IntervalInterval--Valued Intuitionistic Fuzzy Valued ... · IntervalInterval--Valued Intuitionistic Fuzzy Valued Intuitionistic Fuzzy TODIM Renato A. Krohling Department of Production

IntervalInterval--Valued Intuitionistic Fuzzy Valued Intuitionistic Fuzzy TODIMTODIM

Renato A. Krohling

Department of Production Engineering &Graduate Program in Computer Science, PPGI

UFES - Federal University of Espírito SantoVitória – ES - BrazilVitória – ES - Brazil

André G. C. Pacheco

Department of Computer Science, UFESVitória – ES - Brazil

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SummarySummary

1. Interval-Valued IIntuitionistic Fuzzy

2. Interval-Valued Intuitionistic Fuzzy Multi-criteria

Decision Making

3. Interval-Valued Intuitionistic Fuzzy TODIM

4. Simulation Results

5. Conclusions

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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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4. Simulation Results

� Interval-valued intuitionistic fuzzy decision matrix

([0.4,0.5],[0.3,0.4]) ([0.4,0.6],[0.2,0.4]) ([0.1,0.3],[0.5,0.6])

([0.6,0.7],[0.2,0.3]) ([0.6,0.7],[0.2,0.3]) ([0.4,0.8],[0.1,0.2])

([0.3,0.6],[0.3,0.4]) ([0.5,0.6],[0.3,0.4]) ([0.4,0.5],[0.1,0.3])

([0.7,0.8],[0.1,0.2]) ([0.6,0.7],[0.1,0.3]) ([0.3,0.4],[0.1,0.2])

� Ranking of the alternatives

� The order of the alternatives obtained is:

� 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

Zadeh, LA. Fuzzy sets, Information and Control 1965, 8:338-353.

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Atanassov KT, Gargov G. Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 1989, 31: 343-349

Nayagam VLG, Muralikrishnan S, Sivaraman G. Multi-criteria decision making based on interval-valued intuitionistic fuzzy sets. Expert Systems with Applications 2011, 38:1464-1467.

Xu Z. Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making, Fuzzy Optimization and Decision Making 2007, 6:109-121.

Gomes LFAM, Lima MMPP. TODIM: Basics and application to multicriteria ranking of projects with environmental impacts, Foundations of Computing and Decision Sciences 1992, 16:113-127.

Krohling RA, de Souza TTM. Combining prospect theory and fuzzy numbers to multi- criteria decision making, Expert Systems with Applications 2012, 39:11487-11493.

Krohling RA, Pacheco AGC, Siviero ALT. IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making. Knowledge-Based Systems 2013, 53: 142-146.

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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Thank you for your attention

Contact:

krohling.renato@gmail.com

pacheco.comp@gmail.com

Acknowledgements:

Prof. Dr. L.F.A.M. Gomes the developer of TODIM method

for his availability to present this paper

R.A. Krohling would like to thank the financial support of the Brazilian Research agency CNPq