the generalized dice measures for multiple attribute decision making
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The generalized Dice measures for multiple attribute decision making
under simplified neutrosophic environments
Abstract
Keywords:
1. Introduction
E-mail address:
2. The Dice measures of SNSs
Definition 1 S X
XxxvxuxtxS SSS tS x X uS x X vS x X
x S tS x uS x vS x x X
x tS x uS x vS x S sx tx ux vx
tx ux vx tx ux vx
Ux
Lxx ttt U
xLxx uuu
Ux
Lxx vvv U
xUx
Ux vut
Definition 2 S s s s n S s s s n s j t j, u j, v j
s j t j, u j, v j j n j S S
S S
n
j jjjjjj
jjjjjjn
j jj
jjSVNN vutvut
vvuutt
nss
ss
nSSD
s j t j, u j, v j s j t j, u j, v j j n j S S
S S
n
j
Uj
Uj
Uj
Lj
Lj
Lj
Uj
Uj
Uj
Lj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
n
j jj
jjINN
vutvut
vutvut
vvvvuuuutttt
n
ss
ss
nSSD
DSVNN S S DINN S S
DSVNN S S DSVNN S S DINN S S DINN S S
DSVNN S S DINN S S
DSVNN S S DINN S S S S
Uij
Lijij ttt U
ijLijij uuu U
ijLijij vvv i j n
sij i j
n W w w wn sij i j n wj
n
j jw
n
j jjjjjj
jjjjjjj
n
j jj
jjjWSVNN vutvut
vvuuttw
ss
sswSSD
n
j
Uj
Uj
Uj
Lj
Lj
Lj
Uj
Uj
Uj
Lj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
j
n
j jj
jjjWINN
vutvut
vutvut
vvvvuuuuttttw
ss
sswSSD
3. Another form of the Dice measures of SNSs
Definition 3 S s s s n S s s s n s j t j, u j, v j s j
t j, u j, v j j n j S S
S S
n
j jjj
n
j jjj
n
j jjjjjj
SVNNvutvut
vvuutt
SS
SSSSD
s j t j, u j, v j s j t j, u j, v j j n j S S
S S
n
j
Uj
Lj
Uj
Lj
Uj
Lj
n
j
Uj
Lj
Uj
Lj
Uj
Lj
n
j
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
INN
vvuutt
vvuutt
vvvvuuuutttt
SS
SSSSD
DSVNN S S DINN S S
DSVNN S S DSVNN S S DINN S S DINN S S
DSVNN S S DINN S S
DSVNN S S DINN S S S S
Proof:
abba
S S s j s j j n |S | |S | DSVNN S S
DINN S S
sij i j n W
w w wn sij i j n wjn
j jw
n
j jjjj
n
j jjjj
n
j jjjjjjj
ww
wWSVNN
vutwvutw
vvuuttw
SS
SSSSD
n
j
Uj
Lj
Uj
Lj
Uj
Ljj
n
j
Uj
Lj
Uj
Lj
Uj
Ljj
n
j
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Ljj
ww
wWINN
vvuuttw
vvuuttw
vvvvuuuuttttw
SS
SSSSD
4. The generalized Dice measures of SNSs
Definition 4. S s s s n S s s s n s j t j u j v j
s j t j u j v j j n j S S
S S
n
j jjjjjj
jjjjjj
n
j jj
jjSVNN
vutvut
vvuutt
n
ss
ss
nSSG
n
j jjj
n
j jjj
n
j jjjjjj
SVNN
vutvut
vvuutt
SS
SSSSG
n
j jjj
jjjjjjn
j j
jjSVNN vut
vvuutt
ns
ss
nSSG
n
j jjj
jjjjjjn
j j
jjSVNN vut
vvuutt
ns
ss
nSSG
n
j jjj
n
j jjjjjj
SVNNvut
vvuutt
S
SSSSG
n
j jjj
n
j jjjjjj
SVNNvut
vvuutt
S
SSSSG
sij i j n
W w w wn sij i j n wjn
j jw
n
j jjjjjj
jjjjjjj
n
j jj
jjjWSVNN
vutvut
vvuuttw
ss
sswSSG
n
j jjjj
n
j jjjj
n
j jjjjjjj
ww
wWSVNN
vutwvutw
vvuuttw
SS
SSSSG
Definition 5 S s s s n S s s s n s j t j u j v j
s j t j u j v j j n j S S
S S
n
j
Uj
Uj
Uj
Lj
Lj
Lj
Uj
Uj
Uj
Lj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
n
j jj
jjINN
vutvut
vutvut
vvvvuuuutttt
n
ss
ss
nSSG
n
j
Uj
Lj
Uj
Lj
Uj
Lj
n
j
Uj
Lj
Uj
Lj
Uj
Lj
n
j
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
INN
vvuutt
vvuutt
vvvvuuuutttt
SS
SSSSG
Uij
Lijij ttt U
ijLijij uuu
Uij
Lijij vvv i j n
n
jU
jU
jU
jL
jL
jL
j
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
n
j j
jjINN
vutvut
vvvvuuuutttt
n
s
ss
nSSG
n
jU
jU
jU
jLj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
n
j j
jjINN
vutvut
vvvvuuuutttt
n
s
ss
nSSG
n
j
Uj
Lj
Uj
Lj
Uj
Lj
n
j
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
INN
vvuutt
vvvvuuuutttt
S
SSSSG
n
j
Uj
Lj
Uj
Lj
Uj
Lj
n
j
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
INN
vvuutt
vvvvuuuutttt
S
SSSSG
sij i j n
W w w wn sij i j n wj n
j jw
n
j
Uj
Uj
Uj
Lj
Lj
Lj
Uj
Uj
Uj
Lj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
j
n
j jj
jjjWINN
vutvut
vutvut
vvvvuuuuttttw
ss
sswSSG
n
j
Uj
Lj
Uj
Lj
Uj
Ljj
n
j
Uj
Lj
Uj
Lj
Uj
Ljj
n
j
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Lj
Uj
Uj
Lj
Ljj
ww
wWINN
vvuuttw
vvuuttw
vvvvuuuuttttw
SS
SSSSG
5. Decision making-methods based on the generalized Dice measures
S S S Sm R
R R Rn Rj j n wj wj
n
j jw Rj j n
Si i m iniii sssS sij tij uij vij
tij uij vijUij
Uij
Uij vut j n i m
D sij m n
iji
iji
iji
jjjj vutvuts
Uij
i
Lij
i
Uij
i
Lij
i
Uij
i
Lij
ijjjj vvuuttvuts j
n i m
nsssS jjjj vuts j
Si
i m S
Si i m
S Si
6. Decision-making example of manufacturing schemes
S= S S S S
R
R R R
W
Rj j Si i
D
iji
iji
iji
jjjj vutvuts j i
ssssS
Si i S
GWSVNN S S GWSVNN S S GWSVNN S S GWSVNN S S
S �S �S �S
S �S �S �S
S �S �S �S
S �S �S �S
S �S �S �S
GWSVNN S S GWSVNN S S GWSVNN S S GWSVNN S S
S �S �S �S
S �S �S �S
S �S �S �S
S �S �S �S
S �S �S �S
S S S
Si S S
Si S S
S Si S S
Rj j Si i
D
Uij
i
Lij
i
Uij
i
Lij
i
Uij
i
Lij
ijjjj vvuuttvuts
j i
nsssS
Si i S
GWINN S S GWINN S S GWINN S S GWINN S S
S �S �S �S
S �S �S �S
S �S �S �S
S �S �S �S
S �S �S �S
GWINN S S GWINN S S GWINN S S GWINN S S
S �S �S �S
S �S �S �S
S �S �S �S
S �S �S �S
S �S �S �S
S S S
Si S S
Si S S
S Si S
7. Conclusion
References
Applied Mathematical Modelling 37
Multispace
and Multistructure 4
Science World Journal 2014
Neural Computing and Applications
Applied Soft Computing 25
International Journal of Systems
Science
International Journal of General Systems 42
Journal of Intelligent and Fuzzy Systems 26
Journal of Intelligent and Fuzzy Systems 26
Journal of Intelligent and Fuzzy
Systems 27
International Journal of Fuzzy Systems 16
Applied
Mathematical Modelling 38
Journal of Intelligent and
Fuzzy Systems 28
Fuzzy Sets and Systems 20
Fuzzy Sets and Systems 31
Neutrosophic Sets and Systems 1
International Journal of Fuzzy
Systems 16
Neural Computing and Applications 25
Neural Computing and Applications
Journal of Intelligent
and Fuzzy Systems
Neural Computing and Applications
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