Dual Trapezoidal Fuzzy Number and Its Applications

Dual trapezoidal brumous Number and Its Applications Jon Arockiaraj. J, Pathinathan.T, Revathy.SAbstract In this paper, we introduce Convergence of -Cut. We define at which point the -Cut converges to the dazed forms pool it leave be illustrated by warning using doubled trapezoidal blurred repress and Some elementary applications on mensuration are numerically illustrated with approximated determine.KeyWords Fuzzy number, -Cut, Dual trapezoidal wooly number, Defuzzification.IntroductionFuzzy finds have been introduced by Lotfi. A. Zadeh (1965). Fuzzy numbers were first introduce by Zadeh in 1975.There after theory of blear-eyed number was further studied and developed by Dubois and Prade, R.Yager Mizomoto, J.Buckly and Many others. Since and and hence many workers studied the theory of foggy numbers and achieved fruitful results. The fuzziness can be represented by different ways one of the most useful representation is social rank function. Also depending the spi rit and shape of the rank and file function the fuzzy number can be classified in different forms, such as angulate fuzzy number, trapezoidal fuzzy number etc. A fuzzy number is a quantity whose abide bys are imprecise, rather than exact as is the case with genius apprised number. Fuzzy numbers are used in statistics computer programming, engineering and experimental science. So far fuzzy numbers like triangular fuzzy number, trapezoidal fuzzy numbers, pentagonal, hexagonal, octagonal pyramid and diamond fuzzy numbers have been introduced with its membership functions. These numbers have got many applications like non-linear equations, take a chance analysis and reliability. In this paper, we introduce Dual trapezoidal fuzzy numbers with its membership functions and its applications. Section one presents the introduction, section two presents the basic definition of fuzzy numbers section three presents Dual trapezoidal fuzzy numbers and its applications and in the final sectio n we give conclusion.2. Basic DefinitionsDefinition 2.1 (Fuzzy set)A fuzzy set A in a universe of discourse X is defined as the following set of pairs A= (x, A(x)) xX Here A(x) x is a mapping called the degree of membership function of the fuzzy set A and A(x) is called the membership value of xX in the fuzzy set These membership grades are often represented by real numbers ranging from 0, 1.Definition 2.2 (Fuzzy Number)A fuzzy set A defined on the universal set of real number R is said to be a fuzzy number if its membership function has satisfy the following characteristics. ( i) A (x) is a piecewise continual(ii) A is convex, i.e., A (x1 + (1-) x2) min (A(x1), A(x2)) x1 ,x2R 0,1(iii) A is normal, i.e., there exist xo R such that A (xo)=1Definition 2.3 (Trapezoidal Fuzzy Number)A trapezoidal fuzzy number represented with four points as A = (a b c d), Where all a, b, c, d are real numbers and its membership function is presumptuousness below where a b c dA(x)=3. DUAL TRAPEZOID AL FUZZY NUMBERDefinition 3.1 (Dual Trapezoidal Fuzzy Number)A Dual Trapezoidal fuzzy number of a fuzzy set A is defined as ADT= a, b, c, d () Where all a, b, c, d are real numbers and its membership function is given below where abcdDT(x) =where is the base of the trapezoidal and also for the inverted reflection of the above trapezoidal that is to say a b c d Figure Graphical Representation of Dual Trapezoidal fuzzy Number 3.2 DEFUZZIFICATIONlet ADT= (a, b, c, d, ) be a dual trapezoidal fuzzy number .The defuzzification value of ADT is an approximate real number. There are many method for defuzzification such as Centroid rule, Mean of Interval Method , Removal Area Method etc. In this Paper We have used Centroid field of operations method for defuzzification .CENTROID OF AREA METHOEDCentroid of bowl method or centry of gravitational attraction method. It obtains the centre of area (X*)occupied by the fuzzy sets.It can be expressed asX* = Defuzzification Value for dual trapezo idal fuzzy numberLet ADT= a, b, c, d () be a DTrFN with its membership functionDT(x) =Using centroid area method+dx+++dx= + + + + + = = = ++ dx+++dx== = c + d a bDefuzzification = = = 3.3 APPLICATIONIn this section. We have discussed the convergence of -cut using the example of dual trapezoidal fuzzy number.CONVERGENCE OF -CUT Let ADT = a, b, c, d, () be a dual trapezoidal fuzzy number whose membership function function is given asDT(x) =To find -cut of ADT .We first set 0,1 to both left and right reference functions of ADT. Expressing X in toll of which gives -cut of ADT.= x l= a+ (b-a) = x r =d-(d-c) ADT= a+ (b-a) , d-(d-c) In ordinary to find -cut, we give values as 0 or 0.5 or 1 in the interval 0, 1 .Instead of giving these values for . we divide the interval 0,1 as many continuous subinterval. If we give very small values for , the -cut converges to a fuzzy number a, d in the domain of X it will be illustrated by example as given below.ExampleADT = (-6,-4, 3, 6) and its membership function will beDT(x) =- cut of dual Trapezoidal fuzzy Number = (x l + 6)/2Xl = 2-6 = (6 xr)/3Xr = 6-3ADT= 2-6, 6-3 When =1/10 then(prenominal) ADT = -5. 8 , 5.7When =1/102 then ADT =-5.98 , 5.97When =1/103 then ADT = -5.998 , 5.997When =1/104 then ADT=-5.9998 , 5.9997When =1/105 then ADT=-5.99998 , 5.99997 When =1/106 then ADT= -5.999998 , 5.999997 When =1/107 then ADT= -5.9999998 , 5.9999997, When =1/108 then ADT= -5.99999998 , 5.99999997 When =1/109 then ADT= -5.999999998 , 5.999999997When =1/1010 then ADT=-6 , 6When =1/1011 then ADT=-6 , 6When =1/1012 then ADT =-6 , 6When =1/1013 then ADT =-6 , 6 ..etcWhen =1/10n as n then the -cut converges to ADT=-6, 6 Figure Graphical Representation of convergence of -cutWhen =2/10 then ADT= -5.6,5.4 When =2/102 then ADT= -5.96,5.94 When =2/103 then ADT= -5.996,5.994 When =2/104 then ADT= , -5.9996,5.9994 When =2/105 then ADT= , -5.99996,5.99994 When =2/106 then ADT= , -5.999996,5.999994 When =2/107 then ADT=-5.999999 6, 5.9999994 , When =2/108 then ADT= , -5.99999996,5.99999994 When =2/109 then ADT= , -5.999999996,5.999999994 When =2/1010 then ADT= , -6,6 When =2/1011 then ADT= -6,6 When =2/1012 then ADT= -6,6 When =2/1013 then ADT= -6,6etcWhen =2/10n as n then the -cut converges to ADT= -6,6 Simillarly, =3/10n,4/10n,5/10n,6/10n,7/10n,8/10n,9/10n,10/10n upto these value n varies from 1to after 11/10n,12/10n..100/10n as n varies from 2 to and101/10n,102/10n. as n varies from 3 to and the process is goes on like this if we give the value for it will converges to the dual trapezoidal fuzzy number-6,6From the above example we conclude that , In general we have K/10n if we give different values for K as n- varies upto to if we give as n tends to then the values of ADT converges to the fuzzy numbera,d in the domain X. 3.4 APPLICATIONSIn this section we have numerically work some elementary problems of mensuration based on arithmetic carrying out using defuzzified centroid area method1.Perime ter of RectangleLet the length and breadth of a rectangle are two positive dual trapezoidal fuzzy numbers ADT = (10cm, 11cm, 12cm,13cm) and BDT = (4cm, 5cm,6cm,7cm) then perimeter CDT of rectangle is 2ADT+BDTTherefore the perimeter of the rectangle is a dual trapezoidal fuzzy number CDT = (28cm, 32cm,36cm,40cm) and its membership functionsDT(x) =The Perimeter of the rectangle is not less than 28 and not great than 40 .The perimeter value takes between 32 to 36.Centroid area methodX* = = = = = 34The approximate value of the perimeter of the rectangle is 34 cm.2.Length of RodLet length of a rod is a positive DTrFN ADT = (10cm,11cm,12cm, 13cm). If the length BDT = (5cm, 6cm , 7cm, 8cm), a DTrFN is cut off from this rod then the remain length of the rod CDT is ADT(-)BDTThe remaining length of the rod is a DTrFN CDT = (2cm, 4cm, 6cm, 8cm) and its membership functionDT(x) =The remaining length of the rod is not less than 2cm and not greater than 8cm.The length of the rod takes the value between 4cm and 6cm.Centroid area methodX* = = = = = 5The approximate value of the remaining length of the rod is 5cm.3.Length of a Rectangle Let the area and breadth of a rectangle are two positive dual trapezoidal fuzzy numbers ADT=(36cm,40cm,44cm,48cm) and BDT=(3cm,4cm,5cm,6cm) then the length CDT of the rectangle is is ADT()BDT.Therefore the length of the rectangle is a dual trapezoidal fuzzy number CDT=(6cm,8cm,11cm,16cm) and its membership functionsDT(x) = The length of the rectangle is not less than 6cm and not greater than 16cm .The length of the rectangle takes the value between 8cm and 11cm.Centroid area methodX * = = = = 10.38The approximate value of the length of the rectangle is 10.38cm.4. Area of the RectangleLet the length and breadth of a rectangle are two positive dual trapezoidal fuzzy numbers ADT=(3cm,4cm,5cm,6cm) and BDT=(8cm,9cm,10cm,11cm) then the area of rectangle is ADT(.) BDTTherefore the area of the rectangle is a dual trapezoidal fuzzy number CDT= (24cm, 36cm, 50cm, 66cm) and its membership functionsDT(x) =The area of the rectangle not less than 24 and not greater than 66.The area of the reactangle takes the value between 36 and 50.Centroid area methodX * = = = = 44.167sq.cm4.CONCLUSIONIn this paper, we have worked on DTrFN .We have define the Convergence of -Cut to the fuzzy number. We have solved numerically some problems of mensuration based on operations using DTrFN and we have calculated the approximate values. Further DTrFN can be used in various problem of engineering and mathematical science.5. 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