitionistic fuzzy numbers. This paper is organized as follows, in section 2, the representation of trapezoidal intuitionistic fuzzy numbers and, cut sets. In section 3, operations including exponentiation, logarithms and n th root of trapezoidal intuitionistic fuzzy number. 2 Preliminaries Definition 2.1. Let X be the universal set. An intuitionistic fuzzy set (IFS) in X is given by = ( x, µ ( x I ), ν I ( x ) / x X % % where the function µ % I ( x), ν % I ( x) define the degree of { } A A I acceptance and degree of rejection of the element x X to the set A %, for every x X, 0 µ ( x) + ν ( x) 1. I I A% A% A A 125

2 Thangaraj Beaula and M.Priyadharsini Definition 2.2. Let be the Trapezoidal intuitionistic fuzzy number with parameters b a b a a b a b which is denoted by =,,,,,,,.The degree of acceptance and degree of rejection functions are defined as,, = 1, and = 0,,, 0, > 1, > Definition 2.3. The, cut trapezoidal intuitionistic fuzzy numbers is defined as usually, by, =,,[,,+ 1,, [0,1] where = +, = and =, = +. Definition 2.4. A cut sets of =,,,,,,, is a crisp subset of, which is defined as =/. Using and definition of cut it follows that is a closed interval, denoted by =, = [ +, ]. Definition 2.5. A cut sets of =,,,,,,, a crisp subset of, which is defined as =/. Using and definition of cut, it follows that is a closed interval, denoted by =, =[ +, ]. 3 Operations on TrIFNs Theorem 3.1. If =,,,,,,, and =,,,,,,, are two TrIFNs, then + = +, +, +, +, +, +, +, + is also a TrIFN. Proof: For every, [0,1], + = + + +, + + (3.1) + = + +, (3.2) = and = + +, + + 1, + + which gives, =, + + 0, h 126 (3.3)

3 Operations on Intuitionistic Trapezoidal Fuzzy Numbers Using Interval Arithmetic = + + and= which gives, + + +, + + = 0, , + + 1, h So + represented by (3.3) and (3.4) is a trapezoidal shaped IFN. It can be approximated to TrIFN. Thus we have + = +, +, +, +, +, +, +, + is also a TrIFN. Theorem 3.2. If =,,,,,,, and =,,,,,,, are two TrIFNs, then =[,,,,,,, ] is also a TrIFN. Proof. For every, [0,1]. Let the additive image of and be = [ +, ] and =[, ]. = +[ + ], [ + ] (3.5) = [ ], +[ ] (3.6) = +[ + ] and = [ + ], 1, = +, 0, h = [ ] and = +[ ]. Now expressing in terms of and setting =0 and =1 in (3.6) which gives,

4 Thangaraj Beaula and M.Priyadharsini, 0, =, 1, h 3.8 So represented by (3.7) and (3.8) is a trapezoidal shaped IFN. It can be approximated to TrIFN. Thus we have =[,,,,,,, ] is also a TrIFN. Theorem 3.3. If =,,,,,,, and =,,,,,,, are two TrIFNs, then =,,,,,,, is also a TrIFN. Proof. For every, [0,1]. To calculate the multiplication of TrIFNs and, we first multiply the, cuts of and using interval arithmetic. = + +, (3.9) =, + + (3.10) =[ ] and =[ [ + ]+ [a a a +a a a ±[a a a +a a a ] 4a a a a a a x 2a a a a 1, = [ + ±[ + ] 4 2 0, > 3.11 =[ ] [ + ]+ and =[ ] +[ + ]+ So represented by (3.11) and (3.12) is a trapezoidal shaped IFN. It can be approximated to TrIFN. Thus we have =,,,,,,, is also a TrIFN. Theorem 3.4.Let. If =,,,,,,, is a TrIFN then = is also a TrIFN and is given by = =,,,,,,,, >0,,,,,,,, <0 Proof. Let =,,,,,,, be TrIFN and 128,,

5 Operations on Intuitionistic Trapezoidal Fuzzy Numbers Using Interval Arithmetic Case (i) >0 To calculate the scalar multiplication of TrIFN we first multiply and. cuts of gives =[ +, ] (3.13) =[, ] (3.14) = + and=, = = and= + 1,, 0, h, 0, =, 1, h Thus, scalar multiplication can be given as =[,,,,,,, ] which is also a TrIFN when >0. Case (ii) <0 Similarly, it can be shown that = if <0 where, [, ], [, ] 1, [ =, ] 0, [ and =, ], [, ], [, ] 0, h 1, h Thus, scalar multiplication can be given as =[,,,,,,, ] which is also a TrIFN where <0. Theorem 3.5. If =,,,,,,, and =,,,,,,, are two TrIFN, then =,,,,,,,,. Proof. To calculate division of TrIFNs and we first divide the, cuts of and using interval arithmetic. = [, ] [, ] = [, ] [, ] = and = =, =, (3.15) (3.16) 129

6 / = Thangaraj Beaula and M.Priyadharsini, 1,, 0, h = and = (3.17), / = 0, (3.18), 1, h So / represented by (3.17) and (3.18) is a trapezoidal shaped IFN. It can be approximated to TrIFN, / =,,,,,,,,. Theorem 3.6. If =,,,,,,, is a positive TrIFN, then = 1, 1, 1, 1, 1, 1, 1, 1 Proof. Let =,,,,,,, be a positive TrIFN. To calculate inverse of TrIFN we first take the inverse of the, cuts of using interval arithmetic. 1 = =, (3.19) [, ] 1 = =, (3.20) [, ] = = and = 1,,, 0, h (3.21) = and = 130

7 Operations on Intuitionistic Trapezoidal Fuzzy Numbers Using Interval Arithmetic, = 0, (3.22), 1, h So represented by (3.21) and (3.22) is a trapezoidal shaped IFN. It can be approximated to TRIFN =,,,,,,,. Theorem 3.7. If =,,,,,,, is a trapezoidal intuitionistic fuzzy number then =,,,,,,, ) is also a TrIFN. Proof. To calculate exponential of the TrIFN we first take the exponential of the, cut of using interval arithmetic. =,, =, (3.23)& (3.24) = and = =, 1, (3.25), 0, h = and = =, 0, (3.26), 1, h So represented by (3.25) and (3.26) is a trapezoidal shaped IFN. It can be approximated to TrIFN =,,,,,,, ). Theorem 3.8. If =,,,,,,, is a positive TrIFN. Then =[,,,,,,, ] is also a TrIFN. Proof. To calculate logarithm of the TrIFN we first take the logarithm of the, cut of using interval arithmetic. =[ +, ] = +, (3.27) =[, + ] =[, + ] (3.28) 131

8 Thangaraj Beaula and M.Priyadharsini =[ + ] and =[ ]., = 1, (3.29), 0, h =[ ] and =[ + ]. =, 0,, 1, h (3.30) So represented by (3.29) and (3.30) is a trapezoidal shaped IFN. It can be approximated to =[,,,,,,, ] Theorem 3.9. If =,,,,,,, is a positive TrIFN then is also a TrIFN as =,,,,,,,. Proof. To calculate the n th root of the TrIFN we first take the n th root of the, cuts of using interval arithmetic. =[ + ], [ ] (3.31) =[ ], [ + ] (3.32) =[ + ] and =[ ]. =, 0, = (3.33), 1, h =[ ] and=[ + ]., So 0, (3.34), 1, h represented by (3.41) and (3.42) is a trapezoidal shaped IFN by =,,,,,,,. 132

9 Operations on Intuitionistic Trapezoidal Fuzzy Numbers Using Interval Arithmetic 4. Conclusion In this paper we have deliberated many operations of TrIFN based on, cut method including n th root and exponentiation. REFERENCES 1. K.T.Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems, 33 (1989) K.T.Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986) D.De and P.K.Das, A study on ranking of trapezoidal intuitionistic fuzzy numbers, International Journal of Computer Information Systems and Industrial Management Applications, 6 (2014) D.-F.Li, A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems, Computers and Mathematics with Applications, 60 (2010) D.Dubus and H.Prade, Ranking fuzzy numbers in the setting of possibility theory, Inform. Sci., 30 (1983) G.S.Mahapatra and T.K.Roy, Intuitionistic fuzzy number and its arithmetic operation with application on system failure, Journal of Uncertain System, 7(2) (2013)