Renato A. Krohling Department of Production Engineering & Graduate Program in Computer Science, PPGI UFES - Federal University of Espírito Santo

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1 Interval-Valued Intuitionistic Fuzzy TODIM Renato A. Krohling Departent of Production Engineering & Graduate Progra in Coputer Science, PPGI FES - Federal niversity of Espírito Santo Vitória ES - Brazil André G. C. Pacheco Departent of Coputer Science, FES Vitória ES - Brazil

2 Suary 1. Interval-Valued Intuitionistic Fuzzy 2. Interval-Valued Intuitionistic Fuzzy Multi-criteria Decision Making 3. Interval-Valued Intuitionistic Fuzzy TODIM 4. Siulation Results 5. Conclusions

3 1. Interval-Valued Intuitionistic Fuzzy LetX bea non-eptyuniverseofdiscourse, then an interval-valued intuitionistic fuzzy set (IVIFS) over X is defined by: { µ ν } = x, ( x ), ( x ) x X, µ : X [0, 1] : X [0, 1] ν The nubers µ ( x) and ν ( ) stands for the degree of ebership and x non-ebership of xin, respectively, with the conditions: 0 µ ( x) + ν ( x) 1 x X. Each x X, µ ( x) and ν ( x) are closed intervals and their lower and upper bounds are denoted by µ L ( x), µ ( x), ν L ( x), ν ( x) { [ µ ] [ ] } L µ ν L ν Therefore = x, ( x), ( x), ( x), ( x) x X, Let two IVIFN ã = ([ a1, a2],[ a3, a4]) and b ɶ = ([0.2,0.5],[0.3,0.4]), then the distance between the is calculated by 1 d( aɶ, bɶ ) = [ a ] 1/ 2 1 b1 + a2 b2 + a3 b3 + a4 b4 4

4 2. Interval-ValueIntuitionistic Fuzzy Multi-criteria Decision Making Let us consider the fuzzy decision atrix A, which consists of alternatives and criteria, described by: C1... Cn A1 xɶ 11 xɶ 1n A =... A x1 x ɶ ɶn Where A, A,, A are alternatives, C, C,..., C the values xɶ are n interval-valued intuitionistic fuzzy nubers that indicates the rating of the alternative A with respect to criterion C i ( ) The weight vector W = w, w..., w coposed of the individual 1 2 n weights for each criterion satisfying: n j = 1 w j = 1. j

5 3. The TODIM ethod Step 1: Noralization of the decision atrix Step 2: Calculate the doinance aong alternatives where i j c i j c= 1 δ( R, R ) = φ ( R, R ) ( i, j) w ( r r ) rc ic jc if ( r > r ) ic jc w c= 1 rc φ ( R, R ) = 0, if ( ) c i j r = r ic jc -1 ( w )( r r ) c= 1 rc ic jc if ( r < r ) ic jc θ wrc Step 3: Calculate the final value ξ i δ ( i, j) in δ ( i, j) = ax δ ( i, j) in δ ( i, j).

6 3. Interval-Valued Intuitionistic Fuzzy TODIM The interval-valued intuitionistic fuzzy TODIM is described in the following steps: 1) Noralize the interval-valued intuitionisticfuzzy decision atrix with A ɶ = x ɶ with xɶ = L,, L, into the interval-valued intuitionisticfuzzy xn a a b b decision atrix ɶ = ɶ with r ɶ = L L,,, µ µ ν ν R r xn using the following expressions: L a a L µ = and µ = with i = 1,..., ; j = 1,... n, L 2 2 L 2 2 ( k = 1( ( a ) + ( a ) )) k 1( ( a ) ( a ) kj kj = + kj kj ) ( ) L ν L b b = and ν = with i = 1,..., ; j = 1,... n, L 2 2 L 2 2 ( k = 1( ( b ) + ( b ) )) k 1( ( b ) ( b ) kj kj = + kj kj ) ( )

7 3. Interval-Valued Intuitionistic Fuzzy TODIM 2) Calculate the doinance of each alternative over Rɶi each alternative Rɶ j using the following expression: i j c i j c= 1 δ( Rɶ, Rɶ ) = φ ( Rɶ, Rɶ ) ( i, j) where: wrc d( rɶ, rɶ ) if ( rɶ > rɶ ) ic jc ic jc w c= 1 rc φ ( Rɶ, Rɶ ) = 0, if ( ) c i j rɶ = rɶ ic jc -1 ( w ) c= 1 rc d( rɶ, rɶ ) if ( rɶ < rɶ ) ic jc ic jc θ wrc 3) Calculate the global value of the alternative iby ξ i δ ( i, j) in δ ( i, j) = ax δ ( i, j) in δ ( i, j)

8 4. Siulation Results The decision aking proble investigated by Nayaga, Muralikrishnan, and Sivaraan[10] is used as benchark. There are four alternatives to invest the oney: A1 is a car copany, A2is a food copany, A3is a coputer copany, and A4 is an ars copany The alternatives are evaluated according to three criteria: C1is the risk analysis, C2is the growth analysis, and C3is the environental ipact analysis. The weight vector associated to each criterion is W = ( w, w, w, w ) = (0.35, 0.25, 0.3, 0.40) The factor of attenuation of losses, was set to value has also been used. θ = 2.5 θ θ = 1 but the

9 4. Siulation Results Interval-valued intuitionistic fuzzy decision atrix ([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: A A A A is the sae as copared with that reported by Nayaga, Muralikrishnan, and Sivaraan [10]

10 5. Conclusions The interval-valued intuitionistic fuzzy TODIM ethod presented is able to tackle MCDM probles affected by uncertainty represented by interval-valued intuitionistic fuzzy nubers Interval-valued intuitionistic fuzzy nubers is a uch ore natural way to describe rating of the alternatives The IVIF-TODIM ethod has been investigated on two exaples. In both cases, siulation results deonstrate the effectiveness of the presented ethod Applications of the proposed ethod to other challenging MCDM probles are under investigation

11 Zadeh, LA. Fuzzy sets, Inforation and Control 1965, 8: Atanassov KT. Intuitionistic fuzzy sets, Fuzzy Sets and Systes 1986, 20: Atanassov KT, Gargov G. Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systes 1989, 31: Nayaga VLG, Muralikrishnan S, Sivaraan G. Multi-criteria decision aking based on interval-valued intuitionistic fuzzy sets. Expert Systes with Applications 2011, 38: XuZ. Soe siilarity easures of intuitionisticfuzzy sets and their applications to ultiple attribute decision aking, Fuzzy Optiization and Decision Making2007, 6: Goes LFAM, Lia MMPP. TODIM: Basics and application to ulticriteriaranking of projects with environental ipacts, Foundations of Coputing and Decision Sciences1992,16: Krohling RA, de Souza TTM. Cobining prospect theory and fuzzy nubers to ulti-criteria decision aking, Expert Systes with Applications 2012, 39: Krohling RA, Pacheco AGC, Siviero ALT. IF-TODIM: An intuitionisticfuzzy TODIM to ulti-criteria decision aking. Knowledge-Based Systes 2013, 53: LourenzuttiR, KrohlingRA. A Study of TODIM in a intuitionisticfuzzy and rando environent, Expert Systes with Applications 2013, 40: Coplete list of references cited in the paper

12 Thank you for your attention Contact: Acknowledgeents: Prof. Dr. L.F.A.M. Goes the developer of TODIM ethod for his availability to present this paper R.A. Krohlingwould like to thank the financial support of the Brazilian Research agency CNPq