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Proceedings of the 26th Chinese Control ConferenceJuly 26-31, 2007, Zhangjiajie, Hunan, China
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���� ����������� � 530004Email: [email protected]
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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)
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2 �� (Preliminaries)
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3 ����(The Method of Decision-Making)
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− = �"#!S���������
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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 ×= ����
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��������� �������� , ,r lD D D+ + −�
Step 4 O��> , ,r lD D D+ + −a����F��
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1
m
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=
= ∑
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1
m
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=
= ∑
T1 2( , , , )l l l lnd d d d− − − −= � ��a�
1
m
li lijj
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=
= ∑
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()
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= ≠
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= =
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= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦∑ ∑ ∑
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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��#���.
$%&'(References)
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[8] ��*, ���. �������� ��*+�#$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. ��: ���\,
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-
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