Simpliﬁed neutrosophic sets and their applications in multi-criteria group decision-making problems

As a variation of fuzzy sets and intuitionistic fuzzy sets, neutrosophic sets have been developed to represent uncertain, imprecise, incomplete and inconsistent information that exists in the real world. Simpliﬁed neutrosophic sets (SNSs) have been proposed for the main purpose of addressing issues with a set of speciﬁc numbers. However, there are certain problems regarding the existing operations of SNSs, as well as their aggregation operators and the comparison methods. Therefore, this paper deﬁnes the novel operations of simpliﬁed neutrosophic numbers (SNNs) and develops a comparison method based on the related research of intuitionistic fuzzy numbers. On the basis of these operations and the comparison method, some SNN aggregation operators are proposed. Additionally, an approach for multi-criteria group decision-making (MCGDM) problems is explored by applying these aggregation operators. Finally, an example to illustrate the applicability of the proposed method is provided and a comparison with some other methods is made.

IFSs take into account the membership degree, nonmembership degree and degree of hesitation simultaneously.Therefore, they are more flexible and practical when addressing fuzziness and uncertainty than traditional FSs.Moreover, in some actual cases, the membership degree, non-membership degree and hesitation degree of an element in IFSs may be not a specific number.Hence, they were extended to interval-valued intuitionistic fuzzy sets (Atanassov & Gargov, 1989).Furthermore, in order to handle the situations where people are hesitant when expressing their preferences regarding objects in the decision-making process, hesitant fuzzy sets were introduced by Torra and Narukawa (2009) and Torra (2010).
Although the FSs theory has been developed and generalised, it cannot deal with all uncertainty in different reallife problems.For instance, certain types of uncertainty, such as indeterminate and inconsistent information, cannot be dealt with.For example, when an expert is asked for his or her opinion about a certain statement, he or she may say that the possibility that the statement is true is 0.5, that it is false is 0.6 and the degree that he or she is not sure is 0.2 (Wang, Smarandache, Zhang, & Sunderraman, 2010).This issue is beyond the scope of FSs and IFSs, therefore some new theories are required.Smarandache (1999Smarandache ( , 2003) ) proposed neutrosophic logic and neutrosophic sets (NSs).An NS is a set where each element of the universe has the degrees of truth, indeterminacy and falsity and it lies in ]0 − , 1 + [, the nonstandard unit interval (Rivieccio, 2008).Clearly, this is an extension of the standard interval [0, 1] of IFSs.Moreover, the uncertainty presented here, i.e., the indeterminacy factor, is independent of truth and falsity values, while the incorporated uncertainty is dependent on the degree of belongingness and non-belongingness of IFSs (Majumdar & Samant, 2014).Furthermore, the aforementioned example of NSs can be expressed as x(0.5, 0.2, 0.6).
However, without a specific description, NSs are difficult to apply in real-life situations.Hence, single-valued neutrosophic sets (SVNSs) were proposed, which are a variation of NSs (Majumdar & Samant, 2014).Furthermore, the information energy of SVNSs, their correlation and correlation coefficient and the decision-making method that used them were also proposed (Ye, 2013).Additionally, Ye (2014a) also introduced the concept of simplified neutrosophic sets (SNSs), which can be described by three real numbers in the real unit interval [0,1], and proposed an MCDM method using the aggregation operators of SNSs.Moreover, Majumdar and Samant (2014) introduced a measure of entropy of an SVNS.Wang, Smarandache, Zhang, and Sunderraman (2005) and Lupiáñez (2009) proposed the concept of interval neutrosophic sets (INSs) and proposed the set-theoretic operators of INSs.Furthermore, Ye (2013bYe ( , 2014c) ) proposed the similarity measures between SVNSs and INSs, which were based on the relationship between similarity measures and distances.
However, in some cases, the SNSs' operations (Ye, 2014a) may be impractical.For instance, the sum of any element and the maximum value should be equal to the maximum value, but this does not occur when using the SNSs' operations (Ye, 2014a).Therefore, the operations and comparison approach between simplified neutrosophic numbers (SNNs) and the aggregation operators for SNNs are re-defined in this paper.Moreover, a multi-criteria group decision-making (MCGDM) method is subsequently established based on the proposed operators.
The rest of paper is organised as follows.In Section 2, the properties of the t-norm and t-conorm as well as the concepts and operations of NSs and SNSs are briefly introduced.In Section 3, the operations and comparison approach for SNNs are defined on the basis of IFSs.In Section 4, the aggregation operators of SNNs are provided and a decision-making method making use of them is developed for SNSs.In Section 5, an illustrative example is presented to test the proposed method, and a sensitivity analysis and comparison analysis are also provided.Finally, in Section 6, the conclusions are drawn.

Preliminaries
In this section, some basic concepts and definitions related to NSs, including the t-norm and t-conorm, and the definitions and operations of NSs and SNSs are introduced.All of these will be utilised in this paper.

The t-norm and t-conorm
The t-norm and its dual, the t-conorm, play an important role in constructing the operations and average operators of NSs.Here, some related basic concepts are introduced.
Then an algebraic tconorm and t-norm are obtained: . Then an Einstein tconorm and t-norm are obtained: S(x, y) = x+y

NSs and SNSs
In this section, the definitions and operations of NSs and SNSs are introduced.
Definition 4 (Smarandache, 1999): Let X be a space of points (objects), with a generic element in X, denoted by x.An NS A in X is characterised by a truth-membership function T A (x), an indeterminacy-membership function I A (x) and a falsity-membership function F A (x). T A (x), I A (x) and F A (x) are standard or non-standard subsets of ]0 Definition 5 (Smarandache, 1999): Since it is difficult to apply NSs to practical problems, Ye (2014a) reduced NSs of non-standard intervals into SNSs of standard intervals that would preserve the operations of NSs.
Definition 6 (Ye, 2014a): Let X be a space of points (objects), with a generic element in X, denoted by x.An NS A in X is characterised by T A (x), I A (x) and F A (x), which are subintervals/subsets in the standard interval [0, 1], that is, which is called an SNS.In particular, if X has only one element, A = < T A (x), I A (x), F A (x) > is called an SNN.For convenience, an SNN is denoted by A = < T A , I A , F A >. Clearly, SNSs are a subclass of NSs.
Definition 8 (Ye, 2014a): Let A and B be two SNSs.For any x ∈ X, the following operations are defined: (1) There are some limitations related to Definition 8 and these are now outlined.
(1) In some situations, operations, such as A + B and A • B, might be impractical.This can be demonstrated in the example below.
Example 1: Let A = < 0.5, 0.5, 0.5 > and B = < 1, 0, 0 > be two SNNs.Clearly, B = < 1, 0, 0 > is the larger of these SNNs.Theoretically, the sum of any number and the maximum number should be equal to the maximum one.However, according to Definition 8, A + B =< 1, 0.5, 0.5 > = B; therefore, the operation ' + ' cannot be accepted.Similar contradictions exist in other operations of Definition 8, and thus those defined above are incorrect.
(1) The correlation coefficient of SNSs (Ye, 2013), which is based on the operations of Definition 8, cannot be accepted in some special cases.

Example 2:
Let A 1 = < 0.8, 0, 0 > and A 2 = < 0.7, 0, 0 > be two SNNs, and B = < 1, 0, 0 > be the largest one of the SNNs.According to the correlation coefficient of SNSs (Ye, 2013), W 1 (A 1 , B) = W 2 (A 2 , B) = 1 can be obtained, but this does not indicate which one is the best.However, it is clear that A 1 is superior to A 2 .
(1) In addition, the similarity measure for SNSs (Ye, 2014b), which is based on the operations of Definition 8, cannot be accepted in special cases.

The operations and comparison method for SNNs
In this section, the novel operations and comparison method for SNNs are developed based on t-norm and t-conorm.Downloaded by [Central South University], [jian-qiang Wang] at 00:23 17 March 2016

The operations for SNNs
According to the operations of IFNs, which are based on the algebraic t-norm and t-conorm (Beliakov et al., 2011;Xu, 2007) and Einstein t-norm and t-conorm (Wang & Liu, 2011, 2012), it can be seen that these operations are all based on different t-norms and t-conorms; therefore, the novel operations of two SNNs can be defined as follows.
Definition 9: Let A = < T A , I A , F A > and B = < T B , I B , F B > be two SNNs, and λ > 0. The operations for SNNs are defined based on the Archimedean t-conorm and t-norm and are as follows: ( If their generator k is assigned a specific form, then the specific operations of SNNs will be obtained.Let k(x) = − log(x), therefore Example 4: Based on Example 1 and the operations in Definition 9, let k(x) = − log(x), k −1 (x) = e −x , l(x) = −log(1 − x) and l −1 (x) = 1 − e −x , and then the following results are obtained: which can overcome the drawbacks outlined earlier.
Apparently, SNSs are the extension of IFSs.Let I A = 0, I B = 0 and T A + F A ≤ 1.Then two SNNs A =< T A , I A , F A > and B =< T B , I B , F B > are reduced to IFNs.According to Definition 9, if k(x) = − log(x), then the operations coincide with those of IFNs (Beliakov et al., 2011;Xu, 2007Xu, , 2008Xu, , 2010;;Yager, 2009).This indicates that the same principles of SNSs in Definition 9 also apply to IFSs.In fact, when the indeterminacy factor i is replaced by π = 1 − T − F , the NS becomes an IFS.

Theorem 1:
Let A = < T A , I A , F A >, B = < T B , I B , F B > and C = < T C , I C , F C > be three SNNs, therefore the following equations are true: Proof: Equations ( 1) and ( 2) are obvious, but the others can also be proved: (3) (5) The proof is therefore complete.

The comparison method of SNNs
Based on the score function and accuracy function of IFNs (Xu, 2007(Xu, , 2008(Xu, , 2010;;Yager, 2009), the score function, accuracy function and certainty function of an SNN are defined as follows.
Definition 10: Let A =< T A , I A , F A > be an SNN, and then the score function s(A), accuracy function a(A) and certainty function c(A) of an SNN are defined as follows: (1) The score function is an important index in ranking SNNs.For an SNN A, the bigger the truth-membership T A is, the greater the SNN will be; furthermore, the smaller the indeterminacy-membership I A is, the greater the SNN will be; similarly, the smaller the false-membership F A is, the greater the SNN will be.For the accuracy function, the bigger the difference between truth and falsity, the more affirmative the statement is.As for the certainty function, the certainty of any SNN positively depends on the value of truth-membership T A .
On the basis of Definition 10, the method for comparing SNNs can be defined as follows.Definition 11: Let A and B be two SNNs.The comparison method can be defined as follows: ( Example 5: Based on Example 2 and Definition 10, Example 6: Based on Example 3 and Definition 10, s(A 1 ) < s(A 2 ), then A 2 A 1 , i.e., A 2 is superior to A 1 , which also avoids the shortcomings discussed in Example 3.

The aggregation operators of SNNs and their applications to MCGDM problems
In this section, by applying the SNNs' operations, the aggregation operators of SNNs are presented and a method for MCGDM problems that utilises the proposed aggregation operators is proposed.

Theorem 2:
Let A j = < T A j , I A j , F A j > (j = 1, 2, . . ., n) be a collection of SNNs, and w = (w 1 , w 2 , . . ., w n ) be the weight vector of A j (j = 1, 2, . . ., n), with w j ≥ 0 (j = 1, 2, . . ., n) and n j =1 w j = 1.Then their aggregated result using the SNNWA operator is also an SNN, and where k is the additive generator of the Archimedean t-norm that is used in the operations of SNNs and l Thus, the aggregated result using the SNNWA operator in Theorem 2 can be represented by (3) , then Theorem 2 can be represented by Proof: By using the mathematical induction on n: (1) let n = 2, since i.e., Equation (2) holds for n = k + 1.Thus, Equation ( 2) holds for all n.
The proof is therefore complete.It is clear that the SNNWA operator has the following properties.

Property 1:
Let A j = < T A j , I A j , F A j > (j = 1, 2, . . ., n) be a collection of SNNs.
(1) Since (2) For any The proof is therefore complete.

Theorem 3:
Let A j = < T A j , I A j , F A j > (j = 1, 2, . . ., n) be a collection of SNNs, and w = (w 1 , w 2 , . . ., w n ) be the vector of A j (j = 1, 2, . . ., n), with w j ∈ [0, 1] and n j =1 w j = 1.Then their aggregated result using the SNNWG operator is also an SNN, and Let k(x) = − log(x), and then Theorem 3 can be represented by x ), and Theorem 3 can be denoted by Proof: Theorem 3 can be proved by the mathematical induction method, and the process is omitted here.

Property 2:
Let A j = < T A j , I A j , F A j > (j = 1, 2, . . ., n) be a collection of SNNs.

Theorem 4:
Let A j = < T A j , I A j , F A j > (j = 1, 2, . . ., n) be a collection of SNNs, and ω = (ω 1 , ω 2 , . . ., ω n ) be the aggregation-associated vector such that ω j ≥ 0 (j = 1, 2, . . ., n) and n j =1 ω j = 1.Their aggregated result using the SNNOWA operator is also an SNN, and Downloaded by [Central South University], [jian-qiang Wang] at 00:23 17 March 2016 Let k(x) = − log(x), and then Theorem 4 can be represented by x ), then Theorem 4 can be represented by where A σ (j ) is the jth largest value according to the total order Proof: Theorem 4 can be proved by the mathematical induction method, and the process is omitted here.
Property 3: Let A j = <T A j , I A j , F A j > (j =1, 2, . . ., n) be a collection of SNNs.
(1) Idempotency: if all A j (j = 1, 2, . . ., n) are equal, i.e., A j = A, for all j ∈ {1, 2, . . ., n}, then SNNOWA (2) Boundedness: assume A − = < min j T A j , max j I A j , max j F A j > and A + = < max j T A j , min j I A j , min j F A j > for all j ∈ {1, 2, . . ., n}, and then Proof: The proof is similar to that for Property 1.

Definition 15:
Let A j = < T A j , I A j , F A j > (j = 1, 2, . . ., n) be a collection of SNNs, and SNNOWG : and thus the SNNOWG operator is called an simplified neutrosophic number ordered weighted geometric operator of dimension n, where A σ (j ) is the jth largest value and ω = (ω 1 , ω 2 , . . ., ω n ) is the aggregation-associated vector such that .(j = 1, 2, . . ., n) and n j =1 ω j = 1.Theorem 5: Let A j = <T A j , I A j , F A j > (j = 1, 2, . . ., n) be a collection of SNNs, and ω = (ω 1 , ω 2 , . . ., ω n ) be the aggregation-associated vector such that ω j ≥ 0 (j = 1, 2, . . ., n) and n j =1 ω j = 1.Then their aggregated result using the SNNOWG operator is also an SNN, and Assume k(x) = − log(x), and then Theorem 5 can be represented by x ), and then Theorem 5 can be denoted by where A σ (j ) is the jth largest value according to the total order Proof: Theorem 5 can be proved by the mathematical induction method, and the process is omitted here.
(2) Boundedness: assume A − = < min j T A j , max j I A j , max j F A j > and A + = <max j T A j , min j I A j , min j F A j > for all j ∈ {1, 2, . . ., n}, and then Proof: The proof is similar to that for Property 1.
Proof: Theorem 6 can be proved by the mathematical induction method, and the process is omitted here.

Property 5:
Let A j = < T A j , I A j , F A j > (j = 1, 2, . . ., n) be a collection of SNNs.
(1) Idempotency: if all A j (j = 1, 2, . . ., n) are equal, i.e., A j = A, for all j ∈ {1, 2, . . ., n}, then (2) Boundedness: assume A − = <min j T A j , max j I A j , max j F A j > and A + = < max j T A j , min j I A j , min j F A j > for all j ∈ {1, 2, . . ., n}, and then Proof: The proof is similar to that for Property 1.
Proof: Theorem 7 can be proved by the mathematical induction method and the process is omitted here.

Property 6:
Let A j = < T A j , I A j , F A j > (j = 1, 2, . . ., n) be a collection of SNNs.
(2) Boundedness: assume A − = < min j T A j , max j I A j , max j F A j > and A + = < max j T A j , min j I A j , min j F A j > for all j ∈ {1, 2, . . ., n}, and then Proof: The proof is similar to that for Property 1.

Theorem 8:
Let A j = < T A j , I A j , F A j > (j = 1, 2, . . ., n) be a collection of SNNs, and w = (w 1 , w 2 , . . ., w n ) be the weight vector of A j (j = 1, 2, . . ., n), with λ > 0, w j ≥ 0 (j = 1, 2, . . ., n) and n j =1 w j = 1.Then their aggregated result using the SNNHOWG operator is also an SNN, and Assume k(x) = − log(x), and then Theorem 8 can be represented by , and then Theorem 8 can be denoted by where Apparently, if λ = 1, then the GSNNWA operator is reduced to the SNNWA operator.
Similarly, it can be proved that the GSNNWA and GSNNWG operators have the same properties as the SNNWA operator.

The MCGDM method based on the aggregation operators of SNNs
Assume that there are n alternatives A = {a 1 , a 2 , . . ., a n } and m criteria C = {c 1 , c 2 , . . ., c m }, and the weight vector of criteria is w = (w 1 , w 2 , . . ., w m ), where w j ≥ 0 (j = 1, 2, . . ., m), m j =1 w j = 1.Suppose that there are k decision-makers D = {d 1 , d 2 , . . ., d k }, whose corresponding weight vector is λ = (λ 1 , λ 2 , . . ., λ k ).Let R = (a k ij ) n×m be the simplified neutrosophic decision matrix, where is the value of a criterion, denoted by SNNs, where T a k ij indicates the truth-membership function that the alternative a i satisfies the criterion c j for the kth decision-maker, I a k ij indicates the indeterminacymembership function that the alternative a i satisfies the criterion c j for the kth decision-maker, and F a k ij indicates the falsity-membership function that the alternative a i satisfies the criterion c j for the kth decision-maker.This method uses the aggregation operators of SNNs in order to solve the MCGDM problem mentioned above.
In the following steps, a procedure for ranking and selecting the most desirable alternative(s) is provided.
Step 1: Aggregate the SNNs of each decision-maker.
Utilise the GSNNWA or GSNNWG operator to aggregate the SNNs of each decision-maker, and the individual value of the alternative a i (i = 1, 2, . . ., n) can be obtained.
Step 2: Aggregate the SNNs of all decision-makers.
Step 4: Rank the alternatives.
According to Definition 11, an order of priority for all the alternatives a i (i = 1, 2, . . ., m) could be obtained and the best one can be chosen.

An illustrative example
In this section, an example of MCGDM problems is used to demonstrate the applicability and effectiveness of the proposed decision-making method.
Let us consider a decision-making problem adapted from Lupiáñez (2009).There is an investment company, which wants to invest a sum of money in the best option.There are four possible alternatives to invest the money: (1) A 1 is a car company; (2) A 2 is a food company; (3) A 3 is a computer company and (4) A 4 is an arms company.The investment company must make a decision according to the following three criteria: (1) C 1 is the risk analysis; (2) C 2 is the growth analysis and (3) C 3 is the environmental impact analysis, where C 1 and C 2 are of the maximising type, and C 3 is a minimising type.The weight vector of the criteria is given by w = (0.35, 0.25, 0.4).Suppose that there are three decision-makers, {d 1 , d 2 , d 3 } whose corresponding weight vector is λ = (0.5, 0.3, 0.2).The four possible alternatives are to be evaluated under these three criteria and are in the form of SNNs for each decision-maker, as shown in the following simplified neutrosophic decision matrix D:

The decision-making procedure based on SNNs
Step 1: Utilise the GSNNWA or GSNNWG operator to obtain the SNNs for each decision-maker.Because the aggregation results based on the GSNNWA and GSNNWG operators are different, they are calculated separately.For convenience, the operations of SNNs are based on the Algebraic t-conorm and t-norm and λ = 1.
By using the GSNNWA operator, the alternatives matrix A WA can be obtained: Step 2: Aggregate the SNNs of all decision-makers.
Utilise the SNNHOWA or SNNHOWG operator to obtain the overall SNN y i for all alternatives a i (i = 1, 2, . . ., n), with the SNNHOWA operator, the overall SNN can be obtained as follows: Step 3: Calculate the score function value, accuracy function value and certainty function value.
For the alternatives matrix H WA , using Definition 10, the score function of H WA can be obtained: s(H WA ) = (0.708, 0.816, 0.757, 0.842).
For the alternatives matrix H WG , using Definition 10, the score function of H WG is shown as follows: s(H WG ) = (0.713, 0.809, 0.741, 0.819).
It is clear that the score values are different; therefore, there is no need to compute the accuracy function value and certainty function value.
Step 4: Obtain an order of priority for the alternatives and choose the best one.
According to Definition 11 and the results in Step 3, for A WA , the final ranking is A 4 A 2 A 3 A 1 .Clearly, the best alternative is A4.
Similarly, for A WG , the final ranking is A 4 A 2 A 3 A 1 .And the best alternative is A 1 .
If k(x) = log( 2−x x ) for A WA , then the ranking of the four alternatives is still A 4 A 2 A 3 A 1 , and the ranking is

The sensitivity analysis
In the following discussion, the influence of λ on the ranking of alternatives is investigated.The results can be found in Table 1.
From Table 1, it can be seen that the rankings of the alternatives are slightly different as λ changes.However, for the GSNNWA and GSNNWG operators, the best alternative is A 4 , while the worst alternative is A 1 or A 3 .The authors believe that λ can be considered as a reflection of the decision-makers' preferences.Based on Archimedean t-conorm and t-norm, Beliakov et al. (2011) strated that the operators obtained by using Lukasiewicz t-norm are consistent with the ones for ordinary FSs.In order to calculate the actual aggregation values of the alternatives, different aggregation operators can be used.It is also found that those aggregation operators are all based on different t-conorms and t-norms and are used to deal with different relationships of the aggregated arguments, which can provide more choices for decision-makers.

The comparison analysis and discussion
In order to validate the feasibility of the proposed decisionmaking method based on the aggregation operators of SNNs, a comparison study is now conducted.
(1) The similarity measure proposed by Majumdar and Samant (2014) is going to be utilised in this discussion.By using the proposed aggregation operators, the aggregated values that were presented in Step 1 of Subsection 5.1.can be obtained.According to the similarity measure (Majumdar & Samant, 2014), the ideal alternative is A * = < 1, 0, 0 >, thus the following results could be obtained.
With the GSNNWA operator (λ = 1), Therefore, the final ranking is still A 4 A 2 A 3 A 1 , and the best one is A 4 .
(1) Similarly, if the operations and correlation coefficient are used (Ye, 2013a), then the final ranking is A 2 A 4 A 3 A 1 , and the best one is A 2 .
(2) If the operations and cross-entropy are utilised (Ye, 2014b), then the final ranking is A 2 A 4 A 3 A 1 , and the best one is A 2 .
From the analysis above, it can be seen that the result obtained by using the similarity measure (Majumdar & Samant, 2014) is A 4 A 2 A 3 A 1 , which is consistent with that obtained by using the proposed method, while the final ranking obtained by utilising the correlation coefficient and cross entropy (Ye, 2014(Ye, , 2014b) ) is A 2 A 4 A 3 A 1 , which is different from that obtained by using the proposed method.The reason for this phenomenon is that the proposed operations and aggregation operators have been used before utilising the similarity measure (Majumdar & Samant, 2014).However, the operations, correlation coefficient and similarity measure (Ye, 2013(Ye, , 2014b) ) have been proved to be impractical in Section 2. Thereby, the differences were amplified in the aggregation values because of the criteria weights and the final ranking of all alternatives was influenced adversely by the similarity measure or correlation coefficient.By contrast, the proposed operations could overcome these shortcomings as were discussed in Examples 1-3.Therefore, the best alternative is A 4 , which is more precise and reliable.

Conclusions
SNSs can be applied in solving problems with uncertain, imprecise, incomplete and inconsistent information that exist in scientific and engineering situations.However, as a new branch of NSs, there is not enough extant research on SNSs.In particular, the existing literatures have not proposed using the aggregation operators and the MCGDM methods for SNSs.Based on related research achievements in IFSs, the operations of SNSs were defined in this paper, and an approach to solve MCGDM problems with SNNs was proposed.Additionally, the aggregation operators of SNNWA and SNNWG, SNNOWA and SNNOWG, SNNHOWA and SNNHOWG, and GSNNWA and GSNNWG were provided.Thus, an MCGDM method was established based on the proposed operators.By using the comparison method, the ranking of all alternatives can Downloaded by [Central South University], [jian-qiang Wang] at 00:23 17 March 2016 be determined and the best one can easily identified.An illustrative example demonstrated the applicability of the proposed decision-making method.Although there is no consensus on the best way to sequence SNNs, when compared to the MCDM method for SNSs (Majumdar & Samant, 2014;Ye, 2013Ye, , 2014b)), the illustrative example showed that the final result produced by the method proposed in this paper is more precise and reliable than the results produced by existing methods.Therefore, the method proposed in this paper can provide a reliable basis for SNSs.In future research, the relative measures of SNSs will be studied and applied to other fields.

Table 1 .
introduced some operations for IFSs, proposed two general concepts for constructing other types of aggregation operators for IFSs, which extended the existing methods, and demon-Downloaded by [Central South University], [jian-qiang Wang] at 00:23 17 March 2016 The results of the sensitivity analysis.