Interval-valued Intuitionistic Fuzzy TODIM

Procedia Computer Science 31 ( 2014 ) 236 – 244

1877-0509 © 2014 Published by Elsevier B.V. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of the Organizing Committee of ITQM 2014. doi: 10.1016/j.procs.2014.05.265

ScienceDirect

* Corresponding author. Tel.: +55-27 4009-2649; fax: +55-27-4009 2649. E-mail address:[email protected].

Information Technology and Quantitative Management (ITQM2014)

Interval-Valued Intuitionistic Fuzzy TODIM

Renato A. Krohling

a*, André G. C. Pachecob

a_{Department 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

b_{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.

© 2014The Authors. Published by Elsevier B.V.

Selection and/or peer-review under responsibility of the organizers of ITQM 2014

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

1.Introduction

TOPSIS (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 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

© 2014 Published by Elsevier B.V. Open access under CC BY-NC-ND license.

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 Chen}16

proposed score and accuracy functions to rank IVIFN. Xu15_{, Xu and Yager}24 _{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-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: LetXbe a non-empty universe of discourse. An intuitionistic fuzzy set (IFS) A is characterized by a subset of X defined7 _by

^

`

, A( ), A( ) ,

A x P x Q x xX where

P

_A:X o[0, 1] and

Q

_A:X o[0, 1] with the condition 0dP_A( )x Q_A( ) 1x d x X.The numeric values

P

_A( )x and

Q

_A( )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

S

_A( ) 1x

P

_A( )x

Q

_A( )x known as the indeterminacy or hesitation degree of the IFS A in X. The condition 0d

S

_A( ) 1x d for each xXmust also be fulfilled.

Definition 2: LetXbe a non-empty universe of discourse, and [0,1]D be the subset of all closed subinterval of [0,1], then an interval-valued intuitionistic fuzzy set (IVIFS) Ã over X is defined7 _by

^

, _Ã( ), _Ã( )

`

,

à xP x Q x x X where P_Ã:X o[0, 1] and Q_à :Xo[0, 1] with the condition 0dP_Ã( )x Q_Ã( ) 1 x d x X.The intervals P_Ã( )x and Q_Ã( )x stands for the degree of membership and non-membership of x in Ã, respectively. In addition, it is also defined the interval-valued intuitionistic fuzzy index as S_Ã( ) 1x P_Ã( )x Q_Ã( )x known as the indeterminacy or hesitation degree of the IVIFS à in X. The condition 0dS_Ã( ) 1x d for each xXmust also be fulfilled. For each xX P_Ã( )x and Q_Ã( )x are closed

intervals and their lower and upper bounds are denoted by PL_Ã( ),x PU_Ã( ),x Q_ÃL( ),x QU_Ã( )x . So, an alternative way to express the IVIFS is Ã

^

x,_¬ªP_ÃL( ), ( ) ,x PU_à x º ª_{¼ ¬}Q_ÃL( ), ( )x QU_à x _¼º x X

`

, with PU_Ã( )x QU_Ã( ) 1, x d

0dP_ÃL( )x dPU_Ã( ) 1,x d 0dQL_Ã( )x dQU_Ã( ) 1.x d

Considering an IVIFS à and a given x, the pair P_Ã( ),x Q_Ã( )x is an interval-valued intuitionistic fuzzy number (IVIFN) ã, for convenience of notation, denoted by ã ([ , ],[ , ]),a a1 2 a a3 4 where [ , ] [0,1]a a1 2  ,

3 4

[ ,a a ] [0,1] and a2a4d117, 22.

An IVIFN may have a physical interpretation. For instance, if you consider

([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]

ã 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 numberã ([ , ],[ , ])a a1 2 a a3 4 , then its score value is calculated by26:

1 3 2 4

( ) with ( ) [ 1,1] 2

a a a a

S a S a  (1)

Definition 4: Let an interval-valued fuzzy numberã ([ , ],[ , ])a a1 2 a a3 4 , then its accuracy value is calculated

by26_: 1 2 3 4 ( ) with ( ) [0,1] 2 a a a a H a H a  (2)

Next, we introduce a relation of order between IVIFNs.

Definition 5: Let two interval-valued intuitionistic fuzzy numbers ã ([ , ],[ , ])a a_{1 2} a a_{3 4} and

1 2 3 4 ([ , ],[ , ]) b b b b b then26_: If S a( )!S b( ), then a b! . If S a( ) S b( ) and If H a( ) H b( ) then a b. If S a( ) S b( ) and H a( )!H b( ) then a b! .

Definition 6: Let two interval-valued intuitionistic fuzzy numbers ã ([ , ],[ , ])a a1 2 a a3 4 and

1 2 3 4

([ , ],[ , ])

b b b b b then the distance between them is calculated by26_: 1/ 2 1 1 2 2 3 3 4 4 1 ( , ) such that 0 ( , ) 1 4 d a b ª_¬a b a b a b a b º_¼ dd a b d (3)

For instance, consider ã ([0.7,0.8],[0.1,0.2]), and b ([0.2,0.5],[0.3,0.4]), then the distance is

( , ) 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: 1 ... _m A A A 1 11 1 1 ... _n n m mn C C x x x x § · ¨ ¸ ¨ ¸ ¨ ¸ © ¹

where A A₁, ₂, ,A_m are alternatives,C C_{1 2}, ,...,_Cn are criteria, the values x_ij are interval-valued intuitionistic

fuzzy numbers that indicates the rating of the alternative Ai with respect to criterionCj.The weight vector

1, 2..., n

W w w w composed of the individual weights w jj( 1,..., )n for each criterion Cj satisfying

1 1. n j j w

¦

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 _{¬ ¼}ª ºx_{ij mxn} where , , ,

L U L U ij ij ij ij ij

x _¬ªa a _{¼ ¬}º ªb b º_¼ into the interval-valued intuitionistic fuzzy decision matrix R _{¬ ¼}ª ºr_{ij mxn} where , , ,

L U L U ij ij ij ij ij

r _¬ªP P _{¼ ¬}º ªQ Q _¼º with i 1,..., , and m j 1,...,n using the following expressions 33_:

2 2

12

2 2

12 1 1 and with 1,..., ; 1, ... , ( ) ( ) ( ) ( ) L U ij ij L U ij ij m _{L U} m _{L U} kj kj kj kj k k a a i m j n a a a a P P § · § · ¨ ¸ ¨ ¸ ©

¦

¹ ©

¦

¹ (4)

2 2

12

2 2

12 1 1 and with 1,..., ; 1, ... , ( ) ( ) ( ) ( ) L U ij ij L U ij ij m _{L U} m _{L U} kj kj kj kj k k b b i m j n b b b b Q Q § · § · ¨ ¸ ¨ ¸ ©

¦

¹ ©

¦

¹ (5)

Step 2: Calculate the dominance of each alternative R_i over each alternative R_j using the following expression: 1 ( ,_{i j}) m _c( ,_{i j}) ( , ) c R R R R i j G

¦

I (6)

where:

1 1 ( , ) if ( ) ( , ) 0, if ( ) -1 ( , ) if ( ) rc ic jc ic jc m rc c c i j ic jc m rc c ic jc ic jc rc w d r r r r w R R r r w d r r r r w I T ­ ° _{˜ !} ° ° ° ® ° ° ° _˜ ° ¯

¦

¦

(7)

The term

I

_c( ,R R_{i j}),denoted by partial dominance, represents the contribution of the criterion c to the function

G

( ,R R_{i j})when comparing the alternative i with alternative j. The values

r

_ic and

r

jc are the rating of

the alternatives

i

and

j

, respectively with respect to criterion

c

.

The value w_rc represents the weight of criterion

c

divided by the weight of the reference

r

,

i.e., wrc w wc/ r,whereas the latter is the criterion that

has the greater weight. The term d r r( ,ic jc) stands for the distance between the two interval-valued

intuitionistic fuzzy numbers

r

_ic and

r

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

(ric !rjc), it represents a gain; ii) if (ric rjc), it is nil; and iii) if (ricrjc), it represent a loss. Definition

(5) is used in each case. The parameter T 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:

( , ) min ( , ) max ( , ) min ( , ) i i j i j i j i j G G [ G G

¦

¦

¦

¦

(8)

Sorting the values

[

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

[

_i.

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: 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 risk analysis,C₂is the growth analysis, and C₃is the environmental impact analysis. The weight vector associated to each criterion is W ( ,w w w w1 2, 3, 4)= (0.35, 0.25, 0.3, 0.40). The

interval-valued intuitionistic fuzzy decision matrix is listed in Table 1.The factor of attenuation of losses T, was set to 1

T but the value T 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 T 1and T 2.5 is listed in Table 2 and depicted in Fig. 1. The order of the alternatives obtained is A₂ A₄ A₃ A₁,which is the same as compared with that reported by Nayagam, Muralikrishnan, and Sivaraman 10.

Table 1. Interval-valued intuitionistic fuzzy decision matrix10_. 1 C C₂ C₃ 1 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]) 2 A ([0.6,0.7],[0.2,0.3]) ([0.6,0.7],[0.2,0.3]) ([0.4,0.8],[0.1,0.2]) 3 A ([0.3,0.6],[0.3,0.4]) ([0.5,0.6],[0.3,0.4]) ([0.4,0.5],[0.1,0.3]) 4 A ([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 T 1and T 2.5.

Alternative IVIF-TODIM(T 1) IVIF-TODIM(T 2.5)

1 A 0.0000 0.0000 2 A 1.0000 1.0000 3 A 0.4189 0.3993 4 A 0.9702 0.9683

Fig. 1. Ranking of the alternatives using T 1and T 2.5.

Example 2. The decision making problem investigated by Wei and Merigó14 _{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 w w w w1 2, 3, 4, 5)= (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 T 1and T 2.5. The IVIF-TODIM method was applied to the decision matrix given in Table 3. The ranking of the alternatives using

1

T and T 2.5 is listed in Table 4 and depicted in Fig. 2. The order of the alternatives obtained is

2 4 5 3 1.

A A A A A 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 Merigó14_.

Table 3. Interval-valued intuitionistic fuzzy decision matrix14_. 1 C C₂ C₃ C₄ C₅ 1 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]) 2 A ([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]) 3 A ([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]) 4 A ([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]) 5 A ([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 T 1and T 2.5.

Alternative IVIF-TODIM (T 1) IVIF-TODIM (T 2.5) 1 A 0.0000 0.0000 2 A 1.0000 1.0000 3 A 0.3442 0.3141 4 A 0.8514 0.8151 5 A 0.6334 0.6015

Fig. 2. Ranking of the alternatives using T 1and T 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

.

References

1. Hwang CL, Yoon K. Multiple attribute decision making, Berlin: Springer-Verlag, 1981.

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

4. Bellman RE, Zadeh LA. Decision-making in a fuzzy environment, Management Science 1970, 17:141-164.

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

6. Fan Z-P, Zhang X, Chen F-D, Liu Y. Extended TODIM method for hybrid multiple attribute decision making problems, Knowledge-Based systems 2013, 42:40-48.

7. Atanassov KT. Intuitionistic fuzzy sets, Fuzzy Sets and Systems 1986, 20:87-96.

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

9. Chen S-M, Yang M-W, Yang S-W, Sheu T-W, Liau C-J. Multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets, Expert Systems with Applications 2012, 39:12085-12091.

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

11. Park JH, Park IY, Kwun YC, Tan X. Extension of the TOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environment, Applied Mathematical Modelling 2011, 35:2544-2556.

12. Wang Z, Li KW, Wang W. An approach to multiattribute decision making with interval-valued intuitionistic fuzzy assessments and incomplete weights, Information Sciences 2009, 179:3026-3040.

13. Wang Z, Li KW, Xu J. A mathematical programming approach to multiattribute decision making with interval-valued intuitionistic fuzzy assessment information, Expert Systems with Applications 2011, 38:12462-12469.

14. Wei GW, Merigó JM. Methods for strategic decision-making problems with immediate probabilities in intuitionistic fuzzy setting,

Scientia Iranica 2012, 19: 1936-1946.

15. 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.

16. Xu Z, Chen J. An overview of distance and similarity measures of intuitionistic fuzzy sets. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 2008, 16:529-555.

17. Xu Z, Yager RR. Dynamic intuitionistic fuzzy multi-attribute making, International Journal of Approximate Reasoning 2008, 48 :246-262.

18. Ye F. An extended TOPSIS method with interval-valued intuitionistic fuzzy numbers for virtual enterprise partner selection, Expert Systems with Applications 2010, 37: 7050-7055.

19. Ye F. Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment, Expert Systems with Applications 2008, 35: 322-326.

20. Ye F. Fuzzy cross entropy of interval-valued intuitionistic fuzzy sets and its optimal decision-making method based on the weights of alternatives, Expert Systems with Applications 2011, 38: 6179-6183.

21. Atanassov KT, Pasi G, Yager RR. Intuitionistic fuzzy interpretations of multi-criteria multi-person and multi-measurement tool decision making, International Journal of Systems Science 2005, 36:859-868.

22. Chen TY, Wang HP, Lu YY. A multicriteria group decision-making approach based on interval-valued intuitionistic fuzzy sets: a comparative perspective, Expert Systems with Applications 2011, 38:7647-7658.

23. Wei G. Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making, Applied Soft Computing 2010, 10: 423-431.

24. Xu Z, Yager RR. Intuitionistic and interval-valued intuitionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group, Fuzzy Optimization and Decision Making 2009, 8:123-139.

25. Xu Z. A method based on distance measure for interval-valued intuitionistic fuzzy group decision making, Information Sciences 2010,

180: 181-190.

26. Xu Z. A deviation-based approach to intuitionistic fuzzy multiple attribute group decision making, Group Decision and Negotiation

2010, 19: 57-76.

27. Xu Z, Cai X. Recent advances in intuitionistic fuzzy information aggregation, Fuzzy Optimization and Decision Making 2010, 9 :359-381.

28. Xu K, Zhou J, Gu R, Qin H. Approach for aggregating interval-valued intuitionistic fuzzy information and its application to reservoir operation, Expert Systems with Applications 2011, 38:9032-9035.

29. Yue Z. An approach to aggregating interval numbers into interval-valued intuitionistic fuzzy information for group decision making,

Expert Systems with Applications 2011, 38:6333-6338.

30. Wang Z, Li KW, Xu J. A mathematical programming approach to multiattribute decision making with interval-valued intuitionistic fuzzy assessment information, Expert Systems with Applications 2011, 38:12462–12469.

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

32. Lourenzutti R, Krohling RA. A Study of TODIM in a intuitionistic fuzzy and random environment, Expert Systems with Applications

2013, 40:6459-6468.

33. Jahanshahloo GR, Hosseinzadeh LF, Davoodi AR. Extension of TOPSIS for decision-making problems with interval data: Interval efficiency, Mathematical and Computer Modelling 2009, 49:1137-1142.