Multi-Attribute Decision Making Based on Intuitionistic Fuzzy Power Maclaurin Symmetric Mean Operators in the Framework of Dempster-Shafer Theory

: It is well known that there are some unfavorable shortcomings in the ordinary operational rules (OORs) of intuitionistic fuzzy number (IFN), and there exists a close and forceful connection between the intuitionistic fuzzy set (IFS) and Dempster-Shafer Theory (DST). We can utilize this relationship to present a transparent and fruitful semantic framework for IFS in terms of DST. In the framework of DST, an IFN can be converted into a basic probability assignment (BPA) and operations on IFNs can be represented as operations on a belief interval (BI), which can break away from the revealed shortcomings of the OORs of the IFN. Although there are many operators to aggregate the IFN, the operator to aggregate the BPA is rare. The Maclaurin symmetric mean (MSM) operator has the advantage of considering interrelationships among any number of attributes. The power average (PA) operator can reduce the inﬂuences of extreme evaluation values. In addition, for measuring the di ﬀ erence between IFNs, we replace the Hamming distance and Euclidean distance with the Jousselme distance (JD). In this paper, we develop an intuitionistic fuzzy power MSM (IFPMSM DST ) operator and an intuitionistic fuzzy weighted power MSM (IFPWMSM DST ) operator in the framework of the DST and provide their favorable properties. Then, we propose a novel method based on the proposed operators to solve multi-attribute decision-making (MADM) problems without intermediate defuzziﬁcation when their attributes and weights are both IFNs. Finally, some examples are utilized to demonstrate that the proposed methods outperform the previous ones.

Generally, the MADM techniques based on OAs are superior to than the traditional MADM techniques because they can obtain the comprehensive values of alternatives by aggregating all attribute values and then rank the alternatives.Thus, it is significant and valuable to research the OAs and then deal with the MADM problems.
The PA operator, initially investigated by Yager [24], can reduce the influences of the extreme evaluation values to a great extent by given different weights.When extreme evaluation values appear, the PA operator can supply them a smaller weight by allowing attribute values to support and complement each other, so that the influences of extreme evaluation values are reduced greatly.This favorable characteristic is extremely useful in real MADM problems, so the PA operator has attracted considerable attention from researchers.The PA operator has also been successfully extended to IFS.Xu [32] developed an intuitionistic fuzzy weighted PA (IFWPA) operator and intuitionistic fuzzy weighted geometric PA (IFWGPA) operator.He et al. [33] investigated a generalized interaction IFPA (GIIFPA) operator and weighted GIIFPA (WGIIFPA) operator.Jiang and Wei [11] presented an intuitionistic fuzzy evidential power average (IFEPA) operator in the framework of the Dempster-Shafer theory (DST).
In some MADM problems, for obtaining convincing aggregate results, we should be concerned about the interrelationships between attributes.In this case, the BM operator [26] and HM operator [29] were developed to solve these MADM problems.Then, Xu and Yager [27] further extended the BM operator to IFS and presented an intuitionistic fuzzy BM (IFBM) operator and weighted IFBM (WIFBM) operator.He and He [28] proposed the extended intuitionistic fuzzy interaction BM (EIFIBM) operator and weighted EIFIBM (WEIFIBM) operator.Liu and Chen [29] presented the intuitionistic fuzzy Archimedean HM (IFAHM) operator and weighted IFAHM (WIFAHM) operator.However, the BM operator and HM operator just take into account the interrelationships between attributes.In most MADM problems, we should fully incorporate the interrelationships among attributes.Obviously, the MSM operator can help us to solve these problems.The great advantage of the MSM operator is that it can neatly capture interrelationships among attributes by assigning a different value to the parameter κ.Recently, many researchers have payed considerable attention to extending the MSM operator to IFS.Qin and Liu [31] first investigated the intuitionistic fuzzy MSM (IFMSM) operator and the weighted IFMSM (WIFMSM) operator.Liu and Liu [8] developed an intuitionistic fuzzy interaction MSM (IFIMSM) operator and the weighted IFIMSM (WIFIMSM) operator.
Undoubtedly, we usually encounter some extraordinary MADM problems where the attributes are interrelated with each other, and some extreme evaluation values are provided.However, the OAs for the interrelationships among attributes, and reducing the influence of extreme evaluation values, are rare, so it is essential to combine the MSM operator with the PA operator in processing IFNs.However, with respect to the OAs for IFNs, we should not only consider the function, but also pay close attention to the ORs of the IFNs.
In addition, it is noteworthy that the existing OAs of IFNs with attribute weights represented by IFNs have not be defined, and this is also a greater disadvantage of IFS.Because in many real DM problems, it is troublesome for decision-makers to provide the exact significance of attributes by using real number, we can apply the weights denoted by IFNs to decrease the loss of information.
DST, which was initially introduced by Dempster [34] and then developed and expanded by Shafer [35], is a favorable tool for processing inaccurate or ambiguous information.DST gives BPA, which can express the occurrence rate of attributes in basic events.While DST is introduced, the belief function (BF) and plausibility function (PF) are likewise defined, and this constitutes the BI of the focal element (FE).The BI expresses the belief and uncertainty of the FE.In [9,10], we have seen that there is a strong and close connection between IFS and DST.This strong and close connection makes it certainly possible to immediately apply the aggregation rules (ARs) of DST to aggregate the attribute information of the alternatives represented by IFNs in a real MADM process.In the framework of DST, an IFN can be converted into a BPA, and operations on IFNs can be denoted as operations on the belief interval (BI), which can overcome the revealed drawbacks of the OORs of IFNs and get more convincing results.Although there are many operators to aggregate IFNs, the operators to aggregate the BPA is rare.
Based on the above discussions, the main purposes of this paper are as follows: (1) For overcoming the revealed drawbacks of OORs of IFN and getting more convincing aggregate results, we convert an IFN into a BI and replace operations on IFNs with operations on BI; (2) For utilizing ORs of BI to develop some PMSM operator for IFNs in the framework of DST, we convert an IFN into a BPA and replace Hamming distance and Euclidean distance with Jousselme distance (JD); (3) For further reducing the loss of information, we use the presented operators to solve MADM problems without intermediate defuzzification when attributes and their weights are all IFNs.
Therefore, we will firstly propose the intuitionistic fuzzy power MSM (IFPMSM DST ) operator and intuitionistic fuzzy weighted power MSM (IFPWMSM DST ) operator in the framework of DST; then, based on the IFPMSM DST operator and IFPWMSBM DST operator, we develop a new MADM method.By comparing with the previous methods based on a intuitionistic fuzzy evidential power aggregation (IFEPA) operator [11], a weighted intuitionistic fuzzy MSM (WIFMSM) operator [28], and an extended weighted intuitionistic fuzzy interaction Bonferroni mean (EWIFIBM) operator [31], the advantages of the proposed methods are discussed.
The rest of this paper is organized as follows: In Section 2, we review concepts of the IFS and provide a critical analysis of OORs of the IFS, then give an interpretation of the IFS in the framework of DST.In Section 3, we propose IFPMSM DST operator and IFPWMSM DST operator based the PA operator and MSM operator.In Section 4, we develop a novel method with IFNs based on IFPMSM DST operator and IFPWMSM DST operator in the framework of DST.In Section 5, we utilize some numerical examples to demonstrate the reasonability and flexibility of presented operators.In Section 6, we discuss the conclusions.

IFSs
Definition 1 [2].Let A = { α i |i = 1, 2, • • • , t} be a fixed set, and then the IFS B on A can be defined as follows: Usually, we use β = u β , v β to represent an IFN.
(1) The Equation ( 1) is not a constant operation-that is to say, (2) Equation ( 2) is not a constant operation, i.e., β 1 < β 2 cannot always generate (β (4) IFWAM is not always monotone with respect to the SF and AF.In other words, In some studies [9,10], we have found out that there is a very close connection between DST and IFS.With the help of this connection, we can provide transparent and fruitful semantics for IFN in terms of DST.Therefore, to reinforce the performance of operations on IFNs, we rewrite the definition and operations of IFS in the framework of DST, which break away from the above-listed limitations.

IFS in the Framework of Dempster-Shafer Theory
Let us assume Φ to be the set of κ mutually exclusive and exhaustive objects, which corresponds to κ hypotheses or propositions.Φ is called the frame of discernment and defined as follows: Φ = 1, 2, 3, • • • , γ .Γ(Φ) is known as the power set of Φ containing all the possible subsets of Φ and defined as follows: Φ) consists of 2 κ elements representing the event "the object is in X".
Note that Γ(Φ) includes φ and the condition ϑ(φ) = 0 is required, but the subsets of Φ for which the mapping does not assume a 0 value are defined focal elements in classical DST.ϑ(X) denotes the degree of evidence support for the proposition of the object belongs to X.In a word, ϑ(X) is a measure of the belief attributed exactly to X, and to none of the subsets of X.
Definition 4 [35,36].Given a BPA ϑ on Φ, the plausibility function Pl can be defined as: where Υ is the complementary set of X.
Therefore, the belief interval (BI) can be represented by interval [Bel(X), Pl(X)].This may be explained as the interval enclosing the "true probability" of X.
In order to measure the similarity between two sets, we present a detailed description of JD between two bodies of evidence as follows: Definition 5 [35,36].Let Φ be a frame of discernment including κ mutually exclusive and exhaustive hypothesis, and let Λ Γ(Φ) be the space produced by all the subsets of Φ.A BPA is a vector → ϑ of Λ Γ(Φ) with coordinates ϑ(X i ) such that: In the above definition, ϑ(X i ) = 0 is not necessarily required.
Definition 6 [36].Let ϑ 1 and ϑ 2 be two BPAs on the same frame of discernment Φ, including κ mutually exclusive and exhaustive hypotheses.The JD between ϑ 1 and ϑ 2 can be defined as follows: where From Definition 6, another description of d BPA is as follows: where We know that the u β (γ), v β (γ) and π β (γ) of IFN β can represent a BPA of DST, respectively [9,10], so we can completely denote the IFS in the framework of DST based on above information of DST.In practice, when solving the DM problem with the information of IFNs, we implicitly confront three hypotheses as follows: γ ∈ β, γ β and γ ∈ β or γ β (the case of hesitation).Therefore, these three hypotheses can be expressed as True (γ ∈ β), False (γ β), and (True or False) (the case of hesitation).In such a case, u β (γ) signifies the probability or evidence of γ ∈ β, i.e., ϑ(True) = u β (γ).By analogy, Therefore, we rewrite the definition of IFS in the framework of DST.
are the MD and NMD of γ ∈ Υ to B, respectively, and For convenience, we represent an IFN in the framework of DST by In order to enhance the performance of operations on IFNs, Dymova and Sevastjanov [9,10] redefined the operational rules on IFN in the framework of DST. (2) (3 (5) From above operational Rules ( 24)-( 27), we can get the IFWAM DST and IFWGM DST operators in the framework of DST.Let be BI and w i be the weight of β i , For BI β = [u, 1 − v], the SF and AF in the framework of DST are defined by following form: Further, the order relation between h 1 and h 2 is denoted as follows: (1) (2 It is easy to discover that there is a very close connection between Rules ( 7) and ( 8) and Rules ( 32) and (33).However, it is not suitable that we use Rules ( 7) and ( 8) to compare BIs and use Rules (32) and (33) be three Bis.In this way, it is easy to prove that ( 24)-( 29) have the following properties: (1) (2) (3 (5) ( The above new operational rules of IFNs in the framework of DST can overcome the drawbacks and shortcomings of the OORs of IFNs.Theorem 1. Equation ( 24) is a constant operation.
Proof.Theorem 2 is similar to Theorem 1; the proof is omitted here.Theorem 3. Equation ( 26) is persistent under multiplication.
Proof.Theorem 3 is similar to Theorem 1; the proof is omitted here.Theorem 4. The IFWAM DST has monotonicity.
Proof.Theorem 5 is similar to Theorem 4; the proof is omitted here.

PA Operator
Yager [24] initially developed the PA operator, which permits the attribute values to assist and balance each other in the aggregation process.

MSM Operator
Qin and Liu [31] first proposed the MSM operator, which can capture the interrelationships among any number of multi-input attribute arguments by changing parameter values.

The IFPMSM DST Operators
In this part, we initially propose the intuitionistic fuzzy (IF) power average (IFPA DST ) operator, the IF power weighted average (IFPWA DST ) operator, and the IF MSM (IFMSM DST ) operator in the framework of the DST.Subsequently, we combine the MSM operator with the PA operator and extend them to IFNs to present the IF power MSM (IFPMSM DST ) operator and IF power weighted MSM (IFPWMSM DST ) operator.
For simplifying Equation (42), we indicate where (θ 1 , θ 2 , • • • , θ t ) is the power-weighted vector of the β 1 , β 2 , • • • , and β t .Evidentially, θ η ≥ 0 and t η=1 θ η = 1; then, Equation (43) can be simplified as follows: The aggregated result based on the IFPA DST operator is also a BI and Proof.By the operational rules of IFNs in the framework of DST, we get , where 1 < η < t.Then, we get The aggregated result based on the IFPWA DST operator is also a BI and where Sup β η , β τ denotes the support degree for β η from β τ , which satisfied the properties of Definition 9, where 1 < η < t, 1 < τ < t and η τ.
For simplifying Equation (41), we indicate where Equation ( 46) can be simplified as follows: The aggregated result based on the IFPA DST operator is also a BI and The proof of Theorem 7 is similar to that of Theorem 6.

MSM Operator for IFNs in the Framework of DST
The IFMSM DST operator of β 1 , β 2 , • • • , and β t can be defined as follows: The aggregated result based on the IFMSM DST operator is also a BI and Proof.
By the operational rules of IFNs in the framework of DST, we get The IFPMSM DST operator of β 1 , β 2 , • • • , and β t can be defined as follows: where and Sup β η , β τ denotes the support degree for β η from β τ , which satisfied the properties of Definition 9, where 1 < η < t, 1 < τ < t and η τ.
By the operational rules of IFNs in the framework of DST, we get tθ , and then we can get Next, some desirable properties of the IFPMSM DST operator are proposed.
, based on the Theorem 9, we can get Proof. Since From Equation (33), That is We should see that the IFPMSM DST operators do not have an idempotent property.However, if we make a slight modification (multiply by C κ t 1 κ ), the modified IFPMSM DST (MIFPMSM DST ) operator will be an idempotent operator, as follows: (61) A similar expression was first obtained by Xu and Yager in [37].It is evidential that we can obtain the same ordering result of the alternatives by using the IFPMSM DST operator and the MIFPMSM DST operator in a real DM.In addition, the IFPMSM DST operator does not satisfy monotonicity because the d BPA will also make a difference if the attribute values are changed.
Next, some special cases of the IFPMSM DST operator are investigated by considering some diverse values of κ.
(1) When κ = 1, the IFPMSM DST operator become the IFPA DST operator, that is According to Equation (53), we discover that the IFPMSM DST operator has a definite fault, i.e., it does not take the importance of the attributes into account.However, in many practice DM environments, the weights of attributes play a crucial role in the aggregate process.Therefore, we next present the weighted IFPMSM DST (IFPWMSM DST ) operator.
Likewise, the IFPWMSM DST operator has also some desirable properties, shown as follows: Theorem 13 (commutativity).
We can also see that the IFPWMSM DST operators do not have an idempotent property.However, if we make a slight modification (multiply by C κ t 1 κ ), the modified IFPWMSM DST (MIFPMSM DST ) operator will be an idempotent operator, as follows: The advantages of operational rules of IFNs in the framework of DST is that we can solve MADM problems without intermediate defuzzification when input arguments and their weights are both IFNs, which are not defined in ordinary operators of IFNs.Therefore, we firstly define the IFWPMSM DST operator where the attribute weights are IFN as follows.
is the corresponding BI of ωi .
be a decision matrix, where r ij is an evaluation value, which takes the form of a crisp number or IFN, given by decision makers for the alternative e i ∈ E with respect to the attribute c i ∈ C. The purpose of this problem is to select the optimal alternatives.
The method for this MADM problem is shown in detail as follows: Step 1. Normalize the decision matrix R = r ij m×n .Only the cost criterion c j , r ij is normalized by using the converted formula (Note: The value converted using r ij = v ij , u ij is still denoted by r ij ).
Step 2. Convert IFN r ij to BI β ij .
Step 3. Calculate Sup where d BPA β ij , β iε is JD between β ij and β iε in the framework of the DST.
Step 4. Calculate T β ij of β ij by the other β iε , that is, Step 5. Calculate θ i or is not a normalized interval weight vector (NIWV), then convert into a normalized weight vector (see [38]), that is, (1 Step 6. Apply the proposed IFPWMSM DST operator or IFPWMSM DST operator to acquire the comprehensive value Step 7. Calculate the SF DST β i , AF DST β i by Equation (32) and Equation (33), respectively.
Step 8. Rank the alternatives and obtain the best alternative.

Practical Application
In this section, we apply an illustrated example of the share-bike evaluation to demonstrate the process of the novel method.Example 6.By increasing the value of customer experience, a share-bike operation company plans to put new share-bikes into market.Now, there are four different share-bikes (E = {e 1 , e 2 , e 3 , e 4 }) from four different share-bike manufacturers, but the share-bike operation company does not know which one is best.Therefore, they invite a tester to evaluate the four different share-bikes.Suppose four attributes are considered, containing safety (c 1 ), comfortability (c 2 ), convenience (c 3 ) and aesthetic (c 4 ).The weight vector ω of the attribute is ω = (0.4,0.3, 0.2, 0.1) T .The assessment value r ij of criterion c j ( j = 1, 2, 3, 4) with the alternative e i (i = 1, 2, 3, 4) takes the form of the IFN, and the collected and processed decision matrix is constructed, as shown in Table 1.In this section, we present the detailed calculation process of the novel method based on the IFWPMSM DST operator.
Step 1: Normalize the IFN matrix R = r ij m×n .Because the four attributes are beneficial, it is not essential to perform normalization.
Step 8: Rank the alternatives and obtain the best alternative.Because SF DST β 1 > SF DST β 2 > SF DST β 3 > SF DST β 4 , the ranking order is e 1 e 2 e 3 e 4 , and the best share-bike is e 1 .

The Influence of the Parameter κ on Ranking Results
Further, to analyze the influence of parameter κ on the ranking results, we assign a distinct parameter κ in the presented novel method to solve the above example, and the ranking orders are shown in Table 4. From Table 4, we can see that the ranking orders of the four alternatives with different κ parameters are different.However, the best share-bike does not change by a different parameter κ, which is still e 1 .This is, in all probability, because of the fact that the proposed novel method allows for more interacted attributes with an increase of the value of parameter κ.When κ = 1, the proposed novel method does not take interrelationship among attributes into account, and the ranking order is separate from the ones when κ = 2, κ = 3, and κ = 4. Clearly, this can illuminate the significance of taking interrelationship among attributes into account because there is a universal interrelationship among more than two attributes in the practice DM environment.
In addition, we also can find that the SF DST β decreases with an increase of value of parameter κ.Based on this case, parameter κ can be used as the risk preference of the tester.For example, if the tester is risk-averse, then he/she can select a smaller value for parameter κ.Under normal circumstances, κ = [n/2] is a suitable value, where n is the number of attributes and the symbol [•] is the round function.

The Verification of the Effectiveness
To demonstrate the plausibility and validity of the presented novel method based on the IFPWMSM DST operator, we deal with the same share-bike problem in Section 5.1 by applying the four existing methods, Jiang and Wei's method [11], based on the intuitionistic fuzzy evidential power aggregation (IFEPA) operator, He and He's method [28], based on the extended weighted intuitionistic fuzzy interaction Bonferroni mean (EWIFIBM) operator, Qin and Liu's method [31], based on the weighted intuitionistic fuzzy MSM (WIFMSM) operator, where we let κ = 2 for the presented novel method, and Qin and Liu's method [28]; let λ = 1, p = 1, and q = 1 for He and He's method [31].The ranking orders of different methods are shown in Table 5.

Score Values Ranking Orders
Jiang and Wei's method [11]  From Table 5, we get a desirable outcome, that is, the ranking order based on proposed novel method is same as existing three methods [11,28,31].Therefore, the proposed novel method is effective.

The Advantages Compared with the Existing Methods
In Section 5.3, the effectiveness of the proposed novel method is verified.However, owing to the same ranking orders, it is not possible to highlight visually the advantages of the proposed novel method and limitations of the existing some method in [11,28,31].Accordingly, we apply the proposed novel method and the existing three methods in [11,28,31] to deal with new numerical practice examples.

Considering the Interrelationship among Attributes
Example 7. In order to maintain the long-term stability of their high-quality talent, a company wants to lease a dorm to them.Now, there are four alternatives (E = {e 1 , e 2 , e 3 , e 4 }) from four different residential districts.Suppose four attributes are considered, consisting of cost (c 1 ), comfortability (c 2 ), convenience (c 3 ), and living spaces (c 4 ).The weight vector ω of the attribute is ω = (0.4,0.1, 0.2, 0.3) T .The assessment value r ij of criterion c j ( j = 1, 2, 3, 4) with alternative e i (i = 1, 2, 3, 4) takes the form of IFN, and the collected and processed decision matrix is constructed, as shown in Table 6.The ranking orders are shown in Table 7. From Table 6, it is easy to see that when κ = 1 the ranking order based on the presented method is the same as the one based on Jiang and Wei's method [11], i.e., e 3 e 2 e 1 e 4 .To put it other way, when κ = 1, the presented method does not take into consideration the interrelationship of attributes.In this instance, the presented method is similar to Jiang and Wei's method [11], which simply offers a weighted average function.It is obvious that these ranking orders may not be reasonable, because, in this example, there is a dependable interrelationship between cost, comfort, and living spaces.In a real DM environment, we should consider this interaction among attributes.At the same time, it is also easy to find that when κ = 2 and κ = 3, the ranking orders based on presented method is the same as the one based on Qin and Liu's method [31] and He and He's method [28], i.e., e 3 e 2 e 4 e 1 .Obviously, when κ = 2 and κ = 3, the presented method considers this interaction among attributes, which is similar to Qin and Liu's method [31] and He and He's method [28].Undoubtedly, these ranking orders are more reasonable.

Score Values Ranking Orders
Jiang and Wei's method [11] based on IFEPA operator

Reducing the Influence of Extreme Evaluation Values
Example 8.In many cases, due to the preferences of decision makers, extreme evaluation values of attributes may be provided, i.e., values that are too high or too low.Thus, it is possible to get some unreasonable ranking orders.To illustrate this case, based on Example 7, we change the value of r 11 to 0.99, 0.01 and change the value of r 44 to 0.01, 0.01 .Then, we obtain a new decision matrix, which is shown in Table 8.The ranking orders are shown in Table 9.From Table 9, it is easy to see that too high an evaluation value 0.99, 0.01 and too low an evaluation value 0.01, 0.01 have pivotal effects on the ranking orders based on Qin and Liu's method [31] and He and He's method [28].Their ranking orders are changed from e 3 e 2 e 4 e 1 to e 1 e 3 e 2 e 4 , and the best alternative is also changed from e 3 to e 1 .Evidentially, the ranking order may be unreasonable.In other words, in real DM problems, decision-makers may become a manipulator by giving some extreme and biased evaluation values.Nevertheless, the ranking orders based on Jiang and Wei's method [11] and the presented method are still reasonable.Although some ranking orders have changed, their best alternative is still the same as the one acquired in Example 7, i.e., e 3 and the influence of extreme evaluation values is not very strong.
Obviously, the reason for the favorable ranking orders is that the PA operator can significantly reduce the influence of extreme evaluation values by inputting different weights.When decision-makers give extreme evaluation values, the PA operator can give these attributes a relatively smaller weight by the support degrees (SDs) between attributes.Under such circumstances, the influence of extreme evaluation values on ranking orders fades.

Score Values Ranking Orders
Jiang and Wei's method [11] based on IFEPA operator Cannot be counted Cannot be ranked He and He's method [28] based on EWIFIBM operator Cannot be counted Cannot be ranked Qin and Liu's method [31] based on WIFMSM Cannot be counted Cannot be ranked The proposed method based on IFPWMSM DST operator S From Table 10, we can see that only the proposed method based on the IFPWMSM DST operator can give a ranking order, i.e., e 3 e 2 e 4 e 1 .Consequently, the operation laws of the IFS in the framework of DST proposed in this paper can augment the function of AOs.
On the strength of the above three examples, the limitations of the existing methods [11,28,31] are analyzed and summarized as follows: (1) With regard to method [11], on the one hand, this method does not take into account the interrelationships of attributes.In Example 7, we point out that in some real circumstances, it is meaningful to consider the interrelationships of attributes, but this method can only deal with MADM problems in which attributes are independent of each other.On the other hand, because there are not variable parameters, this method cannot manifest the decision-makers' subject preference, so it does not apply to some experts with risk attitudes.

→ ϑ 1 and → ϑ 2
are the BPAs according to Definition 5 and D _ is a 2 κ * 2 κ matrix whose elements are D are the weights of β i and β i , ω is the weight of β, and t i=1

3. 1 .
PA Operator for IFNs in the Framework of DST Definition 11.Let B

and t η=1 θ η = 1 ,
based on Equation (57) and the boundedness of the MSM operator, we can get

4 .
A Novel MADM Method with IFNs in the Framework of DST In this section, a new MADM method based on the proposed IFPWMSM DST operator and IFPWMSM DST operator is developed to solve the MADM problems with IFNs in the framework of DST.Considering a MADM problem, there are m alternatives, denoted by E = {e 1 , e 2 , • • • , e m } and n attributes, denoted by C = {c 1 , c 2 , • • • , c n }.If the attribute weights are crisp numbers, then the weight vector is denoted by ω

5. 4 . 3 .
The Attribute Weights can be Denoted by IFNs Example 9. From Section 2, we know that the ordinary operation laws of IFS may lead to unreasonable ranking orders because of the unfavorable properties and because the AOs presented by IFNs are not developed.Undoubtedly, these are critical defects in the ordinary operation laws of the IFS.However, the operation laws of the IFS in the framework of the DST proposed in this paper can overcome these critical problems by solving the MADM problems without intermediate defuzzification when the attributes and their weights are IFNs.To illustrate the advantage of this proposed method based on the IFPWMSM DST operator, and based on Example 7, we initially change ω = (0.4,0.1, 0.2, 0.3) T to ŵ = ( 0.3, 0.3 , 0.2, 0.6 , 0.1, 0.4 , 0.3, 0.5 ) T .The corresponding interval weights of ŵ are ŵ = ([0.3,0.7], [0.2, 0.4], [0.1, 0.6], [0.3, 0.5]).From Step 5 in Section 4, we know that ŵ is NIWV.The ranking orders are listed in Table

Table 1 .
The decision matrix represented by intuitionistic fuzzy number (IFNs).

Table 2 .
The belief intervals (Bis) from the IFNs.

Table 5 .
The ranking orders of different methods for Example 6.

Table 6 .
The decision matrix represented by IFNs.

Table 7 .
The ranking orders of the different methods for Example 7.

Table 8 .
The changed decision matrix represented by IFNs based on Example 7.

Table 9 .
The ranking orders of the different methods for Example 8.

Table 10 .
The ranking orders of different methods for Example 9.