Some intuitionistic fuzzy Einstein hybrid aggregation operators and their application to multiple attribute decision making

Knowledge-Based Systems 37 (2013) 472–479

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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys

Some intuitionistic fuzzy Einstein hybrid aggregation operators and their application to multiple attribute decision making Xiaofei Zhao, Guiwu Wei ⇑ School of Economics and Management, Chongqing University of Arts and Sciences, Chongqing 402160, PR China

a r t i c l e

i n f o

Article history: Received 6 June 2012 Received in revised form 26 August 2012 Accepted 19 September 2012 Available online 12 October 2012 Keywords: Multiple attribute decision making (MADM) Intuitionistic fuzzy numbers Operational laws Intuitionistic fuzzy Einstein hybrid averaging (IFEHA) operator Intuitionistic fuzzy Einstein hybrid geometric (IFEHG) operator

a b s t r a c t Intuitionistic fuzzy information aggregation plays an important part in intuitionistic fuzzy set theory, which has emerged to be a new research direction receiving more and more attention in recent years. In this paper, we investigate the multiple attribute decision making (MADM) problems with intuitionistic fuzzy numbers. Then, we first introduce some operations on intuitionistic fuzzy sets, such as Einstein sum, Einstein product, and Einstein exponentiation, and further develop some new Einstein hybrid aggregation operators, such as the intuitionistic fuzzy Einstein hybrid averaging (IFEHA) operator and intuitionistic fuzzy Einstein hybrid geometric (IFEHG) operator, which extend the hybrid averaging (HA) operator and the hybrid geometric (HG) operator to accommodate the environment in which the given arguments are intuitionistic fuzzy values. Then, we apply the intuitionistic fuzzy Einstein hybrid averaging (IFEHA) operator and intuitionistic fuzzy Einstein hybrid geometric (IFEHG) operator to deal with multiple attribute decision making under intuitionistic fuzzy environments. Finally, some illustrative examples are given to verify the developed approach and to demonstrate its practicality and effectiveness. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Atanassov [1,2] introduced the concept of intuitionistic fuzzy set (IFS) characterized by a membership function and a non-membership function, which is a generalization of the concept of fuzzy set [3] whose basic component is only a membership function. The intuitionistic fuzzy set has received more and more attention since its appearance [4–33]. Xu [34] developed the intuitionistic fuzzy weighted averaging (IFWA) operator, the intuitionistic fuzzy ordered weighted averaging (IFOWA) operator, and the intuitionistic fuzzy hybrid aggregation (IFHA) operator. Xu [35] developed some geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator and gave an application of the IFHG operator to multiple attribute group decision making with intuitionistic fuzzy information. Xu [36] developed a series of operators for aggregating intuitionistic fuzzy numbers, establish various properties of these power aggregation operators, and then apply them to develop some approaches to multiple attribute group decision making with intuitionistic fuzzy information. Xu and Yager [37] investigated the BM under intuitionistic fuzzy environments and developed an intuitionistic fuzzy BM (IFBM) and discuss its variety of special cases. Then, they applied the weighted IFBM to multicriteria decision making. Xu and Xia [38] studied ⇑ Corresponding author. Tel.: +86 23 49891870; fax: +86 23 49891870. E-mail addresses: [email protected], [email protected] (G. Wei). 0950-7051/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.knosys.2012.09.006

the induced generalized aggregation operators under intuitionistic fuzzy environments. Choquet integral and Dempster–Shafer theory of evidence were applied to aggregate inuitionistic fuzzy information and some new types of aggregation operators were developed, including the induced generalized intuitionistic fuzzy Choquet integral operators and induced generalized intuitionistic fuzzy Dempster–Shafer operators. Xu and Yager [39] investigated the dynamic intuitionistic fuzzy multiple attribute decision making problems and developed the dynamic intuitionistic fuzzy weighted averaging (DIFWA) operator to aggregate dynamic intuitionistic fuzzy information. Moreover, based on the DIFWA and IFWA operators, they have developed the procedure for solving the dynamic intuitionistic fuzzy multiple attribute decision making problems where all the attribute values are expressed in intuitionistic fuzzy numbers. Wei [40] investigated the dynamic intuitionistic fuzzy multiple attribute decision making problems where all the attribute values are expressed in intuitionistic fuzzy numbers and proposed the dynamic intuitionistic fuzzy weighted geometric (DIFWG) operator to aggregate dynamic intuitionistic fuzzy information. Furthermore, they developed the procedure for solving the dynamic intuitionistic fuzzy multiple attribute decision making problems where all the attribute values are expressed in intuitionistic fuzzy numbers based on the DIFWG and IFWG operators. Wei [41] proposed the induced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) operator and some desirable properties of the I-IFOWG operator are studied, such as commutativity, idempotency and monotonicity. An I-IFOWG and IFWG (intuition-

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X. Zhao, G. Wei / Knowledge-Based Systems 37 (2013) 472–479

istic fuzzy weighted geometric) operators-based approach is developed to solve the MAGDM problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of intuitionistic fuzzy numbers. Wei and Zhao [42] investigated some multiple attribute group decision making (MAGDM) problems in which both the attribute weights and the expert weights are usually correlative, attribute values take the form of intuitionistic fuzzy values and developed the induced intuitionistic fuzzy correlated averaging (I-IFCA) operator and induced intuitionistic fuzzy correlated geometric (I-IFCG) operator and some desirable properties of the I-IFCA and I-IFCG operators are studied, such as commutativity, idempotency and monotonicity. An I-IFCA and IFCA (intuitionistic fuzzy correlated averaging) operators-based approach is developed to solve the MAGDM problems in which both the attribute weights and the expert weights usually correlative, attribute values take the form of intuitionistic fuzzy values. Zeng and Su [43] considered the situation with intuitionistic fuzzy information and develop an intuitionistic fuzzy ordered weighted distance (IFOWD) operator. The IFOWD operator is very suitable to deal with the situations where the input data are represented in intuitionistic fuzzy information and includes a wide range of distance measures and aggregation operators. They studied some of its main properties and different families of IFOWD operators and developed an application of the new approach in a group decision-making under intuitionistic fuzzy environment and illustrate it with a numerical example. All the above operators are based on the algebraic operational laws of IFSs for carrying the combination process and are not consistent with the limiting case of ordinary fuzzy sets [44]. Recently, Wang and Liu [45] treated the intuitionistic fuzzy aggregation operators with the help of Einstein operations and developed some intuitionistic fuzzy Einstein aggregation operators, such as the intuitionistic fuzzy Einstein weighted averaging operator and the intuitionistic fuzzy Einstein ordered weighted averaging operator, which extend the weighted averaging operator and the ordered weighted averaging operator to aggregate intuitionistic fuzzy values, respectively. They further established various properties of these operators and analysis the relations between these operators and the existing intuitionistic fuzzy aggregation operators. Wang and Liu [46] further developed some new geometric Einstein aggregation operators, such as the intuitionistic fuzzy Einstein weighted geometric operator and the intuitionistic fuzzy Einstein ordered weighted geometric operator, which extend the weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator to accommodate the environment in which the given arguments are intuitionistic fuzzy values. They also established some desirable properties of these operators, such as commutativity, idempotency and monotonicity, and gave some numerical examples to illustrate the developed aggregation operators. Finally, they applied the intuitionistic fuzzy Einstein weighted geometric operator to deal with multiple attribute decision making under intuitionistic fuzzy environments. We know that the IFEWA and IFEWG operators weights only the intuitionistic fuzzy values, while the IFEOWA and IFEOWG operator weights only the ordered positions of the intuitionistic fuzzy values instead of weighting the intuitionistic fuzzy values themselves. Therefore, weights represent different aspects in both the IFEWA (IFEWG) and IFEOWA (IFEOWG) operators. However, both the operators consider only one of them. To solve this drawback, in the following we shall propose the intuitionistic fuzzy Einstein hybrid averaging (IFEHA) operator and intuitionistic fuzzy Einstein hybrid geometric (IFEHG) operators. Therefore, how to extend the Einstein operations to aggregate the intuitionistic fuzzy information based on the hybrid averaging operators are meaningful works, which are also the focus of this paper. In order to do so, the remainder of this paper is set out as follows: In the next section, we introduce some basic concepts related to

intuitionistic fuzzy sets, some existing intuitionistic fuzzy aggregating operators and Einstein operations on intuitionistic fuzzy sets. In Section 3, we develop some intuitionistic fuzzy Einstein hybrid aggregation operators, such as the intuitionistic fuzzy Einstein hybrid averaging (IFEHA) operator and intuitionistic fuzzy Einstein hybrid geometric (IFEHG) operator, which extend the hybrid averaging (HA) operator [51,52] and the hybrid geometric (HG) operator [51,52] to accommodate the environment in which the given arguments are intuitionistic fuzzy values and study some desired properties of these operators. In Section 4, we apply the intuitionistic fuzzy Einstein hybrid averaging (IFEHA) operator and intuitionistic fuzzy Einstein hybrid geometric (IFEHG) operator to deal with multiple attribute decision making problems under intuitionistic fuzzy environments. In Section 5, some illustrative examples are pointed out. In Section 6 we conclude the paper and give some remarks. 2. Preliminaries 2.1. Intuitionistic fuzzy set In the following, we shall introduce some basic concepts related to intuitionistic fuzzy sets. Definition 1. Let X be an universe of discourse, then a fuzzy set is defined as:

A ¼ fhx; lA ðxÞijx 2 Xg

ð1Þ

Which is characterized by a membership function lA:X ? [0, 1], where lA(x) denotes the degree of membership of the element x to the set A [3]. Atanassov [1,2] extended the fuzzy set to the IFS, shown as follows: Definition 2. An IFS A in X is given by

A ¼ fhx; lA ðxÞ; mA ðxÞijx 2 Xg

ð2Þ

where lA:X ? [0, 1] and mA:X ? [0, 1], with the condition

0 6 lA ðxÞ þ mA ðxÞ 6 1;

8x 2 X

The numbers lA(x) and mA(x) represent, respectively, the membership degree and non-membership degree of the elementxto the set A. Definition 3. For each IFS A in X, if

pA ðxÞ ¼ 1  lA ðxÞ  mA ðxÞ; 8x 2 X:

ð3Þ

Then pA(x) is called the degree of indeterminacy of x to A [1,2]. ~ ¼ ðl; mÞ be an intuitionistic fuzzy number, a Definition 4. Let a score function S of an intuitionistic fuzzy value can be represented as follows [47]:

~ Þ ¼ l  m; Sða

~Þ 2 ½1; 1: Sða

ð4Þ

~ ¼ ðl; mÞ be an intuitionistic fuzzy number, an Definition 5. Let a accuracy function H of an intuitionistic fuzzy value can be represented as follows [27]:

~ Þ ¼ l þ m; Hða

~Þ 2 ½0; 1: Hða

ð5Þ

to evaluate the degree of accuracy of the intuitionistic fuzzy value ~ ¼ ðl; mÞ, where Hða ~Þ 2 ½0; 1. The larger the value of Hða ~Þ, the more a ~. the degree of accuracy of the intuitionistic fuzzy value a As presented above, the score function S and the accuracy function H are, respectively, defined as the difference and the sum of ~ A ðxÞ and the non-membership function the membership function l

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X. Zhao, G. Wei / Knowledge-Based Systems 37 (2013) 472–479

m~A ðxÞ. Hong and Choi [27] showed that the relation between the score function S and the accuracy function H is similar to the relation between mean and variance in statistics. Based on the score function S and the accuracy function H, in the following, Xu [34] give an order relation between two intuitionistic fuzzy values, which is defined as follows:

~_ rðjÞ is the jth largest of the weighted intuitionistic fuzzy values where a ~_ j ða ~j ; j ¼ 1; 2; . . . ; nÞ; x ¼ ðx1 ; x2 ; . . . ; xn ÞT be the weight vec~_ j ¼ nxj a a P ~j ðj ¼ 1; 2; . . . ; nÞ, and xj > 0; nj¼1 xj ¼ 1; w ¼ ðw1 ; w2 ; . . . ; wn ÞT tor of a P is the aggregation-associated vector such that wj 2 ½0; 1; nj¼1 wj ¼ 1 and n is the balancing coefficient. (6) The intuitionistic fuzzy hybrid geometric (IFHG) operator [35]:

~1 ¼ ðl1 ; m1 Þ and a ~2 ¼ ðl2 ; m2 Þ be two intuitionistic Definition 6. Let a ~1 Þ ¼ l1  m1 and sða ~2 Þ ¼ l2  m2 be the scores of a ~ fuzzy values, sða ~ respectively, and let Hða ~1 Þ ¼ l1 þ m1 and Hða ~2 Þ ¼ l2 þ m2 be and b, ~ respectively, then if Sða ~ then ~ and b, ~Þ < SðbÞ, the accuracy degrees of a ~ ~ ~ ~ ~ ~ a is smaller than b, denoted by a < b; if SðaÞ ¼ SðbÞ, then ~ then a ~ represent the same information, ~Þ ¼ HðbÞ, ~ and b (1) if Hða ~ ~ a ~ denoted by ~ ~ ~ is smaller than b, denoted by a ¼ b; (2) if HðaÞ < HðbÞ; ~ ~ < b. a Since its appearance, the ordered weighted averaging (OWA) operator, introduced by Yager [48], has received more and more attention. Xu [34] and Xu and Yager [35] gave some intuitionistic fuzzy aggregation operators as listed below: ~j ¼ ðlj ; mj Þðj ¼ 1; 2; . . . ; nÞ, then. For a collection of IFVs a (1) The intuitionistic fuzzy weighted averaging (IFWA) operator [34]: n

~j Þ ~1 ; a ~2 ; . . . ; a ~n Þ ¼  ðxj a IFWAx ða j¼1

¼

1

n n Y Y x ð1  lj Þxj ; mj j j¼1

! ð6Þ

j¼1

~j ðj ¼ 1; 2; . . . ; where x = (x1, x2, . . . , xn)T be the weight vector of a P nÞ, and xj > 0; nj¼1 xj ¼ 1. (2) The intuitionistic fuzzy weighted geometric (IFWG) operator [35]: n

~1 ; a ~2 ; . . . ; a ~ n Þ ¼  ða ~ j Þx j ¼ IFWGx ða j¼1

n Y

xj

lj ;1 

j¼1

n Y

! ð1  mj Þ

n

j¼1

n Y

wj

ð1  lrðjÞ Þ ;

j¼1

n Y

wj rðjÞ

!

m

ð8Þ

j¼1

where (r(1), r(2), . . . , r(n)) is a permutation of (1, 2, . . . , n), such ~ rðj1Þ P a ~ rðjÞ for all j = 2, . . . , n and w = (w1, w2, . . . , wn)T is that a P the aggregation-associated vector such that wj 2 [0, 1], nj¼1 wj ¼ 1. (4) The intuitionistic fuzzy ordered weighted geometric (IFOWG) operator [35]: n

j¼1

n Y

n Y ð1  mrðjÞ Þwj

j¼1

j¼1

lwrðjÞj ; 1 

ð9Þ

n ~_ rðjÞ Þ ~1 ; a ~2 ; . . . ; a ~n Þ ¼  ðwj a IFHAx;w ða j¼1

¼

1

j¼1

l_ wrðjÞj ; 1 

n Y

j¼1

! ð1  m_ rðjÞ Þwj

j¼1

ð11Þ ~_ rðjÞ is the jth largest of the weighted intuitionistic fuzzy values where a _a ða _ ¼a ~ ~ ~nj xj ; j ¼ 1; 2; . . . ; nÞ, x = (x1, x2, . . . , xn)T be the weight vector j j P ~j ðj ¼ 1; 2; . . . ; nÞ, and xj > 0; nj¼1 xj ¼ 1; w ¼ ðw1 ; w2 ; . . . ; wn ÞT is of a Pn the aggregation-associated vector such that wj 2 [0, 1], j¼1 wj ¼ 1 and n is the balancing coefficient. 2.2. Einstein operations The set theoretical operators have had an important role since in the beginning of fuzzy set theory. All types of the particular operators were included in the general concepts of the t-norms and tconorms, which satisfy the requirements of the conjunction and disjunction operators, respectively. There are various t-norms and t-conorms families can be used to perform the corresponding intersections and unions of IFSs. Einstein operations includes the Einstein product and Einstein sum, which are examples of t-norms and t-conorms, respectively. They are defined as follows [49]: Einstein product e is a t-norm and Einstein sum e is a t-conorm, where aþb ; 1þab

a e b ¼

ab ; 1 þ ð1  aÞ  ð1  bÞ

8ða; bÞ 2 ½0; 12

ð12Þ

n

wj ð1  l_ rðjÞ Þwj ; d m_ rðjÞ j¼1

2.3. Einstein operations of intuitionistic fuzzy set In this section, we shall introduce the Einstein operations on intuitionistic fuzzy sets and analyze some desirable properties of these operations. Motivated by (12), let the t-norm T and t-conorm S be Einstein product T’’ and Einstein sum S’’ respectively, then the generalized intersection and union on two IFSs A and B become the ~1 e a ~2 ) and Einstein sum (denoted Einstein product (denoted by a ~1 e a ~2 ) on two IVIFSs a ~1 and a ~2 , respectively, as follows [45,46]. by a

~1 e a ~2 ¼ a ~2 ¼ ~1 e a a

!

where (r(1), r(2), . . . , r(n)) is a permutation of (1, 2, . . . , n), such ~ rðj1Þ P a ~ rðjÞ for all j = 2, . . . , n and w = (w1, w2, . . . , wn)T is that a P the aggregation-associated vector such that wj 2 [0, 1], nj¼1 wj ¼ 1. (5) The intuitionistic fuzzy hybrid averaging (IFHA) operator [34]:

n Y

n Y

j¼1

~rðjÞ Þ ~1 ; a ~2 ; . . . ; a ~n Þ ¼  ðwj a IFOWAw ða

~1 ; a ~2 ; . .. ; a ~n Þ ¼  ða ~rðjÞ Þwj ¼ IFOWGw ða

¼

ð7Þ

~j ðj ¼ 1; 2; . . . ; where x = (x1, x2, . . . , xn) be the weight vector of a P nÞ, and xj > 0; nj¼1 xj ¼ 1. (3) The intuitionistic fuzzy ordered weighted averaging (IFOWA) operator [34]:

1

j¼1

a e b ¼

xj

T

¼

n ~1 ; a ~2 ; . . . ; a ~n Þ ¼  ða ~_ rðjÞ Þwj IFHGx;w ða

~1 ¼ ka

~1 Þk ¼ ða

 

l1 l2 m1 þ m2 ; 1 þ ð1  l1 Þð1  l2 Þ 1 þ m1 m2 l1 þ l2 m1 m2 ; 1 þ l1 l2 1 þ ð1  m1 Þð1  m2 Þ

ð1 þ l1 Þk  ð1  l1 Þk

; k

 ð13Þ 

2mk1

ð14Þ !

ð1 þ l1 Þk þ ð1  l1 Þ ð2  m1 Þk þ mk1 2lk1

;

ð1 þ m1 Þk  ð1  m1 Þk

ð2  l1 Þk þ lk1 ð1 þ m1 Þk þ ð1  m1 Þk

k > 0;

;

ð15Þ

! ;

k > 0:

ð16Þ

3. Intuitionistic fuzzy Einstein hybrid aggregation operators 3.1. Intuitionistic fuzzy Einstein hybrid averaging operators

! ð10Þ

In the section, we shall introduce the intuitionistic fuzzy arithmetic aggregation operators with the help of the Einstein operations.

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X. Zhao, G. Wei / Knowledge-Based Systems 37 (2013) 472–479

~j ¼ ðlj ; mj Þðj ¼ 1; 2; . . . ; nÞ be a collection of Definition 7 [45]. Let a intuitionistic fuzzy values, and let IFEWA: Qn ? Q, if

 0 0  ~1 ; a ~1 ; a ~2 ; . . . ; a ~n Þ 6 IFEOWAw a ~2 ; . . . ; a ~0n IFEOWAw ða

n

~j Þ ~1 ; a ~2 ;. .. ; a ~n Þ ¼ e ðxj a IFEWAx ða j¼1

! Q x 2 nj¼1 mj j j¼1 ð1 þ lj Þ  j¼1 ð1  lj Þ Q ; ¼ Qn ð17Þ Q Q xj n xj xj n xj n j¼1 ð1 þ lj Þ þ j¼1 ð1  lj Þ j¼1 ð2  mj Þ þ j¼1 mj Qn

xj

Qn

~j ðj ¼ 1; 2; . . . ; nÞ are equal, i.e. Theorem 1 (Idempotency). If all a ~j ¼ a ~ for all j, then a

~1 ; a ~2 ; . . . ; a ~n Þ ¼ a ~ IFEWAx ða

ð18Þ

~j ðj ¼ 1; 2; . . . ; nÞ be a collection of Theorem 2 (Boundedness). Let a IFVN, and let

j

ð24Þ

xj

~j ðj ¼ 1; 2; . . . ; where x = (x1, x2, . . . , xn)T be the weight vector of a P nÞ, and xj > 0; nj¼1 xj ¼ 1, then IFEWA is called the intuitionistic fuzzy Einstein weighted averaging (IFEWA) operator. It can be easily proved that the IFEWA operator has the following properties [45].

~ ¼ mina ~j ; a

~j ðj ¼ 1; 2; . . . ; nÞ and a ~0j ðj ¼ 1; 2; Theorem 6 (Monotonicity). Let a 0 ~ ~ . . . ; nÞ be two set of IFVNs, if aj 6 aj , for all j, then

~j ðj ¼ 1; 2; . . . ; nÞ and a ~0j ðj ¼ 1; 2; Theorem 7 (Commutativity). Let a . . . ; nÞ be two set of IFVNs, then

 0 0  ~1 ; a ~1 ; a ~2 ; . . . ; a ~n Þ ¼ IFEOWAw a ~2 ; . . . ; a ~0n IFEOWAw ða

ð25Þ

~0j ðj ¼ 1; 2; . . . ; nÞ is any permutation of a ~j ðj ¼ 1; 2; . . . ; nÞ. where a From Definitions 7 and 8, we know that the IFEWA operator weights only the intuitionistic fuzzy values, while the IFEOWA operator weights only the ordered positions of the intuitionistic fuzzy values instead of weighting the intuitionistic fuzzy values themselves. Therefore, weights represent different aspects in both the IFEWA and IFEOWA operators. However, both the operators consider only one of them. To solve this drawback, in the following we shall propose an intuitionistic fuzzy Einstein hybrid averaging (IFEHA). Definition 9. An intuitionistic fuzzy Einstein hybrid averaging (IFEHA) operator of dimension n is a mapping IFEHA: Qn ? Q, that has an associated vector w = (w1, w2, . . . , wn)T such that wj > 0 and Pn j¼1 wj ¼ 1. Furthermore,

~j ~þ ¼ maxa a j

Then

~ 6 IFEWAx ða ~1 ; a ~2 ; . . . ; a ~n Þ 6 a ~þ a

ð19Þ

n

~1 ; a ~2 ; . .. ; a ~n Þ ¼ e ðxj a ~_ rðjÞ Þ IFEHAx;w ða j¼1

Qn

~j ðj ¼ 1; 2; . . . ; nÞ and Theorem 3 (Monotonicity). Let a ~j 6 a ~0j , for all j, then . . . ; nÞ be two set of IFVNs, if a

~0j ðj a

¼ 1; 2;

 0 0  ~1 ; a ~1 ; a ~2 ; . . . ; a ~n Þ 6 IFEWAx a ~2 ; . . . ; a ~0n IFEWAx ða

ð20Þ

Furthermore, Wang and Liu [45] developed the intuitionistic fuzzy Einstein ordered weighted averaging (IFEOWA) operator. ~j ¼ ðlj ; mj Þðj ¼ 1; 2; . . . ; nÞ be a collection of Definition 8 [45]. Let a intuitionistic fuzzy numbers. An intuitionistic fuzzy Einstein ordered weighted averaging (IFEOWA) operator of dimension n is a mapping IFEOWA:Qn ? Q, that has an associated vector w = (w1,w2, . . . , wn)T P such that wj > 0 and nj¼1 wj ¼ 1. Furthermore,

Q

Q

w

j 2 nj¼1 m_ rðjÞ l_ rðjÞ Þwj  nj¼1 ð1  l_ rðjÞ Þwj Qn Qn wj wj w j ; Qn w j _ _ _ _ j¼1 ð1 þ lrðjÞ Þ þ j¼1 ð1  lrðjÞ Þ j¼1 ð2  mrðjÞ Þ þ j¼1 mrðjÞ

j¼1 ð1 þ

¼ Qn

! ð26Þ

~_ rðjÞ is the jth largest of the weighted intuitionistic fuzzy where a ~_ j ða ~_ j ¼ nxj a ~j ; j ¼ 1; 2; . . . ; nÞ; x ¼ ðx1 ; x2 ; . . . ; xn ÞT be the values a P ~ weight vector of aj ðj ¼ 1; 2; . . . ; nÞ, and xj > 0; nj¼1 xj ¼ 1, and n is the balancing coefficient. Let w = (1/n, 1/n, . . . , 1/n), then the IFEWA operator is a special case of the IFEHA operator; Let x = (1/n, 1/n, . . . , 1/n), then the IFEOWA operator is a special case of the IFEHA operator. So we know that the IFEHA operator generalizes both the IFEWA and IFEOWA operators, and reflects the importance degrees of both the given arguments and their ordered positions. From Definition 9, we know that:

n

~rðjÞ Þ ~1 ; a ~2 ;...; a ~n Þ ¼ e ðxj a IFEOWAw ða Qn

j¼1 ð1 þ

¼ Qn

j¼1 ð1 þ

j¼1

wj

Qn

wj

! wj j¼1 rðjÞ Q wj wj n rðjÞ Þ þ j¼1 rðjÞ

Qn

2 lrðjÞ Þ  j¼1 ð1  lrðjÞ Þ Q ;Q lrðjÞ Þwj þ nj¼1 ð1  lrðjÞ Þwj nj¼1 ð2  m

m

m

ð21Þ

where (r(1), r(2), . . . , r(n)) is a permutation of (1, 2, . . . , n), such ~ rðj1Þ P a ~ rðjÞ for all j = 2, . . . , n. that a It can be easily proved that the IFEOWA operator has the following properties [45]. ~j ðj ¼ 1; 2; . . . ; nÞ are equal, i.e. Theorem 4 (Idempotency). If all a ~j ¼ a ~ for all j, then a

(1) The IFEHA operator first weights the given arguments, and then reorders the weighted arguments in descending order and weights these ordered arguments by the IFEHA weights, and finally aggregates all the weighted arguments into a collective one. (2) The IFEHA operator generalizes both the IFEWA and IFEOWA operators, and reflects the importance degrees of both the given arguments and their ordered positions. 3.2. Intuitionistic fuzzy Einstein hybrid geometric operators

ð22Þ

In this section, we shall introduce the intuitionistic fuzzy geometric aggregation operators with the help of the Einstein operations.

~j ðj ¼ 1; 2; . . . ; nÞ be a collection of Theorem 5 (Boundedness). Let a IFVNs, and let

~j ¼ ðlj ; mj Þðj ¼ 1; 2; . . . ; nÞ be a collection Definition 10 [46]. Let a of intuitionistic fuzzy values, and let IFEWG: Qn ? Q, if

~1 ; a ~2 ; . . . ; a ~n Þ ¼ a ~ IFEOWAw ða

~ ¼ minj a ~j ; a

~j ~þ ¼ maxj a a

n

~1 ; a ~2 ; . . . ; a ~n Þ ¼ e ða ~j Þxj IFEWGx ða

Then

~ 6 IFEOWAw ða ~1 ; a ~2 ; . . . ; a ~n Þ 6 a ~þ a

¼

ð23Þ

Qn

j¼1 ð2

Q x 2 nj¼1 lj j  lj Þxj þ

j¼1

Qn

j¼1

Qn

lxj j

j¼1 ð1

þ mj Þxj 

j¼1 ð1

þ mj Þxj þ

; Qn

Qn

j¼1 ð1

 mj Þxj

j¼1 ð1

 mj Þxj

Qn

! ð27Þ

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X. Zhao, G. Wei / Knowledge-Based Systems 37 (2013) 472–479

~j ðj ¼ 1; 2; . . . ; where x = (x1, x2, . . . , xn)T be the weight vector of a P nÞ, and xj > 0; nj¼1 xj ¼ 1, then IFEWG is called the intuitionistic fuzzy Einstein weighted geometric (IFEWG) operator. It can be easily proved that the IFEWG operator has the following properties [46]. ~j ðj ¼ 1; 2; . . . ; nÞ are equal, i.e. Theorem 8 (Idempotency). If all a ~j ¼ a ~ for all j, then a

~1 ; a ~2 ; . . . ; a ~n Þ ¼ a ~ IFEWGx ða

ð28Þ

~j ðj ¼ 1; 2; . . . ; nÞ be a collection of Theorem 9 (Boundedness). Let a IFVN, and let

~ ¼ minj a ~j ; a

~j ~þ ¼ maxj a a

Then

~ 6 IFEWGx ða ~1 ; a ~2 ; . . . ; a ~n Þ 6 a ~þ a

ð29Þ

~j ðj ¼ 1; 2; . . . ; nÞ and a ~0j ðj ¼ 1; 2; Theorem 10 (Monotonicity). Let a 0 ~ ~ . . . ; nÞ be two set of IFVNs, if aj 6 aj , for all j, then

 0 0  ~1 ; a ~1 ; a ~2 ; . . . ; a ~n Þ 6 IFEWGx a ~2 ; . . . ; a ~0n IFEWGx ða

ð30Þ

Furthermore, Wang and Liu [46] shall develop the intuitionistic fuzzy Einstein ordered weighted geometric (IFEOWG) operator. ~j ¼ ðlj ; mj Þðj ¼ 1; 2; . . . ; nÞ be a collection of Definition 11 [46]. Let a intuitionistic fuzzy numbers. An intuitionistic fuzzy Einstein ordered weighted geometric (IFEOWG) operator of dimension n is a mapping IFEOWG:Qn ? Q, that has an associated vector w = P (w1, w2, . . . , wn)T such that wj > 0 and nj¼1 wj ¼ 1. Furthermore,

~0j ðj ¼ 1; 2; . . . ; nÞ is any permutation of a ~j ðj ¼ 1; 2; . . . ; nÞ. where a From Definitions 10 and 11, we know that the IFEWG operator weights only the intuitionistic fuzzy values, while the IFEOWG operator weights only the ordered positions of the intuitionistic fuzzy values instead of weighting the intuitionistic fuzzy values themselves. Therefore, weights represent different aspects in both the IFEWG and IFEOWG operators. However, both the operators consider only one of them. To solve this drawback, in the following we shall propose an intuitionistic fuzzy Einstein hybrid geometric (IFEHG). Definition 12. An intuitionistic fuzzy Einstein hybrid geometric (IFEHG) operator of dimension n is a mapping IFEHG: Qn ? Q, that has an associated vector w = (w1,w2, . . . , wn)T such that wj > 0 and Pn j¼1 wj ¼ 1. Furthermore, n ~1 ; a ~2 ;. . . ; a ~ n Þ ¼  e ða ~_ rðjÞ Þwj IFEHGx;w ða j¼1

! Q Qn Qn wj wj wj _ _ 2 nj¼1 l_ rðjÞ j¼1 ð1 þ mrðjÞ Þ  j¼1 ð1  mrðjÞ Þ Qn ¼ Qn Qn w j ; Qn wj wj wj _ _ _ _ j¼1 ð1 þ mrðjÞ Þ þ j¼1 ð1  mrðjÞ Þ j¼1 ð2  lrðjÞ Þ þ j¼1 lrðjÞ

ð36Þ

~_ rðjÞ is the jth largest of the weighted intuitionistic fuzzy valwhere a _ ~j ða ~_ j ¼ a ~^j e nxj ; j ¼ 1; 2; . . . ; nÞ; x ¼ ðx1 ; x2 ; . . . ; xn ÞT be the ues a P ~j ðj ¼ 1; 2; . . . ; nÞ, and xj > 0; nj¼1 xj ¼ 1, and nis weight vector of a the balancing coefficient. Letw = (1/n, 1/n, . . . , 1/n), then the IFEWG operator is a special case of the IFEHG operator; Let x = (1/n, 1/ n, . . . , 1/n), then the IFEOWG operator is a special case of the IFEHG operator. So we know that the IFEHG operator generalizes both the IFEWG and IFEOWG operators, and reflects the importance degrees of both the given arguments and their ordered positions. From Definition 12, we know that:

n

~1 ; a ~2 ; . .. ; a ~n Þ ¼ e ða ~rðjÞ Þwj IFEOWGx ða ¼ Qn

2

j¼1 ð2 

Qn

j¼1

Qn

wj j¼1 lrðjÞ

lrðjÞ Þwj þ

Qn

j¼1

j¼1 ð1 þ

lwrðjÞj

; Qn

j¼1 ð1 þ

wj

Qn

wj

mrðjÞ Þ  j¼1 ð1  mrðjÞ Þ Q mrðjÞ Þwj þ nj¼1 ð1  mrðjÞ Þwj

! ð31Þ

where (r(1), r(2), . . . , r(n)) is a permutation of (1, 2, . . . , n), such ~ rðj1Þ P a ~ rðjÞ for all j = 2, . . . , n. that a It can be easily proved that the IFEOWG operator has the following properties [46]. ~j ðj ¼ 1; 2; . . . ; nÞ are equal, i.e. Theorem 11 (Idempotency). If all a ~j ¼ a ~ for all j, then a

~1 ; a ~2 ; . . . ; a ~n Þ ¼ a ~ IFEOWGw ða

ð32Þ

~j ðj ¼ 1; 2; . . . ; nÞ be a collection of Theorem 12 (Boundedness). Let a IFVNs, and let

~ ¼ minj a ~j ; a

~j ~þ ¼ maxj a a

Then

~ 6 IFEOWGw ða ~1 ; a ~2 ; . . . ; a ~n Þ 6 a ~þ a

ð33Þ

~j ðj ¼ 1; 2; . . . ; nÞ and a ~0j ðj ¼ 1; 2; Theorem 13 (Monotonicity). Let a ~j 6 a ~0j , for all j, then . . . ; nÞ be two set of IFVNs, if a

 0 0  ~1 ; a ~1 ; a ~2 ; . . . ; a ~n Þ 6 IFEOWGw a ~2 ; . . . ; a ~0n IFEOWGw ða

ð34Þ

~j ðj ¼ 1; 2; . . . ; nÞ and a ~0j ðj ¼ 1; 2; Theorem 14 (Commutativity). Let a . . . ; nÞ be two set of IFVNs, then

 0 0  ~1 ; a ~1 ; a ~2 ; . . . ; a ~n Þ ¼ FEOWGw a ~2 ; . . . ; a ~0n IFEOWGw ða

ð35Þ

(1) The IFEHG operator first weights the given arguments, and then reorders the weighted arguments in descending order and weights these ordered arguments by the IFEHG weights, and finally aggregates all the weighted arguments into a collective one. (2) The IFEHG operator generalizes both the IFEWG and IFEOWG operators, and reflects the importance degrees of both the given arguments and their ordered positions. 4. An approach to multiple attribute decision making with intuitionistic fuzzy information In this section, we shall investigate the multiple attribute decision making (MADM) problems based on the IFEHA (or IFEHG) operator in which the attribute weights take the form of real numbers, attribute values take the form of intuitionistic fuzzy numbers. Let A = {A1, A2, . . . , Am} be a discrete set of alternatives, and G = {G1, G2, . . . , Gn} be the set of attributes. The information about attribute weights is completely known. Let x = (x1, x2, . . . , xn) 2 H be the P weight vector of attributes, where wj P 0; j ¼ 1; 2; . . . ; n; nj¼1 wj e ¼ ð~r ij Þ ¼ 1. Suppose that R mn ¼ ðlij ; mij Þmn is the intuitionistic fuzzy decision matrix, where lij indicates the degree that the alternative Ai satisfies the attribute Gj given by the decision maker, mij indicates the degree that the alternative Ai does not satisfy the attribute Gj given by the decision maker, lij  [0, 1], mij  [0, 1], lij + mij 6 1, i = 1, 2, . . . , m, j = 1, 2, . . . , n. In the following, we apply the IFEHA (or IFEHG) operator to multiple attribute decision making based on intuitionistic fuzzy information. The method involves the following steps: e and the Step 1. Utilize the decision information given in matrix R, IVIFEHG operator

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X. Zhao, G. Wei / Knowledge-Based Systems 37 (2013) 472–479

~ri ¼ ðli ; mi Þ ¼ IFEHAx;w ð~r i1 ; ~r i2 ; . . . ; ~r in Þ;

i ¼ 1; 2; . . . ; m:

Then by score functions, we can get:

ð37Þ Or

~ri ¼ ðli ; mi Þ ¼ IFEHGx;w ð~r i1 ; ~r i2 ; . . . ; ~rin Þ;

i ¼ 1; 2; . . . ; m: ð38Þ

to derive the overall preference values ~r i ði ¼ 1; 2; . . . ; mÞ of the alternative Ai, where x = (x1, x2, . . . , xn)T is the weighting vector of the attributes and w = (w1, w2, . . . , wn)T is the associated vector of the IFEHA operator, such that wj > 0 P and nj¼1 wj ¼ 1. Step 2. calculate the scores Sð~ri Þði ¼ 1; 2; . . . ; mÞ of the collective overall values ~ri ði ¼ 1; 2; . . . ; mÞ to rank all the alternatives Ai(i = 1, 2, . . . , m) and then to select the best one(s) if there is no difference between two scores Sð~ri Þ and Sð~r j Þ, then we need to calculate the accuracy degrees Hð~r i Þ and Hð~rj Þ of the collective overall preference values ~r i and ~rj , respectively, and then rank the alternatives Ai and Aj in accordance with the accuracy degrees Hð~ri Þ and Hð~r j Þ. Step 3. Rank all the alternatives Ai (i = 1, 2, . . . , m) and select the best one (s) in accordance with Sð~ri Þ and Hð~ri Þði ¼ 1; 2; . . . ; mÞ. Step 4. End.

In order to demonstrate the application of the developed method, we will consider an example where there is an investment company, which wants to invest a sum of money in the best option (adapted from [50]). There is a panel with five possible alternatives to invest the money: r A1 is a car company; s A2 is a food company; t A3 is a computer company; u A4 is an arms company; v A5 is a TV company. The investment company must take a decision according to the following four attributes: r G1 is the risk analysis; s G2 is the growth analysis; t G3 is the social–political impact analysis; u G4 is the environmental impact analysis. The five possible alternatives Ai(i = 1, 2, 3, 4, 5) are to be evaluated using the intuitionistic fuzzy information by the decision maker under the above four attributes (whose weighting vector x = (0.2, 0.1, 0.3, 0.4)T), as listed in the following matrix.

ð0:4; 0:5Þ 6 ð0:6; 0:4Þ 6 e¼6 R 6 ð0:5; 0:5Þ 6 4 ð0:7; 0:2Þ ð0:5; 0:3Þ

ð0:5; 0:4Þ ð0:6; 0:3Þ ð0:4; 0:5Þ ð0:5; 0:4Þ ð0:3; 0:4Þ

ð0:2; 0:7Þ ð0:6; 0:3Þ ð0:4; 0:4Þ ð0:2; 0:5Þ ð0:6; 0:2Þ

3

ð0:2; 0:5Þ ð0:3; 0:6Þ 7 7 7 ð0:5; 0:4Þ 7 7 ð0:3; 0:7Þ 5 ð0:4; 0:4Þ

In the following, we apply the IFEHA operator to multiple attribute decision making based on intuitionistic fuzzy information. The method involves the following steps: e and Step 1. Utilize the decision information given in matrix R, ~ij , we get: ~_ ij ¼ nxj a a

~_ 11 a ~_ 13 a ~_ 21 a ~_ 23 a ~_ 31 a ~_ 33 a ~_ 41 a ~_ 43 a ~_ 51 a ~_ a 53

~_ rð12Þ ¼ ð0:3265; 0:5109Þ a ~_ rð14Þ ¼ ð0:2386; 0:7406Þ a ~_ rð22Þ ¼ ð0:5039; 0:4319Þ a ~_ a ¼ ð0:2704; 0:4396Þ

~_ rð31Þ ¼ ð0:7059; 0:2975Þ; a ~_ rð33Þ ¼ ð0:6004; 0:2561Þ; a ~_ rð41Þ ¼ ð0:4132; 0:5109Þ; a ~_ rð43Þ ¼ ð0:1679; 0:5171Þ; a ~_ ¼ ð0:6814; 0:1535Þ; a

~_ rð32Þ ¼ ð0:4687; 0:3659Þ a ~_ rð34Þ ¼ ð0:2163; 0:4814Þ a ~_ rð42Þ ¼ ð0:2386; 0:4850Þ a ~_ rð44Þ ¼ ð0:4584; 0:8210Þ a ~_ a ¼ ð0:5901; 0:2975Þ

~_ rð53Þ ¼ ð0:4132; 0:3478Þ; a

~_ rð54Þ ¼ ð0:1232; 0:4814Þ a

rð23Þ

rð51Þ

¼ ð0:3265; 0:5109Þ; ¼ ð0:2386; 0:7406Þ; ¼ ð0:5039; 0:4319Þ; ¼ ð0:6814; 0:2548Þ; ¼ ð0:4132; 0:5109Þ; ¼ ð0:4687; 0:3659Þ; ¼ ð0:6004; 0:2561Þ; ¼ ð0:2386; 0:4850Þ; ¼ ð0:4132; 0:3478Þ; ¼ ð0:6814; 0:1535Þ;

~_ 12 a ~_ 14 a ~ a_ 22 ~_ 24 a ~_ 32 a ~_ 34 a ~_ 42 a ~_ 44 a ~_ 52 a ~_ a 54

¼ ð0:2163; 0:4814Þ ¼ ð0:3135; 0:4458Þ ¼ ð0:2704; 0:4396Þ ¼ ð0:4584; 0:6213Þ ¼ ð0:1679; 0:5171Þ ¼ ð0:7059; 0:2975Þ ¼ ð0:2163; 0:4814Þ ¼ ð0:4584; 0:8210Þ ¼ ð0:1232; 0:4814Þ ¼ ð0:5901; 0:2975Þ

rð24Þ

rð52Þ

e and the Step 2. Utilize the decision information given in matrix R, IFEHA operator which has associated weighting vector w = (0.20, 0.30, 0.30, 0.2)T, we obtain the overall preference values ~ri of the alternatives Ai (i = 1, 2, . . . , 5). ~r 1 ¼ ð0:2434; 0:5477Þ; ~r 2 ¼ ð0:4534;0:4360Þ; ~r 3 ¼ ð0:4273;0:4119Þ ~r 4 ¼ ð0:3109; 0:5072Þ; ~r 5 ¼ ð0:4440;0:2941Þ

Step 3. Calculate the scores Sð~r i Þði ¼ 1; 2; . . . ; 5Þ of the overall intuitionistic fuzzy preference values ~r i ði ¼ 1; 2; . . . ; 5Þ

Sð~r 1 Þ ¼ 0:3043; Sð~r 4 Þ ¼ 0:1964;

5. Illustrative example

2

~_ rð11Þ ¼ ð0:3135; 0:4458Þ; a ~_ rð13Þ ¼ ð0:2163; 0:4814Þ; a ~_ rð21Þ ¼ ð0:6814; 0:2548Þ; a ~_ a ¼ ð0:4584; 0:6213Þ;

Sð~r 2 Þ ¼ 0:0174; Sð~r 5 Þ ¼ 0:1499

Sð~r 3 Þ ¼ 0:0154

Step 4. Rank all the alternatives Ai(i = 1, 2, 3, 4, 5) in accordance with the scores Sð~r i Þði ¼ 1; 2; . . . ; 5Þ of the overall preference values ~r i ði ¼ 1; 2; . . . ; 5Þ : A5 A2 A3 A4 A1 , and thus the most desirable alternative is A5. Based on the IFEHG operator, then, in order to select the most desirable alternative, we can develop an approach to multiple attribute decision making problems with intuitionistic fuzzy information, which can be described as following: e and Step 10 . Utilize the decision information given in matrix R, ~_ ij ¼ a ~^ij e nxj , we get: a

~_ 11 a ~_ 13 a ~_ 21 a ~_ 23 a ~_ 31 a ~_ 33 a ~ a_ 41 ~_ 43 a ~_ 51 a ~_ a 53

¼ ð0:4319; 0:4132Þ; ¼ ð0:1535; 0:7782Þ; ¼ ð0:5863; 0:3265Þ; ¼ ð0:6103; 0:3552Þ; ¼ ð0:5109; 0:4132Þ; ¼ ð0:3659; 0:4687Þ; ¼ ð0:6593; 0:1608Þ; ¼ ð0:1535; 0:5778Þ; ¼ ð0:5109; 0:2427Þ; ¼ ð0:6103; 0:2386Þ;

~_ 12 a ~_ 14 a ~_ 22 a ~_ 24 a ~_ 32 a ~_ 34 a ~ a_ 42 ~_ 44 a ~_ 52 a ~_ a 54

¼ ð0:5171; 0:1679Þ ¼ ð0:0875; 0:7059Þ ¼ ð0:5490; 0:1232Þ ¼ ð0:1779; 0:8037Þ ¼ ð0:4814; 0:2163Þ ¼ ð0:4458; 0:5901Þ ¼ ð0:5171; 0:1679Þ ¼ ð0:1779; 0:8827Þ ¼ ð0:4396; 0:1679Þ ¼ ð0:2975; 0:5901Þ

Then by score functions, we can get:

~_ rð11Þ ¼ ð0:5171; 0:1679Þ; a ~_ rð13Þ ¼ ð0:0875; 0:7059Þ; a ~_ rð21Þ ¼ ð0:5490; 0:1232Þ; a ~_ a ¼ ð0:6103; 0:3552Þ;

~_ rð12Þ ¼ ð0:4319; 0:4132Þ a ~_ rð14Þ ¼ ð0:1535; 0:7782Þ a ~_ rð22Þ ¼ ð0:5863; 0:3265Þ a ~_ a ¼ ð0:1779; 0:8037Þ

~_ rð31Þ ¼ ð0:4814; 0:2163Þ; a ~_ rð33Þ ¼ ð0:3659; 0:4687Þ; a ~_ rð41Þ ¼ ð0:6593; 0:1608Þ; a ~_ a ¼ ð0:1535; 0:5778Þ;

~_ rð32Þ ¼ ð0:5109; 0:4132Þ a ~_ rð34Þ ¼ ð0:4458; 0:5901Þ a ~_ rð42Þ ¼ ð0:5171; 0:1679Þ a ~_ a ¼ ð0:1779; 0:8827Þ

~_ rð51Þ ¼ ð0:6103; 0:2386Þ; a ~_ rð53Þ ¼ ð0:5109; 0:2427Þ; a

~_ rð52Þ ¼ ð0:4396; 0:1679Þ a ~_ rð54Þ ¼ ð0:2975; 0:5901Þ a

rð23Þ

rð43Þ

rð24Þ

rð44Þ

478

X. Zhao, G. Wei / Knowledge-Based Systems 37 (2013) 472–479

e and the Step 20 . Utilize the decision information given in matrix R, IFEHG operator which has associated weighting vector w = (0.20, 0.30, 0.30, 0.2)T, we obtain the overall preference values ~ri of the alternatives Ai (i = 1, 2, . . . , 5). ~r 1 ¼ ð0:2411;0:4664Þ; ~r 2 ¼ ð0:5195; 0:2964Þ; ~r 3 ¼ ð0:4694; 0:3597Þ ~r 4 ¼ ð0:3325;0:3525Þ; ~r 5 ¼ ð0:4761; 0:2238Þ

Step 30 . Calculate the scores Sð~r i Þði ¼ 1; 2; . . . ; 5Þ of the overall intuitionistic fuzzy preference values ~r i ði ¼ 1; 2; . . . ; 5Þ

Sð~r 1 Þ ¼ 0:2253; Sð~r 4 Þ ¼ 0:0200;

Sð~r2 Þ ¼ 0:2231; Sð~r 5 Þ ¼ 0:2523

Sð~r3 Þ ¼ 0:1097

Step 40 . Rank all the alternatives Ai (i = 1, 2, 3, 4, 5) in accordance with the scores Sð~r i Þði ¼ 1; 2; . . . ; 5Þ of the overall preference values ~r i ði ¼ 1; 2; . . . ; 5Þ : A5 A2 A3 A4 A1 , and thus the most desirable alternative is A5. 6. Conclusion In this paper, we investigate the multiple attribute decision making (MADM) problems with intuitionistic fuzzy numbers Then, we first introduce some operations on intuitionistic fuzzy sets, such as Einstein sum, Einstein product, and Einstein exponentiation, and further develop some new Einstein hybrid aggregation operators, such as the intuitionistic fuzzy Einstein hybrid averaging (IFEHA) operator and intuitionistic fuzzy Einstein hybrid geometric (IFEHG) operator, which extend the hybrid averaging (HA) operator [51,52] and the hybrid geometric (HG) operator [51,52] to accommodate the environment in which the given arguments are intuitionistic fuzzy values. Then, we apply the intuitionistic fuzzy Einstein hybrid averaging (IFEHA) operator and intuitionistic fuzzy Einstein hybrid geometric (IFEHG) operator to deal with multiple attribute decision making under intuitionistic fuzzy environments. Finally, some illustrative examples are given to verify the developed approach and to demonstrate its practicality and effectiveness. In the future, we shall continue working in the extension and application of the developed operators to other domains [53–59]. Acknowledgment The work was supported by the National Natural Science Foundation of China under Grant No. 61174149, Natural Science Foundation Project of CQ CSTC of the People’s Republic of China (No. CSTC, 2011BA0035), the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China under Grant Nos. 11XJC630011, 12YJC630314 and the Science and Technology Research Foundation of Chongqing Education Commission under Grant KJ111214. References [1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87– 96. [2] K. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems 33 (1989) 7–46. [3] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–356. [4] R. Yager, Some aspects of intuitionistic fuzzy sets, Fuzzy Optimization and Decision Making 8 (2009) 67–90. [5] Z.S. Xu, Models for multiple attribute decision making with intuitionistic fuzzy information, International Journal of Uncertainty Fuzziness and KnowledgeBased Systems 15 (2007) 285–297. [6] Z.S. Xu, Multi-person multi-attribute decision making models under intuitionistic fuzzy environment, Fuzzy Optimization and Decision Making 6 (2007) 221–236. [7] M.M. Xia, Z.S. Xu, B. Zhu, Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm, Knowledge-Based Systems 31 (2012) 78–88.

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