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Most of existing studies done using generalized fuzzy numbers were based on Chen’s 1985 arithmetic operations.. In order to overcome the drawbacks of Chen’s method, this paper develo

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., A=( , , , ;a a a a w1 2 3 4 A) and

( , , , ; B),

B = b b b b w where a a a a b b b1, , , , , ,2 3 4 1 2 3 and b4

are real values, 0 ≤ wA ≤ 1 and 0 ≤ wB ≤ 1.

Some arithmetic operators between the generalized fuzzy numbers A and B are defined as follows:

(i) Generalized trapezoidal fuzzy numbers addition ( ) :+

A + = B a b a b a b a b + + + + w w (1) where a a a a b b b1, , , , , ,2 3 4 1 2 3 and b4 are real values

(ii) Generalized trapezoidal fuzzy numbers subtraction

A − = B a b a b a b a − − − − b w w (2) where a a a a b b b1, , , , , ,2 3 4 1 2 3 and b4 are real values

(iii) Generalized trapezoidal fuzzy numbers multiplication (x) :

(x) ( , , , ; min( ,A B))

A B= a b a b a b a b× × × × w w (3) where a a a a b b b1, , , , , ,2 3 4 1 2 3 and b4 are all positive real

numbers

(iv) Generalized trapezoidal fuzzy numbers division (/) : Let a a a a b b b1, , , , , ,2 3 4 1 2 3 and b4 be non-zero positive

real numbers Then,

(/) ( / , / , / , / ; min( ,A B)),

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 A=(0.5,0.6,0.7;0.5) and B=(0.6,0.7,0.8;0.9)

shown in Fig 1 It is observed from Fig 2 that,

min(w A =0.5,w B=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

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 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) as in Fig 2 Intuitively, the

order of fuzzy numbers A2 and A3 is A2∈A3 Then, we

should have A12=A1( )+ A2∈A13=A1( ) + A3 However,

using the Chen’s method, we have

12 (0.7,1.1,1.5;0.5)

A = and A13=(0.7,1.1,1.5;0.5)

Thus, the additions between the generalized fuzzy

numbers A1 and A2, and A1 and A3 are the same, i.e.,

12 13

A ∼A Therefore, Chen’s method cannot consistency

calculate the arithmetic operations between generalized

fuzzy numbers

1

A

2

A

3

A

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 A a a a a w = ( , , , ; )1 2 3 4 A and B = ( , , , ; b b b b w1 2 3 4 B) are two generalized trapezoidal fuzzy numbers with membership function f x A( ) and f x B( ), respectively, which can be written in the following form:

( )

A

A A

A

f x

⎧

⎪

= ⎨

⎪

⎪⎩

(5)

and

( )

B

B B

B

f x

⎧

⎪

= ⎨

⎪

⎪⎩

(6)

where, a a a a b b b1, , , , , ,2 3 4 1 2 3 and b4 are real values,

0 ≤ wA≤ 1 and 0 ≤ wB≤ 1 w w = A ≤ w wB, A 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, take w w = A < wB cut of fuzzy number

( , , , ; B),

B = b b b b w then B will transform into a new generalized trapezoidal fuzzy number as

( , , , ; ),

B

B = b b b b w where * ,

B

w = w and the values of

* 2

b and *

3

b are determined as *

b = + b w b b − w and

*

b = − b w b b − w respectively

Then, the α-cuts of generalized fuzzy numbers

( , , , ; A)

A a a a a w = and * ( , , , ;1 2* 3* 4 *)

B

B = b b b b w are given as:

α

∀ ∈ < ≤ (7)

α

∀ ∈ < ≤ (8)

4.1 Addition of two generalized trapezoidal fuzzy numbers

Theorem 1 Addition of two generalized fuzzy numbers

( , , , ; A)

A a a a a w = and B = ( , , , ; ), b b b b w1 2 3 4 B with different confidence levels generates a trapezoidal fuzzy number as follows:

( ) ( , , , ; min( ,A B))

C = + = A B c c c c w = w w where,

1 1 1;

c = + a b

c = + + b a w b − b w

c = + −b a w b −b w

c = +a b

and w w = A≤ wB; a a a a b b b1, , , , , ,2 3 4 1 2 3 and b4 are any

real numbers

Proof: Suppose that A ( ) + = B C where

[ ( ), ( )] [0, ],

Cα = C α C α ∀ ∈α w 0< ≤w 1,

min( ,A B).

w = w w Then,

*

( )

[ ( ) ( ), ( ) ( )]

α

(9)

Let Cα ={ :x x∈[ ( ),C1 α C2( )]α ∀ ∈α [0, ]w

We now have two equations to solve - namely:

*

a b + + α b b − w + α a − a w − = x (10)

*

a + − b α b b − w − α a − a w − = x (11)

From Equations (10) and (11), the left and right

membership functions f x L( ) and f x R( ) of C can

be calculated as:

*

L

C

A B

w x a b

w b b w w a a w

(12)

*

[ ( )]

R

C

A B

− +

We have *

,

b = +b w b −b w and

*

4 ( 4 3) / B,

b = −b w b −b w then Equations (12) and (13)

become:

*

( )

,

L

C

B

B

w x a b

f 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)

*

( )

,

R

C

B B

w x a b

f x

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 A=( , , , ;a a a a w1 2 3 4 A) and

( , , , ; B)

B= b b b b w is a generalized trapezoidal fuzzy

number as follows:

( )

C=A + B=( , , ,c c c c w1 2 3 4; =min( ,w w A B)) where,

c = +a b (16)

c = + +b a w b −b w (17)

c = + −b a w b −b w (18)

Notably, when w A=w B =w, formulae (16)-(19) are the

same as in Chen [6]

Theorem 2 Addition of two generalized triangular fuzzy

numbers A a a a w = ( , , ;1 2 3 A) and B = ( , , ; b b b w1 2 3 B) with different confidence levels generates a trapezoidal fuzzy numbers as follows:

( ) ( , , , ; min( ,A B))

D= +A B= d d d d w= w w where,

d = + +b a w b −b w (21)

d = + +b a w b −b w (22)

and w w = A≤ wB; a a a b b1, , , , ,2 3 1 2 and b3 are any real

numbers

Proof: The proof is similar to Theorem 1

Notably, when w A=w B =w , we will have

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 A a a a a w = ( , , , ; )1 2 3 4 A and

( , , , ; )B

B = b b b b w with different confidence levels generates a trapezoidal fuzzy number as follows:

( ) ( , , , ; min( ,A B))

E = − = A B e e e e w = w w where,

e = − +a b w b −b w (25)

e = − −a b w b −b w (26)

and w w = A ≤ wB; a a a a b b b1, , , , , ,2 3 4 1 2 3 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 ( ) ( )( ),

A− B= + −A B where − = − − − −B ( b4, b3, b2, b1) Hence, the proof is similar to Theorem 1

Notably, when w A=w B =w, then formulae (24-27) are the same as in Chen [6]

Theorem 4 Subtraction operation of two generalized

triangular fuzzy numbers A a a a w = ( , , ; )1 2 3 A and

( , , ; )B

generates a trapezoidal fuzzy number as follows:

( ) ( , , , ; min( ,A B))

F= − =A B f f f f w= w w where,

f = − +a b w b −b w (29)

f = − +a b w b b− w (30)

and w w = A≤ wB; a a a b b1, , , , ,2 3 1 2 and b3 are any real

numbers

Proof: The proof is similar to Theorem 1

Notably, when wA= wB = w , we will have

f = f = −a b then formulae (28-31) are the same as in

Chen [6]

4.3 Multiplication of two generalized trapezoidal fuzzy numbers

Theorem 5 Multiplication of two generalized fuzzy

numbers A a a a a w = ( , , , ; )1 2 3 4 A and B = ( , , , ; ), b b b b w1 2 3 4 B

with different confidence levels generates a fuzzy number

as follows:

(x) ( , , , ; min( ,A B))

1 1 1;

g =a b

g =w a b −a b w +a b

g =w a b −a b w +a b

4 4 4,

g =a b

and w w = A≤ wB; a a a a b b b1, , , , , ,2 3 4 1 2 3 and b4 are

non-zero and positive real numbers

Proof: Suppose that A( )× =B G where,

[ ( ), ( )] [0, ],0 1, min( ,A B)

Gα = G α G α ∀ ∈ α w < ≤ w w = w w

{

}

*

*

(x) [ ( ) ( ), ( ) ( )]

*

*

⎣

⎤

(32)

We now have two equations to solve - namely:

2

2

U α − T α + − = V x (34)

where,

*

*

a a b b

a a b b

*

B

T =a b −b w +b a −a w

*

2 4( 4 3) / B* 4( 4 3) / A,

T = a b − b w + b a − a w

V = a b V = a b

We have *

,

b = +b w b −b w

and *

b = −b w b −b w then U U T1, 2, ,1 and T2

become:

,

a a b b

a a b b

a b b b a a

a b b b a a

Only the roots in [0,1] will be retained in (33) and (34)

The left and right membership functions L( )

G

f x and

( )

R

G

f x of G can be calculated as:

L

G

f x = − + T T + U x V − U g ≤ ≤ x g (35)

R

G

f x = T − T + U x V − U g ≤ ≤ x g (36)

L

G

df x dx = T + U x V − > and

( ) / 1/ [ 4 ( )] 0,

R

G

df x dx= − T + U x V− < then L( )

G

f x

and L( )

G

f x are increasing and decreasing functions in x,

respectively

The values of g g g1, , ,2 3 and g4 are determined

respectively as follow:

L G

L

x V a b

R G

R

x V a b

L G

x w a b a b w a b

⇔ = − + (39)

R G

x w a b a b w a b

Thus, the multiplication operation between two generalized fuzzy numbers A=( , , , ;a a a a w1 2 3 4 A) and

( , , , ; B)

B= b b b b w is a fuzzy number:

(x) ( , , , ; min( ,A B))

G A = B = g g g g w = w w where,

1 1 1;

4 4 4,

Notably, when w A=w B =w, formulae (41-44) are the same as in Chen [6]

Theorem 6 Multiplication of two triangular fuzzy

numbers A a a a w = ( , , ; )1 2 3 A and B = ( , , ; ), b b b w1 2 3 B with different confidence levels generates a fuzzy number as follows:

(x) ( , , , ; min( ,A B))

H =A B= h h h h w= w w where,

1 1 1;

4 3 3

and w w = A≤ wB; a a a b b1, , , , ,2 3 1 2 and b3 are non-zero

and positive real numbers

Proof: The proof is similar to Theorem 5

Notably, when wA = wB = w , we will have

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 A a a a a w = ( , , , ; )1 2 3 4 A and B = ( , , , ; ), b b b b w1 2 3 4 B

with different confidence levels generates a fuzzy number

as follows:

1 2 3 4

(/) ( , , , ; min( ,A B))

I =A B= i i i i w= w w where,

2 ( /2 3 2 / ) /4 B 2/ ;4

3 ( /3 2 3/ ) /1 B 3/ ;1

i =w a b −a b w +a b (51)

and w w = A≤ wB; a a a a b b b1, , , , , ,2 3 4 1 2 3 and b4 are

non-zero and positive real numbers

Proof: Consider two generalized fuzzy numbers

( , , , ; A)

A= a a a a w and B=( , , , ; ).b b b b w1 2 3 4 B 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

where 1/B=(1/ ,1/ ,1/ ,1/ ;b4 b3 b2 b w1 B) Hence, the

division operation between A and B,can be obtained

Notably, when w A=w B =w, formulae (49-52) are the

same as in Chen [6]

Theorem 8 Division operation between two triangular

fuzzy numbers A=( , , ;a a a w1 2 3 A) and B=( , , ;b b b w1 2 3 B),

with different confidence levels generates a fuzzy number

as follows:

(/) ( , , , ; min( ,A B))

J = A B = j j j j w = w w where,

1 1/ ;3

2 ( /2 2 2/ ) /3 B 2/ ;3

3 ( /2 2 2/ ) /1 B 2/ ;1

and w w = A≤ wB; a a a b b1, , , , ,2 3 1 2 and b3 are non-zero

and positive real numbers

Proof: The proof is similar to Theorem 7

Notably, when wA = wB = w , we will have

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., A=(0.5, 0.6, 0.7;0.5) 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),

(x) (0.3, 0.393, 0.447, 0.56;0.5),

H =A B=

and I=A(/)B=(0.625, 0.81,1.01,1.167;0.5)

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., A1=(0.2, 0.4, 0.6;0.5),

2 (0.5, 0.7, 0.9;0.7),

A = and A3 = (0.5,0.7,0.9;0.9) in

Example 2 According to Theorem 1, the addition

operations between A1, A2, and A3 are

12 1( ) 2 (0.7,1.043,1.157,1.5;0.5),

13 1( ) 3 (0.7,1.011,1.189,1.5;0.5),

A =A + A = 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 5 Consider the two generalized trapezoidal

fuzzy numbers A=(0.1, 0.2, 0.3, 0.4;0.6) 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),

(x) (0.3, 0.09, 0.2025, 0.36;0.6),

H =A B= and I=A(/)B=(0.111, 0.296, 0.733,1.333;0.6). 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 A=(0.2, 0.3, 0.5;0.5) and generalized 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), ( ) ( 0.8, 0.475, 0.15, 0.1;0.6),

(x) (0.08, 0.135, 0.36, 0.4;0.5),

H =A B= 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

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 chosen for further evaluation A committee of three decision makers, D D1, 2, 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

personal qualification ( ), C3 personality factors ( ), C4

activity in professional society ( ), C5 classroom teaching

6

( ), C 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 1 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

1, , ,9

C … C from three decision makers can be obtained as shown in Tables 1a-c

Table 1a The linguistic ratings evaluated by decision makers

Crit

eria

Can

dida

tes

Decision makers

R ij

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

R ij

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)

Table 1c The linguistic ratings evaluated by decision makers

Crit

eria

Can

dida

tes

Decision makers

R ij

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 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 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 D1 D2 D3 w ij

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 10

Thus, the best selection is candidate A9 having the largest distance

Table 4 Distance between the centroid point and the minimum point of

each candidate Candidates Distances Ranking order

A3 0.0582 3

A4 0.0668 2

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

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