Power Muirhead mean operators of interval-valued intuitionistic fuzzy values in the framework of Dempster–Shafer theory for multiple criteria decision-making

Multiple criteria decision-making (MCDM) based on interval intuitionistic fuzzy value (IVIFV) is a process of aggregating decision criteria represented by multiple interval-valued intuitionistic fuzzy numbers to select the optimal alternative. Among them, an aggregation operator is an indispensable tool, and the properties of an aggregation operator directly affect the decision results. Existing aggregation operators based on IVIFV have satisfactory results in eliminating the correlation between criteria and removing the influence of outliers on the results. However, there are some unreasonable results due to some undesired properties of IVIFVs. In this paper, IVIFV operation under the Dempster-Shafer theory (DST) framework is applied to combine the power average and Muirhead mean operators and interval intuitionistic fuzzy power Muirhead mean operators under DST framework are presented. Then a method based on the presented operators for MCDM problems is proposed. Finally, a set of numerical experiments are conducted to demonstrate the proposed method. The experimental results suggest that the proposed method not only retains the robustness of the power average operator and the capability of the Muirhead mean operator, but also eliminates a shortcoming that existing interval intuitionistic fuzzy operators cannot handle the case where the weights are in IVIFVs.


Introduction
Multiple criteria decision-making (MCDM) is the process to rank desirable alternatives through explicitly evaluating multiple criteria. To determine desirable alternatives, MCDM generally includes two critical tasks (Greco et al. 2016;Qin et al. 2020), specifically (1) quantifying the considered criteria of each alternative and (2) synthetically aggregating the performance of each alternative with respect to the quantified criteria.
For the first problem of multi-criteria decision-making, Zedeh (1996) proposed an important tool, named fuzzy set, which introduces membership degrees to quantify the considered criteria and is sufficient for certain applications (Fu et al. 2021;Pramanik et al. 2021). However, only using membership degrees as criteria cannot separately express the negative and refusal degrees. To this end, Atanassov (1986) proposed an intuitionistic fuzzy set (IFS), which also considers a non-membership degree to further extend the fuzzy set. Due to the increasing complexity of decisionmaking problems, more uncertain information is introduced into the real world, so it is difficult to quantify the membership and non-membership of each alternative with accurate numbers. Based on this, Atanassov (1999) introduced an interval-valued intuitionistic fuzzy set (IVIFS), where membership degree and non-membership degree are both represented by intervals. Such interval can measure objectively the neutral hesitancy degree of the evaluators. Because of such capability, various uncertain decisionmaking fields based on IVIFS have emerged, e.g., multiobjective planning , and sentiment classification (Einstein et al. 1935), and cluster analysis (Xu 2009). Many works are focusing on the theory and application of IVIFS, which mainly include the following three aspects: (1) basic theory, such as operation criterion (Xu 2007;Xu 2007), score function (Xu 2007;Hung and Wu 2002), distance measurement (Xu 2010), and similarity measurement (Ren and Wang 2015). (2) Extended MCDM methods for IVIFS, such as multi-objective linear/nonlinear programming (Mesa et al. 2017) and statistical convergent sequence spaces. (3) Aggregation operators to aggregating IVIFVs, such as power average operator (He et al. 2013), Muirhead mean operator (Muirhead 1902), Maclaurin symmetric mean operator (Sun et al. 2016), power Bonferroni mean operator , and Heronian mean operator (Yu and Wu 2012). Although interval-valued intuitionistic fuzzy number can be a good candidate standard of quantization scheme, its operation has several undesired properties (Qinet al. 2020;Wang et al. 2022). Firstly, the inequality addition does not meet the monotonicity, secondly, the semantics of the operation result is poor, and finally, the interval intuitionistic fuzzy number cannot be brought into the operation as a power. The undesired properties of interval intuitionistic fuzzy numbers may lead to unreasonable results (Wang et al. 2022;Qin et al. 2020), and the operator based on IVIFV is not reliable in theory. Therefore, Dymova et al. (2016) extended interval intuitionistic fuzzy numbers to the framework of Dempster-Shafer theory (DST) to solve the defects of its operation. This paper will also use interval intuitionistic fuzzy numbers based on the DST framework to quantify the criteria of each scheme.
Dempster-Shafer theory (DST) was first proposed by Dempster (Shafer and Glenn 1976;Dempster 1967) and was further extended by Shafer (Dempster and Arthur 2008). DST has a great advantage in dealing with inaccurate or vague information Gao et al. 2019;Qin et al. 2020;Zhong et al. 2021). Under the DST framework, the basic probability assignment (BPA) is used to represent the probability of occurrence of the standard in a basic event, while the confidence function and likelihood function are used to form a belief interval. The confidence function and likelihood function represent the focal element confidence and uncertainty in the event, respectively. As mentioned in (Dymova et al. 2016), interval intuitionistic fuzzy values (IVIFVs) in the DST framework still contain original fuzzy information (i.e., membership degree, non-membership degree, and a hesitancy degree). IVIFVs under the DST framework are found to get rid of the shortcomings and limitations of IVIFPMM. Thus, it is a good way to aggregate IVIFVs under the DST framework and thus obtain more reasonable aggregation results.
The second key task of the MCDM method is to use an aggregation operator to aggregate the quantitative criteria of each alternative and sort the alternatives according to the aggregation results. Thus it can be seen that aggregation operators are of great importance for solving MCDM problems, which perform better than conventional decision-making methods (e.g., TOPSIS Li et al. 2020;Zhang et al. 2020;Chen, 2015, VIKOR Zeng et al. 2019Jiang 2011, andELECTRE Veeramachaneni andKandikonda 2016). When using aggregation operators, it is necessary to reduce the effect of unreasonable input data on the aggregation's results. Yager (2001) originally proposed the power average (PA) operator. In the PA operator, the distance between parameters is used to calculate the positive correlations of parameters, and these positive correlations are accumulated to adjust final weights. Through such an operation, the impact of unreasonable data on the sorting results can be weakened. In addition to the influence of unreasonable data, the relationship between parameters is also an important factor to be considered. The Muirhead mean (MM) operator is well known for having the capability to capture the interrelationships among all aggregation parameters (Muirhead 1902). It reduces the influence of relevant factors on ranking results by eliminating the overlapping influence of non-orthogonal terms. Although the above operators can deal with the problem of outliers and the correlation between attributes separately, they cannot solve the problem of both outliers and attribute correlations. At the same time, the above operators are based on interval intuitionistic fuzzy numbers and are not extended to the DST framework. Therefore, while the operator has the above single advantages, it has the disadvantages of unreliable theory, unclear calculation semantics, and limited calculation range of the operator. Based on the background described above, the combination of PA operator and MM operator under the DST framework (Khan, Hassan and amp;Mahmoud, 2018) can meet the advantages of the above operators and operations under the DST framework at the same time.
The motivations of this paper are summarized as follows: (1) To measure the membership and non-membership of variables more comprehensively, we use intervalvalued intuitionistic fuzzy sets (IVIFS) in the field of MCDM. To remove the influence of the undesired operation properties between interval intuitionistic fuzzy numbers on the results of aggregation operators, PA and MM operators are extended to the DST framework in this paper. PA and MM operators based on the DST framework can make their operations more concise and semantics clearer while maintaining their original advantages. At the same time, due to the advantages of operation under the DST framework, PA and MM operators under the DST framework can expand the weights from real numbers to interval intuitionistic fuzzy numbers. (2) To retain the advantages of the PA operator, MM operator, and operation under the DST framework, the PA operator and MM operator are fused under the DST framework to generate IVIFPMM DST operator under the DST framework and IVIFWPMM DST operator with interval intuitionistic fuzzy values as weights. In the MCDM problem, the IVIFPMM operator under the DST framework is more reliable than the PA operator based on IVIFVs (Qin et al. 2020;Liu and Gao 2019). (3) To deal with the situation of multi-criteria decisionmaking with less expert experience, abnormal criteria data, and correlation between criteria, this paper will propose an MCDM method based on theIVIFPMM DST operator. This method retains the advantages of the strong robustness of the PA operator and the advantages of the MM operator to eliminate the correlation between criteria. Finally, this method can deal with the situation where the weights are in IVIFVs.
Based on the above motivations, this paper aims to construct IVIFWPMM DST to deal with the MCDM problem based on the operation laws of IVIFV under the DST framework. This aim is achieved via combining the MM operator and the PA operator to assess the weighted criteria of IVIFVs under the DST framework. The major contributions of the paper are summarized as follows: (1) The PA operator and MM operator are extended to interval intuitionistic fuzzy set under the DST framework, which not only retains the advantages of the original operator, but also makes its operation more semantic and its theoretical results more credible; (2) based on PA operator, MM operator, and DST framework, three operators (IVIFPMM DST ; IVIFWPMM DST ; IVIFWPMM DST ) are proposed and their properties are proved, which can not only have the advantages of PA and MM operator but also make up for the disadvantages of interval intuitionistic fuzzy number operations; (3) a method based on the proposed operators to solve the MCDM problems with IVIFVs is proposed. The proposed method is compared with other existing MCDM methods to show its advantages.
The rest of the paper is organized as follows: In Sect. 2, some basic concepts and related operations of IVIFVs are introduced, the limitations of the basic algorithm of IVIFVs are highlighted, and the definition and operations of IVIFVs under the DST framework are recalled. Section 3 presents the IVIFPA, IVIFMM, IFIVPMM, and IVIFWPMM operators under the DST framework. In Sect. 4, a method based on the presented operators to solve the MCDM problems with IVIFVs is proposed. In Sect. 5, a set of comparative experiments for illustrating the rationality and advantages of the proposed method are documented. Section 6 ends the paper with a conclusion.

IVIFSs
Definition 2.1 (Atanassov and Krassimir 1999) Let E be the set of natural numbers, then an IVIFS named G, defined on E, can be expressed as: Xu (2007) proposed a set of arithmetic operations of IVIFVs, which are defined below: } be any two IVIFVs, then: where k [ 0.
Definition 2.5 (Xu 2007) The score function and accuracy function of an IVIFV G are, respectively, defined as follows: The order relation of any two IVIFVs (e.g., G and H) can be obtained from the following rules: (2) if SF(G) = SF(H), then: Let G 1 ; G 2 be two IVIFVs, it is easy to prove that Eqs.
The above four examples show some limitations of the arithmetic operations in Eqs. (2)-(5), which could lead to irrational results when dealing with MCDM problems (Wang et al. 2022). Further, [20] shows that there is a very close relationship between DST and IVIFV. Under the DST framework, an IVIFV has transparent and fruitful semantics. More importantly, the arithmetic operations of IVIFVs under this framework can overcome the limitations above.
where S denotes the complementary collection of X.
From the above definition of Bel and Pl, a belief interval (BI) can be formed, that is, BI(X) = [Bel(X), Pl(X)]. Specifically, BI is composed of the ''true, probability'' interval of X.
Based on the definitions above, the definition of IVIFS under the DST framework is as follows: Definition 2.8 (Dymova et al. 2016) Let G be an IVIFS based on a generic finite set s, then: where is a belief interval with a bound of BIs: where lðsÞ; vðsÞ & ½0; 1 and inf lðsÞ þ supvðsÞ 1; suplðsÞ þ infvðsÞ 1.
Definition 2.9 (Dymova et al. 2016) The expression form of an IVIFV G under the DST framework is: where On the basis of the definition above, the arithmetic operations of IVIFVs under the DST framework are defined as follows: Pl] be a BI, then: From the arithmetic operations above, an IVIFWAM DST operator and an IVIFWGM DST operator in the DST framework can be deduced below (Dymovaet al. 2016): Let nÞ be a set of BIs. w ¼ ðw 1 ; w 2 ; . . .; w nÀ1 ; w n Þ T is the weight corresponding to G i ði ¼ 1; 2; 3. . .; nÞ, which satisfies: w i 2 0; 1 ½ and P n i¼1 w i ¼ 1.Then the IVIFWAM operator in the DST framework is given as follows: The IVIFWGM operator in the DST framework is given as follows: An accuracy function and a score function are important tools for comparing IVIFVs in the DST framework, which are defined as follows (Dymova et al. 2016): AFðGÞ ; HðGÞ 2 ½0; 1: Let G 1 ; G 2 be two BIs. It is easy to prove that Eqs. (24)-(30) have the following properties: ( Definition 2.11 (Xu 2010) The Hamming distance of any two IVIFVs g 1 ¼ f½a 1 ; b 1 ; ½c 1 ; s 1 gg 2 ¼ f½a 2 ; b 2 ; ½c 2 ; s 2 g is defined as follows: 2.3 MM operator and PMM operator Definition 2.12 (Yager and Ronald 2001) The power average (PA) operator of the evaluation value set G ¼ g k ðk ¼ 1; 2; :::; sÞ is defined as follows: where Tðg k Þ ¼ P s k¼1k6 ¼j Supðg i ; g j Þ indicates the degree of support of all evaluation values to g i , Supðg j ; g k Þ ¼ 1 À dðg j ; g k Þ and satisfies Supðg i ; g j Þ 2 ½0; 1 and Supðg i ; g j Þ = Supðg j ; g i Þ.
Definition 2.14 (Xu et al. 2019) The definition of the power Muirhead mean (PMM) operator of IVIFVs based on a positive number set G ¼ g i ði ¼ 1; 2; :::; nÞ and a parameter set P ¼ ðp 1 ; p 2 ; :::; p n Þ is as follows: where Tðg k Þ ¼ P s k¼1k6 ¼j Supðg i ; g j Þ indicates the degree of support of all evaluation values to g i , Supðg j ; g k Þ ¼ 1 À dðg j ; g k Þ and satisfies Supðg i ; g j Þ 2 ½0; 1 and Supðg i ; g j Þ = Supðg j ; g i Þ.

IVIFPA DST operator
The IVIFPA operator under the DST framework is defined as follows: Þfor convenience. Then the IVIFPA DST operator is defined as follows: where For the detailed definition of Supðe g j ; e g k Þ and the properties ofSupðe g j ; e g k Þ, see Definition 2.12.
Theorem 3.1 Based on Definition 3.1, the value aggregated by the IVIFPA DST operator is still a BI, as shown below: where Supðe g k ; e g u Þ = 1-d(e g k ; e g u ) and Supðe g k ; e g u Þ denotes the support degree for e g k from e g u . relationship between ð e g 0 1 ; e g 0 2 ; :::; e g 0 n Þ and ð e g 1 ; e g 2 ; :::; e g n Þ, it is easy to obtain the following expression: DST ð e g 1 ; e g 2 ; :::; e g n Þ\e g þ .
Proof From Definition 2.11, we can easily obtain that: h Therefore, it can be obtained that: e g À \IVIFPA S DST e g 1 ; e g 2 ; :::; e g n ð Þ \e g þ :

IVIFMM DST operator
Þfor convenience. Then the IVIFPMM DST operator is defined as follows: Theorem 3.4 Based on Definition 3.1, the value aggregated by the IVIFPMM DST operator is still a BI, as shown below: Proof Based on Eqs. (24)-(30) we can get ð e h Thus the equation can be derived as Eqs. (49). 1 ; e g 0 2 ; :::; e g 0 n Þ and ð e g 1 ; e g 2 ; :::; e g n Þ, it is easy to obtain the following expression: DST ð e g 1 ; e g 2 ; :::; e g n Þ\e g þ .
Proof From Definition 2.11, we can easily obtain that: Therefore n j¼1BNP ðg nðjÞ Þ s j n j¼1BNP ðg þ Þ s j : h Therefore, it can be obtained that e g À \IVIFMM S DST e g 1 ; e g 2 ; :::; e g n ð Þ \e g þ :

IVIFPMM DST operator
The IFIVPMM operator under the DST framework is defined as follows: . . .; ð nÞ for convenience. Then the IVIFPMM DST operator is defined as follows: Theorem 3.7 (Idempotence) Based on Definition 3.1, the value aggregated by the IVIFPMM DST operator is still a BI, as shown below: Proof Based on Eqs. (24)-(30) we can get ðnx i e  :; e g 0 n Þ and ð e g 1 ; e g 2 ; :::; e g n Þ, it is easy to obtain the following expression: h Theorem 3.9 (Boundedness). I f e g þ ¼ max n i¼1 f½Bel L i ; Pl L i ; ½Bel U i ; Pl U i g and e g À ¼ min n i¼1 f½Bel L i ; Pl L i ; ½Bel U i ; Pl U i g, then e g À \IVIFPMM S DST ð e g 1 ; e g 2 ; :::; e g n Þ\e g þ .
Proof From Definition 2.11, we can easily obtain that: h Therefore, it can be obtained that: e g À \IVIFPMM S DST e g 1 ; e g 2 ; :::; e g n ð Þ \e g þ :

IVIFWPMM DST operator
The IVIFPMM DST operator calculates the average power between any number of criteria but does not consider the weights of the criteria. The IVIFWPMM operator under the DST framework which contains the weight of the criteria is defined as follows: . . .; n ð Þ . w i ði ¼ 1; 2; :::; nÞ is the weight corresponding tog i ði ¼ 1; 2; 3. . .; nÞ, where P n j¼1 x j ¼ 1and0 x j 1. The IVIFWPMM DST operator of ð e g 1 ; e g 2 ; :::; e g n Þ is defined as follows: where Tð e g s Þ ¼ P n l¼1;l6 ¼s Supð e g s ; f g l Þ, Supð e g s ; f g l Þ = 1d( e g s ; f g l ) and Supð e g s ; f g l Þ denotes the support degree for f g l from e g s .
To simplify the above operator, let d j ¼ x nðjÞ ð1 þ Tð g g nðjÞ ÞÞ= P n j¼1 x j ð1 þ Tð e g j ÞÞ,then the above equation can be expressed as:

IVIFWPMM S
DST ðg 1 ;g 2 ; :::;g n Þ ¼ È Theorem 3.10 (Idempotence) Based on Definition 3.7, the value aggregated by the IVIFWPMM DST operator is still a BI, as shown below: Proof The proof here is similar to that of Eq. (47). It will not be repeated here. h Proof The proof here is similar to that of Theorem 3.8. It is not repeated here. h Theorem 3.12 (Boundedness) If e g þ ¼ max n i¼1 f½Bel L i ; Pl L i ; ½Bel U i ; Pl U i g and e g À ¼ min n i¼1 f½Bel L i ; Pl L i ; ½Bel U i ; Pl U i g, then e g À \IVIFWPMM S DST ð e g 1 ; e g 2 ; :::; e g n Þ\e g þ .

Proof
The proof here is similar to that of Theorem 3.9. It will not be repeated here. h The weights of the IVIFWPMM DST operator are positive numbers. Using positive numbers to express the weights sometimes cannot accurately express the decision maker's preference. Therefore, it is necessary to define an IVIFWPMM DST operator, where the weights are IVIFVs, shown as follows.
Pl U j g be the BI corresponding to the weight w j represented by the IVIFV. When W ¼ ðw 1 ; w 2 ; :::; w n Þ satisfies the following two conditions, W is normalized IVIFN vector (1) There is at least one normalized vectora 2 I, where I ¼ The IVIFWPMM DST operator of ð e g 1 ; e g 2 ; :::; e g n Þ is defined as follows: IVIFWPMM S DST ðg 1 ;g 2 ; . . .;g n Þ ¼ where Tð e g s Þ ¼ P n l¼1;l6 ¼s Supð e g s ; f g l Þ,Supð e g s ; f g l Þ = 1d( e g s ; f g l ) and Supð e g s ; f g l Þ denotes the support degree for f g l from e g s . To simplify the above operator, let d j ¼ x nðjÞ ð1 þ Tð g g nðjÞ ÞÞ= P n j¼1 x j ð1 þ Tð e g j ÞÞ,then the above equation can be expressed as the following form: Theorem 3.13 (Idempotence) Based on Definition 3.7, the value aggregated by the IVIFVWPMM DST operator is still [BI], as shown below: Proof The proof here is similar to that of Theorem 3.1, so it will not be repeated here. h The proof here is similar to that of Theorem 3.8, so it will not be repeated here. h Theorem 3.15 (Boundedness). If e g þ ¼ max n i¼1 f½Bel L i ; Pl L i ; ½Bel U i ; Pl U i g and e g À ¼ min n i¼1 f½Bel L i ; Pl L i ; ½Bel U i ; Pl U i g, then e g À \IVIFWPMM S DST ð e g 1 ; e g 2 ; :::; e g n Þ\e g þ .

Proof
The proof here is similar to that of Theorem 3.9, so it will not be repeated here. h

MCDM method with IVIFVs in the DST framework
An MCDM method based on a fuzzy operator has the advantages of fast speed and strong robustness in the fuzzy unknown environment (Devara et al. 2021; Alyami et al. 2022). Herein, a novel MCDM method based on the proposed operators is proposed to solve the MCDM problems based on IVIFVs. Considering the following MCDM problem: There are q local criteria, denoted by R ¼ fr 1 ; r 2 ; . . .; r q g and p alternatives, denoted by T ¼ ft 1 ; t 2 ; . . .; t p g. If the weights of criteria are positive numbers such that w l [ 0ðl ¼ 1; 2; . . .; qÞ and P q l¼1 w l ¼ 1. Else, if criteria weights are IVIFNs, then weight vector is denoted by an interval intuitive weight vector w ¼ ðw 1 ; w 2 ; . . .; w q Þ with w l ¼ f½Bel L l ; Pl L l ; ½Bel U l ; Pl U l g; ðl ¼ 1; 2; . . .; qÞ. The MCDM method aims to sort the columns of a decision matrix B ¼ ðb jk Þ pÂq , where b jk denotes the IVIFV preference value of the alternative t j with respect to the criterion r k .
The detailed steps of the proposed MCDM method are as follows: Step 1 Normalize the decision matrix B ¼ ðb jk Þ pÂq . The higher the value of b jk , the higher the positive preference.
Step 4 Calculate T f h jk of f h jk by the support of other . . .; qÞ: Step and then Step 6 Use the IVIFWPMM DST operator (i.e., Eq. 48) or IVIFWPMM DST operator (i.e., Eq. 56) to calculate the comprehensive value of each alternative.
Step and AF DST e h j are abbreviated as S j and A j , respectively.
Step 8 Sort the alternatives by using S j and A j , ðj ¼ 1; 2; . . .; pÞ to get the optimal alternative.

Application and verification
In this section, a practical example is firstly introduced to illustrate the proposed MCDM method. A sensitivity analysis experiment is then conducted to explore the influence of different s values on the ranking results. Finally, comparisons to the existing methods are carried out to demonstrate the effectiveness and advantages of the proposed method.

Application of the proposed MCDM method
The proposed method is used to solve a practical MCDM problem (Xu et al. 2019) that which a research team optimally selects a smart device alternative.
Example 5.1 A university research team plans to buy a new batch of smart devices for the team. After preliminary screening, there are four alternatives T ¼ ft 1 ; t 2 ; t 3 ; t 4 g to meet the requirements. To select the most suitable alternative, there are four criteria R ¼ fr 1 ; r 2 ; r 3 ; r 4 g need to be considered, i.e., equipment price (r 1 ), team demand (r 2 ), compatibility with the team's current smart equipment (r 3 ), and after-sale service quality (r 4 ). The weight vector of the criteria r i ði ¼ 1; 2; 3; 4Þ is w ¼ ð0:3; 0:25; 0:25; 0:2Þ T . Ptðt ¼ 1; 2; 3Þ are three experts who evaluate the four programs.c ¼ 0:32; 0:45; 0:23 ð Þ T is the weight vector of the decision results of the three experts. The evaluation value G k ¼ g l jk ðk; j ¼ 1; 2; 3; 4; l ¼ 1; 2; 3Þ is in the form of IVIFVs. Tables 1, 2, 3 show three decision matrices.
Using the proposed MCDM method, the MCDM problems can be solved via the following steps: Step 1 Since all attributes are beneficial types, there is no need to normalize the decision matrices.   Step 2 Convert IVIFV g l jk to [BI] f g l jk . For example, all IVIFVs in Table 1 are converted into BIs, which are shown in Table 4.

Effect of the parameters in s on ranking results
Section 5.1 shows that parameter s is an important variable in the decision-making process. To illustrate the influence of parameter s on the decision result, we will use a different parameter s to rank the alternatives in the example in Sect. 5.1. The ranking results obtained by setting s with different values are shown in Table 6.   Since the parameter vector s represents the concern degree of the decision maker on different criteria (Xu et al. 2019), so more elements in parameter s are not 0, means that decision makers consider the correlation between more elements. For example, s = [1,1,0,0], decision makers consider the correlation between two factors. Therefore, as shown in Table 6, IVIFWPMM DST operator obtains different comprehensive evaluation values and rankings under different parameter settings. In addition, when decision makers consider less correlation between attributes, the comprehensive evaluation value increases because of less correlation elimination. Therefore, when making decisions, a more appropriate parameter s can be set according to the correlation between actual factors and actual demands.

Verification of the effectiveness
In Sect. 5.1, we gave a brief demonstration of the proposed method through an example of selecting an optimal smart device scheme. In Sect. 5.2, we analyzed the influence of parameter vector s on decision-making results. Next, we will demonstrate the effectiveness of the proposed method via a set of comparisons.
Example 5.2 (Sun et al. 2016) A company needs to purchase office computers for employees working in a new office building. There are currently five computer brands and models t i ði ¼ 1; 2; 3; 4; 5Þ as possible alternatives. To select a more suitable alternative, the company needs to evaluate the five alternatives from the four attributes, i.e., computer price (r 1 ), hardware performance (r 2 ), after-sales service (r 3 ), and service life (r 4 ). A weight vector w ¼ ð0:2; 0:1; 0:3; 0:4Þ T denotes the importance of each attribute. The company uses IVIFVs g ij ¼ ð½a ij ; b ij ; ½c ij ; d ij Þ to evaluate the attribute r j of the alternative t i . The decision matrix composed of IVIFVs g ij is shown in Table 7.
Since there is no ranking by an authoritative expert for Example 5.2, we use the ranking results of multiple diachronically tested MCDM methods (Xu 2007;He et al. 2013;Sun et al. 2016;Yu and Wu 2012;Xu and Chen 2011;Liu 2017;Liu et al. 2018) to verify the effectiveness of the presented MCDM method. Specifically, we use the ranking results of the following operators, i.e., the interval-valued intuitionistic fuzzy weighted average (IVIFWA) operator (He et al. 2013), the interval-valued intuitionistic fuzzy power weighted average (IVIFPWA) operator (Xu 2007)[8], the interval-valued intuitionistic fuzzy weighted average weighted Bonferroni mean (IVIFWBM) operator (Xu and Chen 2011), the generalized interval-valued intuitionistic fuzzy weighted Heronian mean (GIVIFWHM) operator (Yu and Wu 2012), and the interval-valued intuitionistic fuzzy weighted Maclaurin symmetric mean (IVIFWMSM) operator (Sun and Xia 2016), the interval-valued intuitionistic fuzzy weighted power Bonferroni mean (IVIFWPBM) operator , the interval-valued intuitionistic fuzzy weighted power Heronian mean (IVIFWPHM) operator (Liu 2017), and the interval-valued intuitionistic fuzzy weighted Maclaurin symmetric mean (IVIFWPMSM) operator (Liu et al. 2018). As shown in Table 8, although the ranking results are different, the first ranking is t 4 , so the design method in this paper is feasible and effective.

Further comparative analysis
From the results of Sect. 5.3, we can demonstrate the effectiveness of the proposed method. However, since almost all methods have the same sorting results, the limitations of the existing methods and the advantages of the proposed method cannot be seen from the comparison in Sect. 5.3. To this end, we will illustrate the advantages of our method via further comparisons. The methods based on IVIFPWA (Xu 2007), IVIFWA (He et al. 2013), and IVIFMM are selected as comparison objects.

Comparison of robustness
In MCDM methods without human intervention, unreasonable decision values may result from data transmission failure, decision makers' preferences, or other accidents. In this case, it may be difficult to obtain a reasonable sorting result via these methods. The robustness of the proposed method is evaluated via Example 5.3.

Example 5.3
To simulate the situation with extreme evaluation values, the value of g 42 ¼ ð½0:6; 0:7; ½0:1; 0:3Þ in Example 5.2 is changed to g 42 ¼ ð½0:01; 0:7; ½0:1; 0:3Þ. Then, the new decision matrix and the ranking results are shown in Tables 9 and 10, respectively.  IVIFWA (He et al. 2013) and IVIFWMM operators are not robust to an extreme value such that they get unreasonable sorting results in this case. Therefore, two operators may not get the reasonable ranking results in an unstable system (e.g., with abnormal decision values), when there is no intervention from experts. In contrast, the IVIFVWPMM DST operator and the IVIWPA operator can get reasonable sorting results. This is because the PA operator calculates the distance among criteria, and then the weight of each criterion is obtained by this distance. Therefore, unreasonable opinions are assigned to a smaller weight, thus reducing the negative effect of these unreasonable opinions on the final sorting results. The proposed operator is well integrated with the PA operator under the DST framework and therefore inherits the advantage of the PA operator to reduce the negative effect of extreme values.

Comparison of correlation elimination ability between attributes
The correlation between attributes is one of the important factors to be considered in the MCDM problem (Xian et al. 2021). Example 5.4 shows the performance and result analysis of different operators when there is a correlation between attributes.
Example 5.4 (Xu et al. 2011) A tank unit of an army is evaluating the operational deployment scheme, in which the commander gives four evaluation schemes t i ði ¼ 1; 2; 3; 4Þ, each of which has five attributes r i ði ¼ 1; 2; 3; 4; 5Þ, i.e., concealment(r 1 ); Mobility(r 2 );Availability of bunkers(r 3 ); Air defense capability(r 4 ); Fire support(r 5 ). The weight of each attribute is w ¼ ð0:2; 0:2; 0:2; 0:2; 0:2Þ T , the decision matrix is shown in Table 11, and the ranking results are shown in Table 12:  (Xu et al. 2011), the rationality of the sorting results of the IVIFWPA operator and IVIFWA operator is weak. Attribute r 5 has an obvious correlation with attributes r 1 , r 2 , and r 3 , but the IVIFWPA operator and IVIFWA operator cannot eliminate the relationship between factors, so it obtains unreasonable results. MM operator can reduce the repeated influence of non-orthogonal elements (Xian et al. 2021;Wang 2021). Therefore, when there is an obvious correlation between attributes, the MM operator can get more reasonable ranking results. In the experiment, the ranking results obtained by using this operator are consistent with the results of the IVIFWPMM DST operator and literature (Xu et al. 2011). Therefore, the operator of this paper also can eliminate the correlation between attributes.

Comparison of form of weights
In some real MCDM applications, due to the lack of human experience, experts cannot provide accurate real weight and can only use the weight represented by IVIFV. The operator based on IVIFV cannot directly calculate such cases due to the defects of operation between IVIFVs (Liu 2017;Zhong et al. 2021), but the operator in this paper is based on the DST framework that can deal with this problem. Example 5.5 shows the performance of different operators when the weight is IVIFV.
Example 5.5 In the path planning of an autonomous underwater vehicle (AUV), AUV has preliminarily preset five forward directions t i ði ¼ 1; 2; 3; 4; 5Þ. There are three attributes r i ði ¼ 1; 2; 3Þ that determine the specific forward direction of the machine: the deviation degree between the direction of advance and the preset route(r 1 ); the confrontation degree between the direction of advance and the ocean current(r 2 ); the safety of the direction of advance(r 3 ). Experts can only provide the weight represented by IVIFV, and the weight of each attribute is w ¼ ð½0:2; 0:4; ½0:4; 0:5Þ; ð½0:4; 0:6; ½0:1; 0:3Þ; ð½0:1; 0:3; ½0:6; 0:7Þ T , the decision matrix is shown in Table 13, and the ranking results are shown in Table 14: It can be seen from Table 14 that the IVIFPWA operator (Xu 2007) and IVIFWA operator (He et al. 2013) cannot directly calculate the case that the weight is IVIFV. This is because IVIFV cannot be brought into the operation as a power, but IVIFV under the DST framework has no such limitation. Therefore, the IVIFWPMM operator under the DST framework in this paper can directly solve the problem of calculating the real-time path selection of AUV above.

Summary of the comparisons
All in all, through Examples 5.3-5.5, we can further summarize the limitations of the IVIFPWA operator (Xu 2007), IVIFWA operator (He et al. 2013), and IVIFWMM operator and the advantages of the operator in this paper: (1) The IVIFPWA operator does not perform the operation for eliminating parameter correlation, and secondly, it cannot handle the case that the decision weights are IVIFVs.
(2) Similarly, the IVIFWA operator (He et al. 2013) also has the above two limitations as the IVIFPWA operator (Xu 2007). At the same time, the operator cannot reduce the weight of abnormal opinions through the distance between criteria, which may lead to unreasonable ranking results. (3) The IVIFWMM operator has the limitation that the weights cannot be adjusted according to the existing information, so it cannot deal with the data distortion in an unstable system or extreme situation. In addition, it also cannot handle the case that the decision weights are IVIFVs. The operator in this paper inherits both the advantages of the IVIFPWA operator (Xu 2007) in dealing with outliers and the advantages of the IVIFWMM operator in dealing with correlation between attributes. At the same time, due to the advantages of IVIFV in the DST framework, the operator can also deal with the weights of interval intuitionistic fuzzy numbers. To make the summary looks more intuitive, Table 13 lists the comparison between the proposed methods with several existing methods (Table 15).

Conclusions
In this paper, we mainly propose two operators, i.e., IVIFPMM DST operator and IVIFWPMM DST operator, of IVIFVs in the DST framework. An interpretation of IVIFS in the DST framework is provided, and a set of brief and intuitive operation laws based on the interpretation are developed. Based on the developed operation laws, PA operator and MM operator are extended in the DST framework, which are named IVIFPA DST and IVIFMM DST operator, respectively. Moreover, two novel operators, IVIFPMM DST and IVIFWPMM DST operator, are successively constructed. Subsequently, the complete properties of these two operators are discussed and proved. On the basis of the constructed operators, we present a method to solve the MCDM problems with IVIFVs. Finally, we also construct practical examples to illustrate the proposed method and report comparisons with the existing methods to demonstrate its effectiveness, robustness, and advantages. These examples show that: (1) The proposed method can maintain the advantage (eliminating parameter correlation) of the existing MM operator. (2) The sorting results of the presented method are robust to the abnormal decision values. (3) The proposed operator is capable to solve the MCDM problems where the criterion weights are in the form of IVIFVs.
Future work will focus especially on improving the operator and the proposed MCDM method from two aspects. On the one hand, the parameter vector in the proposed operators needs preset. However, it is easy to set unreasonable parameter vector when there is no thorough understanding of the correlation of decision attributes. To address this issue, it is essential to explore an adaptive method so that these parameters can be adaptively adjusted to the appropriate parameter value according to the correlation of specific decision attributes. In addition, we will also consider the interpretation and situation of extending parameter vector s into range of [0, 1]. On the other hand, the proposed MCDM method will be applied in more actual decision-making applications, like in the field of underwater autonomous robot. In the underwater multiobjective path planning, due to real-time changes of current and ocean current, the underwater information is uncertain and fuzzy. Therefore, it is generally accepted that IVIFVs are used to quantify the considered criteria of each alternate. On the basis of the IVIFVs, the proposed MCDM method will be used to decide the multi-objective path planning of an autonomous robot system. Data availability Enquiries about data availability should be directed to the authors.