Proceedings of 2013 International Conference on Fuzzy Theory and Its Application National Taiwan University of Science and Technology, Taipei, Taiwan, Dec 6-8, 2013 Improved arithmetic operations on generalized fuzzy numbers Luu Quoc Dat Canh Chi Dung 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 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 Shuo-Yan Chou Department of Industrial Management, National Taiwan University of Science and Technology Taipei, Taiwan, ROC E-mail: sychou2@me.com 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 operations, Fuzzy MCDM fuzzy numbers, Arithmetic 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 978-1-4799-0386-3/13/$31.00 ©2013 IEEE 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 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., and A = ( a1 , a2 , a3 , a4 ; wA ) B = (b1 , b2 , b3 , b4 ; wB ), where a1 , a2 , a3 , a4 , b1 , b2 , b3 and b4 are real values, ≤ wA ≤ and ≤ wB ≤ Some arithmetic operators between the generalized fuzzy numbers A and B are defined as follows: (i) Generalized trapezoidal fuzzy numbers addition ( + ) : A(+)B = (a1 + b1 , a2 + b2 , a3 + b3 , a4 + b4 ;min(wA , wB )), (1) where a1 , a2 , a3 , a4 , b1 , b2 , b3 and b4 are real values (ii) Generalized trapezoidal fuzzy numbers subtraction ( −) : A(−) B = (a1 − b4 , a2 − b3 , a3 − b2 , a4 − b1 ; min(wA , wB )), (2) where a1 , a2 , a3 , a4 , b1 , b2 , b3 and b4 are real values (iii) Generalized multiplication (x) : trapezoidal fuzzy numbers A(x) B = (a1 × b1 , a2 × b2 , a3 × b3 , a4 × b4 ; min( wA , wB )) (3) where a1 , a2 , a3 , a4 , b1 , b2 , b3 and b4 are all positive real numbers (iv) Generalized trapezoidal fuzzy numbers division (/) : Let a1 , a2 , a3 , a4 , b1 , b2 , b3 and b4 be non-zero positive real numbers Then, A(/) B = (a1 / b4 , a2 / b3 , a3 / b2 , a4 / b1 ; min( wA , wB )), (4) 407 Proceedings of 2013 International Conference on Fuzzy Theory and Its Application National Taiwan University of Science and Technology, Taipei, Taiwan, Dec 6-8, 2013 Fig Additions between the generalized fuzzy numbers in Example 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], 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 Example 1: Consider the generalized triangular fuzzy numbers A = (0.5, 0.6, 0.7;0.5) and B = (0.6, 0.7, 0.8;0.9) shown in Fig It is observed from Fig that, min( wA = 0.5, wB = 0.9) = 0.5 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 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 αcuts of fuzzy number The revised arithmetical operations between generalized trapezoidal fuzzy numbers are described as follows: Let A = (a1 , a2 , a3 , a4 ; wA ) and B = (b1 , b2 , b3 , b4 ; wB ) are two generalized trapezoidal fuzzy numbers with membership function f A ( x ) and f B ( x ), respectively, which can be written in the following form: ⎧ wA ( x − a1 ) / (a2 − a1 ), ⎪w , ⎪ f A ( x) = ⎨ A ⎪ wA ( x − a4 ) / (a3 − a4 ), ⎪⎩0, ≤ wA ≤ and ≤ wB ≤ Fig Generalized fuzzy numbers A and B in Example 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 Example 2: Consider the generalized triangular fuzzy numbers A1 = (0.2, 0.4, 0.6; 0.5), A2 = (0.5, 0.7, 0.9; 0.7), cut of fuzzy number w = wA < wB B = (b1 , b2 , b3 , b4 ; wB ), then B will transform into a new take generalized A12 = (0.7,1.1,1.5;0.5) method, and we fuzzy A13 = (0.7,1.1,1.5; 0.5) given as: Aα = [ a1 + α (a2 − a1 ) / wA , a4 − α (a4 − a3 ) / wA ] , ∀ α ∈ [0, wA ], < wA ≤ calculate the arithmetic operations between generalized fuzzy numbers Bα* = ⎡⎣b1 + α (b2* − b1 ) / wB* , b4 − α (b4 − b3* ) / wB* ⎤⎦ , ∀ α ∈ [0, wB* ], < wB* ≤ (7) (8) 4.1 Addition of two generalized trapezoidal fuzzy numbers A3 A12 , A13 A2 as Then, the α-cuts of generalized fuzzy numbers A = (a1 , a2 , a3 , a4 ; wA ) and B* = (b1 , b2* , b3* , b4 ; wB* ) are A12 ∼ A13 Therefore, Chen’s method cannot consistency A1 number b2* and b3* are determined as b2* = b1 + w(b2 − b1 ) / wB and b3* = b4 − w(b4 − b3 ) / wB , respectively have Thus, the additions between the generalized fuzzy numbers A1 and A2 , and A1 and A3 are the same, i.e., trapezoidal B* = (b1 , b2* , b3* , b4 ; wB* ), where wB* = w, and the values of A12 = A1 ( + ) A2 ∈ A13 = A1 ( + ) A3 However, Chen’s w = wA ≤ wB , wA and wB 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, and A3 = (0.5, 0.7, 0.9; 0.9) as in Fig Intuitively, the order of fuzzy numbers A2 and A3 is A2 ∈ A3 Then, we the otherwise, b1 ≤ x ≤ b2 , ⎧ wB ( x − b1 ) / (b2 − b1 ), ⎪w , b2 ≤ x ≤ b3 , (6) ⎪ f B ( x) = ⎨ B b3 ≤ x ≤ b4 , ⎪ wB ( x − b4 ) / (b3 − b4 ), ⎪⎩0, otherwise, where, a1 , a2 , a3 , a4 , b1 , b2 , b3 and b4 are real values, A using a2 ≤ x ≤ a3 , (5) a3 ≤ x ≤ a4 , and B should have a1 ≤ x ≤ a2 , Theorem Addition of two generalized fuzzy numbers A = (a1 , a2 , a3 , a4 ; wA ) and B = (b1 , b2 , b3 , b4 ; wB ), with different confidence levels generates a trapezoidal fuzzy number as follows: C = A(+) B = (c1 , c2 , c3 , c ; w = min( wA , wB )) where, 408 Proceedings of 2013 International Conference on Fuzzy Theory and Its Application National Taiwan University of Science and Technology, Taipei, Taiwan, Dec 6-8, 2013 c1 = a1 + b1 ; c2 = b1 + a2 + w(b2 − b1 ) / wB ; c3 = b4 + a3 − w(b4 − b3 ) / wB ; c4 = a4 + b4 , and w = wA ≤ wB ; a1 , a2 , a3 , a4 , b1 , b2 , b3 and b4 are any real numbers Proof: Suppose that A( +) B = C Cα = [C1 (α ), C2 (α )]∀α ∈[0, w], < w ≤ 1, (9) We now have two equations to solve - namely: a1 + b1 + α (b2 − b1 ) / wB* + α (a2 − a1 ) / wA − x = a4 + b4 − α (b4 − b3 ) / wB* − α (a4 − a3 ) / wA − x = (10) (11) From Equations (10) and (11), the left and right membership functions f CL ( x ) and f CR ( x ) of C can be calculated as: w[ x − (a1 + b1 )] , c1 ≤ x ≤ c2 , (12) fCL ( x) = * w(b2 − b1 ) / wB* + w(a2 − a1 ) / wA w[ x − (a4 + b4 )] , c3 ≤ x ≤ c4 , (13) f CR ( x) = * w(b3 − b4 ) / wB* + w(a3 − a4 ) / wA We have w = wA = wB* , b2* = b1 + w(b2 − b1 ) / wB , and b3* = b4 − w(b4 − b3 ) / wB , then Equations (12) and (13) become: w[ x − ( a1 + b1 )] b2* + a2 − ( a1 + b1 ) w[ x − (a1 + b1 )] = , [b1 + a2 + w(b2 − b1 ) / wB ] − ( a1 + b1 ) a1 + b1 ≤ x ≤ b1 + a2 + w(b2 − b1 ) / wB , w[ x − (a4 + b4 )] f CR ( x) = * (b3 + a3 ) − ( a4 + b4 ) w[ x − ( a4 + b4 )] = , [b4 + a3 − w(b4 − b3 ) / wB ] − ( a4 + b4 ) b4 + a3 − w(b4 − b3 ) / wB ≤ x ≤ a4 + b4 , f CL ( x ) = fuzzy d1 = a1 + b1 ; d = b1 + a2 + w(b2 − b1 ) / wB ; d = b3 + a2 + w(b2 − b3 ) / wB ; d = a3 + b3 , where w = min( wA , wB ) Then, Cα = Aα (+) Bα* = [ AL (α ) + Bα* (α ), AR (α ) + BR* (α )] = [a1 + b1 + α (b2* − b1 ) / wB* + α (a2 − a1 ) / wA , a4 + b4 − α (b4 − b3* ) / wB* − α (a4 − a3 ) / wA ] Let Cα = { x : x ∈ [C (α ), C (α )]∀ α ∈ [0, w] Theorem Addition of two generalized triangular fuzzy numbers A = (a1 , a2 , a3 ; wA ) and B = (b1 , b2 , b3 ; wB ) with different confidence levels generates a trapezoidal fuzzy numbers as follows: D = A( + ) B = ( d1 , d , d , d ; w = min( wA , wB )) where, (21) (22) (23) and w = wA ≤ wB ; a1 , a2 , a3 , b1 , b2 , and b3 are any real numbers Proof: The proof is similar to Theorem Notably, when wA = wB = w , we will have d = d = a2 + b2 , then formulae (20-23) are the same as in Chen [6] 4.2 Subtraction of two generalized trapezoidal fuzzy numbers Theorem Subtraction operation of two generalized and fuzzy numbers A = (a1 , a2 , a3 , a4 ; wA ) B = (b1 , b2 , b3 , b4 ; wB ) with different confidence levels generates a trapezoidal fuzzy number as follows: E = A(−) B = (e1 , e2 , e3 , e ; w = min( wA , wB )) where, e1 = a1 − b4 ; e2 = a2 − b4 + w(b4 − b3 ) / wB ; e3 = a3 − b1 − w(b2 − b1 ) / wB ; e4 = a4 − b1 , (24) (25) (26) (27) and w = wA ≤ wB ; a1 , a2 , a3 , a4 , b1 , b2 , b3 and b4 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 where − B = ( −b4 , −b3 , −b2 , −b1 ) A( − ) B = A( + )( − B ), Hence, the proof is similar to Theorem Notably, when wA = wB = w , then formulae (24-27) are the same as in Chen [6] Theorem Subtraction operation of two generalized triangular fuzzy numbers A = (a1 , a2 , a3 ; wA ) and (14) (15) B = (b1 , b2 , b3 ; wB ) with different confidence levels Thus, the addition of two generalized trapezoidal numbers and A = ( a1 , a2 , a3 , a4 ; wA ) generates a trapezoidal fuzzy number as follows: F = A(−) B = ( f1 , f , f , f ; w = min( wA , wB )) where, f1 = a1 − b3 ; f = a2 − b3 + w(b3 − b2 ) / wB ; f = a2 − b1 + w(b1 − b2 ) / wB ; f = a3 − b1 , B = (b1 , b2 , b3 , b4 ; wB ) is a generalized trapezoidal fuzzy number as follows: C = A( + ) B = (c1 , c2 , c3 , c ; w = min( wA , wB )) where, (16) c1 = a1 + b1 ; (17) c2 = b1 + a2 + w(b2 − b1 ) / wB ; (18) c3 = b4 + a3 − w(b4 − b3 ) / wB ; (19) c4 = a4 + b4 , Notably, when wA = wB = w , formulae (16)-(19) are the same as in Chen [6] (20) (28) (29) (30) (31) and w = wA ≤ wB ; a1 , a2 , a3 , b1 , b2 , and b3 are any real numbers Proof: The proof is similar to Theorem Notably, when wA = wB = w, we will have f = f = a2 − b2 , then formulae (28-31) are the same as in Chen [6] 4.3 Multiplication of two generalized trapezoidal fuzzy numbers 409 Proceedings of 2013 International Conference on Fuzzy Theory and Its Application National Taiwan University of Science and Technology, Taipei, Taiwan, Dec 6-8, 2013 Theorem Multiplication of two generalized fuzzy numbers A = (a1 , a2 , a3 , a4 ; wA ) and B = (b1 , b2 , b3 , b4 ; wB ), f GL ( x) = {−T1 + [T1 + 4U ( x − V1 )]1/2 } / 2U = with different confidence levels generates a fuzzy number as follows: G = A(x) B = ( g1 , g , g , g ; w = min( wA , wB )) where, f GR ( x) = {T2 − [T22 + 4U ( x − V2 )]1/2 } / 2U = ⇔ xL = V1 = a1b1 ⇔ xR = V2 = a4b4 f GL ( x) = {−T1 + [T1 + 4U1 ( x − V1 )]1/2 } / 2U1 = w ⇔ xL = w(a2b2 − a2b1 ) / wB + a2b1 g1 = a1b1 ; g = w(a2b2 − a2b1 ) / wB + a2b1 ; g = w(a3b3 − a3b4 ) / wB + a3b4 ; g = a4b4 , and w = wA ≤ wB ; a1 , a2 , a3 , a4 , b1 , b2 , b3 and b4 are nonzero and positive real numbers Proof: Suppose that A(×) B = G where, f GR ( x) = {T2 − [T22 + 4U ( x − V2 )]1/2 } / 2U = w ⇔ xR = w(a3b3 − a3b4 ) / wB + a3b4 (37) (38) (39) (40) Thus, the multiplication operation between two generalized fuzzy numbers A = ( a1 , a2 , a3 , a4 ; wA ) and B = (b1 , b2 , b3 , b4 ; wB ) is a fuzzy number: Gα = [G1 (α ), G2 (α )]∀α ∈[0, w],0 < w ≤ 1, w = min(wA , wB ) G = A(x) B = ( g1 , g , g3 , g ; w = min(wA , wB )) where, Gα = Aα (x) Bα* = [ AL (α ) Bα* (α ), AR (α ) BR* (α )] (41) g = ab; * = [a1 + α ( a2 − a1 ) / wA ][b1 + α (b2 − b1 ) / wB* ], (42) g = w(a b − a b ) / w + a b ; { } [a4 − α ( a4 − a3 ) / wA ][b4 − α (b4 − b3* ) / wB* ] ⎡(a − a ) (b* −b ) ⎛a ⎞ b , = ⎢ 2 α2 + ⎜ (b2* −b1 ) + (a2 − a1 ) ⎟α + ab 1 ⎜ w* ⎟ wB* w ⎢⎣ wA A ⎝ B ⎠ (32) * ⎤ ⎛ ⎞ (a4 − a3 ) (b4 −b3 ) a4 b α − ⎜ (b4 −b3* ) + (a4 − a3 ) ⎟α + a4b4 ⎥ ⎜ ⎟ wA wB* wA ⎥⎦ ⎝ wB* ⎠ (33) U 2α − T2α + V2 − x = (34) T2 = a4 (b4 − b3* ) / wB* + b4 ( a4 − a3 ) / wA , V1 = a1b1 , V2 = a4b4 We have w = wA = wB* , b2* = b1 + w(b2 − b1 ) / wB , and b = b4 − w(b4 − b3 ) / wB , then U , U , T1 , and T2 * become: (a − a )(b − b ) ( a2 − a1 )(b2 − b1 ) , U2 = 4 , wwB wwB a1 (b2 − b1 ) b1 (a2 − a1 ) a4 (b4 − b3 ) b4 (a4 − a3 ) + + T1 = , T2 = wB w wB w U1 = Only the roots in [0,1] will be retained in (33) and (34) The left and right membership functions f GL ( x ) and fGR ( x) = {T2 − [T22 + 4U2 ( x −V2 )]1/2 B g = a4 b4 , (44) = a1b1 ; = w(a2 b2 − a2 b1 ) / wB + a2 b1 ; = w(a2 b2 − a2 b3 ) / wB + a2b3 ; = a3b3 (45) (46) (47) with different confidence levels generates a fuzzy number as follows: I = A(/) B = (i1 , i2 , i3 , i4 ; w = min( wA , wB )) where, f ( x ) of G can be calculated as: } / 2U , } / 2U , g = g = a2 b2 , then formulae (45-48) are the same as in Chen [6] 4.4 Division of two generalized trapezoidal fuzzy numbers Theorem Division operation of two generalized fuzzy numbers A = (a1 , a2 , a3 , a4 ; wA ) and B = (b1 , b2 , b3 , b4 ; wB ), R G f (x) = {−T1 + [T1 + 4U1 ( x −V1 )] (48) and w = wA ≤ wB ; a1 , a2 , a3 , b1 , b2 , and b3 are non-zero and positive real numbers Proof: The proof is similar to Theorem Notably, when wA = wB = w, we will have T1 = a1 (b2* − b1 ) / wB* + b1 ( a2 − a1 ) / wA , 1/2 (43) h1 h2 h3 h4 ( a − a ) (b4 − b3* ) ( a − a ) (b * − b ) U1 = 2 , U = , wA wB* wA wB* 2 g3 = w(a3b3 − a3b4 ) / wB + a3b4 ; different confidence levels generates a fuzzy number as follows: H = A(x) B = (h1 , h2 , h3 , h4 ; w = min( wA , wB )) where, where, L G Notably, when wA = wB = w, formulae (41-44) are the same as in Chen [6] Theorem Multiplication of two triangular fuzzy numbers A = (a1 , a2 , a3 ; wA ) and B = (b1 , b2 , b3 ; wB ), with We now have two equations to solve - namely: U1α + T1α + V1 − x = g1 ≤ x ≤ g2 , (35) g3 ≤ x ≤ g4 , (36) dfGL ( x) / dx = 1/ [T1 + 4U1 ( x − V1 )]1/2 > and df GR ( x ) / dx = −1/ [T22 + 4U ( x − V2 )]1/2 < 0, then f GL ( x ) and f GL ( x ) are increasing and decreasing functions in x, Since, respectively The values of g1 , g , g , and g are determined respectively as follow: i1 = a1 / b4 ; (49) i2 = w(a2 / b3 − a2 / b4 ) / wB + a2 / b4 ; i3 = w( a3 / b2 − a3 / b1 ) / wB + a3 / b1 ; (50) (51) i4 = a4 / b1 (52) and w = wA ≤ wB ; a1 , a2 , a3 , a4 , b1 , b2 , b3 and b4 are nonzero and positive real numbers Proof: Consider two generalized fuzzy numbers A = (a1 , a2 , a3 , a4 ; wA ) and B = (b1 , b2 , b3 , b4 ; wB ) In order to determine the division operation between A and B , the value of A(/) B can be defined as A(/) B = A(x)(1 / B ), 410 Proceedings of 2013 International Conference on Fuzzy Theory and Its Application National Taiwan University of Science and Technology, Taipei, Taiwan, Dec 6-8, 2013 where 1/ B = (1/ b4 ,1 / b3 ,1/ b2 ,1 / b1 ; wB ) Hence, the division operation between A and B , can be obtained Notably, when wA = wB = w, formulae (49-52) are the same as in Chen [6] Theorem Division operation between two triangular fuzzy numbers A = ( a1 , a2 , a3 ; wA ) and B = (b1 , b2 , b3 ; wB ), with different confidence levels generates a fuzzy number as follows: J = A(/) B = ( j1 , j2 , j3 , j4 ; w = min( wA , wB )) where, j1 = a1 / b3 ; j2 = w(a2 / b2 − a2 / b3 ) / wB + a2 / b3 ; j3 = w(a2 / b2 − a2 / b1 ) / wB + a2 / b1 ; (53) j4 = a3 / b1 (56) (54) (55) and w = wA ≤ wB ; a1 , a2 , a3 , b1 , b2 , and b3 are non-zero and positive real numbers Proof: The proof is similar to Theorem Notably, when wA = wB = w, we will have j2 = j3 = a2 / b2 , then formulae (53-56) are the same as in Chen [6] A and B are D = A( + ) B = (0.4, 0.65, 0.975,1.3;0.6), F = A( − ) B = ( −0.8, −0.475, −0.15, 0.1; 0.6), H = A(x) B = (0.3, 0.09, 0.2025, 0.36;0.6), and I = A(/) B = (0.111, 0.296, 0.733,1.333; 0.6) Again, the arithmetic operations between the arithmetic operations between generalized trapezoidal fuzzy numbers obtained by the proposed approach can overcome the shortcomings of Chen’s approach Example Consider the generalized triangular fuzzy and generalized number A = (0.2, 0.3, 0.5;0.5) trapezoidal fuzzy number B = (0.4, 0.5, 0.7, 0.8;1) Using the proposed approach, the arithmetic operations between A and B are D = A( + ) B = (0.6, 0.75,10.05,1.3; 0.5), F = A( − ) B = ( −0.8, −0.475, −0.15, 0.1; 0.6), H = A(x) B = (0.08, 0.135, 0.36, 0.4; 0.5), and I = A(/) B = (0.25, 0.402, 0.675,1.25;0.5) 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 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 Re-consider the two generalized triangular fuzzy numbers, i.e., A = (0.5, 0.6, 0.7; 0.5) and B = (0.6, 0.7, 0.8; 0.9) in example Using the proposed approach, the arithmetic operations between and are fuzzy numbers A B D = A( + ) B = (1.1,1.256,1.344,1.5; 0.5), F = A( − ) B = ( −0.3, −0.144, −0.056, 0.1; 0.5), H = A(x) B = (0.3, 0.393, 0.447, 0.56;0.5), and I = A(/) B = (0.625, 0.81,1.01,1.167; 0.5) chosen for further evaluation A committee of three decision makers, D1 , D2 , and D3 , conducts the evaluation and selection of the ten candidates Nine selection criteria are considered including number of publications (C1 ), quality of publications (C2 ), 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 Re-consider the three generalized triangular fuzzy numbers, i.e., A1 = (0.2, 0.4, 0.6; 0.5), A2 = (0.5, 0.7, 0.9; 0.7), and A3 = (0.5, 0.7, 0.9;0.9) in Example According to Theorem 1, the addition operations between A2 , and A3 are A1 , A12 = A1 (+ ) A2 = (0.7,1.043,1.157,1.5; 0.5), 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 A1 ,…, A9 , and A10 , are and A13 = A1 (+ ) A3 = (0.7,1.011,1.189,1.5; 0.5), respectively Clearly, the results show that A12 ∈ A13 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 Consider the two generalized trapezoidal and fuzzy numbers A = (0.1, 0.2, 0.3, 0.4; 0.6) B = (0.3, 0.5, 0.6, 0.9;0.8) Using the proposed approach, personal qualification (C3 ), personality factors (C4 ), activity in professional society (C5 ), classroom teaching (C6 ), student advising (C7 ), research and/or creative activity (independent of publication) (C8 ), and fluency in a foreign language (C9 ) [24-26] The computational procedure is summarized as follows: Step Aggregate ratings of alternatives versus criteria Assume that the decision makers use the linguistic rating set S = {VL,L,M,H,VH}, 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 A1 ,… , A10 versus nine criteria, i.e C1 ,…, C9 , from three decision makers can be obtained as shown in Tables 1a-c 411 Proceedings of 2013 International Conference on Fuzzy Theory and Its Application National Taiwan University of Science and Technology, Taipei, Taiwan, Dec 6-8, 2013 Table 1a The linguistic ratings evaluated by decision makers Crit eria C1 C2 C3 Can dida tes D1 D2 D3 A1 G G VG A2 F G G A3 G F G A4 G G G A5 F G G A6 G VG G A7 G G VG A8 F F G A9 VG G G A10 F F F A1 G G F A2 F F G A3 G VG G A4 G G VG A5 G G G A6 VG G G A7 F F F A8 VG VG VG A9 G F G A10 G G G A1 VG VG G A2 G F F A3 F G G A4 F F F A5 F G G A6 G F G A7 G VG G A8 G G G A9 VG VG G A10 F G G Decision makers Table 1b The linguistic ratings evaluated by decision makers Crit eria Rij (0.667, 0.763, 0.797, 0.933; 0.9) (0.500, 0.685, 0.752, 0.833; 0.8) (0.500, 0.685, 0.752, 0.833; 0.8) (0.600, 0.800, 0.867, 0.900; 0.9) (0.500, 0.685, 0.752, 0.833; 0.8) (0.667, 0.830, 0.897, 0.933; 0.9) (0.667, 0.830, 0.863, 0.933; 0.9) (0.400, 0.593, 0.659, 0.767; 0.8) (0.667, 0.830, 0.897, 0.933; 0.9) (0.300, 0.500, 0.567, 0.700; 0.8) (0.500, 0.685, 0.700, 0.833; 0.8) (0.400, 0.593, 0.659, 0.767; 0.8) (0.667, 0.830, 0.897, 0.933; 0.9) (0.667, 0.830, 0.863, 0.933; 0.9) (0.600, 0.800, 0.867, 0.900; 0.9) (0.667, 0.830, 0.897, 0.933; 0.9) (0.300, 0.500, 0.567, 0.700; 0.8) (0.800, 0.900, 0.933, 1.000; 1.0) (0.500, 0.685, 0.752, 0.833; 0.8) (0.600, 0.800, 0.867, 0.900; 0.9) (0.733, 0.874, 0.867, 0.967; 0.9) (0.400, 0.593, 0.659, 0.767; 0.8) (0.500, 0.685, 0.752, 0.833; 0.8) (0.300, 0.500, 0.567, 0.700; 0.8) (0.500, 0.685, 0.752, 0.833; 0.8) (0.500, 0.685, 0.752, 0.833; 0.8) (0.667, 0.830, 0.897, 0.933; 0.9) (0.600, 0.800, 0.867, 0.900; 0.9) (0.733, 0.860, 0.867, 0.967; 0.9) (0.500, 0.685, 0.752, 0.833; 0.8) C4 C5 C6 412 Can dida tes D1 D2 D3 A1 F F G A2 G G F A3 VG G VG A4 G G G A5 G VG G A6 F G G A7 G F F A8 F F F A9 G F G A10 G VG G A1 VG VG G A2 G VG G A3 G F F A4 G G G A5 F G G A6 F F G A7 G F F A8 F F F A9 G G G A10 G VG G A1 F F G A2 F F F A3 G G VG A4 VG VG G A5 G F F A6 F F F A7 VG G G A8 G G G A9 G F F A10 F F F Decision makers Rij (0.400, 0.593, 0.659, 0.767; 0.8) (0.500, 0.685, 0.700, 0.833; 0.8) (0.733, 0.860, 0.893, 0.967; 0.9) (0.600, 0.800, 0.867, 0.900; 0.9) (0.667, 0.830, 0.897, 0.933; 0.9) (0.500, 0.685, 0.752, 0.833; 0.8) (0.400, 0.593, 0.659, 0.767; 0.8) (0.300, 0.500, 0.567, 0.700; 0.8) (0.500, 0.685, 0.752, 0.833; 0.8) (0.667, 0.830, 0.897, 0.933; 0.9) (0.733, 0.860, 0.867, 0.967; 0.9) (0.667, 0.830, 0.897, 0.933; 0.9) (0.400, 0.593, 0.659, 0.767; 0.8) (0.600, 0.800, 0.867, 0.900; 0.9) (0.500, 0.685, 0.752, 0.833; 0.8) (0.400, 0.593, 0.659, 0.767; 0.8) (0.400, 0.593, 0.659, 0.767; 0.8) (0.300, 0.500, 0.567, 0.700; 0.8) (0.600, 0.800, 0.867, 0.900; 0.9) (0.667, 0.830, 0.897, 0.933; 0.9) (0.400, 0.593, 0.659, 0.767; 0.8) (0.300, 0.500, 0.567, 0.700; 0.8) (0.667, 0.830, 0.863, 0.933; 0.9) (0.733, 0.860, 0.867, 0.967; 0.9) (0.400, 0.593, 0.659, 0.767; 0.8) (0.300, 0.500, 0.567, 0.700; 0.8) (0.667, 0.830, 0.897, 0.933; 0.9) (0.600, 0.800, 0.867, 0.900; 0.9) (0.400, 0.593, 0.659, 0.767; 0.8) (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 Table 1c The linguistic ratings evaluated by decision makers Crit eria C7 C8 C9 Can dida tes D1 D2 D3 A1 G G VG A2 VG G G A3 G G G A4 F G G A5 G F F A6 F F G Decision makers A7 G F F A8 F F F A9 G G VG A10 F F G A1 G F G A2 G G VG A3 VG G G A4 G G G A5 F G F A6 F F F A7 F G F A8 F F G A9 G VG G A10 F F G A1 G G VG A2 VG G G A3 F F G A4 G F G A5 F F G A6 F G G A7 G F G A8 G G G A9 G VG G A10 F G F Rij (0.667, 0.830, 0.863, 0.933; 0.9) (0.667, 0.830, 0.897, 0.933; 0.9) (0.600, 0.800, 0.867, 0.900; 0.9) (0.500, 0.685, 0.752, 0.833; 0.8) (0.400, 0.593, 0.659, 0.767; 0.8) (0.400, 0.593, 0.659, 0.767; 0.8) (0.400, 0.593, 0.659, 0.767; 0.8) (0.300, 0.500, 0.567, 0.700; 0.8) (0.667, 0.830, 0.863, 0.933; 0.9) (0.400, 0.593, 0.659, 0.767; 0.8) (0.500, 0.685, 0.752, 0.833; 0.8) (0.667, 0.830, 0.863, 0.933; 0.9) (0.667, 0.830, 0.897, 0.933; 0.9) (0.600, 0.800, 0.867, 0.900; 0.9) (0.400, 0.593, 0.659, 0.767; 0.8) (0.300, 0.500, 0.567, 0.700; 0.8) (0.400, 0.593, 0.659, 0.767; 0.8) (0.400, 0.593, 0.659, 0.767; 0.8) (0.667, 0.830, 0.897, 0.933; 0.9) (0.400, 0.593, 0.659, 0.767; 0.8) (0.667, 0.830, 0.863, 0.933; 0.9) (0.667, 0.830, 0.897, 0.933; 0.9) (0.400, 0.593, 0.659, 0.767; 0.8) (0.500, 0.685, 0.752, 0.833; 0.8) (0.400, 0.593, 0.659, 0.767; 0.8) (0.500, 0.685, 0.752, 0.833; 0.8) (0.500, 0.685, 0.752, 0.833; 0.8) (0.600, 0.800, 0.867, 0.900; 0.9) (0.667, 0.830, 0.897, 0.933; 0.9) (0.400, 0.593, 0.659, 0.767; 0.8) Step Aggregate the importance weights Also assumes that the decision makers employ a linguistic weighting set Q = {UI,OI,I,VI,AI}, 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 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 Table The importance weights of the criteria evaluated by decision makers Criteria C1 C2 C3 C4 C5 C6 C7 C8 C9 Decision makers D1 D2 D3 AI AI VI AI VI VI I VI I I VI I VI VI AI AI AI VI I I VI I VI VI VI I I wij (0.733, 0.860, 0.867, 0.967; 0.9) (0.667, 0.830, 0.830, 0.933; 0.9) (0.400, 0.593, 0.593, 0.767; 0.8) (0.400, 0.593, 0.593, 0.767; 0.8) (0.667, 0.830, 0.833, 0.933; 0.9) (0.733, 0.860, 0.867, 0.967; 0.9) (0.400, 0.593, 0.600, 0.767; 0.8) (0.500, 0.685, 0.693, 0.833; 0.8) (0.400, 0.593, 0.593, 0.767; 0.8) Step 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 shows the weighted ratings of each candidate Table Weighted ratings of each candidate Candidates A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 Weighted ratings (0.316, 0.520, 0.566,0.753; 0.8) (0.280, 0.491, 0.551,0.723; 0.8) (0.310, 0.521, 0.584,0.748; 0.8) (0.320, 0.539, 0.589,0.752; 0.8) (0.264, 0.474, 0.536,0.704; 0.8) (0.259, 0.472, 0.520,0.695; 0.8) (0.270, 0.473, 0.535,0.705; 0.8) (0.265, 0.470, 0.530,0.699; 0.8) (0.320, 0.534, 0.598,0.761; 0.8) (0.252, 0.460, 0.523,0.693; 0.8) Step 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 According to Table 4, the ranking order of the ten candidates is: A9 A4 A3 A1 A2 A7 A5 A8 A6 A10 Thus, the best selection is candidate A9 having the largest distance 413 Proceedings of 2013 International Conference on Fuzzy Theory and Its Application National Taiwan University of Science and Technology, Taipei, Taiwan, Dec 6-8, 2013 Table Distance between the centroid point and the minimum point of each candidate Candidates A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 Distances 0.0578 0.0292 0.0582 0.0668 0.0124 0.0041 0.0140 0.0090 0.0704 0.0000 Ranking order 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) Finally, we applied the proposed arithmetic operations to deal with university academic staff evaluation and selection problem It can be seen that the proposed algorithms is efficient and easy to implement So in future, the proposed method can be applied to solve the problems that involve the generalized fuzzy number REFERENCES [1] L A Zadeh, “Fuzzy sets,” Inf Control, Vol 8, No 3, pp 338-353, 1965 [2] D Dubois, H Prade, “Operations on fuzzy numbers,” Int J Syst Sci., Vol 9, No 6, pp 613-626, 1978 [3] G Klir and B Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications Prentice-Hall, Englewood Cliffs, New York, 1995 [4] M Mizumoto and K Tanaka, “Fuzzy sets and their operations,” Inf Control, Vol 48, No 1, pp 30-38, 1981 [5] H J Zimmermann, Fuzzy set theory and its applications Kluwer Academic Publishers, Boston, 1991 [6] S H Chen, “Operations on fuzzy numbers with function principal,” Tamkang J Manage Sci., Vol 6, No 1, pp 13-25, 1985 [7] C.H Hsieh and S.H Chen, “Similarity of generalized fuzzy numbers with graded mean integration representation,” Proc 8th Int fuzzy Syst Association World Congress, Taipei, Taiwan, Republic of China, 2, 551-555, 1999 [8] S.H Chen and C.C Wang, “Backorder fuzzy inventory model under function principle,” Inf Sci., Vol 95, No 1-2, pp 71-79, 1996 [9] S.J Chen and S.M Chen, “Fuzzy risk analysis based on the ranking of generalized trapezoidal fuzzy numbers,” Appl Intell Vol 26, No 1, pp 1-11, 2007 [10] S.M Chen and J.H Chen, “Fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads,” Expert Syst Appl., Vol 36, No 3, pp 6833-6842, 2009 [11] S.H Chen, C.C Wang, and S.M Chang, “Fuzzy economic production quantity model for items with imperfect quality,” Int J Innovative Comput Inf Control, Vol 3, No 1, pp 85-95, 2007 [12] S.M Chen and K Sanguansat, “Analyzing fuzzy risk based on a new fuzzy ranking method between generalized fuzzy numbers,” Expert Syst Appl Vol 38, No 3, pp 2163-2171, 2011 [13] S.M Chen, A Munif, G.S Chen, H.C Liu, and B.C Kuo, “Fuzzy risk analysis based on ranking generalized fuzzy numbers with different left heights and right heights,” Expert Syst Appl Vol 39, No 7, pp 6320-6334, 2012 [14] C.H Hsieh and S.H Chen, “A model and algorithm of fuzzy product positioning,” Inf Sci Vol 121, No 1-2, pp 61-82, 1999 [15] S Islam and T.K Roy, “A new fuzzy multi-objective programming: Entropy based geometric programming and its application of transportation problems,” Eur J Oper Res Vol 173, No 2, pp 387-404, 2006 [16] A Kaur and A Kumar, “A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers,” Appl Soft Comput Vol 12, No 3, pp 1201-1213, 2012 [17] A Kumar, P Singh, P Kaur, and A Kaur, “A new approach for ranking of L-R type generalized fuzzy numbers,” Expert Syst Appl., Vol 38, No 9, pp 10906-10910, 2011 [18] G S Mahapatra and T K Roy, “Fuzzy multi-objective mathematical programming on reliability optimization model,” Appl Math Comput Vol 174, No 1, pp 643-659, 2006 [19] L Qi, X Jia, and D Yong, “A subjective methodology for risk quantification based on generalized fuzzy numbers,” Int J Gen Syst., Vol 37, No 2, pp 149-165, 2008 [20] S.H Wei and S.M Chen, “A new approach for fuzzy risk analysis based on similarity measures of generalized fuzzy numbers,” Expert Syst Appl., Vol 36, No 1, pp 589-598, 2009 [21] Z Xu, S Shang, W Qian, and W Shu, “A method for fuzzy risk analysis based on the new similarity of trapezoidal fuzzy numbers,” Expert Syst Appl., Vol 37, No 3, pp 1920-1927, 2010 [22] D Yong, S Wenkang, D Feng, and L Qi, “A new similarity measure of generalized fuzzy numbers and its application to pattern recognition,” Pattern Recognit Lett Vol 25, No 8, pp 875-883, 2004 [23] D Chakraborty, and D Guha, “Addition of two generalized fuzzy numbers,” Int J Ind Syst Eng Math., Vol 2, No 1, pp 9-20, 2010 [24] J A Centra, How Universities Evaluate Faculty Performance: A Survey of Department Heads, Graduate Record Examinations Program Educational Testing Service Princeton, NJ 08540, 1977 [25] F Wood, “Factors Influencing Research Performance of University Academic Staff,” Higher Education, Vol 19, No 1, pp 81-100, 1990 [26] M Dursun and E E Karsak, “A fuzzy MCDM approach for personnel selection,” Expert Syst Appl., Vol 37, No 6, pp 43244330, 2010 [27] V F Yu, H T X Chi, L Q Dat, P N K Phuc, C W Shen, “Ranking generalized fuzzy numbers in fuzzy decision making based on the left and right transfer coefficients and areas,” Appl Math Model., 2013 Doi:10.1016/j.apm.2013.03.022 [28] L Q Dat, V F Yu, and S Y Chou, “An Improved Ranking Method for Fuzzy Numbers Based on the Centroid-Index,” Int J Fuzzy Syst, Vol 14, No 3, pp 413-419, 2012 414 ... 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. .. prove that the arithmetic operations between generalized fuzzy numbers, proposed by Chen [6], not satisfy the reasonable properties for the arithmetic operations of fuzzy numbers In 2010, Chakraborty... 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