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Procedía Computer Science 31 (2014) 236 - 244

Information Technology and Quantitative Management (ITQM2014) Interval-Valued Intuitionistic Fuzzy TODIM

Renato A. Krohlinga*, André G. C. Pachecob

aDepartment of Production Engineering & Graduate Program in Computer Science, PPGI, UFES - Federal University of Espirito Santo,

Av. Fernando Ferrari, 514, CEP 29075-910, Vitória, Espírito Santo, Brazil ^Department of Informatics, UFES - Federal University of Espirito Santo, Av. Fernando Ferrari, 514, CEP 29075-910, Vitória, Espírito

Santo, Brazil

Abstract

The TODIM (an acronym in Portuguese for Interative Multi-criteria Decision Making) method has recently been extended firstly to fuzzy and next to intuitionistic fuzzy environments with promising results. In this paper, we further consider the extension of TODIM to interval-valued intuitionistic fuzzy (IVIF) environments. Two case studies are used to illustrate the approach to multi-criteria decision making. Experimental results show the effectiveness of the presented method.

© 2014 Publishedby ElsevierB.V.Thisis anopen access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of the Organizing Committee of ITQM 2014.

Keywords: Multi-criteria decision making (MCDM), interval-valued intuitionistic fuzzy numbers, IVIF-TODIM, TODIM.

1. Introduction

TOP SIS (The Technique for Order Preference by Similarity to Ideal Solution) developed by Hwang and Yoon1 is based on a distance metric, i.e., the Euclidean distance of each alternative to positive ideal solution (PIS) and to negative ideal solution (NIS). Another interesting technique in the context of multi-criteria decision making (MCDM) is known as TODIM (an acronym in Portuguese for Interative Multi-criteria Decision Making) proposed by Gomes and Lima2 is based on dominance of an alternative i over alternative j. The standard TODIM as originally proposed2 is only applicable to crisp decision matrices.

The theory of fuzzy sets and fuzzy logic developed by Zadeh3 has been used to model vagueness, lack of knowledge and ambiguity inherent in the human decision making process. Bellman and Zadeh4 introduced the theory of fuzzy sets in multi-criteria decision making (MCDM) problems, which is known as fuzzy multi -criteria decision making (FMCDM). Krohling and de Souza5 proposed a fuzzy TODIM, where the partial dominance of an alternative i over an alternative j is calculated using the distance between fuzzy numbers. A clear advantage of the fuzzy TODIM is its ability to treat uncertain information using fuzzy numbers. Fan et al.6 have also presented an extension of the TODIM method, whereas the attribute values (crisp numbers, interval numbers, and fuzzy numbers) are expressed in the format of random variables with cumulative

* Corresponding author. Tel.: +55-27 4009-2649; fax: +55-27-4009 2649. E-mail address:krohling.renato@gmail.com.

1877-0509 © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of the Organizing Committee of ITQM 2014. doi: 10.1016/j.procs.2014.05.265

distribution functions, and so the standard TODIM with crisp decision matrix can be applied to solve such MCDM problems.

Atanasov7 proposed a more general theory for fuzzy sets, known as intuitionistic fuzzy sets (IFS), which is described by a real-valued membership function and a non-membership function. In some decision cases involving uncertainty and imprecise judgment, it may be difficult to define membership and non-membership functions by exact values. Later, Atanasov, and Gargov8 introduced the concept of interval-valued intuitionistic fuzzy sets (IVIFS) as a further generalization of IFS. The main characteristic of IVIFS is that the membership as well as the non-membership functions are intervals instead of exact values. In recent years, there is a growing research interest in IVIFS, which have been applied to solve MCDM problems 9 10, 11 12, 13, 14, 15, 16, 17, 18, 19, 20, 30 and also to group decision making 21, 22, 23, 24, 25, 26, 27, 28, 29. Recently, an intuitionistic fuzzy TODIM method, for short, IF-TODIM to handle uncertain MCDM problems has been developed31 as well as an IFR-TODIM 32 to intuitionistic fuzzy and random environments.

In various real-life situations ranking of alternatives is an important issue in decision making. In cases where the raw data are provided by interval-valued intuitionistic fuzzy numbers (IVIFN), Xu 15, Xu and Chen16 proposed score and accuracy functions to rank IVIFN. Xu15, Xu and Yager24 developed formulae to calculate distance between IVIFNs. Based on our previous works31, 32 and using the definitions of distance between IVIFNs, score and accuracy functions, and the order relation between IVIFNs, we present an interval-valued intuitionistic fuzzy TODIM, named IVIF-TODIM, for short, which is an interesting approach to handle uncertain MCDM problems.

The remainder of this paper is organized as follows. In section 2, some preliminary background on interval-valued intuitionistic fuzzy numbers is provided. In section 3, an interval-valued intuitionistic fuzzy TODIM method is developed. In section 4, two case studies are presented to illustrate the method and the results show the feasibility of the approach. In section 5, some conclusions and directions for future work are given.

2. Interval-valued Intuitionistic Fuzzy Multi-criteria Decision Making

Next, some basic definitions of intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets are

provided 7, 8.

Definition 1: Let X be a non-empty universe of discourse. An intuitionistic fuzzy set (IFS) A is characterized by a subset of X defined7 by A = {(x, ¡uA (x),^A (x)) |x e X}, where ¡j.a : X ^ [0, 1] and va : X ^ [0, 1] with the condition 0 < ha (x) + vA (x) < 1 |Vx e X .The numeric values ¡j.a (x) and va (x) stands for the degree of membership and non-membership of x in A, respectively. In addition, it is also defined the intuitionistic fuzzy index as ^a(x) = 1 ~Ma(x) ~va(x) known as the indeterminacy or hesitation degree of the IFS A in X. The condition 0 < &a(x) ^ 1 for each x e X must also be fulfilled.

Definition 2: LetXbe a non-empty universe of discourse, and D[0,1] be the subset of all closed subinterval of [0,1], then an interval-valued intuitionistic fuzzy set (IVIFS) A over X is defined7 by A ={(x, /¿a (x),^a (x))|x e X J, where /ja : X ^ [0, 1] and vA : X ^ [0, 1] with the condition

0 < ij.A (x) + va (x) < 1 |Vx e X. The intervals ^a (x) and va (x) stands for the degree of membership and non-

membership of x in A, respectively. In addition, it is also defined the interval-valued intuitionistic fuzzy index as TiA (x) = 1 - ha (x) -va (x) known as the indeterminacy or hesitation degree of the IVIFS A in X. The

condition 0 < na (x) < 1 for each x e X must also be fulfilled. For each x e X ^A (x) and va (x) are closed

intervals and their lower and upper bounds are denoted by ¡¿A (x), /JA (x),vA (*), ylj (*) • So, an alternative way

to express the IVIFS is A = ux,

j^x,(x), ¡JUÁ (x)],\v\(x), vUÁ(x)]^\x e xj, with juU(x) + vU (x) < 1

0 (x) <MUA (x) < 1, 0 <v} (x) <vUA (x) < 1.

Considering an IVIFS A and a given x, the pair juj (x),vj (x) is an interval-valued intuitionistic fUzzy number (IVIFN) a, for convenience of notation, denoted by a = ([^1,^4]), where [ai, a2] e [0,1], [a3,a4] g [0,1] and a2 + a4 < 117, 22.

An IVIFN may have a physical interpretation. For instance, if you consider a = ([0.5,0.6],[0.2,0.3]) = ([0.5,0.6],[1 - 0.3,1 - 0.2]) = ([0.5,0.6],[0.7,0.8]it may be interpreted as "the vote for resolution is between 5 and 6 in favor, between 2 and 3 against and between 7 and 8 abstentions9.

Definition 3: Let an interval-valued fuzzy number a = ([a1, a2],[a3, a4]), then its score value is calculated by26:

<?(a) = ai~a3+a2~a4 with 5(5) e [-1,1] (1)

Definition 4: Let an interval-valued fuzzy number a = ([a1, a2],[a3, a4]), then its accuracy value is calculated by26:

H(a) = -1+^2+^3+^4 with//(5)e[0,i] (2)

Next, we introduce a relation of order between IVIFNs.

Definition 5: Let two interval-valued intuitionistic fuzzy numbers a = ([ai,a2], [<23, <24]) and b=([^b2UbM) then26:

If S(a) > S(b), then a > b.

If S(a) = S(b) and If H(a) = H(b) then 5 = 6.

If S(a) = S(b) and H(a) > H(b) then a > b.

Definition 6: Let two interval-valued intuitionistic fuzzy numbers a = ([ai,a2], [<23, <24]) and b = ([bi,b2],[63,64]) then the distance between them is calculated by26:

d{a,b) = — [1^ - b^| + \a2 - b2\ + \a3 -b31 + \a4 -64|]1/2 such that 0 < d{a,b) < 1 (3)

For instance, consider a = ([0.7,0.8],[0.1,0.2]), and b = ([0.2,0.5], [0.3,0.4]), then the distance is d(a,b) = 0.273.

3. Decision making problem with uncertain decision matrix

Let us consider the interval-valued fuzzy decision matrix A, which consists of alternatives and criteria, described by:

A= ...

where Ai,A2,---,Am are alternatives, С\,С2,...,Сп are criteria, the values Xy are interval-valued intuitionistic fuzzy numbers that indicates the rating of the alternative At with respect to criterionCj.The weight vector W = {w1,w2...,wn) composed of the individual weights Wj(j = 1,...,n) for each criterion Cj satisfying

I ^j=1.

For information on the TODIM method the reader is referred to Gomes and Lima2. The interval-valued intuitionistic fuzzy TODIM method, named IVIF-TODIM, which is an extension of the intuitionistic fuzzy TODIM method31, 32 is described in the following steps:

Step 1: Normalize the interval-valued intuitionistic fuzzy decision matrix A= ] where

L J -imxn

, into the interval-valued intuitionistic fuzzy decision matrix R= [^-J where ry=\juy, ju^, ^vjj, v^J with/ = l,...,m, and j = l,...,n using the following expressions 33:

z:, ((4 )2+(^ )2)

and jlE = -

z:, ((4 )2 - (4 )2)

— with i = j = 1, ...я,

vjj =-j-f and vH =-j-f with i = 1,...,m; j = 1, ...w, (5)

(I:, k )2 - (b )2 ))5 (I 1, ((^ )2+«g )2

Step 2: Calculate the dominance of each alternative Rt over each alternative Rj using the following expression:

S(Rt,Rj) = ^c(Rt,Rj) V(/J)

where:

d(riC9rJC)

S m=1 wrc )

if (ric > rjc)

if(fic =ïjc)

d{rlcJjc) if (ñc < ?ic)

ic JC)

The term ^{R^Rj), denoted by partial dominance, represents the contribution of the criterion c to the function S(Rj,Rj) when comparing the alternative i with alternative j. The values ric and rJC are the rating of the alternatives i and j, respectively with respect to criterion c. The value wrc represents the weight of criterion c divided by the weight of the reference r, i.e., wrc =wc/ wr, whereas the latter is the criterion that has the greater weight. The term d(ric,rjc) stands for the distance between the two interval-valued intuitionistic fuzzy numbers ric and

calculated by Eq. (3). Three cases can occur in Eq. (7): i) if

(ric > ?jC), it represents a gain; ii) if (ric = rJC), it is nil; and iii) if (ric < fJC), it represent a loss. Definition

(5) is used in each case. The parameter 6 represents the attenuation factor of the losses. The final matrix of dominance is obtained by summing up the partial matrices of dominance for each criterion.

Step 3: Calculate the global value of the alternative i by normalizing the final matrix of dominance according to the following expression:

j) - min , j) max ^ S (i, j ) - min ^ S (i, j )

Sorting the values ^ provides the rank of each alternative. The best alternatives are those that have higher value ^.

4. Simulation results

Next we present two examples to illustrate the method.

Example 1. The decision making problem investigated by Nayagam, Muralikrishnan, and Sivaraman10 is used as benchmark. There are four alternatives to invest the money: A, 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: C, is the risk analysis, C2 is the growth analysis, and C3 is the environmental impact analysis. The weight vector associated to each criterion is W = (w,, w2, W3, w4)= (0.35, 0.25, 0.3, 0.40). The interval-valued intuitionistic fuzzy decision matrix is listed in Table l.The factor of attenuation of losses 6, was set to 6 = 1 but the value 6 = 2.5 has also been used. The IVIF-TODIM method was applied to the decision matrix given in Table 1. The ranking of the alternatives using 6 = 1 and 6 = 2.5 is listed in Table 2 and depicted in Fig. 1. The order of the alternatives obtained is A2 > A^ >- A3 > which is the same as compared with that reported by Nayagam, Muralikrishnan, and Sivaraman 10.

Table 1. Interval-valued intuitionistic fuzzy decision matrix10.

C1 C2 C3

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

A2 ([0.6,0.7],[0.2,0.3]) ([0.6,0.7],[0.2,0.3]) ([0.4,0.8],[0.1,0.2])

A3 ([0.3,0.6],[0.3,0.4]) ([0.5,0.6],[0.3,0.4]) ([0.4,0.5],[0.1,0.3])

A4 ([0.7,0.8],[0.1,0.2]) ([0.6,0.7],[0.1,0.3]) ([0.3,0.4],[0.1,0.2])

Table 2. Ranking of the alternatives using 0 = 1 and 0 = 2.5.

Alternative IVIF-TODIM (0 = 1) IVIF-TODIM (0 = 2.5)

A1 0.0000 0.0000

A2 1.0000 1.0000

A3 0.4189 0.3993

A4 0.9702 0.9683

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

i H 0 = \ ^m 0=2.5

Fig. 1. Ranking of the alternatives using 9 = 1 and 9 = 2.5.

Example 2. The decision making problem investigated by Wei and Merigo14 is used as the second benchmark. There are five alternatives, which are evaluated according to five criteria. The weight vector associated to each criterion is W = (w\, w2, w3, w4, w^)= (0.20, 0.25, 0.15, 0.30, 0.10). The interval-valued intuitionistic fuzzy decision matrix is listed in Table 3.

Similar to the previous example, the experiment was carried out using 6 = 1 and 6 = 2.5. The IVIF-TODIM method was applied to the decision matrix given in Table 3. The ranking of the alternatives using 6 = 1 and 6 = 2.5 is listed in Table 4 and depicted in Fig. 2. The order of the alternatives obtained is A2 >- A^ >- A5 >- A3 yA1. It is noticed from the results that the ranking of the alternatives provided by IVIF-TODIM is the same as compared with that reported by Wei and Merigo14.

Table 3. Interval-valued intuitionistic fuzzy decision matrix14.

C1 C2 C3 C4 C5

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

A2 ([0.6,0.8],[0.1,0.2]) ([0.6,0.7],[0.2,0.3]) ([0.7,0.8],[0.1,0.2]) ([0.6,0.8],[0.1,0.2]) ([0.5,0.6 ],[0.3,0.4])

A3 ([0.4,0.5],[0.3,0.4]) ([0.3,0.4],[0.2,0.4]) ([0.3,0.4],[0.4,0.5]) ([0.4,0.5],[0.1,0.3]) ([0.4,0.5],[0.3,0.4])

A4 ([0.4,0.6],[0.1,0.3]) ([0.5,0.7],[0.1,0.2]) ([0.5,0.6],[0.2,0.4]) ([0.4,0.5],[0.1,0.2]) ([0.5,0.6 ],[0.2,0.4])

A5 ([0.2,0.4],[0.3,0.4]) ([0.4,0.5],[0.1,0.2]) ([0.4,0.5],[0.3,0.5]) ([0.5,0.6],[0.1,0.2]) ([0.5,0.6 >],[0.1,0.2])

Tab le 4. Ranking of the alternatives using 0 = 1 and 0 = 2.5.

Alternative IVIF-TODIM (9 = 1) IVIF-TODIM (0 = 2.5)

A1 0.0000 0.0000

A2 1.0000 1.0000

A3 0.3442 0.3141

A4 0.8514 0.8151

A5 0.6334 0.6015

Fig. 2. Ranking of the alternatives using 0 = 1 and 6 = 2.5.

For both examples investigated we notice the effectiveness of the IVIF-TODIM for MCDM problems described by IVIF numbers.

5. Conclusions

In this paper, the interval-valued intuitionistic fuzzy TODIM method (IVIF-TODIM for short) has been proposed, which is able to tackle MCDM problems affected by uncertainty represented by interval-valued intuitionistic fuzzy numbers. Since in many MCDM problems may be difficult to describe the rating of the

alternatives by intuitionistic fuzzy numbers, this approach using interval-valued intuitionistic fuzzy numbers is a much more natural way. The IVIF-TODIM method has been investigated on two examples. In both cases, simulation results demonstrate the effectiveness of the presented method. Applications of the proposed method to other challenging MCDM problems are under investigation.

Acknowledgements

R.A. Krohling would like to thank the financial support of the Brazilian Research agency CNPq under Grant No. 303577/2012-6.

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