Reverse order Triangular, Trapezoidal and Pentagonal Fuzzy Numbers

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1 nnals of Pure and pplied Mathematics Vol. 9, No., 205, 07-7 ISSN: X (P), (online) Pulished on 20 January nnals of Reverse order Triangular, Trapezoidal and Pentagonal Fuzzy Numers T.Pathinathan and K.Ponnivalavan 2 Departmentof Mathematics, Loyola College, Chennai ,India. pathiloyolamaths@gmail.com; 2 ponnivalavan986@gmail.com Received 27 Decemer 204; accepted 6 January 205 stract. In this paper we define reverse order triangular fuzzy numer and reverse order pentagonal fuzzy numer with the help of triangular fuzzy numer. We also define reverse order trapezoidal fuzzy numer with the help of trapezoidal fuzzy numer. we include asic arithmetic operations like addition, sutraction for reverse order triangular fuzzy numer and reverse order trapezoidal fuzzy numer. We have also verified with examples for the aove mentioned operations. Keywords: Fuzzy numers,triangular fuzzy numer, reverse order triangular fuzzy numer, reverse order trapezoidal fuzzy numer, reverse order pentagonal fuzzy numer, fuzzy operations MS Mathematics Suject Classification (200): 94Dxx. Introduction Zadeh introduced fuzzy set theory in 965. Different types of fuzzy sets [3] are defined in order to clear the vagueness of the existing prolems. fuzzy numer [9], is a quantity whose values are imprecise, rather than exact as in the case with single-valued function. The concept of fuzzy numers is the generalization of the concept of real numers. D.Duois and H.Prade has defined fuzzy numer as a fuzzy suset of the real line [5,2]. So far fuzzy numers like triangular fuzzy numers [4], trapezoidal fuzzy numers [2,0], Pentagonal fuzzy numers [], Hexagonal, Octagonal, pyramid fuzzy numers and Diamond fuzzy numer [4] have een introduced with its memership functions. These numers have got many applications [7] like non-linear equations, risk analysis and reliaility. Many operations [8,,6] were done using fuzzy numers. In the case, of triangular fuzzy numer, the memership function characterizing a fuzzy numer has one maximum on the supporting interval ut in the case of trapezoidal fuzzy numer, the requirement of one maximum has een relaxed allowing a flat segment at level α =. In order to gain more diversity and to descrie large numer of linguistic variales [3], we allow minimum instead of maximum or a flat segment at level α = 0. We give three functions, with respect to the axis α defined on ounded supporting interval. If a ounded interval is more suitale we can restrict x etween its end points. shift of the function to the left or right can e easily made. lso if the interest is in a large 07

2 T.Pathinathan and K.Ponnivalavan variale which is either positive or negative. Then correspondingly only values, x 0 or x 0 are to e considered.when the supporting interval considered is unounded we get Bell shaped fuzzy numers. In this paper, we introduce reverse order triangular fuzzy numer (rotfn)with its memership functions, reverse order trapezoidal fuzzy numer and reverse order pentagonal fuzzy numer with its memership functions. Section one is the introduction and section two presents the asic definitions of fuzzy numers; section three presents reverse order triangular fuzzy numer and its arithmetic operations; section four presents reverse order trapezoidal fuzzy numer and its arithmetic operations, and finally section five presents reverse order pentagonal fuzzy numer. 2. Preliminaries and notations Definition 2..(Fuzzy set): fuzzy set is characterized y a memership function mapping the elements of a domain, space or universe of discourse X to the unit interval [0, ]. fuzzy set in a universe of discourse X is defined as the following set of pairs: = {( x,µ ( x ); x X } Here : µ X [0,] fuzzy set and µ ( x ) is a mapping called the degree of memership function of the is called the memership value of x X in the fuzzy set. These memership grades are often represented y real numers ranging from [0,]. Definition 2.2. (Fuzzy Numer): Fuzzy numer is a fuzzy set on the real line R, must satisfy the following conditions. (i) µ ( ) x is piecewise continous o (ii) There exist atleast one xo R with µ ( x 0) = (iii) must e normal and convex Definition 2.3. (Triangular Fuzzy Numer): Triangular Fuzzy Numer is defined as = {a,,c}, where all a,, c are real numers and its memership function is given elow. ( x a) for a x ( a) ( c x) µ ( x) = for x c ( c ) 0 otherwise Definition 2.4. (Trapezoidal Fuzzy Numer): fuzzy set = (a,, c, d) is said to trapezoidal fuzzy numer if its memership function is given y where a c d 08

3 Reverse order Triangular, Trapezoidal and Pentagonal Fuzzy Numers µ ( x) = 0 for x < a ( x a) for a x ( a) for x c ( d x) for c x d ( d c) 0 for x > d Definition: 2.5. (Pentagonal Fuzzy Numer): Pentagonal Fuzzy Numer (PFN) of a fuzzy set is defined as = {a,, c, d, e}, and its memership function is given y, P µ ( x) = P 0 for x < a, ( x a ) fo r a x ( a ) ( x ) for x c ( c ) x = c ( d x ) for c x d ( d c ) ( e x ) fo r d x e ( e d ) 0 for x > e Definition: 2.6 (Diamond Fuzzy Numer) Diamond Fuzzy Numer (DFN) of a fuzzy set is defined as D = {a,, c, ( α µ β )}, and its memership function is given y, 0 f o r x < a, ( x a ) f o r a x ( a ) ( c x ) f o r x c ( c ) ( x ) = - a s e α D ( a x ) f o r a x ( a ) ( x c ) f o r x c ( c ) x = β 0 o t h e r w i s e 09

4 where α is the ase of the triangle a aove triangle, namely ac. T.Pathinathan and K.Ponnivalavan 3. Reverse order Triangular fuzzy numer β c and also for the reverse order reflection of the Definition 3.. (Reverse order triangular fuzzy numer). fuzzy set is defined as t = {-a,0,} is said to e reverse order triangular fuzzy numer(rotfn) if its memership function is given y t for x a x for a x 0 a x for 0 x for x α a 0 Figure : Graphical representation of reverse order triangular fuzzy numer (RoTFN). Example: n reverse order triangular fuzzy numer, memership is given as, µ t = (-3,0,2) and its t for x 3 x for 3 x 0 3 x for 0 x 2 2 for x 2 α- Cut of reverse order triangular fuzzy numer 3 x = α x = 3α 0

5 Reverse order Triangular, Trapezoidal and Pentagonal Fuzzy Numers 2 x = α x = 2α t α = [- a, ] α α ]=[ 3 α,2α α cut Figure 2: Graphical representation ofα- Cut of reverse order triangular fuzzy numer(rotfn). When (α =0.5), we get 0.5 = [-.5,], lso when (α =0), t 0 = 0 0 [ a, ] = [-3,2]. 3.2 Conditions on reverse order triangular fuzzy numer. n reverse order triangular Fuzzy Numer should satisfy the following conditions; (i) µ t ( x ) is a continuous function in the interval [,0] (ii) µ ( x ) is strictly decreasing and continuous function on [-a, 0] (iii) t µ ( x ) is strictly increasing and continuous function on [0,] t 3.3. rithmetic operations on reverse order triangular fuzzy numer (ROTFN) ddition of two reverse order triangular fuzzy numers If, = (-a,0, ) and B = (-a,0, ); are two reverse order fuzzy numers then, t t B = ( a t t a 2, + 2). For example; If, = ( 2,0,); B = ( 3,0,2); t t. t t t Then, + B = ( 5,0,3 ) Suraction of two reverse order triangular fuzzy numers If t, = a,0, and B = a t Bt = ( a 2, a2) ( ) (,0, ); t 2 2 are two reverse order fuzzy numers then,

6 T.Pathinathan and K.Ponnivalavan For example; If, t= ( 4,0,2 ); B t= ( 3,0,) then, B = (,0,) t t Perfect reverse order triangular fuzzy numer perfect reverse order triangular fuzzy numer(protfn) is definsd as P = ( a,o,a ), Where the numerical value of -a and a oth are equal. In other words Perfect Reverse order triangular fuzzy numer(protfn) preserves symmetry. For example; P t = ( 2,0,2 ) t Figure 3: Graphical representation of perfect reverse order triangular fuzzy numer (PRoTFN). 4. Reverse order Trapezoidal fuzzy numer. Definition: 4.. (Reverse order trapezoidal fuzzy numer). fuzzy set is defined as tr = {-a,-,c,d}is said to e reverse order trapezoidal fuzzy numer(rotrfn) if its memership function is given y tr for x a ( x+ ) for a x a 0 for x c ( x c ) for c x d d c for x d 2

7 Reverse order Triangular, Trapezoidal and Pentagonal Fuzzy Numers a 0 c d Figure 4: Graphical representation of reverse order trapezoidal fuzzy numer (RoTRFN). Example: n reverse order trapezoidal fuzzy numer, memership is given as, µ tr = (-3,-2,,3,) and its tr α- Cut of reverse order trapezoidal fuzzy numer ( x + 2) = α x = α 2, 3 2 ( x ) = α x = 2α + 3 tr α =[ α 2,2α + ] When (α =0.5), we get 0.5 = [-2.5,2], lso when (α =0), t 0 = [-3,-2,,3]. for x 3 ( x+ 2 ) for 3 x for 2 x ( x ) for x 3 3 for x 3 3

8 T.Pathinathan and K.Ponnivalavan α cut Figure 5: Graphical representation ofα- Cut of reverse order trapezoidal fuzzy numer (RoTRFN) Conditions on reverse order trapezoidal fuzzy numer. n reverse order trapezoidal Fuzzy Numer should satisfy the following conditions; (i) µ tr ( x ) is a continuous function in the interval [,0] (ii) µ ( x ) is strictly decreasing and continuous function on [-a,-] (iii) tr µ ( x ) is strictly increasing and continuous function on [c,d] tr 4.3. rithmetic operations on reverse order trapezoidal fuzzy numer (ROTRFN) ddition of two reverse order trapezoidal fuzzy numers = ( a, ) and = (-a, ); If,, c t, d Bt 2, c 2 2,, d r r 2 are two reverse order trapezoidal fuzzy numers then, + B = ( a t t a 2, 2, c + c2, d + d2). r r For example; If, = ( 4, 2,,2); B = ( 3,,2,3); tr tr Then, tr+ B tr= ( 7, 3,3,5 ) Suraction of two reverse order trapezoidal fuzzy numers If tr, = ( a,, c, d) and B = (-a t 2,, c 2 2,, d2); are two reverse order r trapezoidal fuzzy numers then, tr Btr = ( a d2, c2, c + 2, d + a2). For example; If, tr= ( 4,,3,4 ); B tr= ( 3, 2,2,3 ) then, tr Btr = ( 7, 3,5,7). 4 tr

9 Reverse order Triangular, Trapezoidal and Pentagonal Fuzzy Numers Perfect reverse order trapezoidal fuzzy numer perfect reverse order trapezoidal fuzzy numer(prtrfn) is definsd as P ( a,,,a ), Where the numerical value of -a and a oth are equal. In other words Perfect Reverse order trapezoidal fuzzy numer(protrfn) preserves symmetry. For example; P tr = ( 2,,,2 ) α tr = Figure 6: Graphical representation of perfect reverse order trapezoidal fuzzy numer (PRoTRFN). 5. Reverse order Pentagonal fuzzy numer. Definition 5.. (Reverse order Pentagonal fuzzy numer). fuzzy set is defined as P = {-a,-,0,c,d}is said to e reverse order Pentagonal fuzzy numer(ropfn) if its memership function is given y P for x a x for a x a x for x 0 x for 0 x c c x for c x d d for x d 5

10 T.Pathinathan and K.Ponnivalavan k Figure 7: Graphical representation of reverse order pentagonal fuzzy numer (RoPFN). Example: perfect reverse order pentagonal fuzzy numer, its memership is given as, µ P = (-5,-2,0,2,5,) and P for x 5 x for 5 x 2 5 x for 2 x 0 2 x for 0 x 2 2 x for 2 x 5 5 for x 5 k Figure 8: Graphical representation of perfect reverse order pentagonal fuzzy numer (RoPFN). 6

11 Reverse order Triangular, Trapezoidal and Pentagonal Fuzzy Numers 6. Conclusion The concept of fuzzy numers and their utility aspects have een studied y researchers in recent times. Categorization of fuzzy numers and their related properties have initiated new notions and approaches. In this paper, the Reverse Order Triangular Fuzzy Numer and Reverse Order Trapezoidal Fuzzy Numer has een introduced with arithmetic operations. Using a few examples,we have explained the relevant arithmetic operations. lso we defined Reverse Order Pentagonal Fuzzy Numer. We oserve that fuzzy numer concepts could e applied to many real life prolem. REFERENCES..Bansal, Some non linear arithmetic operations on triangular fuzzy numer (m,α,β), dvances in Fuzzy Mathematics, 5 (200) C.Wu, G.Wang, Convergence of sequences of fuzzy numers and fixed point theorems for inceasing fuzzy mappings and application, Fuzzy Sets and Systems, 30 (2002) Barnaas Bede, Mathematics of fuzzy sets and fuzzy logic, Springer Pulications, 295, (992) D.Dinagar and K.Latha, Some types of Type-2 Triangular Fuzzy Matrices, International Journal of Pure and pplied Mathematics, 82() (203) D.Duois and H.Prade, Operations on fuzzy numers, International Journal of Systems Science, 9(6) (978) P.Dutta and..tazid, Fuzzy arithmetic with and without α-cut method; a comparative study, International Journal of latest trends in Computing, 3 (20) K.H.Lee, First course on Fuzzy Theory and pplications, Springer, M.Hanss, pplied Fuzzy rithmetic: n Introduction with Engineering pplications, Springer, G.J.Klir and B.Yuan, Fuzzy Sets and Fuzzy logic, Prentice Hall of India Private Limited, C.Parvathi and C.Malathi, rithmetic operations on symmetric trapezoidal intuitionistic fuzzy numers, International Journal of Soft Computing and Engineering, 2 (202) T.Pathinathan and K.Ponnivalavan, Pentagonal fuzzy numers, International Journal of Computing lgorithm, 3 (204) B.hinav, Trapezoidal fuzzy numers (a,,c,d); arithmetic ehavior, International Journal of Physical and Mathematical Sciences, 2, (20) G.Bojadziev and M.Bojadzier, Fuzzy sets and fuzzy logic applications, World Scientific Pulications, 5 (995) (dvances in Fuzzy systems) 4. T.Pathinathan and K.Ponnivalavan, Diamond fuzzy numers, to appear in Journal of Fuzzy Set Valued nalysis. 7