Interval Interval-Valued Intuitionistic Fuzzy Valued Intuitionistic Fuzzy TODIM TODIM Renato A. Krohling Department of Production Engineering & Graduate Program in Computer Science, PPGI UFES - Federal University of Espírito Santo Vitória – ES - Brazil Vitória – ES - Brazil André G. C. Pacheco Department of Computer Science, UFES Vitória – ES - Brazil
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IntervalInterval--Valued Intuitionistic Fuzzy Valued ... · IntervalInterval--Valued Intuitionistic Fuzzy Valued Intuitionistic Fuzzy TODIM Renato A. Krohling Department of Production
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1. Interval-Valued Intuitionistic Fuzzy
{ }, ( ), ( ) ,Ã ÃÃ x x x x Xµ ν= ∈
• Let X be a non-empty universe of discourse, then an interval-valued intuitionistic fuzzy set (IVIFS) Ã over X is defined by:
: [0, 1]Ã Xµ → : [0, 1]Ã Xν →
• The numbers and stands for the degree of membership and non-membership of x in Ã, respectively, with the conditions:
( )Ã
xµ ( )Ã
xν
non-membership of x in Ã, respectively, with the conditions:
0 ( ) ( ) 1 .Ã Ãx x x Xµ ν≤ + ≤ ∀ ∈
Each are closed intervals and their lower and upper bounds are denoted by
, ( ) and ( )Ã Ã
x X x xµ ν∈
[ ] [ ]{ }Therefore , ( ), ( ) , ( ), ( ) ,L U L U
à à à Ãà x x x x x x Xµ µ ν ν→ = ∈
( ), ( ), ( ), ( )L U L U
à à à Ãx x x xµ µ ν ν
[ ]1/ 21 1 2 2 3 3 4 4
1( , )
4d a b a b a b a b a b= − + − + − + −ɶɶ
Let two IVIFN and
then the distance between them is calculated by1 2 3 4([ , ],[ , ])ã a a a a= ([0.2,0.5],[0.3,0.4]),b =ɶ
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2. Interval-Value Intuitionistic Fuzzy Multi-criteria Decision Making
• Let us consider the fuzzy decision matrix , which consists of alternatives and criteria, described by:
1
...
A
A
A
=
1
11 1
... n
n
C C
x x
x x
ɶ ɶ…
⋮ ⋱ ⋮
ɶ ɶ⋯
A
mA 1m mnx x ɶ ɶ⋯
• Where are alternatives, the values are interval-valued intuitionistic fuzzy numbers that indicates the rating of the alternative with respect to criterion
1 2, , ,
mA A A⋯
1 2, ,...,
nC C C
ijxɶ
iA
jC
• The weight vector composed of the individual weights for each criterion satisfying:
( )1 2, ...,
nW w w w=
1
1.n
jj
w=
=∑
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3. The TODIM method
.Step 2: Calculate the dominance among alternatives
Step 1: Normalization of the decision matrix
( )w r r −
1
( , ) ( , ) ( , )m
i j c i jc
R R R R i jδ φ=
= ∀∑
where
Step 3: Calculate the final value
( )
1
1
( ) if ( )
( , ) 0, if ( )
( )-1 if ( )
rc ic jc
m ic jc
rcc
c i j ic jc
m
rc ic jcc
ic jc
rc
w r rr r
w
R R r r
w r rr r
w
φ
θ
=
=
− >
= =
−<
∑
∑
( , ) min ( , )
max ( , ) min ( , )i
i j i j
i j i j
δ δξ
δ δ
−=
−
∑ ∑∑ ∑
where
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3. Interval-Valued Intuitionistic Fuzzy TODIM
• The interval-valued intuitionistic fuzzy TODIM is described in the following steps:
1) Normalize the interval-valued intuitionistic fuzzy decision matrix with with into the interval-valued intuitionistic fuzzy decision matrix with
ij mxn
A x= ɶ ɶ , , ,
L U L U
ij ij ij ij ijx a a b b= ɶ
ij mxn
R r= ɶ ɶ
, , , L U L U
ij ij ij ij ijr µ µ ν ν= ɶ using the following expressions:, , ,
ij ij ij ij ijr µ µ ν ν= ɶ
( )( ) ( )( )1 1
2 22 2 2 2
1 1
and with 1,..., ; 1, ... ,
( ) ( ) ( ) ( )
L U
L Uij ij
ij ijL U L Um m
k kkj kj kj kj
a ai m j n
a a a aµ µ
= =∑ ∑= = = =
+ +
( )( ) ( )( )1 1
2 22 2 2 2
1 1
and with 1,..., ; 1, ... ,
( ) ( ) ( ) ( )
L U
L Uij ij
ij ijL U L Um m
k kkj kj kj kj
b bi m j n
b b b bν ν
= =∑ ∑= = = =
+ +
using the following expressions:
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3. Interval-Valued Intuitionistic Fuzzy TODIM
2) Calculate the dominance of each alternative over each alternative using the following expression:
iRɶ
jRɶ
1
( , ) ( , ) ( , )m
i j c i jc
R R R R i jδ φ=
= ∀∑ɶ ɶ ɶ ɶ
where:
( , ) if ( )rc
m ic jc ic jc
rc
wd r r r r
w
⋅ >∑
ɶ ɶ ɶ ɶ
( )
1
1
( , ) 0, if ( )
-1 ( , ) if ( )
rcc
c i j ic jc
m
rcc
ic jc ic jc
rc
w
R R r r
wd r r r r
w
φ
θ
=
=
= =
⋅ <
∑
∑
ɶ ɶ ɶ ɶ
ɶ ɶ ɶ ɶ
3) Calculate the global value of the alternative i by
( , ) min ( , )
max ( , ) min ( , )i
i j i j
i j i j
δ δξ
δ δ
−=
−
∑ ∑∑ ∑
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4. Simulation Results
� The decision making problem investigated by Nayagam, Muralikrishnan, and Sivaraman [10] is used as benchmark.
� There are four alternatives to invest the money: A1 is a car company, A2 is a food company, A3 is a computer company, and A4 is an arms company
� The alternatives are evaluated according to three criteria: C1 is � The alternatives are evaluated according to three criteria: C1 is the risk analysis, C2 is the growth analysis, and C3 is the environmental impact analysis.
� The factor of attenuation of losses was set to but the value has also been used.
1 2 3 4( , , , ) (0.35, 0.25, 0.3, 0.40)W w w w w= =
� The weight vector associated to each criterion is
,θ 1θ =
2.5θ =
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� is the same as compared with that reported by Nayagam, Muralikrishnan, and Sivaraman[10]
2 4 3 1A A A A≻ ≻ ≻
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5. Conclusions
� The interval-valued intuitionistic fuzzy TODIM method presented is able to tackle MCDM problems affected by uncertainty represented by interval-valued intuitionistic fuzzy numbers
� Interval-valued intuitionistic fuzzy numbers is a much more natural way to describe rating of the alternatives
� The IVIF-TODIM method has been investigated on two examples. In both cases, simulation results demonstrate the effectiveness of the presented method
natural way to describe rating of the alternatives
� Applications of the proposed method to other challenging MCDM problems are under investigation
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Lourenzutti R, Krohling RA. A Study of TODIM in a intuitionistic fuzzy and random environment, Expert Systems with Applications 2013, 40:6459-6468..Complete list of references cited in the paper
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