Some series of intuitionistic fuzzy interactive averaging aggregation operators.

In this paper, some series of new intuitionistic fuzzy averaging aggregation operators has been presented under the intuitionistic fuzzy sets environment. For this, some shortcoming of the existing operators are firstly highlighted and then new operational law, by considering the hesitation degree between the membership functions, has been proposed to overcome these. Based on these new operation laws, some new averaging aggregation operators namely, intuitionistic fuzzy Hamacher interactive weighted averaging, ordered weighted averaging and hybrid weighted averaging operators, labeled as IFHIWA, IFHIOWA and IFHIHWA respectively has been proposed. Furthermore, some desirable properties such as idempotency, boundedness, homogeneity etc. are studied. Finally, a multi-criteria decision making method has been presented based on proposed operators for selecting the best alternative. A comparative concelebration between the proposed operators and the existing operators are investigated in detail.

Background

MCDM is one of the process for finding the optimal alternative from the set of feasible alternatives according to some criteria. Traditionally, it has been generally assumed that all the information which access the alternative in terms of criteria and their corresponding weights are expressed in the form of crisp numbers. But in day-today life, uncertainties play a crucial role in the decision making process. Due to complexities of the system, the decision maker may give their preferences corresponding to each alternative to some certain degree. However, it is obvious that much knowledge in the real world is fuzzy rather than precise and thus their corresponding analysis contains a lot of uncertainties and hence does not give the correct information to the practicing. Such kind of situations is suitably expressed with intuitionistic fuzzy sets (IFSs) (Attanassov 1986) rather than exact numerical values. These days IFSs are one of the most permissible theories to handle the uncertainties and impreciseness in the data than the crisp or probability theory (Garg 2013, 2016a, d; Garg et al. 2014; He et al. 2014b; Li and Nan 2009; Wan et al. 2016a; Xu 2007a, b; Yu 2015a). In the field of MCDM, the primary objective is of the information aggregation process. For this, Yager (1988) proposed the ordered weighted average (OWA) operator by giving some weights to all the inputs according to their ranking positions. Based on its pioneer work, many extensions have been appearing over it and applied it to solve the problems of multi-criteria decision making problems. For instance, Xu and Yager (2006) developed some geometric and Xu (2007a) proposed averaging aggregation operators on IFSs environment including weighted, ordered weighted and hybrid weighted operators. Zhao et al. (2010) combined Xu and Yager’s operators and developed their corresponding generalized aggregation operators. Xia and Xu (2010) proposed a series of intuitionistic fuzzy point aggregation operators based on the generalized aggregation operators (Zhao et al. 2010). He et al. (2014a) proposed an operations based on the principle of probability membership, non-membership and probability heterogenous functions operators. Wang and Liu (2011) and Wang and Liu (2012) proposed some geometric as well as averaging aggregation operator based on weighted and ordered weighted operators for different IFNs under Einstein operations. Zhao and Wei (2013) extended their aggregation operators by using the hybrid average and geometric operators. Apart from them, the various authors have addressed the problem of MCDM by using the different aggregation operators (Fei 2015; Garg 2015, 2016a, b, c, e; Garg et al. 2015; Liu 2014; Li and Ren 2015; Li and Wan 2014; Li 2014; Nan et al. 2016; Robinson and Amirtharaj 2015; Wan and Dong 2015; Wan et al. 2016a, b; Wang and Liu 2011; Xu and Yager 2006; Yu 2013a, b, 2015b; Yu and Shi 2015; Zhou et al. 2012).

It has been observed from the above aggregator operators that they have some drawbacks. For example, if there is an IFS whose at least one grade of non-membership function is zero, then the aggregated IFSs corresponding to the aggregator operators as described by Liu (2014), Wang and Liu (2011, 2013), Xu (2007a), Zhang and Yu (2014), Zhao et al. (2014) etc., have a zero degree of non-membership. This means that the role of the other grades of non-zero non-membership functions does not play any dominant role during the aggregation process. Similarly, if there is at least one degree of membership function to be zero then their corresponding IFSs obtained through geometric aggregator operators have a zero degree of membership functions. In other words, we can say that the effects of the other grades of either membership or non-membership on a corresponding geometric or an averaging aggregator operator does not play any significant role during the aggregation process. Further, it has been observed from above operators that the grades of overall membership (non-membership) functions are independent of their corresponding grades of non-membership (membership) functions. Thus, under such circumstances, the results corresponding to these operators are undesirable and hence does not give the reasonable preference order of the alternative.

Thus the objective of this manuscript is to present some new averaging aggregation operators under the IFSs environment. For this, some new operational laws on IFSs has been defined by considering the degree of hesitation between the grades of membership functions. Based on it, some series of different averaging aggregating operators including weighted average, ordered weighted averaging and hybrid weighted averaging have been proposed. It has been observed from these operators that the existing operators can be deduce from the proposed operators by giving a parameters to be a special numbers. Finally, a MCDM method based on these proposed aggregation operators are presented to show the applicability, utility and validity of the proposed ones. From the studies, it has been concluded that it can properly handle the shortcoming of the existing work and hence give an alternative way to finding the best alternative using an aggregation operators.

Preliminaries

Intuitionistic fuzzy set

An intuitionistic fuzzy set (IFS) A in a finite universe of discourse is given by (Attanassov 1986)where , respectively, be the membership and non-membership degree of the element x to the set A with the conditions , and . For convenience, the pair is called an intuitionistic fuzzy number (IFN) (Xu 2007a). Based on it, a score and accuracy function is defined as and , respectively. In order to compare two two IFNs, and , an order relation between them are summarized as follows (Wang et al. 2009; Xu 2007a).

If then .

If then

If then ;

If then .

t-norm and t-conorm

t-norm (T) and t-conorm () operations are widely used for finding the various arithmetic operations in the IFSs environment. For instance, Xu (2007a) defined the algebraic product, sum, scalar and power operations for three IFNs , and and be a real number, by using t-norm () and t-cornorm () as follows

On the other hand, if we define and then the operations on IFN are known as Einstein t-norm and t-conorm respectively which are defined as below (Wang and Liu 2012)

Hamacher (1978) proposed a more generalized t-norm and t-conorm by defining as and respectively. It is clear from these operations that when then they will reduce to algebraic t-norm and t-cornorm and . Similarly when , they will reduce to Einstein t-norm and t-cornorm respectively as and . Thus, based on these operations, Hamacher sum and product operations are defined for two IFNs and as

and their corresponding aggregation operators have been proposed by Liu (2014) for different IFNs ’s by using weight vector of and and as

The intuitionistic fuzzy Hamacher weighted averaging (IFHWA) operator

The intuitionistic fuzzy Hamacher ordered weighted averaging (IFHOWA) operator where is a permutation of such that for all .

The intuitionistic fuzzy Hamacher hybrid averaging (IFHHA) operator where is the ith largest of the weighted intuitionistic fuzzy values (.

The above operations are very concise and have been widely used by the various authors (He et al. 2014a, b; Liu 2014; Wang and Liu 2012; Xu 2007a; Zhao et al. 2010), but the above operations have several drawbacks. Few of them have listed as below.

Example 1

Let , , , be four IFNs and , is the standardized weight vector corresponding to these IFNs. By utilizing the IFHWA operator to aggregate all these numbers corresponding to we get and for , we get IFHWA . From these results it has been seen that the degree of non-membership is zero and is independent of the parameter . Furthermore, this degree is independent of the degree of other non-membership (those which are nonzero in ’s) and hence these plays an insignificant role during the aggregation process.

Example 2

Let , , and be four IFNs and , is the standardized weight vector of these numbers. Then based on IFHWA operator we get the aggregated IFNs are by taking and when . On the other hand, if we replace and IFNs with and then their corresponding aggregated IFN become when and when . Hence, it has been seen that the degree of non-membership values of aggregated IFN becomes independent of the change of the degree of membership values. Therefore, it is inconsistent and hence does not give a correct information to the decision maker.

Therefore, the existing operators, as proposed by Liu (2014) are invalid to rank the alternative and hence there is a need to pay more attention on these issues.

Some improved weighted averaging aggregator operators

In this section, we have define some improved aggregation operator by using an improved operational laws defined as below.

Definition 1

Let and , be three IFNs and be a real number then some basic arithmetic operations between them have been defined by using Hamacher norms as follows

Weighted average aggregation operator

Definition 2

Let is the set of IFNs and be its weight vector such that and , and , ifthen IFHIWA is called an intuitionistic fuzzy Hamacher interactive weighting averaging operator.

Theorem 1

Letbe the collection of IFNs, then

Proof

When then , and henceThus, results hold for . Assume that result holds for , i.e.,By using the operational laws as given in Definition 1 for we haveHence complete the proof.

Lemma 1

(Xu 2007a) Letforand, thenwith equality holds if and only if.

Corollary 1

Letbe a collections of IFNs then the operators IFHWA and IFHIWA have the following relation:

Let and , and be its corresponding weight vectors thenandThus, where equality holds if and only if and .

Therefore,If then for every , we haveIf i.e. then by the condition , we have and , thus the accuracy function . Thus in this case, from the definition of score function, it follows thatHence,where that equality holds if and only if .

From this corollary it has been concluded that the proposed IFHIWA operator shows the decision maker’s more optimistic attitude than the existing IFHWA operator (Liu 2014) in aggregation process.

Example 3

Let and be four IFNs and be the weight vector of ’s, i.e. , , , , , , ; then for , we haveThus, it has been concluded that

Theorem 2

Ifbe an IFNs, , then the aggregated value by using IFHIWA operator is also an IFN i.e.

Since be an IFNs for , then by definition of IFN, we haveTake, , we haveAlsoThus . On the other hand,

AlsoThus .

Finally,Hence, . Therefore, the aggregated IFN obtained by IFHIWA operator is again an IFN.

Example 4

If we apply the proposed IFHIWA operator on Example 1 then corresponding to , we get the aggregated IFNs as while for we have . Therefore, it has been seen that there is a non-zero degree of non-membership of the overall aggregated IFNs even if at least one of their corresponding grades of IFNs is zero. Thus, the others grades of non-membership function of IFNs play a dominant role during the aggregation process in the proposed operator.

Example 5

If we apply the proposed IFHIWA operator to aggregate the different IFNs as given in Example 2 then we get aggregated IFN are when and when . On the other hand, if we apply proposed aggregated operator on modified IFNs then we get for and for . Thus, the change of membership function will affect on the degree of non-membership functions and is non-zero. Therefore, there is a proper interaction between the degree of membership and non-membership functions and hence the results are consistent and more practical than the existing operators results.

Now, based on Theorem 1, we have some properties of the proposed IFHIWA operator for a collection of IFNs and is the associated weighted vector satisfying and .

Property 1

(Idempotency) If for alli, then

Since and , so by Theorem 1, we have

Property 2

(Boundedness) Letandthen

Let then ; thus, f(x) is decreasing function. Since , for all then for all i, i.e. , for all i. Let is the associated weighted vector satisfying and , then for all i, we have

Thus,On the other hand, let then so g(y) is decreasing function on (0,1]. Since for all i then i.e. for all . Let is the associated weighted vector satisfying and , then for all i, we haveThus,Also

Take , , and . Let then Eqs. (3) and (4) are transformed into .

So, and . If and then by order relation between two IFNs, we have

Property 3

(Monotonicity) Ifand,be two collections of IFNs such thatfor alli, then

Proof of this property is similar to above, so we omit here.

Property 4

(Shift-invariance) Ifbe another IFN, then

As IFNs, soTherefore,

Property 5

(Homogeneity) Ifbe a real number, then

Since be an IFNs for . Therefore, for , we haveTherefore,Hence,

Property 6

Letandbe two collections of IFNs , then

As and be two collections of IFNs, thenTherefore,Hence,

Property 7

Let,be an IFNs and Ifbe any real number, then

By using the Properties 1, 5 and 6, we get the required proof, so it is omitted here.

Ordered weighted averaging operator

Definition 3

Suppose there is a family of IFNs for and , ifwhere is the weight vector associated with IFHIOWA, is a permutation of such that for any i. Then IFHIOWA is called intuitionistic fuzzy Hamacher interactive ordered weighted averaging operator.

Theorem 3

Letbe the collection of IFNs, then based on IFHIOWA operator, the aggregated IFN can be expressed as

Especially, for i.e all are reduced to , respectively then Eq. (5) is reduced to the following formwhich becomes a fuzzy OWA operator of dimension n to aggregate fuzzy information.

The proof is similar to Theorem 1.

Corollary 2

The IFHIOWA operator and IFHOWA operator have the following relation for a collections of IFNs

As similar to those of the IFHIWA operator, the IFHIOWA operator has some properties as follows.

Property 8

Letbe a collection of IFNs andbe the weighting vector associated with IFHIOWA operator,andthen we have the following.

Idempotency: If allare equal i.e.,for alli, then

Boundedness: whereand

Monotonicity: Ifand,be two IFNs such thatfor alli, then

Shift-invariance: Ifbe another IFN, then

Homogeneity: If be a real number, then

The proof is similar to IFHIWA properties, so we omit.

Example 6

Let , and be three IFNs and be the position weighted vector then based on their score functions, we get their ordering as and hence , and . Then for different value of , the aggregated IFNs by the proposed and existing operators are summarized in Table 1.

Table 1

Comparison with IFHIOWA and existing operators

	γ=1		γ=2		γ=3
	IFOWA (Xu 2007a)	Proposed	IFHOWA (Wang and Liu 2012)	Proposed	IFHOWA (Liu 2014)	Proposed
IFN	⟨0.1865,0.2555⟩	⟨0.1865,0.2570⟩	⟨0.1836,0.2561⟩	⟨0.1836,0.2579⟩	⟨0.1812,0.2564⟩	⟨0.1812,0.2586⟩
Score	0.0690	−0.0705	−0.0726	−0.0743	−0.0752	−0.0774

Thus, it is clear from these results that

Hybrid weighted averaging operator

Definition 4

Suppose there is a family of IFNs, and , ifwhere is the weighted vector associated with IFHIHWA, is the weight vector of such that . Let is the ith largest of the weighted IFNs ), n is the number of IFNs and is a permutation of , such that for any i, then, function IFHIHWA is called intuitionistic fuzzy Hamacher interactive hybrid weighted averaging operator.

From the Definition 4, it has been concluded that

It firstly weights the IFNs by the associated weights and multiplies these values by a balancing coefficient n and hence get the weighted IFNs .

It reorders the weighted arguments in descending order , where is the ith largest of .

It weights these ordered weighted IFNs by the IFHIWA weights and then aggregates all these values into a collective one.

Theorem 4

Letbe an IFNs,then by IFHIHWA operator, the aggregated IFN becomes

The proof is similar to Theorem 1, so it is omitted here.

Corollary 3

The IFHIHWA and IFHWA operators satisfies the following inequalityfor a collections of IFNs’s.

Similar to those of the IFHIWA and IFHIOWA operators, the IFHIHWA operator has also follows the same properties as described in Property 8.

Decision making approach using proposed operators

MCDM problem is one of the fast and challenging method for every decision maker for finding the best alternative among the set of feasible one. For this, let be a set of n different alternatives which have been evaluate under the set of m different criteria by the decision maker(s). Assume that the decision maker(s) give their preferences in terms of IFNs , where and represents the degree that the alternative satisfies and doesn’t satisfies the attribute given by the decision maker respectively such that and . Hence, MCDM problem can be concisely expressed in an intuitionistic fuzzy decision matrix . Various steps used in the proposed methodology for MCDM are explained as follows:

Obtain the normalized intuitionistic fuzzy decision matrix. In this step, if there are different types of criteria namely benefit (B) and cost (C) then we may transform the rating values of B into C by using the following normalization formula: where is the complement of .

Aggregated assessment of alternatives. Based on the decision matrix, as obtained from step 1, the overall aggregated value of alternative , under the different choices of criteria is obtained by using IFHIWA or IFHIOWA or IFHIHWA operator and get the overall value .

Compare each alternative: Based on the overall assessment of each alternative , a score value of each index are computed.

Ranking the alternative: Rank the alternative according to the descending value of their score values and hence select the most desirable alternative.

Illustrative example

The above mentioned approach has been illustrated through a case study on multiple criteria decision making problem. For this, assume that a certain company has a sum of money and they want to invest it somewhere. After carefully looking in the market scenario they have decided to invest the money in the following three companies.according to the following four major criteria:The weight vector corresponding to each criteria is given by the committee as . Assume that these alternatives are being assessed by the decision makers and give their preferences in the form of the IFNs. Then following are the step as followed by the proposed approach for accessing the best company.

is a car company,

is a food company, and

is a computer company.

: The risk analysis,

: The growth analysis,

: The social-political impact analysis,

: The environmental impact analysis and

: The development of the society.

By IFHIWA operator

As, it has been observed that there are different types of criteria so the preferences corresponding to each alternative , w.r.t. each criteria , are obtained in the form of normalized intuitionistic fuzzy decision matrix as given below.

Utilize the IFHIWA operator corresponding to to compute the overall assessment of each alternative as

The scores values corresponding to each is.

Since thus we have . Hence, the best financial strategy is i.e. to invest in the food company.

By IFHIHWA operator

In order to aggregate these different IFNs by using IFHIHWA operator, the following steps are utilize.

Use the normalized fuzzy decision matrix as given in Eq. (7).

Compute the IFNs , where we get

Now, reorders these IFNs based on their score function, and get ordered weighted IFNs as

Thus, finally utilize these ordered weighted IFNs and the weight vector corresponding to each criteria, the aggregated value have been obtained corresponding to each alternative as

The score values corresponding to above is

Thus, and their corresponding alternative order are . Therefore, the best company for investing the money is (car company).

Comparison with the existing methodologies

By Xu (2007a) approach

If we utilize IFWA (Xu 2007a) operator to aggregate these IFNs then we get their corresponding aggregated values asand hence order relation is which corresponds to .

By Wang and Liu (2012) approach

If we utilize IFEWA (Wang and Liu 2012) operator to aggregate these IFNs then we get their corresponding aggregated valuesand hence ranking of these alternatives are and thus their corresponding alternative ranking order are .

Sensitivity analysis

To analyze the effect of on the most desirable alternatives on the given attributes, we use the different values of in the proposed approach to rank the alternatives. The corresponding score values and their ranking order are summarized in Table 2 along with the results as obtained by Liu (2014) approach. From this table, it has been analyzed that with the increase of the parameter , their score values corresponding to each alternative is decrease which is in accordance with the results of as obtained from Liu (2014) approach. The variations of the ranking of these three companies with respect to the value of parameter by the proposed IFHIWA and IFHIHWA operator are shown in Figs. 1 and 2 respectively. Furthermore, it has been obtained that the score value of each alternative by the proposed approach is less than the existing approach which shows the optimistic attitude nature to the decision makers’ which validates the Corollary 1.

Table 2

Ordering of the alternatives for different

γ	Approach	Score function by IFHIWA				Score function by IFHIOWA				Score function by IFHIHWA
		S(x_1)	S(x_2)	S(x_3)	Ranking	S(x_1)	S(x_2)	S(x_3)	Ranking	S(x_1)	S(x_2)	S(x_3)	Ranking
0.1	Proposed	0.1386	0.1298	0.0060	x_1≻x_2≻x_3	0.1386	0.1298	0.0060	x_1≻x_2≻x_3	0.1790	0.1260	0.0294	x_1≻x_2≻x_3
	Liu (2014)	0.1984	0.1878	0.0464	x_1≻x_2≻x_3	0.1984	0.1878	0.0464	x_1≻x_2≻x_3	0.2280	0.2227	0.0811	x_1≻x_2≻x_3
0.5	Proposed	0.1214	0.1183	−0.0039	x_1≻x_2≻x_3	0.1214	0.1183	−0.0039	x_1≻x_2≻x_3	0.1625	0.1107	0.0211	x_1≻x_2≻x_3
	Liu (2014)	0.1722	0.1627	0.0288	x_1≻x_2≻x_3	0.1722	0.1627	0.0288	x_1≻x_2≻x_3	0.2105	0.2072	0.0746	x_1≻x_2≻x_3
1.0	Proposed	0.1099	0.1099	−0.0114	x_2≻x_1≻x_3	0.1099	0.1099	−0.0114	x_2≻x_1≻x_3	0.1530	0.0996	0.0157	x_1≻x_2≻x_3
	Liu (2014)	0.1588	0.1498	0.0178	x_1≻x_2≻x_3	0.1588	0.1498	0.0178	x_1≻x_2≻x_3	0.2023	0.2007	0.0721	x_1≻x_2≻x_3
1.5	Proposed	0.1029	0.1045	−0.0163	x_2≻x_1≻x_3	0.1029	0.1045	−0.0163	x_2≻x_1≻x_3	0.1479	0.0925	0.0127	x_1≻x_2≻x_3
	Liu (2014)	0.1514	0.1425	0.0112	x_1≻x_2≻x_3	0.1514	0.1425	0.0112	x_1≻x_2≻x_3	0.1982	0.1978	0.0715	x_1≻x_2≻x_3
2.0	Proposed	0.0981	0.1007	−0.0199	x_2≻x_1≻x_3	0.0981	0.1007	−0.0199	x_2≻x_1≻x_3	0.1443	0.0876	0.0108	x_1≻x_2≻x_3
	Liu (2014)	0.1467	0.1377	0.0068	x_1≻x_2≻x_3	0.1467	0.1377	0.0068	x_1≻x_2≻x_3	0.1960	0.1964	0.0716	x_1≻x_2≻x_3
3.0	Proposed	0.0918	0.0957	−0.0247	x_2≻x_1≻x_3	0.0918	0.0957	−0.0247	x_2≻x_1≻x_3	0.1400	0.0810	0.0087	x_1≻x_2≻x_3
	Liu (2014)	0.1408	0.1316	0.0011	x_1≻x_2≻x_3	0.1408	0.1316	0.0011	x_1≻x_2≻x_3	0.1937	0.1941	0.0725	x_1≻x_2≻x_3
5.0	Proposed	0.0851	0.0902	−0.0300	x_2≻x_1≻x_3	0.0851	0.0902	−0.0300	x_2≻x_1≻x_3	0.1362	0.0738	0.0073	x_1≻x_2≻x_3
	Liu (2014)	0.1349	0.1253	−0.0049	x_1≻x_2≻x_3	0.1349	0.1253	−0.0049	x_1≻x_2≻x_3	0.1926	0.1923	0.0753	x_1≻x_2≻x_3
10	Proposed	0.0785	0.0847	−0.0354	x_2≻x_1≻x_3	0.0785	0.0847	−0.0354	x_2≻x_1≻x_3	0.1337	0.0663	0.0074	x_1≻x_2≻x_3
	Liu (2014)	0.1293	0.1192	−0.0107	x_1≻x_2≻x_3	0.1293	0.1192	−0.0107	x_1≻x_2≻x_3	0.1941	0.1925	0.0783	x_1≻x_2≻x_3
25	Proposed	0.0736	0.0804	−0.0396	x_2≻x_1≻x_3	0.0736	0.0804	−0.0396	x_2≻x_1≻x_3	0.1340	0.0596	0.0105	x_1≻x_2≻x_3
	Liu (2014)	0.1252	0.1147	−0.0151	x_1≻x_2≻x_3	0.1252	0.1147	−0.0151	x_1≻x_2≻x_3	0.2006	0.1962	0.0803	x_1≻x_2≻x_3
50	Proposed	0.0717	0.0788	−0.0412	x_2≻x_1≻x_3	0.0717	0.0788	−0.0412	x_2≻x_1≻x_3	0.1355	0.0563	0.0108	x_1≻x_2≻x_3
	Liu (2014)	0.1237	0.1130	−0.0167	x_1≻x_2≻x_3	0.1237	0.1130	−0.0167	x_1≻x_2≻x_3	0.2074	0.1918	0.0761	x_1≻x_2≻x_3
100	Proposed	0.0707	0.0779	−0.0421	x_2≻x_1≻x_3	0.0707	0.0779	−0.0421	x_2≻x_1≻x_3	0.1369	0.0535	0.0112	x_1≻x_2≻x_3
	Liu (2014)	0.1229	0.1122	−0.0175	x_1≻x_2≻x_3	0.1229	0.1122	−0.0175	x_1≻x_2≻x_3	0.2052	0.1888	0.0731	x_1≻x_2≻x_3

Fig. 1

Score value versus parameter by IFHIWA operator

Fig. 2

Score value versus parameter by IFHIHWA operator

Conclusion

In this article, the objective of the work is to present some series of an averaging aggregation operators by using hamacher operations. For this, firstly shortcoming of the various existing operations and their corresponding aggregator operators is highlighted. These shortcoming has been resolved by defining a new set of operational laws on the intuitionistic fuzzy set environment by considering the degree of interaction or hesitation between the grades of functions. Based on these laws, some series of an averaging aggregation operators namely IFHIWA, IFHIOWA and IFHIHWA have been proposed. The desirable properties corresponding to each operator has been discussed. It has been observed from the operators that some existing operators IFWA and IFEWA are taken as a special case of the proposed operators. These operators have been applied to solve the MCDM problem for showing the substantiality and effectiveness of the approach. From the proposed approach, it has been concluded that it contain almost all of arithmetic aggregation operators for IFNs based on different and hence proposed operators are more general and flexible.