Improved arithmetic operations on generalized fuzzy numbers

Improved arithmetic operations on generalized fuzzy numbers Luu Quoc Dat University of Economics and Business, Vietnam National University Hanoi, Vietnam Department of Industrial Management, National Taiwan University of Science and Technology Taipei, Taiwan, ROC Email: luuquocdat_84@yahoo.com; datlq@vnu.edu.vn Shuo-Yan Chou Department of Industrial Management, National Taiwan University of Science and Technology Taipei, Taiwan, ROC E-mail: sychou2@me.com Canh Chi Dung University of Economics and Business, Vietnam National University Hanoi, Vietnam Email: canhchidung@gmail.com; dungcc@vnu.edu.vn Vincent F. Yu Department of Industrial Management, National Taiwan University of Science and Technology Taipei, Taiwan, ROC e-mail: vincent@mail.ntust.edu.tw Abstract- Determining the arithmetic operations of fuzzy numbers is a very important issue in fuzzy sets theory, decision process, data analysis, and applications. In 1985, Chen formulated the arithmetic operations between generalized fuzzy numbers by proposing the function principle. Since then, researchers have shown an increased interest in generalized fuzzy numbers. Most of existing studies done using generalized fuzzy numbers were based on Chen’s (1985) arithmetic operations. Despite its merits, there were some shortcomings associated with Chen’s method. In order to overcome the drawbacks of Chen’s method, this paper develops the extension principle to derive arithmetic operations between generalized trapezoidal (triangular) fuzzy numbers. Several examples demonstrating the usage and advantages of the proposed method are presented. It can be concluded that the proposed method can effectively resolve the issues with Chen’s method. Finally, the proposed extension principle is applied to solve a multi-criteria decision making (MCDM) problem. Keywords: Generalized fuzzy numbers, Arithmetic operations, Fuzzy MCDM I. INTRODUCTION In 1965, Zadeh [1] introduced the concept of fuzzy sets theory as a mathematical way of representing impreciseness or vagueness in real life. Thereafter, many studies have presented some properties of operations of fuzzy sets and fuzzy numbers [2-5]. Chen [6] further 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 and Chen [7] indicated that arithmetic operators on fuzzy numbers presented in Chen [6] does not only change the type of membership function of fuzzy numbers after arithmetic operations, but they can also reduce the troublesomeness and tediousness of arithmetical operations. Recently, researchers have shown an increased interest in generalized fuzzy numbers [8-22]. Most of existing studies done using generalized fuzzy numbers were based on Chen’s arithmetic operations. Despite its merits, in some special cases, the arithmetic operations between generalized fuzzy numbers proposed by Chen [6] led to some misapplications and inconsistencies as pointed out by Chakraborty and Guha [23]. In addition, it is also found that using Chen’s [6] method the arithmetic operations between generalized fuzzy numbers are the same when we change the degree of confidence w of generalized fuzzy numbers. Due to this reason, it has been observed that arithmetic operations between generalized fuzzy numbers proposed by Chen [6] cause the loss of information and do not give exact results. In order to overcome the drawbacks of Chen’s method, this paper develops new arithmetic operations between generalized trapezoidal fuzzy numbers. We then applied the proposed extension principle to solve a multi-criteria decision making problem. II. PRELIMINARIES Chen [6] presented arithmetical operations between generalized trapezoidal fuzzy numbers based on the extension principle. Let A and B are two generalized trapezoidal fuzzy numbers, i.e., 1 2 3 4( , , , ; )AA a a a a w= and 1 2 3 4( , , , ; ),BB b b b b w= where 1 2 3 4 1 2 3, , , , , ,a a a a b b b and 4b are real values, 0 1Aw≤ ≤ and 0 1.Bw≤ ≤ Some arithmetic operators between the generalized fuzzy numbers A and B are defined as follows: (i). Generalized trapezoidal fuzzy numbers addition ( ) :+ 1 1 2 2 3 3 4 4( ) ( , , , ;min( , )),A BA B a b a b a b a b w w+ = + + + + (1) where 1 2 3 4 1 2 3, , , , , ,a a a a b b b and 4b are real values. (ii). Generalized trapezoidal fuzzy numbers subtraction ( ) :− 1 4 2 3 3 2 4 1( ) ( , , , ;min( , )),A BA B a b a b a b a b w w− = − − − − (2) where 1 2 3 4 1 2 3, , , , , ,a a a a b b b and 4b are real values. (iii). Generalized trapezoidal fuzzy numbers multiplication (x) : 1 1 2 2 3 3 4 4(x) ( , , , ;min( , )).A BA B a b a b a b a b w w= × × × × (3) where 1 2 3 4 1 2 3, , , , , ,a a a a b b b and 4b are all positive real numbers. (iv). Generalized trapezoidal fuzzy numbers division (/) : Let 1 2 3 4 1 2 3, , , , , ,a a a a b b b and 4b be non-zero positive real numbers. Then, 1 4 2 3 3 2 4 1(/) ( / , / , / , / ; min( , )),A BA B a b a b a b a b w w= (4) Proceedings of 2013 International Conference on Fuzzy Theory and Its Application National Taiwan University of Science and Technology, Taipei, Taiwan, Dec. 6-8, 2013 407 III. SHORTCOMINGS WITH CHEN’S FUZZY ARITHMETIC OPERATIONS BETWEEN GENERALIZED FUZZY NUMBERS In this section, shortcomings of Chen’s [6] arithmetic operations are pointed out. Several examples are chosen to prove that the arithmetic operations between generalized fuzzy numbers, proposed by Chen [6], do not satisfy the reasonable properties for the arithmetic operations of fuzzy numbers. In 2010, Chakraborty and Guha [23] indicated that Chen’s [6] addition (subtraction) operation does not give the exact values. This drawback is shown in example 1. Example 1: Consider the generalized triangular fuzzy numbers (0.5,0.6,0.7;0.5)A = and (0.6,0.7,0.8;0.9)B = shown in Fig. 1. It is observed from Fig. 2 that, min( 0.5, 0.9) 0.5.A Bw w= = = If we take 0.5 (since 0.5 < 0.9) cut of ,B then B is transformed into a generalized trapezoidal (flat) fuzzy number. Therefore, it is necessary to conserve this flatness into the resultant generalized fuzzy number. In this respect Chen’s [6] approach is incomplete and hence loses its significance. A B Fig. 1. Generalized fuzzy numbers A and B in Example 1 In addition, using Chen’s method, the results of generalized fuzzy numbers arithmetic operations are the same when we change the degree of confidence w of generalized fuzzy numbers. This shortcoming is illustrated in example 2. Example 2: Consider the generalized triangular fuzzy numbers 1 (0.2,0.4,0.6;0.5),A = 2 (0.5,0.7,0.9;0.7),A = and 3 (0.5,0.7,0.9;0.9)A = as in Fig. 2. Intuitively, the order of fuzzy numbers 2A and 3A is 2 3.A A∈ Then, we should have 12 1 2 13 1 3( ) ( ) .A A A A A A= + ∈ = + However, using the Chen’s method, we have 12 (0.7,1.1,1.5;0.5)A = and 13 (0.7,1.1,1.5;0.5).A = Thus, the additions between the generalized fuzzy numbers 1A and 2 ,A and 1A and 3A are the same, i.e., 12 13.A A∼ Therefore, Chen’s method cannot consistency calculate the arithmetic operations between generalized fuzzy numbers. 1A 2A 3A 12 13,A A Fig. 2. Additions between the generalized fuzzy numbers in Example 2 IV. PROPOSED ARITHMETIC OPERATIONS BETWEEN GENERALIZED FUZZY NUMBERS To overcome these shortcomings of Chen’s [6] method, this paper proposes new arithmetical operations between generalized trapezoidal fuzzy numbers using α- cuts of fuzzy number. The revised arithmetical operations between generalized trapezoidal fuzzy numbers are described as follows: Let 1 2 3 4( , , , ; )AA a a a a w= and 1 2 3 4( , , , ; )BB b b b b w= are two generalized trapezoidal fuzzy numbers with membership function ( )Af x and ( ),Bf x respectively, which can be written in the following form: 1 2 1 1 2 2 3 4 3 4 3 4 ( ) / ( ), , , , ( ) ( ) / ( ), , 0, otherwise, A A A A w x a a a a x a w a x a f x w x a a a a x a − − ≤ ≤⎧⎪ ≤ ≤⎪ = ⎨ − − ≤ ≤⎪⎪⎩ (5) and 1 2 1 1 2 2 3 4 3 4 3 4 ( ) / ( ), , , , ( ) ( ) / ( ), , 0, otherwise, B B B B w x b b b b x b w b x b f x w x b b b b x b − − ≤ ≤⎧⎪ ≤ ≤⎪ = ⎨ − − ≤ ≤⎪⎪⎩ (6) where, 1 2 3 4 1 2 3, , , , , ,a a a a b b b and 4b are real values, 0 1Aw≤ ≤ and 0 1.Bw≤ ≤ ,A B Aw w w w= ≤ and Bw denote the degree of confidence with respect to the decision-makers’ opinions A and ,B respectively. To find the arithmetical operations between two generalized trapezoidal fuzzy numbers A and ,B firstly, take A Bw w w= < cut of fuzzy number 1 2 3 4( , , , ; ),BB b b b b w= then B will transform into a new generalized trapezoidal fuzzy number as * * * 1 2 3 4 *( , , , ; ),BB b b b b w= where * ,Bw w= and the values of * 2b and * 3b are determined as * 2 1 2 1( ) / Bb b w b b w= + − and * 3 4 4 3( ) / ,Bb b w b b w= − − respectively. Then, the α-cuts of generalized fuzzy numbers 1 2 3 4( , , , ; )AA a a a a w= and * * * 1 2 3 4 *( , , , ; )BB b b b b w= are given as: [ ]1 2 1 4 4 3( ) / , ( ) / , [0, ],0 1 A A A A A a a a w a a a w w w α α α α = + − − − ∀ ∈ < ≤ (7) * * * 1 2 1 * 4 4 3 * * * ( ) / , ( ) / , [0, ],0 1 B B B B B b b b w b b b w w w α α α α ⎡ ⎤= + − − −⎣ ⎦ ∀ ∈ < ≤ (8) 4.1. Addition of two generalized trapezoidal fuzzy numbers Theorem 1. Addition of two generalized fuzzy numbers 1 2 3 4( , , , ; )AA a a a a w= and 1 2 3 4( , , , ; ),BB b b b b w= with different confidence levels generates a trapezoidal fuzzy number as follows: 1 2 3 4( ) ( , , , ; min( , ))A BC A B c c c c w w w= + = = where, Proceedings of 2013 International Conference on Fuzzy Theory and Its Application National Taiwan University of Science and Technology, Taipei, Taiwan, Dec. 6-8, 2013 408 1 1 1;c a b= + 2 1 2 2 1( ) / ;Bc b a w b b w= + + − 3 4 3 4 3( ) / ;Bc b a w b b w= + − − 4 4 4 ,c a b= + and ;A Bw w w= ≤ 1 2 3 4 1 2 3, , , , , ,a a a a b b b and 4b are any real numbers. Proof: Suppose that ( )A B C+ = where 1 2[ ( ), ( )] [0, ],C C C wα α α α= ∀ ∈ 0 1,w< ≤ min( , ).A Bw w w= Then, * * * * * 1 1 2 1 2 1 * * 4 4 4 3 4 3 ( ) [ ( ) ( ), ( ) ( )] [ ( ) / ( ) / , ( ) / ( ) / ] L R R B A B A C A B A B A B a b b b w a a w a b b b w a a w α α α α α α α α α α α α = + = + + = + + − + − + − − − − (9) Let 1 2{ : [ ( ), ( )] [0, ]C x x C C wα α α α= ∈ ∀ ∈ We now have two equations to solve - namely: * 1 1 2 1 2 1( ) / ( ) / 0B Aa b b b w a a w xα α+ + − + − − = (10) * 4 4 4 3 4 3( ) / ( ) / 0B Aa b b b w a a w xα α+ − − − − − = (11) From Equations (10) and (11), the left and right membership functions ( )LCf x and ( ) R Cf x of C can be calculated as: 1 1 1 2* 2 1 * 2 1 [ ( )]( ) , , ( ) / ( ) / L C AB w x a bf x c x c w b b w w a a w − + = ≤ ≤ − + − (12) 4 4 3 4* 3 4 * 3 4 [ ( )]( ) , , ( ) / ( ) / R C AB w x a bf x c x c w b b w w a a w − + = ≤ ≤ − + − (13) We have * ,A Bw w w= = *2 1 2 1( ) / ,Bb b w b b w= + − and * 3 4 4 3( ) / ,Bb b w b b w= − − then Equations (12) and (13) become: 1 1 * 2 2 1 1 1 1 1 2 2 1 1 1 1 1 1 2 2 1 [ ( )]( ) ( ) [ ( )] , [ ( ) / ] ( ) ( ) / , L C B B w x a bf x b a a b w x a b b a w b b w a b a b x b a w b b w − + = + − + − + = + + − − + + ≤ ≤ + + − (14) 4 4 * 3 3 4 4 4 4 4 3 4 3 4 4 4 3 4 3 4 4 [ ( )]( ) ( ) ( ) [ ( )] , [ ( ) / ] ( ) ( ) / , R C B B w x a bf x b a a b w x a b b a w b b w a b b a w b b w x a b − + = + − + − + = + − − − + + − − ≤ ≤ + (15) Thus, the addition of two generalized trapezoidal fuzzy numbers 1 2 3 4( , , , ; )AA a a a a w= and 1 2 3 4( , , , ; )BB b b b b w= is a generalized trapezoidal fuzzy number as follows: ( )C A B= + 1 2 3 4( , , , ; min( , ))A Bc c c c w w w= = where, 1 1 1;c a b= + (16) 2 1 2 2 1( ) / ;Bc b a w b b w= + + − (17) 3 4 3 4 3( ) / ;Bc b a w b b w= + − − (18) 4 4 4 ,c a b= + (19) Notably, when A Bw w w= = , formulae (16)-(19) are the same as in Chen [6]. Theorem 2. Addition of two generalized triangular fuzzy numbers 1 2 3( , , ; )AA a a a w= and 1 2 3( , , ; )BB b b b w= with different confidence levels generates a trapezoidal fuzzy numbers as follows: 1 2 3 4( ) ( , , , ; min( , ))A BD A B d d d d w w w= + = = where, 1 1 1;d a b= + (20) 2 1 2 2 1( ) / ;Bd b a w b b w= + + − (21) 3 3 2 2 3( ) / ;Bd b a w b b w= + + − (22) 4 3 3 ,d a b= + (23) and ;A Bw w w= ≤ 1 2 3 1 2, , , , ,a a a b b and 3b are any real numbers. Proof: The proof is similar to Theorem 1. Notably, when A Bw w w= = , we will have 2 3 2 2 ,d d a b= = + then formulae (20-23) are the same as in Chen [6]. 4.2. Subtraction of two generalized trapezoidal fuzzy numbers Theorem 3. Subtraction operation of two generalized fuzzy numbers 1 2 3 4( , , , ; )AA a a a a w= and 1 2 3 4( , , , ; )BB b b b b w= with different confidence levels generates a trapezoidal fuzzy number as follows: 1 2 3 4( ) ( , , , ; min( , ))A BE A B e e e e w w w= − = = where, 1 1 4 ;e a b= − (24) 2 2 4 4 3( ) / ;Be a b w b b w= − + − (25) 3 3 1 2 1( ) / ;Be a b w b b w= − − − (26) 4 4 1 ,e a b= − (27) and ;A Bw w w= ≤ 1 2 3 4 1 2 3, , , , , ,a a a a b b b and 4b are any real numbers. Proof: In order to determine the subtraction operation between A and ,B the value of ( )A B− can be defined as ( ) ( )( ),A B A B− = + − where 4 3 2 1( , , , ).B b b b b− = − − − − Hence, the proof is similar to Theorem 1. Notably, when A Bw w w= = , then formulae (24-27) are the same as in Chen [6]. Theorem 4. Subtraction operation of two generalized triangular fuzzy numbers 1 2 3( , , ; )AA a a a w= and 1 2 3( , , ; )BB b b b w= with different confidence levels generates a trapezoidal fuzzy number as follows: 1 2 3 4( ) ( , , , ; min( , ))A BF A B f f f f w w w= − = = where, 1 1 3 ;f a b= − (28) 2 2 3 3 2( ) / ;Bf a b w b b w= − + − (29) 3 2 1 1 2( ) / ;Bf a b w b b w= − + − (30) 4 3 1,f a b= − (31) and ;A Bw w w= ≤ 1 2 3 1 2, , , , ,a a a b b and 3b are any real numbers. Proof: The proof is similar to Theorem 1. Notably, when ,A Bw w w= = we will have 2 3 2 2 ,f f a b= = − then formulae (28-31) are the same as in Chen [6]. 4.3. Multiplication of two generalized trapezoidal fuzzy numbers Proceedings of 2013 International Conference on Fuzzy Theory and Its Application National Taiwan University of Science and Technology, Taipei, Taiwan, Dec. 6-8, 2013 409 Theorem 5. Multiplication of two generalized fuzzy numbers 1 2 3 4( , , , ; )AA a a a a w= and 1 2 3 4( , , , ; ),BB b b b b w= with different confidence levels generates a fuzzy number as follows: 1 2 3 4(x) ( , , , ; min( , ))A BG A B g g g g w w w= = = where, 1 1 1;g a b= 2 2 2 2 1 2 1( ) / ;Bg w a b a b w a b= − + 3 3 3 3 4 3 4( ) / ;Bg w a b a b w a b= − + 4 4 4 ,g a b= and ;A Bw w w= ≤ 1 2 3 4 1 2 3, , , , , ,a a a a b b b and 4b are non- zero and positive real numbers. Proof: Suppose that ( )A B G× = where, 1 2[ ( ), ( )] [0, ],0 1, min( , )A BG G G w w w w wα α α α= ∀ ∈ < ≤ = { } * * * * 1 2 1 1 2 1 * * 4 4 3 4 4 3 * (x) [ ( ) ( ), ( ) ( )] [ ( ) / ][ ( ) / ], [ ( ) / ][ ( ) / ] L R R A B A B G A B A B A B a a a w b b b w a a a w b b b w α α α α α α α α α α α α = = = + − + − − − − − * 2 *2 1 2 1 1 1 2 1 2 1 1 1 * * * 2 *4 3 4 3 4 4 4 3 4 3 4 4 * * ( ) ( ) ( ) ( ) , ( ) ( ) ( ) ( ) A AB B A AB B a a b b a bb b a a ab w w w w a a b b a bb b a a ab w w w w α α α α ⎡ ⎛ ⎞ − − = + − + − +⎢ ⎜ ⎟⎜ ⎟⎢ ⎝ ⎠⎣ ⎤⎛ ⎞ − − − − + − + ⎥⎜ ⎟⎜ ⎟ ⎥⎝ ⎠ ⎦ (32) We now have two equations to solve - namely: 2 1 1 1 0U T V xα α+ + − = (33) 2 2 2 2 0U T V xα α− + − = (34) where, ** 4 3 4 32 1 2 1 1 2 * * ( ) ( )( ) ( ) , , A AB B a a b ba a b bU U w w w w − −− − = = * 1 1 2 1 * 1 2 1( ) / ( ) / ,ABT a b b w b a a w= − + − * 2 4 4 3 * 4 4 3( ) / ( ) / ,ABT a b b w b a a w= − + − 1 1 1 2 4 4,V ab V a b= = We have * ,A Bw w w= = *2 1 2 1( ) / ,Bb b w b b w= + − and *3 4 4 3( ) / ,Bb b w b b w= − − then 1 2 1, , ,U U T and 2T become: 4 3 4 32 1 2 1 1 2 4 4 3 4 4 31 2 1 1 2 1 1 2 ( )( )( )( ) , , ( ) ( )( ) ( ) , B B B B a a b ba a b bU U ww ww a b b b a aa b b b a aT T w w w w − −− − = = − −− − = + = + Only the roots in [0,1] will be retained in (33) and (34). The left and right membership functions ( )LGf x and ( )RGf x of G can be calculated as: { }2 1/21 1 1 1 1 1 2( ) [ 4 ( )] / 2 , ,LGf x T T U x V U g x g= − + + − ≤ ≤ (35) { }2 1/22 2 2 2 2 3 4( ) [ 4 ( )] / 2 , ,RGf x T T U x V U g x g= − + − ≤ ≤ (36) Since, 2 1/21 1 1( ) / 1/ [ 4 ( )] 0LGdf x dx T U x V= + − > and 2 1/2 2 2 2( ) / 1/ [ 4 ( )] 0,RGdf x dx T U x V= − + − < then ( ) L Gf x and ( )LGf x are increasing and decreasing functions in ,x respectively. The values of 1 2 3, , ,g g g and 4g are determined respectively as follow: { }2 1/21 1 1 1 1 1 1 1 ( ) [ 4 ( )] / 2 0LG L f x T T U x V U x V a b = − + + − = ⇔ = = (37) { }2 1/22 2 2 2 2 2 4 4 ( ) [ 4 ( )] / 2 0RG R f x T T U x V U x V a b = − + − = ⇔ = = (38) { }2 1/21 1 1 1 1 2 2 2 1 2 1 ( ) [ 4 ( )] / 2 ( ) / L G L B f x T T U x V U w x w a b a b w a b = − + + − = ⇔ = − + (39) { }2 1/22 2 2 2 2 3 3 3 4 3 4 ( ) [ 4 ( )] / 2 ( ) / R G R B f x T T U x V U w x w a b a b w a b = − + − = ⇔ = − + (40) Thus, the multiplication operation between two generalized fuzzy numbers 1 2 3 4( , , , ; )AA a a a a w= and 1 2 3 4( , , , ; )BB b b b b w= is a fuzzy number: 1 2 3 4(x) ( , , , ; min( , ))A BG A B g g g g w w w= = = where, 1 1 1;g a b= (41) 2 2 2 2 1 2 1( ) / ;Bg w a b a b w a b= − + (42) 3 3 3 3 4 3 4( ) / ;Bg w a b a b w a b= − + (43) 4 4 4 ,g a b= (44) Notably, when ,A Bw w w= = formulae (41-44) are the same as in Chen [6]. Theorem 6. Multiplication of two triangular fuzzy numbers 1 2 3( , , ; )AA a a a w= and 1 2 3( , , ; ),BB b b b w= with different confidence levels generates a fuzzy number as follows: 1 2 3 4(x) ( , , , ; min( , ))A BH A B h h h h w w w= = = where, 1 1 1;h a b= (45) 2 2 2 2 1 2 1( ) / ;Bh w a b a b w a b= − + (46) 3 2 2 2 3 2 3( ) / ;Bh w a b a b w a b= − + (47) 4 3 3.h a b= (48) and ;A Bw w w= ≤ 1 2 3 1 2, , , , ,a a a b b and 3b are non-zero and positive real numbers. Proof: The proof is similar to Theorem 5. Notably, when ,A Bw w w= = we will have 2 3 2 2 ,g g a b= = then formulae (45-48) are the same as in Chen [6]. 4.4. Division of two generalized trapezoidal fuzzy numbers Theorem 7. Division operation of two generalized fuzzy numbers 1 2 3 4( , , , ; )AA a a a a w= and 1 2 3 4( , , , ; ),BB b b b b w= with different confidence levels generates a fuzzy number as follows: 1 2 3 4(/) ( , , , ; min( , ))A BI A B i i i i w w w= = = where, 1 1 4/ ;i a b= (49) 2 2 3 2 4 2 4( / / ) / / ;Bi w a b a b w a b= − + (50) 3 3 2 3 1 3 1( / / ) / / ;Bi w a b a b w a b= − + (51) 4 4 1/ .i a b= (52) and ;A Bw w w= ≤ 1 2 3 4 1 2 3, , , , , ,a a a a b b b and 4b are non- zero and positive real numbers. Proof: Consider two generalized fuzzy numbers 1 2 3 4( , , , ; )AA a a a a w= and 1 2 3 4( , , , ; ).BB b b b b w= In order to determine the division operation between A and ,B the value of (/)A B can be defined as (/) (x)(1 / ),A B A B= Proceedings of 2013 International Conference on Fuzzy Theory and Its Application National Taiwan University of Science and Technology, Taipei, Taiwan, Dec. 6-8, 2013 410 where 4 3 2 11 / (1/ ,1 / ,1/ ,1 / ; ).BB b b b b w= Hence, the division operation between A and ,B can be obtained. Notably, when ,A Bw w w= = formulae (49-52) are the same as in Chen [6]. Theorem 8. Division operation between two triangular fuzzy numbers 1 2 3( , , ; )AA a a a w= and 1 2 3( , , ; ),BB b b b w= with different confidence levels generates a fuzzy number as follows: 1 2 3 4(/) ( , , , ; min( , ))A BJ A B j j j j w w w= = = where, 1 1 3/ ;j a b= (53) 2 2 2 2 3 2 3( / / ) / / ;Bj w a b a b w a b= − + (54) 3 2 2 2 1 2 1( / / ) / / ;Bj w a b a b w a b= − + (55) 4 3 1/ .j a b= (56) and ;A Bw w w= ≤ 1 2 3 1 2, , , , ,a a a b b and 3b are non-zero and positive real numbers. Proof: The proof is similar to Theorem 7. Notably, when ,A Bw w w= = we will have 2 3 2 2/ ,j j a b= = then formulae (53-56) are the same as in Chen [6]. V. NUMERICAL EXAMPLES In this section, numerical examples are used to illustrate the validity and advantages of the proposed arithmetic operations approach. Examples show that the proposed can effectively resolve the drawbacks with Chen’s [6] method. Example 3. Re-consider the two generalized triangular fuzzy numbers, i.e., (0.5,0.6,0.7;0.5)A = and (0.6, 0.7,0.8;0.9)B = in example 1. Using the proposed approach, the arithmetic operations between fuzzy numbers A and B are ( ) (1.1,1.256,1.344,1.5;0.5),D A B= + = ( ) ( 0.3, 0.144, 0.056,0.1;0.5),F A B= − = − − − (x) (0.3,0.393,0.447,0.56;0.5),H A B= = and (/) (0.625,0.81,1.01,1.167;0.5).I A B= = Obviously, the arithmetic operations between generalized triangular fuzzy numbers obtained by the proposed approach is more reasonable than the outcome obtained by Chen’s [6] approach. Example 4. Re-consider the three generalized triangular fuzzy numbers, i.e., 1 (0.2,0.4,0.6;0.5),A = 2 (0.5,0.7,0.9;0.7),A = and 3 (0.5,0.7,0.9;0.9)A = in Example 2. According to Theorem 1, the addition operations between 1,A 2 ,A and 3A are 12 1 2( ) (0.7,1.043,1.157,1.5;0.5),A A A= + = and 13 1 3( ) (0.7,1.011,1.189,1.5;0.5),A A A= + = respectively. Clearly, the results show that 12 13.A A∈ Thus, this example shows that the proposed approach can overcome the shortcomings of the inconsistency of Chen’s [6] approach in addition between generalized fuzzy numbers. Example 5. Consider the two generalized trapezoidal fuzzy numbers (0.1,0.2,0.3, 0.4;0.6)A = and (0.3, 0.5,0.6,0.9;0.8).B = Using the proposed approach, the arithmetic operations between A and B are ( ) (0.4,0.65, 0.975,1.3;0.6),D A B= + = ( ) ( 0.8, 0.475, 0.15,0.1;0.6),F A B= − = − − − (x) (0.3,0.09,0.2025,0.36;0.6),H A B= = and (/) (0.111, 0.296,0.733,1.333;0.6).I A B= = Again, the arithmetic operations between generalized trapezoidal fuzzy numbers obtained by the proposed approach can overcome the shortcomings of Chen’s approach. Example 6. Consider the generalized triangular fuzzy number (0.2,0.3, 0.5;0.5)A = and generalized trapezoidal fuzzy number (0.4, 0.5,0.7, 0.8;1).B = Using the proposed approach, the arithmetic operations between A and B are ( ) (0.6, 0.75,10.05,1.3;0.5),D A B= + = ( ) ( 0.8, 0.475, 0.15,0.1;0.6),F A B= − = − − − (x) (0.08, 0.135,0.36, 0.4;0.5),H A B= = and (/) (0.25,0.402,0.675,1.25;0.5).I A B= = This example demonstrates one of the advantages of the proposed approach, that is, it can determine the arithmetic operations between a mix of various types of fuzzy numbers (normal, non-normal, triangular, and trapezoidal). VI. IMPLEMENTATION OF PROPOSED ARITHMETIC OPERATIONS TO SOLVE A MULTI-CRITERIA DECISION MAKING PROBLEM In this section, we apply the proposed arithmetic operations to deal with university academic staff evaluation and selection problem. Suppose that a university needs to evaluate and sort their teaching staffs’ performance. After preliminary screening, ten candidates, namely 1 9, , ,A A and 10 ,A are chosen for further evaluation. A committee of three decision makers, 1 2, ,D D and 3,D conducts the evaluation and selection of the ten candidates. Nine selection criteria are considered including number of publications 1( ),C quality of publications 2( ),C personal qualification 3( ),C personality factors 4( ),C activity in professional society 5( ),C classroom teaching 6( ),C student advising 7( ),C research and/or creative activity (independent of publication) 8( ),C and fluency in a foreign language 9( )C [24-26]. The computational procedure is summarized as follows: Step 1. Aggregate ratings of alternatives versus criteria Assume that the decision makers use the linguistic rating set {VL,L,M,H,VH},S = where VL = Very Low = (0.0, 0.0, 0.2), L = Low = (0.1, 0.3, 0.5), M = Medium = (0.3, 0.5, 0.7), H = High = (0.6, 0.8, 1.0), and VH = Very High = (0.8, 0.9, 1.0), to evaluate the suitability of the candidates under each criteria. Using proposed arithmetic operations and Yu et al.’s [28] procedure, the aggregated suitability ratings of ten candidates, i.e. 1 10, ,A A versus nine criteria, i.e. 1 9, , ,C C from three decision makers can be obtained as shown in Tables 1a-c. Proceedings of 2013 International Conference on Fuzzy Theory and Its Application National Taiwan University of Science and Technology, Taipei, Taiwan, Dec. 6-8, 2013 411 Table 1a. The linguistic ratings evaluated by decision makers Crit eria Can dida tes Decision makers Rij D1 D2 D3 C1 A1 G G VG (0.667, 0.763, 0.797, 0.933; 0.9) A2 F G G (0.500, 0.685, 0.752, 0.833; 0.8) A3 G F G (0.500, 0.685, 0.752, 0.833; 0.8) A4 G G G (0.600, 0.800, 0.867, 0.900; 0.9) A5 F G G (0.500, 0.685, 0.752, 0.833; 0.8) A6 G VG G (0.667, 0.830, 0.897, 0.933; 0.9) A7 G G VG (0.667, 0.830, 0.863, 0.933; 0.9) A8 F F G (0.400, 0.593, 0.659, 0.767; 0.8) A9 VG G G (0.667, 0.830, 0.897, 0.933; 0.9) A10 F F F (0.300, 0.500, 0.567, 0.700; 0.8) C2 A1 G G F (0.500, 0.685, 0.700, 0.833; 0.8) A2 F F G (0.400, 0.593, 0.659, 0.767; 0.8) A3 G VG G (0.667, 0.830, 0.897, 0.933; 0.9) A4 G G VG (0.667, 0.830, 0.863, 0.933; 0.9) A5 G G G (0.600, 0.800, 0.867, 0.900; 0.9) A6 VG G G (0.667, 0.830, 0.897, 0.933; 0.9) A7 F F F (0.300, 0.500, 0.567, 0.700; 0.8) A8 VG VG VG (0.800, 0.900, 0.933, 1.000; 1.0) A9 G F G (0.500, 0.685, 0.752, 0.833; 0.8) A10 G G G (0.600, 0.800, 0.867, 0.900; 0.9) C3 A1 VG VG G (0.733, 0.874, 0.867, 0.967; 0.9) A2 G F F (0.400, 0.593, 0.659, 0.767; 0.8) A3 F G G (0.500, 0.685, 0.752, 0.833; 0.8) A4 F F F (0.300, 0.500, 0.567, 0.700; 0.8) A5 F G G (0.500, 0.685, 0.752, 0.833; 0.8) A6 G F G (0.500, 0.685, 0.752, 0.833; 0.8) A7 G VG G (0.667, 0.830, 0.897, 0.933; 0.9) A8 G G G (0.600, 0.800, 0.867, 0.900; 0.9) A9 VG VG G (0.733, 0.860, 0.867, 0.967; 0.9) A10 F G G (0.500, 0.685, 0.752, 0.833; 0.8) Table 1b. The linguistic ratings evaluated by decision makers Crit eria Can dida tes Decision makers Rij D1 D2 D3 C4 A1 F F G (0.400, 0.593, 0.659, 0.767; 0.8) A2 G G F (0.500, 0.685, 0.700, 0.833; 0.8) A3 VG G VG (0.733, 0.860, 0.893, 0.967; 0.9) A4 G G G (0.600, 0.800, 0.867, 0.900; 0.9) A5 G VG G (0.667, 0.830, 0.897, 0.933; 0.9) A6 F G G (0.500, 0.685, 0.752, 0.833; 0.8) A7 G F F (0.400, 0.593, 0.659, 0.767; 0.8) A8 F F F (0.300, 0.500, 0.567, 0.700; 0.8) A9 G F G (0.500, 0.685, 0.752, 0.833; 0.8) A10 G VG G (0.667, 0.830, 0.897, 0.933; 0.9) C5 A1 VG VG G (0.733, 0.860, 0.867, 0.967; 0.9) A2 G VG G (0.667, 0.830, 0.897, 0.933; 0.9) A3 G F F (0.400, 0.593, 0.659, 0.767; 0.8) A4 G G G (0.600, 0.800, 0.867, 0.900; 0.9) A5 F G G (0.500, 0.685, 0.752, 0.833; 0.8) A6 F F G (0.400, 0.593, 0.659, 0.767; 0.8) A7 G F F (0.400, 0.593, 0.659, 0.767; 0.8) A8 F F F (0.300, 0.500, 0.567, 0.700; 0.8) A9 G G G (0.600, 0.800, 0.867, 0.900; 0.9) A10 G VG G (0.667, 0.830, 0.897, 0.933; 0.9) C6 A1 F F G (0.400, 0.593, 0.659, 0.767; 0.8) A2 F F F (0.300, 0.500, 0.567, 0.700; 0.8) A3 G G VG (0.667, 0.830, 0.863, 0.933; 0.9) A4 VG VG G (0.733, 0.860, 0.867, 0.967; 0.9) A5 G F F (0.400, 0.593, 0.659, 0.767; 0.8) A6 F F F (0.300, 0.500, 0.567, 0.700; 0.8) A7 VG G G (0.667, 0.830, 0.897, 0.933; 0.9) A8 G G G (0.600, 0.800, 0.867, 0.900; 0.9) A9 G F F (0.400, 0.593, 0.659, 0.767; 0.8) A10 F F F (0.300, 0.500, 0.567, 0.700; 0.8) Proceedings of 2013 International Conference on Fuzzy Theory and Its Application National Taiwan University of Science and Technology, Taipei, Taiwan, Dec. 6-8, 2013 412 Table 1c. The linguistic ratings evaluated by decision makers Crit eria Can dida tes Decision makers Rij D1 D2 D3 C7 A1 G G VG (0.667, 0.830, 0.863, 0.933; 0.9) A2 VG G G (0.667, 0.830, 0.897, 0.933; 0.9) A3 G G G (0.600, 0.800, 0.867, 0.900; 0.9) A4 F G G (0.500, 0.685, 0.752, 0.833; 0.8) A5 G F F (0.400, 0.593, 0.659, 0.767; 0.8) A6 F F G (0.400, 0.593, 0.659, 0.767; 0.8) A7 G F F (0.400, 0.593, 0.659, 0.767; 0.8) A8 F F F (0.300, 0.500, 0.567, 0.700; 0.8) A9 G G VG (0.667, 0.830, 0.863, 0.933; 0.9) A10 F F G (0.400, 0.593, 0.659, 0.767; 0.8) C8 A1 G F G (0.500, 0.685, 0.752, 0.833; 0.8) A2 G G VG (0.667, 0.830, 0.863, 0.933; 0.9) A3 VG G G (0.667, 0.830, 0.897, 0.933; 0.9) A4 G G G (0.600, 0.800, 0.867, 0.900; 0.9) A5 F G F (0.400, 0.593, 0.659, 0.767; 0.8) A6 F F F (0.300, 0.500, 0.567, 0.700; 0.8) A7 F G F (0.400, 0.593, 0.659, 0.767; 0.8) A8 F F G (0.400, 0.593, 0.659, 0.767; 0.8) A9 G VG G (0.667, 0.830, 0.897, 0.933; 0.9) A10 F F G (0.400, 0.593, 0.659, 0.767; 0.8) C9 A1 G G VG (0.667, 0.830, 0.863, 0.933; 0.9) A2 VG G G (0.667, 0.830, 0.897, 0.933; 0.9) A3 F F G (0.400, 0.593, 0.659, 0.767; 0.8) A4 G F G (0.500, 0.685, 0.752, 0.833; 0.8) A5 F F G (0.400, 0.593, 0.659, 0.767; 0.8) A6 F G G (0.500, 0.685, 0.752, 0.833; 0.8) A7 G F G (0.500, 0.685, 0.752, 0.833; 0.8) A8 G G G (0.600, 0.800, 0.867, 0.900; 0.9) A9 G VG G (0.667, 0.830, 0.897, 0.933; 0.9) A10 F G F (0.400, 0.593, 0.659, 0.767; 0.8) Step 2. Aggregate the importance weights Also assumes that the decision makers employ a linguistic weighting set {UI,OI,I,VI,AI},Q = where UI = Unimportant = (0.0, 0.0, 0.3), OI = Ordinary Important = (0.2, 0.3, 0.4), I = Important = (0.3, 0.5, 0.7), VI = Very Important = (0.6, 0.8, 0.9), and AI = Absolutely Important = (0.8, 0.9, 1.0), to assess the importance of all the criteria. Table 2 displays the importance weights of nine criteria from the three decision-makers. Using proposed arithmetic operations and Yu et al.’s [27] procedure, the aggregated weights of criteria from the decision making committee can be obtained as presented in Table 2. Table 2. The importance weights of the criteria evaluated by decision makers. Criteria Decision makers wij D1 D2 D3 C1 AI AI VI (0.733, 0.860, 0.867, 0.967; 0.9) C2 AI VI VI (0.667, 0.830, 0.830, 0.933; 0.9) C3 I VI I (0.400, 0.593, 0.593, 0.767; 0.8) C4 I VI I (0.400, 0.593, 0.593, 0.767; 0.8) C5 VI VI AI (0.667, 0.830, 0.833, 0.933; 0.9) C6 AI AI VI (0.733, 0.860, 0.867, 0.967; 0.9) C7 I I VI (0.400, 0.593, 0.600, 0.767; 0.8) C8 I VI VI (0.500, 0.685, 0.693, 0.833; 0.8) C9 VI I I (0.400, 0.593, 0.593, 0.767; 0.8) Step 3. Determine the weighted fuzzy decision matrix This matrix can be obtained by multiplying each aggregated rating by its associated fuzzy weight using proposed arithmetic operation of generalized fuzzy numbers. Table 3 shows the weighted ratings of each candidate. Table 3. Weighted ratings of each candidate Candidates Weighted ratings A1 (0.316, 0.520, 0.566,0.753; 0.8) A2 (0.280, 0.491, 0.551,0.723; 0.8) A3 (0.310, 0.521, 0.584,0.748; 0.8) A4 (0.320, 0.539, 0.589,0.752; 0.8) A5 (0.264, 0.474, 0.536,0.704; 0.8) A6 (0.259, 0.472, 0.520,0.695; 0.8) A7 (0.270, 0.473, 0.535,0.705; 0.8) A8 (0.265, 0.470, 0.530,0.699; 0.8) A9 (0.320, 0.534, 0.598,0.761; 0.8) A10 (0.252, 0.460, 0.523,0.693; 0.8) Step 4. Defuzzification Using Dat et al.’s [28] ranking method, the distance between the centroid point and the minimum point can be obtained, as shown in Table 4. According to Table 4, the ranking order of the ten candidates is: 9 4 3 1 2 7 5 8 6 10A A A A A A A A A A; ; ; ; ; ; ; ; ; Thus, the best selection is candidate A9 having the largest distance. Proceedings of 2013 International Conference on Fuzzy Theory and Its Application National Taiwan University of Science and Technology, Taipei, Taiwan, Dec. 6-8, 2013 413 Table 4. Distance between the centroid point and the minimum point of each candidate Candidates Distances Ranking order A1 0.0578 4 A2 0.0292 5 A3 0.0582 3 A4 0.0668 2 A5 0.0124 7 A6 0.0041 9 A7 0.0140 6 A8 0.0090 8 A9 0.0704 1 A10 0.0000 10 VII. CONCLUSIONS This paper proposed an extension principle to derived arithmetic operations between generalized fuzzy numbers to overcome the shortcomings of Chen’s approach. Several examples were given to illustrate the usage, applicability, and advantages of the proposed approach. It shows that the arithmetic operations between generalized fuzzy numbers obtained by the proposed method are more consistent than the original method. Thus, utilizing the proposed method is more reasonable than using Chen’s method. In addition, the proposed method can effectively determine the arithmetic operations between a mix of various types of fuzzy numbers (normal, non-normal, triangular, and trapezoidal). 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