4
408 Proceedings of the 26 th Chinese Control Conference July 26-31, 2007, Zhangjiajie, Hunan, China 530004 Email: [email protected] A New Method for Triangular Fuzzy Number Multiple Attribute Decision Making Qin Juying, Meng Fanyong, Zeng Xuelan Mathematics and Information Science College, Guangxi University, Nanning 530004, Guangxi, P. R. China Email: [email protected] Abstract: On the basis of the new distance formula for triangular fuzzy numbers given by the paper, a priority method for triangular fuzzy numbers multiple attribute decision making is given. A new assembly method and decision method are given for triangular fuzzy numbers multiple attribute group decision-making. Finally, a numerical example is given to show its ef- fectiveness and practicability. Key Words: Triangular Fuzzy Numbers, Group Decision-making, Multiple Attribute Decision Making 1 (Introduction) [3-16] λ 2 (Preliminaries) 2.1 [1] ( , , ) l m u a aa a = 0 l m a a < u a l a u a a m a a a , () , 0, l l m m l u a m u m u x a a x a a a x a x a x a a a μ = (1) [2] ( , , ) l m u a aa a = ( , , ) l m u b bb b = ( , , ) ( , , ) ( , , ) l m u l m u l l m m u u a b aa a bb b a ba b a b + = + = + + + ( , , ) ( , , ) ( , , ) l m u l m u l u m m u l a b aa a bb b a b a b a b = = 1/ (1/ ,1/ ,1/ ) u m l a a a a ( , , ) ( , , ) ( , , ) l m u l m u l l m m u u a b aa a bb b ab a b ab × = × = a b = , , l l m m u u a ba b a b = = = 2.2 ( , , ) l m u a aa a = b = ( , l b , ) m u b b (,) dab a b 2 1 0 2 () () () () (,) 2 2 () () () () d 2 2 a a b b dab a a b b λ λ λ λ λ λ λ λ λ + + + + + + = + (2)

[IEEE 2007 Chinese Control Conference - Zhangjiajie, China (2007.07.26-2007.06.31)] 2007 Chinese Control Conference - A New Method for Triangular Fuzzy Number Multiple Attribute Decision

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Page 1: [IEEE 2007 Chinese Control Conference - Zhangjiajie, China (2007.07.26-2007.06.31)] 2007 Chinese Control Conference - A New Method for Triangular Fuzzy Number Multiple Attribute Decision

408

Proceedings of the 26th Chinese Control ConferenceJuly 26-31, 2007, Zhangjiajie, Hunan, China

��������� �������������

���� ����������� � 530004Email: [email protected]

� ��������������������� �!�"#$�����%&'()�*+,-./1���

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A New Method for Triangular Fuzzy Number Multiple AttributeDecision Making

Qin Juying, Meng Fanyong, Zeng XuelanMathematics and Information Science College, Guangxi University, Nanning 530004, Guangxi, P. R. China

Email: [email protected]

Abstract: On the basis of the new distance formula for triangular fuzzy numbers given by the paper, a priority method fortriangular fuzzy numbers multiple attribute decision making is given. A new assembly method and decision method are givenfor triangular fuzzy numbers multiple attribute group decision-making. Finally, a numerical example is given to show its ef-fectiveness and practicability.Key Words: Triangular Fuzzy Numbers, Group Decision-making, Multiple Attribute Decision Making

1 �(Introduction)

IJK%&'()LMN()OP�QR6S

TO��UVHWXYZ[F\]^_Z[`a.�

GbVHa�,c�IJK&'d^eH���@�

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)a�VH�����%��1�������"D

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#$:;<GH�*+,-.

2 �� (Preliminaries)

�� 2.1[1] = ( , , )l m ua a a a= ��a 0 l ma a< >

ua> �? la F ua O@ a*AB� CF)C�< ma a�ad�DE a#'�������"FG�H��

,

( ) ,

0,

ll m

m l

ua m u

m u

x a a x aa ax ax a x aa a

μ

−−−

=−

> >

> >

�I

(1)

",J�KL��)MD�������N>[2].

O ( , , )l m ua a a a= � ( , , )l m ub b b b= �D

P( , , ) ( , , )( , , )

l m u l m u

l l m m u u

a b a a a b b ba b a b a b

+ = += + + +

Q( , , ) ( , , )( , , )

l m u l m u

l u m m u l

a b a a a b b ba b a b a b

− = −= − − −

R 1/ (1/ ,1/ ,1/ )u m la a a a≈

S( , , ) ( , , )( , , )

l m u l m u

l l m m u u

a b a a a b b ba b a b a b

× = ×=

T a b= U?VU , ,l l m m u ua b a b a b= = =

�� 2.2 O ( , , )l m ua a a a= �b = ( ,lb , )m ub b LW

X&'������E ( , )d a b ����� a�b�����a

21

0

2

( ) ( ) ( ) ( )( , )2 2

( ) ( ) ( ) ( ) d2 2

a a b bd a b

a a b b

λ λ λ λ

λ λ λ λ λ

+ − + −

+ − + −

⎧⎡ ⎤+ +⎪= −⎨⎢ ⎥⎣ ⎦⎪⎩

⎫⎡ ⎤− − ⎪+ − ⎬⎢ ⎥⎣ ⎦ ⎪⎭

(2)

Page 2: [IEEE 2007 Chinese Control Conference - Zhangjiajie, China (2007.07.26-2007.06.31)] 2007 Chinese Control Conference - A New Method for Triangular Fuzzy Number Multiple Attribute Decision

409

�a� ( ), ( )a aλ λ− + O@L����� a� λ −(7

[ ( ), ( )]a a aλ λ λ− += �YZ[\#] ( ), ( )b bλ λ− + O@L

����� b� λ −(7 [ ( ), ( )]b b bλ λ λ− += �YZ[

\#.

�� 2.1 ^K_� ( , )d a b `a������ ( , ) 0d a b � � ( , ) 0d a b = � a b= .

� ( , ) ( , )d a b d b a= .

( , ) ( , ) ( , )d a c d a b d b c+� .

��B����[3].�ABK� 1.2a�W��������������������.

3 ����(The Method of Decision-Making)

��,��� 1,2, ,N n= � � 1,2, ,M m= � .

�D n�,c�m�&'(I�#�'�� D&'�LE��)� 1 2( , , , )mw w w w= � ��&'�Q�

� a ( , , )l m uj j j jw w w w= ( j M∈ ) � ( ) �

( )ij n mA a ×= ��a ( , , )l m uij ij ij ija a a a= �� i�,c�� j

�&' QR'Y�.

��3.1 ja+� rja

+F lja− ( j M∈ )O���� j�

&' �����������F�������a�

1 1 1(max ,max ,max )

n n nm m m

j ij ij iji i ia a a a+

= = ==

1 1 1(max ,max ,max )

n n nu u u

rj ij ij iji i ia a a a+

= = ==

1 1 1(min ,min ,min )

n n nl l l

lj ij ij iji i ia a a a−

= = ==

�� �GbVHa������������

��Gb��@� .

�� 3.2 � 2 2 2|| || l m ua a a a= + + ������

( , , )l m ua a a a= ��.

��3.3 �!ij jij a a

d d ++ = �"#!S���

����� ( )ij n mD d+ +×= $!

ij rjrij a ad d +

+ = �"#!

S��������� ( )r rij n mD d+ +×= $!

ij ljlij a ad d −

− = �"#!S���������

( )l lij n mD d− −×= .

%��()&'(

Step 1 ��() ( )ij n mA a ×= ��a ( ,lij ija a=

, )m uij ija a $

Step 2 !"������� ( )ij n mC c ×= �a

1 1 1( / max || ||, / max || ||, / max || ||)ij lij ij mij ij uij ijj m j m j m

c b b b b b b=

ijb = ij ja w �Step 3 �� &'��� ( )ij n mC c ×= ����

���������HK� 2.2�2.3 O��>( )ij n mC c ×= a���V&'��� �����

��������� �������� , ,r lD D D+ + −�

Step 4 O��> , ,r lD D D+ + −a����F��

� T1 2( , , , )nd d d d+ + + += � ��a�

1

m

i ijj

d d+ +

=

= ∑

T1 2( , , , )r r r rnd d d d+ + + += � ��a�

1

m

ri rijj

d d+ +

=

= ∑

T1 2( , , , )l l l lnd d d d− − − −= � ��a�

1

m

li lijj

d d− −

=

= ∑

Step 5 �> /( )i i i ri lic d d d d− − + += + + �� ic ( i N∈ )���1,c��*+.

4 ���������(The Synthesis Meth-

od of Group-Making)�� 2TO��� ���D s�� �2()

�7S,-�� 1,2, ,K s= � , ( )kk ij n mA a ×= ��

( )k k K∈ �� ���()����a ( ,k kij lija a=

, )k kmij uija a .78�!��"�>#�� ��$�

��������`F�%�1/ kd ( k K∈ )��;�>�D� ����������%�`F

11/

s

kk

d=∑ ��H1/ kd �

11/

s

kk

d=∑ �&d'�#� ��

�#��(Q.#$,-�!��")()$��

$()$��,c�()�������,c�*

+()�a(Q,��.�-�(Q,��.�.

/�0�L�"�1���� ���()��2

3.

�� 4.1 4 | |k pkpij ij ijd a a= − ( , )k p K∈ �� k�

()$�� p�()$5W� i�,c�� j�&'� � � � � � a kpij lkpij mkpij ukpijd d d d= + + �

| |k plkpij lij lijd a a= − � | |k p

mkpij mij mijd a a= − � | |k pukpij uij uijd a a= − .

��4.2 4 kijd �� k�()$5W� i�,c�� j�&'������a

1,

1, 1, 1,

( , , )

( , , )

s

kij kpij lkij kmij kuijp p k

s s s

lkpij mkpij ukpijp p k p p k p p k

d d d d d

d d d

= ≠

= ≠ = ≠ = ≠

= =

=

∑ ∑ ∑(6)

�� 4.3 4 ( )ij n mR r ×= �+S()����a

1 1 1

( , , )

( , , )

ij lij mij uij

K K Kk k k k k klij lij mij mij uij uij

k k k

r r r r

w a w a w a= = =

=

= ∑ ∑ ∑(7)

1 1 1

[ , , ]

1/ 1/ 1/, ,

1/ 1/ 1/

k k k kij lij mij uij

klij kmij kuijs s s

klij kmij kuijk k k

w w w w

d d d

d d d= = =

=

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦∑ ∑ ∑

(8)

�� [ , , ]k k k kij lij mij uijw w w w= IL#�������

�L#����6.<= 6,-78��� �

*+9:����;�;� 2TO��� <�!�

Page 3: [IEEE 2007 Chinese Control Conference - Zhangjiajie, China (2007.07.26-2007.06.31)] 2007 Chinese Control Conference - A New Method for Triangular Fuzzy Number Multiple Attribute Decision

410

,c�:"*+.

5 ����(Example Illustration)

D 3 �� 1#$I=>$�=?%> 1 2, ,x x

3 4,x x ��@A'�7B'�GH'FCD'��EF�@:"JKGHI$%>.� J����&'

(QO��" 1 2 3 4( , , , ) ((0.2,0.3,0.4)w w w w w= = �

(0.2,0.25,0.3)� (0.2,0.25,0.3)� (0.15,0.2,0.25))��� � 1K9L&��Md����()��N�:

1

(7,8,9) (5,6,7) (4,5,6) (2,3,4)(6,7,8) (5,6,7) (6,7,8) (5,6,7)(5,6,7) (6,7,8) (4,5,6) (6,7,8)(6,7,8) (6,7,8) (4,5,6) (4,5,6)

A

⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎝ ⎠

2

(4,5,6) (6,7,8) (2,3,4) (3,4,5)(6,7,8) (6,7,8) (4,5,6) (6,7,8)(4,5,6) (7,8,9) (4,5,6) (4,5,6)(4,5,6) (7,8,9) (2,3,4) (6,7,8)

A

⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎝ ⎠

3

(6,7,8) (7,8,9) (4,5,6) (4,5,6)(4,5,6) (6,7,8) (4,5,6) (5,6,7)(7,8,9) (6,7,8) (6,7,8) (4,5,6)(6,7,8) (5,6,7) (4,5,6) (4,5,6)

A

⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎝ ⎠

Step 1 �;� 3TO���78,-��� �*+9:���"

(5.8,6.8,7.8) (6,7,8) (3.6,4.6,5.6) (3,4,5)(5.6,6.6,7.6) (5.8,6.8,7.8) (4.4,5.4,6.4) (5.2,6.2,7.2)(5.2,6.2,7.2) (6.2,7.2,8.2) (4.4,5.4,6.4) (4.4,5.4,6.4)(5.6,6.6,7.6) (6,7,8) (3.6,4.6,5.6) (4.4,5.4,6.4)

A

⎛ ⎞⎜⎜=⎜⎜⎝ ⎠

⎟⎟⎟⎟

Step 2 OP�(�Q���

(0.2971,0.5225,0.7992) (0.3647,0.5318,0.7293)(0.2869,0.5072,0.7787) (0.3525,0.5166,0.7111)(0.2664,0.4764,0.7377) (0.3768,0.5470,0.7475)(0.2869,0.5072,0.7787) (0.3647,0.5318,0.7293)(0.2872,0.458

A

⎛⎜⎜=⎜⎜⎝

8,0.6702) (0.1939,0.3347,0.5386)(0.3511,0.5386,0.7660) (0.3361,0.5343,0.7756)(0.3511,0.5386,0.7660) (0.2844,0.4654,0.6894)(0.2872,0.4588,0.6702) (0.2844,0.4654,0.6894)

⎞⎟⎟⎟⎟⎠

Step 3 R��&'��� ( )ij n mC c ×= ���

�������(IS'�&'���� ������(O�� (1,1,1), (1,1,1),(0,0,0)��HK� 1.2 O��> ( )ij n mC c ×= a���V&'��� ��

������������ ��������

, ,r lD D D+ + −�

0.3630 0.2898 0.3678 0.48480.3716 0.3014 0.3043 0.31970.3905 0.2786 0.3043 0.37020.3716 0.2898 0.3678 0.3702

rD D+ +

⎛ ⎞⎜ ⎟⎜ ⎟= =⎜ ⎟⎜ ⎟⎝ ⎠

0.4337 0.3685 0.3053 0.19570.4115 0.3498 0.4013 0.40980.3690 0.3878 0.4013 0.33240.4115 0.3685 0.3053 0.3324

lD−

⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎝ ⎠

Step 4 O��> , ,r lD D D+ + −a����F

1 2( , , , )nd d d d+ + + += �

1 2( , , , )r r r rnd d d d+ + + += �

1 2( , , , )l l l lnd d d d− − − −= ��"

T(1.5054,1.297,1.3436,1.3994)rd d+ += =T(1.3032,1.5724,1.4850,1.4077)ld

− =

Step 5 �> /( )i i i ri lic d d d d− − + += + + ( 1,2,3,4i = )��,c�%#�(Q�"

1 2 3 4( , , , )x x x xv v v v v= =

(0.2206,0.2756,0.2594,0.2444)D�,c�*+�"

2 0.0162 3 0.015 4 0.0238 1x x x x� � �&)'%> 2x .

6 !"#(Conclusions)

����"#$�������*+,-F�

����%&'()�*+,-�T(")D���

��%&'(),-�UV�<=��OPF>?A

B"C,-LDE7��.0*5W�����*+

,-�UV+IWOX,�Y���D#����,

-��W��� �D��#���.

$%&'(References)

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[2] Chang D Y. Applications of the extent analysis method onfuzzy AHP. European Journal of Operational Research, 1996,95(3): 649-655.

[3] ��� . �W��-�����*+ , ./����,2004, 39(2): 30-36.

[4] Mohammad Modarres, Soheil S adi-Nezhad. Ranking fuzzynumbers by preference ratio. Fuzzy Sets and Systems, 2001,118(3): 429-436.

[5] Chen Chen-Tung. Extensions of the TOPSIS for group deci-sion-making under fuzzy environment. Fuzzy Sets and Sys-tems, 2000, 14(1): 1-9.

[6] Zhang Jijun, Wu Desheng, Olson D L. The method of greyrelated analysis to multiple attribute decision making prob-lems with interval numbers. Mathematics and Compu-terModeling, 2005, 42(9/10): 991-998.

[7] Cheng Chibin. Group opinion aggregation based on a grad-ing process: A method for constructing triangular fuzzynumbers. Computers & Mathematics, 2004, 48(10/11):1619-1632.

[8] ��*, ���. �������� ��*+�#$GH,-. Z[XY, 1996, 20(2): 89-92.

[9] !��. ����������*+�#$*+,-.��Z[���, 2002, 16(1): 47-50.

[10] Facchinetti G, Ricci R G, Muzzioli S. Note on ranking fuzzytriangular numbers. International Journal of Intelligent Sys-tems, 1998, 3: 613-622.

[11] ��. ��% �()�+�VH. ��: ���\,

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411

2002.[12] Wang Wenjune, Chiu Chihhui. The entropy of fuzzy num-

bers with arithmetic operations. Fuzzy Sets and Systems,2000, 111(3): 357-366.

[13] Zhu Kejun, Yu Jing, Chang Dayong. A discussion on extentanalysis method and applications of fuzzy AHP. EuropeanJournal of Operational Research, 1999, 169(2): 450-456.

[14] Moon Joohyun, Kang Changsun. Use of fuzzy set theory inthe aggregation ofexpert judgements. Annals of Nuclear En-

ergy, 1999, 26(6): 461-469.[15] Cengiz Kahraman, Da Ruan. Ibrahim Doan, Fuzzy group

decision-making for facility location Selection. InformationSciences, 2003, 57: 135-153.

[16] Irion Albrecht. Fuzzy rules and fuzzy functions: A combina-tion of logic and arithmetic operations for fuzzy numbers.Fuzzy Sets and Systems, 1998, 99(1): 49-56.