Arithmetic Operations on Generalized Trapezoidal Fuzzy Number and its Applications

TJFS: Turkish Journal of Fuzzy Systems (eissn: 1309 1190) An Official Journal of Turkish Fuzzy Systems Association Vol.3, No.1, pp. 16-44, 2012. Arithmetic Operations on Generalized Trapezoidal Fuzzy Number and its Applications Sanhita Banerjee * Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103, West Bengal, India. Email: sanhita.banerjee88@gmail.com *Corresponding author Tapan Kumar Roy Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103, West Bengal, India. Email: roy_t_k@yahoo.co.in Abstract In this paper, we have studied difuzzification method for generalized trapezoidal fuzzy numbers (GTrFNs) and the four basic arithmetic operations of two GTrFNs based on the Zadeh's extension principle method, interval method and vertex method. Based on these operations, some elementary applications on mensuration are numerically illustrated with approximated values. Keywords: Trapezoidal Fuzzy Number (TrFN), generalized fuzzy number (GFN), generalized trapezoidal fuzzy number (GTrFN), generalized trapezoidal shaped fuzzy number. 1. Introduction At present situation, in science and technology most of the mathematical problems are characterized as complex process for which complete information is not always available. To handle this, the problems need to be set up with the approximately available data. To make this possible Zadeh (1965)[13] introduced fuzzy set theory. In recent years this subject has become an interesting branch of pure and applied sciences. In 1985 Chen [9] further developed the theory and applications of Generalized Fuzzy Number (GFN). Chen (1985) also proposed the function principle, which could be used as the fuzzy numbers arithmetic operations between generalized fuzzy numbers, where these fuzzy arithmetic operations can deal with the generalized fuzzy numbers. Hsieh et al.(1999)[12] pointed out that the arithmetic operators on fuzzy numbers presented in Chen(1985) are not only changing the type of membership function of fuzzy numbers after arithmetic operations, but also they can reduce the troublesomeness of arithmetic operations. In 1987 Dong and Shah[24] introduced Vertex Method using which the value of the functions of interval variable and fuzzy variable can be easily evaluated. Recently GFN has also used in many fields such as risk analysis [19], maximal flow 16

[5], similarity measure [7], reliability [6,11] etc. Already there are several papers [3,6,8,10,20] on arithmetic behaviors of GFN. The difference between the arithmetic operations on generalized fuzzy numbers and the traditional fuzzy numbers is that the former can deal with both non-normalized and normalized fuzzy numbers, but the later with normalized fuzzy numbers. In this paper, we have discussed four arithmetic operations for two GTrFNs in Section-3 based on extension principle method, interval method and vertex method. In section-4 we have compared these three methods based on an example. In section-5 based on these operations we have solved some elementary problems of mensuration and have calculated required approximated values. 2. Mathematical Preliminaries Definition 2.1: Fuzzy Set: A fuzzy set in a universe of discourse X is defined as the following set of pairs Here : X [0,1] is a mapping called the membership value of x X in a fuzzy set Definition 2.2: -cut of a fuzzy set:the -level set (or interval of confidence at level or -cut) of the fuzzy set of X is a crisp set that contains all the elements of X that have membership values in A greater than or equal to i.e. Definition 2.3: Convex fuzzy set: A fuzzy set is called convex fuzzy set if all are convex sets i.e. for every element and and for every Otherwise the fuzzy set is called non convex fuzzy set.. Definition 2.4: Interval Number: An interval number is a closed and bounded set of real numbers. The addition of two interval numbers and denoted by and is defined by The scalar multiplication of a interval number scalar and is defined by is denoted by ka where k is a The subtraction of two interval numbers and denoted by The product of two interval numbers and denoted by and is defined by where, The division of two interval numbers and denoted by and is defined by if empty interval if 17

if if otherwise Definition 2.5: Extension Principle: Let be a mapping from a set X to a set Y. Then the extension principle allows us to define the fuzzy set in Y induced by the fuzzy set in X through f as follows: with where is the inverse image of y. Definition 2.6: Fuzzy Number: A fuzzy number is an extension of a regular number in the sense that it does not refer to one single value but rather to a connected set of possible values, where each possible value has its own weight between 0 and 1. This weight is called the membership function. Thus a fuzzy number is a convex and normal fuzzy set. If is a fuzzy number then is a fuzzy convex set and if then is non decreasing for and non increasing for. Definition 2.7: Trapezoidal Fuzzy Number: A Trapezoidal fuzzy number (TrFN) denoted by defined as where the membership function is or, Definition 2.8: Generalized Fuzzy number (GFN): A fuzzy set ;, defined on the universal set of real numbers R, is said to be generalized fuzzy number if its membership function has the following characteristics: (i) : R [0, 1]is continuous. (ii) for all (iii) is strictly increasing on [, ] and strictly decreasing on [, ]. (iv) for all, where. Definition 2.9: Generalized Trapezoidal Fuzzy number (GTrFN): A Generalized Fuzzy Number ;, is called a Generalized Trapezoidal Fuzzy Number if its membership function is given by 18

or,. Fig-2.1:Comparison between membership function of TrFN and GTrFN Definition 2.10: Equality of two GTrFN: Two Generalized Trapezoidal Fuzzy Number (GTrFN) = and = is said to be equal i.e. if and only if and. Definition 2.11: A GTrFN = is said to be non negative (non positive) i.e. Symmetric ( if and only if. Type of GTrFN ; ) or in central form Table2.1:- different types of GTrFN Conditions Rough sketch of membership function Non symmetric type 1 ( Non symmetric type2 ( ) 19

Left GTrFN Right GTrFN Definition 2.12: Vertex Method [24]: When is continuous in the n-dimensional rectangular region, and also no extreme point exists in this region (including the boundaries), then the value of interval function can be obtained by where is the ordinate of the j-th vertex and are intervals of real numbers. Example2.1: Determine. Given,, The ordinate of vertices are From those ordinates, we obtain Then Definition 2.13: Defuzzification: Let = be a GTrFN. The defuzzification value of is an approximated real number. There are many methods for defuzzication such as Centroid Method, Mean of Interval Method, Removal Area Method etc. In this paper we have used Removal Area Method for defuzzification. Removal Area Method [1]: Let us consider a real number, and a generalized fuzzy number. The left side removal of with respect to, is defined as the area bounded by and the left side of the generalized fuzzy number. Similarly, the right side removal,, is defined. The removal of the generalized fuzzy number with respect to is defined as the mean of and. Thus,, relative to, is equivalent to an ordinary representation of the generalized fuzzy number. 20

Fig-2.2: Left removal area Fig-2.3: Right removal area, The defuzzification value or approximated value of i.e., Defuzzification value for GTrFN: Let ; be a GTrFN with its membership function and -cuts,, Fig-2.4: Left removal area Fig-2.5: Right removal area or,, or, The defuzzification value or approximated value of i.e., 21

3. Arithmetic operations of GTrFNs In this section we discuss four operations (addition, subtraction, multiplication, division) for two generalized trapezoidal fuzzy numbers based on extension principle method, interval method and vertex method. Let = and = be two positive generalized trapezoidal fuzzy numbers and their membership functions are and their -cuts be,,,, 3.1 Addition of two GTrFNs a) Addition of two GTrFNs based on extension principle Let Let where (3.1.1) (3.1.2) [Note-3.1: ] 22

(3.1.3) The addition of two GTrFNs is another GTrFN with membership function given at equation (3.1.3) Fig-3.1:- Rough sketch of Membership function of b) Addition of two GTrFNs based on interval method Let where,,.(3.1.4) [Note-3.2: ] The addition of two GTrFNs shown in Fig-3.1. is another GTrFN with membership function given at equation (3.1.3) and c) Addition of two GTrFNs based on vertex method Let Now the ordinate of the vertices are 23

It can be shown that So Now following Note-3.2 we get that the addition of two GTrFNs is another GTrFN with membership function given at equation (3.1.3) and shown in Fig-3.1. 3.2 Scalar multiplication of a GTrFN a) Scalar multiplication of a GTrFN based on extension principle method Let where Case1: When (3.2.1) (3.2.2) (3.2.3) The positive scalar (k) multiplication of a GTrFN is another GTrFN with membership function given at equation (3.2.3) 24

Fig-3.2:- Rough sketch of Membership function of Case2: When (3.2.4) (3.2.5) (3.2.6) The negative scalar (k) multiplication of a GTrFN is another GTrFN with membership function given at equation (3.2.6) Fig-3.3:- Rough sketch of Membership function of 25

b) Scalar multiplication of a GTrFN based on interval method Let where, Case1: When [Note-3.3: ] The positive scalar (k) multiplication of a GTrFN is another GTrFN with membership function given at equation (3.2.3) and shown in Fig- 3.2. Case2: When [Note-3.4: ] The negative scalar (k) multiplication of a GTrFN is another GTrFN with membership function given at equation (3.2.6) an shown in Fig-3.3. c) Scalar multiplication of a GTrFN based on vertex method Let Now the ordinate of the vertices are, Case1: When, 26

So Following Note-3.3 we get that the positive scalar (k) multiplication of a GTrFN is another GTrFN with membership function given at equation (3.2.3) and shown in Fig-3.2. Case2: When, So Following Note-3.4 we get that the negative scalar (k) multiplication of a GTrFN is another GTrFN with membership function given at equation (3.2.6) an shown in Fig-3.3. 3.3 Subtraction of two GTrFNs a) Subtraction of two GTrFNs based on extension principle method Let where Let (3.3.1) (3.3.2) [Following Note 3.1] (3.3.3) Thus we get that the subtraction of two GTrFNs is another GTrFN with membership function given at equation (3.3.3) 27

Fig-3.4:-Rough sketch of Membership function of b) Subtraction of two GTrFNs based on interval method Let where,, Following Note-3.2 we get that the subtraction of two GTrFNs is another GTrFN with membership function given at equation (3.3.3) and shown in Fig-3.4. c) Subtraction of two GTrFNs based on vertex method Let Now the ordinate of the vertices are It can be shown that So Now following Note-3.2 we get that the subtraction of two GTrFNs is another GTrFN with membership function given at equation (3.3.3) and shown in Fig-3.4. 28

3.4 Multiplication of two GTrFNs a) Multiplication of two GTrFNs based on extension principle method Let Let where (3.4.1) (3.4.2) Let, such that sup [Note-3.5: Let is an increasing function in z.] [Note-3.6: Let 29

is a decreasing function in z. Again, and and ] (3.4.3) Where,. We get that the multiplication of two GTrFNs is a generalized trapezoidal shaped fuzzy number with membership function given at equation (3.4.3). 30

Fig-3.5:-Rough sketch of Membership function of b) Multiplication of two GTrFNs based on interval method Let where,, Now following Note-3.5 and Note-3.6 we get that the multiplication of two GTrFNs is a generalized trapezoidal shaped fuzzy number with membership function given at equation (3.4.3) and shown in Fig-3.5. c) Multiplication of two GTrFNs based on vertex method Let Now the ordinate of the vertices are It can be shown that So 31

Now following Note-3.5 and Note-3.6 we get that the multiplication of two GTrFNs is a generalized trapezoidal shaped fuzzy number with membership function given at equation (3.4.3) and shown in Fig-3.5. 3.5 Division of two GTrFNs a) Division of two GTrFNs based on extension principle method Let where,, and (3.5.1) (3.5.2) Let, sup such that Similarly, sup [Note-3.7: for is an increasing function with z. for is an decreasing function with z. Again, and and ] 32

(3.5.3) Thus we get that the division of two GTrFNs with membership function given at equation (3.5.3) is a generalized trapezoidal shaped fuzzy number Fig-3.6:- Rough sketch of Membership function of b) Division of two GTrFNs based on interval method Again Now following Note-3.7 we get that the division of two GTrFNs is a generalized trapezoidal shaped fuzzy number with membership function given at equation (3.5.3) and shown in Fig-3.6. c) Division of two GTrFNs based on vertex method Let Now the ordinate of the vertices are 33

,, It can be shown that So Now following Note-3.7 we get that the division of two GTrFNs is a generalized trapezoidal shaped fuzzy number with membership function given at equation (3.5.3) and shown in Fig-3.6. Table-3.1:- Arithmetic operations of two Left GTrFNs = and = Arithmetic operations Membership function of i.e. Rough sketch of Nature of Addition Left Generalized Trapezoidal Fuzzy Number Subtraction Generalized Trapezoidal Fuzzy Number Multiplication Left Generalized Trapezoidal shaped Fuzzy Number Division Generalized Trapezoidal shaped Fuzzy Number Remarks:- From the table-3.1 we see that the addition and multiplication of two Left GTrFNs is a Left GTrFN and Left Generalized Trapezoidal shaped Fuzzy Number respectively but the subtraction 34

and the division of two Left GTrFNs is a GTrFN and Generalized Trapezoidal shaped Fuzzy Number respectively. Table-3.2:- Arithmetic operations of two Right GTrFNs = and = Arithmetic operations Membership function of i.e. Rough sketch of Nature of Addition Right Generalized Trapezoidal Fuzzy Number Subtraction Generalized Trapezoidal Fuzzy Number Multiplication Right Generalized Trapezoidal shaped Fuzzy Number Division Generalized Trapezoidal shaped Fuzzy Number Remarks:- From the table-3.2 we see that the addition and multiplication of two Right GTrFNs is a Right GTrFN and Right Generalized Trapezoidal shaped Fuzzy Number respectively but the subtraction and the division of two Right GTrFNs is a GTrFN and Generalized Trapezoidal shaped Fuzzy Number respectively. 4. Comparison among three methods based on an example We consider an expression where more than one arithmetic operation is used. Here =, = and = be three positive GTrFNs and their -cuts be 35

, and Let In vertex method, let Now the ordinate of the vertices are,,,, From the above we see that So -cut of i.e. and the rough sketch of membership function of is shown in Fig-4.1. Fig-4.1:- Rough sketch of Membership function of 36

In extension principle method Now and the rough sketch of membership function of is shown in Fig-4.1. In interval arithmetic method if we consider the given expression as then we get and the rough sketch of membership function of is shown in Fig-4.1. And if we consider the expression as then its -cut and the rough sketch of membership function of is shown in Fig-4.2. Fig-4.2:- Rough sketch of Membership function of Here we get one required value of an expression in vertex method and extension principle method while in interval method we get two possible values for the same expression. So it can be said that vertex method or extension principle method is more useful than interval method in the case of expressions with two or more arithmetic operations. 37

5. Applications In this section we have numerically solved some elementary problems of mensuration based on arithmetic operations described in section-3. a) Perimeter of a Rectangle Let the length and breadth of a rectangle are two GTrFNs The perimeter of the rectangle is a GTrFN generalized fuzzy set with the membership function, then the perimeter of the rectangle is and which is a Fig-5.1: Rough sketch of membership function of Thus we get that the perimeter of the rectangle is not less than 36cm and not greater than 48cm. The value of perimeter is increased from 36cm to 40cm at constant rate 0.2 and is decreased from 44cm to 48cm also at constant rate 0.2. There are 80% possibilities that the perimeter takes the value between 40cm and 44cm. Fig-5.2: Left removal area Fig-5.3: Right removal area,, The approximated value of the perimeter of the rectangle is 42 cm. b) Length of a Rod Let the length of a rod is a GTrFN =. If the length rod =, a GTrFN, is cut off from this rod then the remaining length of the is The remaining length of the rod is a GTrFN fuzzy set with the membership function which is a generalized 38

Fig-5.4: Rough sketch of membership function of Here we get that the remaining length of the rod is not less than 4cm and not greater than 10cm. The value of this length is increased from 4cm to 6cm at constant rate 0.35 and is decreased from 8cm to 10cm also at constant rate 0.35. There are 70% possibilities that the length takes the value between 6cm and 8cm. Fig-5.5: Left removal area Fig-5.6: Right removal area,, The approximated value of the remaining length of the rod is 7 cm. c) Area of a Triangle Let the base and the height of a triangle are two GTrFNs = and = then the area of the triangle is The area of the triangle is a generalized trapezoidal shaped (concave-convex type) fuzzy number function which is a generalized fuzzy set with the membership 39

Fig-5.7: Rough sketch of membership function of Thus we get that the area of the triangle is not less than 5sqcm and not greater than 20sqcm. The value of area is increased from 5sqcm to 9sqcm at nonlinear increasing rate and is decreased from 14sqcm to 20sqcm at nonlinear decreasing rate the value between 9sqcm and 14sqcm.. There are 70% possibilities that the area takes Fig-5.8: Left removal area Fig-5.9: Right removal area The approximated value of the area of the triangle is 14 sqcm. d) Length of a Rectangle Let the area and breadth of a rectangle are two GTrFNs = and =, then the length of the rectangle is, The length of the rectangle is a generalized trapezoidal shaped (concave-convex type) fuzzy number which is a generalized fuzzy set with the membership function Fig-5.10: Rough sketch of membership function of 40

we get that the length of the rectangle is not less than 7cm and not greater than 17cm.The value of length is increased from 7cm to 9cm at nonlinear increasing rate and is decreased from 12cm to 17cm at nonlinear decreasing rate between 9cm and 12cm.. There are 80% possibilities that the length takes the value Fig-5.11: Left removal area Fig-5.12: Right removal area, The approximated value of the length of the rectangle is 11.1 cm. e) Area of an annulus Let the outer radius and inner radius of an annulus are two GTrFNs = and =, then the area of the annulus is The area of the annulus is a generalized trapezoidal shaped (concave-convex type) fuzzy number fuzzy set with the membership function which is a generalized Fig-5.13: Rough sketch of membership function of We get that the area of the annulus is not less than 201.14 sqcm and not greater than 792 sqcm. The value of area is increased from 201.14 sqcm to 374 sqcm at nonlinear increasing rate 41

502.86 sqcm. and is decreased from 502.86 sqcm to 792 sqcm at nonlinear decreasing rate. There are 70% possibilities that the area takes the value between 374 sqcm and Fig-5.14: Left removal area Fig-5.15: Right removal area, The approximated value of the area of the annulus is 378 sqcm. 6. Conclusion and future work In this paper, we have worked on GTrFN. We have described four operations for two GTrFNs based on extension principle, interval method and vertex method and compared three methods with an example. We have solved numerically some problems of mensuration based on these operations using GTrFN and we have calculated the approximated values. Further GTrFN can be used in various problems of engineering and mathematical sciences. Acknowledgement The authors would like to thank to the Editors and the two Referees for their constructive comments and suggestions that significantly improve the quality and clarity of the paper. References A. Kaufmann and M.M. Gupta, Introduction to fuzzy Arithmetic Theory and Application (Van Nostrand Reinhold, New York, 1991). A. Kaufmann and M. M. Gupta, Fuzzy Mathematical Model in Engineering and Management Science, North-Holland, 1988. Abhinav Bansal, Trapezoidal Fuzzy Numbers (a,b,c,d): Arithmetic Behavior, International Journal of Physical and Mathematical Sciences (2011). Amit Kumar, Pushpinder Singh, Amarpreet Kaur, Parmpreet Kaur, Ranking of Generalized Trapezoidal Fuzzy Numbers Based on Rank, Mode, Divergence and Spread, Turkish Journal of Fuzzy Systems (ISSN: 1309 1190), Vol.1, No.2, pp. 141-152, 2010. 42

Amit Kumar and Manjot Kaur, An Algorithm for Solving Fuzzy Maximal Flow Problems Using Generalized Trapezoidal Fuzzy Numbers, International Journal of Applied Science and Engineering2010. 8, 2: 109-118. Angel Garrido, On Generalized Fuzzy Numbers, AMO-Advanced Modeling and Optimization, Volume 13, Number 3, 2011. Ashok Kumar Shaw and Tapan Kumar Roy, Generalized Trapezoidal Fuzzy Number with its Arithmetic Operations and its Application in Fuzzy System Reliability Analysis, Int. J. Pure Appl. Sci. Technol., 5(2) (2011), pp. 60-76. B. Sridevi and R. Nadarajan, Fuzzy Similarity Measure for Generalized Fuzzy Numbers, Int. J. Open Problems Compt. Math., Vol. 2, No. 2, June 2009. BongJu Lee and Yong Sik Yun, The Generalized Trapezoidal Fuzzy Sets, Journal of the Chungcheong Mathematical Society, Volume 24, No. 2, June 2011. C. Veeramani, C. Duraisamy, and A. Rathinasamy, Nearest Symmetric Trapezoidal Approximation of Fuzzy Numbers, Int. J. Open Problems Compt. Math., Vol. 4, No. 1, March 2011. Chen, S. H. 1985. Operations on fuzzy numbers with function principal. Tamkang Journal of Management Sciences,6: 13-25. D. Chakraborty, D. Guha, Addition of Two Generalized Fuzzy Numbers, International journal of Industrial Mathematics, Vol. 2,No. 1(2010)9-20. G. S. Mahapatra and T. K. Roy, Optimal Redundancy Allocation in Series-Parallel System using GeneralizedFuzzy Number, Tamsui Oxford Journal of Information and Mathematical Sciences 27(1) (2011) 1-20,Aletheia University. Hsieh, C. H. and Chen, S. H. 1999.Similarity of generalized fuzzy numbers with graded mean integration representation. In: Proceedings of the Eighth International Fuzzy System Association World Congress, Taipei, Taiwan, Republic of China, 2: 551-555. L. A. Zadeh, Fuzzy sets, Information and Control, 8(1965), 338-353. L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, I Information Sciences. 8 (1975), 199-249. L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, II Information Sciences. 8 (1975), 301-357. L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, III Information Sciences. 9 (1975), 43-80. M. Mizumoto and K. Tanaka, Algebraic properties of fuzzy numbers, in: M.M. Gupta,R.K. Ragade and R.R. Yager, eds., Advances in Fuzzy Set Theory and Applications (North-Holland, Amsterdam, 1982) Przemysław Grzegorzewski, Karolina Pasternak-Winiarska, Weighted Trapezoidal Approximations of Fuzzy Numbers, IFSA-EUSFLAT 2009. Rahim Saneifard and Rasoul Saneifard, A Method for Defuzzification Based on Radius of Gyration, Journal of Applied Sciences Research, 7(3): 247-252, 2011. 43

S. Heilpern, Representation and application of fuzzy numbers, Fuzzy sets and Systems 91(2): 259-268, 1997. Shan-Huo Chen, Chien-Chung Wang, Fuzzy Distance of Trapezoidal Fuzzy Numbers, International Journal of Innovative Computing, Information and Control ICIC International c 2008 ISSN 1349-4198 Volume 4, Number 6, June 2008. Weimin Dong and Haresh C. Shah, Vertex Method For Computing Functions Of Fuzzy Variables, Fuzzy Sets and Systems 24 (1987) 65-78. Xiaoyan Su, Wen Jiang, Jianling Xu, PeidaXu, Yong Deng, A New Fuzzy Risk Analysis Method based on Generalized Fuzzy Numbers, Journal of Software, vol. 6, no. 9, September 2011. Y. S. Yun, S. U. Ryu, and J. W. Park, The generalized triangular fuzzy sets, Journal of The Chungcheong Mathematical Society. 22 (2009), no. 2, 161-170. 44