Multiple Attribute Group Decision-Making Based on Power Heronian Aggregation Operators under Interval-Valued Dual Hesitant Fuzzy Environment

In this paper, we focus on new methods to deal with multiple attribute group decision-making (MAGDM) problems and a new comparison law of interval-valued dual hesitant fuzzy elements (IVDHFEs). More explicitly, the interval-valued dual hesitant fuzzy 2nd-order central polymerization degree (IVDHFCP2) function is introduced, for the case that score values of different IVDHFEs are identical. )is function can further compare different IVDHFEs. )en, we develop a series of interval-valued dual hesitant fuzzy power Heronian aggregation operators, i.e., the interval-valued dual hesitant fuzzy power Heronian mean (IVDHFPHM) operator, the interval-valued dual hesitant fuzzy power geometric Heronian mean (IVDHFPGHM) operator, and their weighted forms. Some desirable properties and their special cases are discussed.)ese proposed operators can simultaneously reflect the interrelationship of aggregated arguments and reduce the influence of unreasonable evaluation values. Finally, two approaches for interval-valued dual hesitant fuzzyMAGDMwith known or unknownweight information are presented. An illustrative example and comparative studies are given to verify the advantages of our methods. A sensitivity analysis of the decision results is analyzed with different parameters.

Aggregation operators are the widely used tools for aggregating individual preference information into collective ones.Recently, some operators have been proposed to aggregate interval-valued dual hesitant fuzzy information.Ju et al. [25] defined a series of interval-valued dual hesitant fuzzy aggregation operators, such as the interval-valued dual hesitant fuzzy average (IVDHFA) operator and the intervalvalued dual hesitant fuzzy geometric (IVDHFG) operator.
ese operators assume that all attributes are completely independent.More and more scholars give attention to propose aggregation operators to measure and integrate the effects of attributes interrelationships on the results of MAGDM problems, such as Choquet integral (CI), Power average (PA), Heronian mean (HM), and Maclaurin symmetric mean (MSM) [32].Qu et al. [28,29] extended CI to interval-valued dual hesitant fuzzy environment and gave some interval-valued dual hesitant fuzzy Choquet integral aggregation operators.
HM [33] is a useful tool to capture the interrelationship of evaluation information.It has been successfully used in decision-making under various uncertain environments [6,12,18,20,27,34].Yu [6] defined the geometric form of the HM (GHM) operator and extended to IFSs.Liu et al. [20] proposed the q-rung orthopair fuzzy HM operator, the q-rung orthopair fuzzy partitioned HM operator, and their weighted forms.PA [35] is another effective tool that can be used to relieve the negative influence of unreasonable evaluation values on decision result.In real decision-making problems, decision makers should consider the interrelationship of aggregated arguments and may give some awkward attribute values due to their own preferences.Combining PA and MSM, Liu et al. [10] developed the power Maclaurin symmetric mean operator in the intervalvalued intuitionistic fuzzy environment.Taking advantages of PA and HM, Liu et al. [9,36] proposed interval-valued intuitionistic fuzzy power Heronian mean (PHM) and linguistic neutrosophic PHM, respectively.
Motivated by the abovementioned ideas, we propose interval-valued dual hesitant fuzzy power Heronian aggregation operators, due to the simultaneous combination of PA and HM (or GHM).It is necessary to synchronously consider the following demands in the decision-making process: (1) Due to the lack of expertise or insufficient knowledge, decision makers are usually willing to express membership degrees and nonmembership degrees with several interval values.IVDHFS can hold the flexibility of interval number when assigning possible membership degrees and nonmembership degrees.
(2) Decision makers use IVDHFEs to express their opinions and might provide some unreasonable attribute values of alternatives.In order to relieve these influences, we can select PA to overcome some effects of awkward data.At the same time, we need to consider the interrelationship of input values, the better selection is HM.In a word, we can use power Heronian aggregation operators to capture the interrelationship of aggregated arguments and also overcome some effects of awkward data given by predispose decision makers.
Based on the abovementioned analysis, the purpose of this paper is to propose interval-valued dual hesitant fuzzy power Heronian aggregation operators and develop new MAGDM approaches with known or unknown weight information.e main advantages of our approaches can be summarized as follows: ( e paper is organized as follows.In Section 2, we review some basic concepts of IVDHFS and IVDHFE,-give a new comparison law by introducing the IVDHFCP 2 function, which can be used to compare different IVDHFEs.Section 3 investigates a variety of interval-valued dual hesitant fuzzy power Heronian aggregation operators and discusses some desirable properties and special cases of these operators.Section 4 presents two approaches on the basis of our proposed operators to MAGDM with interval-valued dual hesitant fuzzy information.In Section 5, a practical example about two cases is illustrated to verify the effectiveness and practicality of our approaches.A conclusion and further research studies are given in Section 6.

IVDHFS
Definition 1 (see [25]).Let X be a reference set, and D[0, 1] be the set of all closed subintervals of [0, 1].en, an interval-valued dual hesitant fuzzy set (IVDHFS) on X is where  h(x): X ⟶ D[0, 1] and  g(x): X ⟶ D[0, 1] denote all possible interval-valued membership degrees and nonmembership degrees of the element x ∈ X to the set  D, respectively, with the conditions: where Definition 2 (see [25]).For three IVDHFEs, , and λ > 0, some basic operational laws are defined as follows: (1) Definition 3 (see [25]).Let  d �  h,  g   be an IVDHFE, score and accuracy functions of  d are described as follows: where #  h and # g are numbers of interval values in  h and  g, respectively.
Theorem 1 (see [25]).For two IVDHFEs To find the distance measure between IVDHFEs, Zang et al. [26] extended the shorter one until the membership degrees and nonmembership degrees of both IVDHFEs have the same length, respectively.To extend the shorter one, the best way is to add the same interval value several times in it.e selection of this interval value mainly depends on decision makers' risk preferences.Optimists anticipate desirable outcomes and add the maximum interval value of membership degrees and minimum interval value of nonmembership degrees, while pessimists expect unfavorable outcomes and add the minimum of membership degrees and the maximum of nonmembership degrees.
e interval-valued dual hesitant normalized Hamming distance between two IVDHFEs is defined as follows: where In order to overcome this flaw, we define the IVDHFCP 2 function as follows.
} be a finite and nonempty IVDHFE, where is called the interval-valued dual hesitant fuzzy 2nd-order central polymerization degree (IVDHFCP 2 ) function of  d.For the sake of simplicity, we letv( us, equation ( 5) can be denoted as follows:

Interval-Valued Dual Hesitant Fuzzy Power Heronian Aggregation Operators
In this section, we develop some new aggregation operators under interval-valued dual hesitant fuzzy environment, i.e., IVDHFPHM, IVDHFPGHM, and their weighted forms.

IVDHFPHM Operator
Definition 6.Let  d i (i � 1, 2, . . ., n) be a collection of IVDHFEs, p, q ≥ 0 and p, q do not take the value 0 simultaneously.An interval-valued dual hesitant fuzzy power Heronian mean (IVDHFPHM) operator is defined as follows: where d s , which satisfies the following three properties: (1) Sup( where d is a distance measure between two IVDHFEs and calculated by equation ( 4).
Theorem 3. Let  d i (i � 1, 2, . . ., n) be a collection of IVDHFEs, p, q ≥ 0 and p , q do not take the value 0 simultaneously.en, the aggregated value using the IVDHFPHM operator is still an IVDHFE, and where Specially, if Sup( ) for all i ≠ s, i.e., ω i � 1, then the IVDHFPHM operator reduces to the interval-valued dual hesitant fuzzy Heronian mean (IVDHFHM) operator:

Theorem 4 (boundedness). For a collection of IVDHFEs
. ., n), p, q ≥ 0 and p, q do not take the value 0 simultaneously.Let we have where Theorem 5 (commutativity).
, (i � 1, 2, . . ., n) be a collection of IVDHFEs, and p � q > 0. us, where e proofs of eorems 3-5 are shown in Appendixes A-C, respectively.However, the IVDHFPHM operator has no properties of idempotency and monotonicity, as illustrated in the following example.
According to equation ( 2), scores of the above aggregated IVDHFEs are , which shows the IVDHFPHM operator is not monotonic.Now, we can discuss some special cases of the IVDHFPHM operator by assigning different values of parameters p and q.
Case 1: if q ⟶ 0, then the IVDHFPHM operator reduces to the interval-valued dual hesitant fuzzy power descending average (IVDHFPDA) operator: Obviously, the weight vector of (ω i  d i ) p is (n, n− 1, . . ., 1).Case 2: if p ⟶ 0, then the IVDHFPHM operator reduces to the interval-valued dual hesitant fuzzy power ascending average (IVDHFPAA) operator: Obviously, the weight vector of 19) and ( 20), we can see that the weight vectors of (ω i  d i ) p and (ω i  d i ) q are different.Hence, parameters p and q of the IVDHFPHM operator are not interchangeable.

IVDHFPGHM Operator.
In this section, we introduce the IVDHFPGHM operator by incorporating PA into GHM.
) be a collection of IVDHFEs, p, q ≥ 0 and p, q do not take the value 0 simultaneously.An interval-valued dual hesitant fuzzy power geometric Heronian mean (IVDHFPGHM) operator is defined as follows: where Based on operational laws of IVDHFEs and mathematical induction on n, we can derive the following theorem.Theorem 6.Let  d i (i � 1, 2, . . ., n) be a collection of IVDHFEs, p, q ≥ 0 and p, q do not take the value 0 simultaneously.e aggregated result of the IVDHFPGHM operator is also an IVDHFE as follows: 6 Mathematical Problems in Engineering where Specially, if Sup( ) for all i ≠ s, then the IVDHFPGHM operator reduces to the interval-valued dual hesitant fuzzy geometric Heronian mean (IVDHFGHM) operator: .  (24) It can be easily proved that the IVDHFPGHM operator has following properties.

Theorem 7 (boundedness). For a collection of IVDHFEs
. ., n), p, q ≥ 0 and p, q do not take the value 0 simultaneously.Let we have where Theorem 8 (commutativity).Let . ., n) be a collection of IVDHFEs, and p � q > 0. us,

Mathematical Problems in Engineering
where e IVDHFPGHM operator is also neither idempotent nor monotonic similar to the IVDHFPHM operator.
Several special cases of the IVDHFPGHM operator can be obtained by taking different values of p and q, which are shown as follows: Case 1: if q ⟶ 0, then the IVDHFPGHM operator reduces to the interval-valued dual hesitant fuzzy power geometric descending average (IVDHFPGDA) operator: Case 2: if p ⟶ 0, then the IVDHFPGHM operator reduces to the interval-valued dual hesitant fuzzy power geometric ascending average (IVDHFPGAA) operator: Equations ( 31) and ( 32) also show that parameters p and q are not interchangeable, according to the differences between weight vectors of p  d ω i i and q  d ω i i .

IVDHFWPHM and IVDHFWPGHM Operators.
In what follows, we propose the interval-valued dual hesitant fuzzy weighted power Heronian mean (IVDHFWPHM) operator and the interval-valued dual hesitant fuzzy weighted power geometric Heronian mean (IVDHFWPGHM) operator by considering the importance of aggregated arguments.
(1) e IVDHFWPHM operator is defined as follows: (2) e IVDHFWPGHM operator is defined as follows: , then IVDHFWPHM and IVDHFWPGHM operators reduce to IVDHFPHM and IVDHFPGHM operators, respectively.Theorem 9.For a collection of IVDHFEs  d i (i � 1, 2, . . ., n), p, q ≥ 0 and p, q do not take the value 0 simultaneously; λ � (λ 1 , λ 2 , . . ., λ n ) T is the associated weight vector of e aggregated value using theIVDHFWPHM or IVDHFWPGHM operator is still an IVDHFE, and where

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where Using IVDHFE operations and mathematical induction on n, the proof of eorem 9 is similar to that of eorem 3.

Approaches to MAGDM under Interval-Valued Dual Hesitant Fuzzy Environment
In this section, we apply the developed interval-valued dual hesitant fuzzy power Heronian aggregation operators to construct approaches for MAGDM.Let A � A 1 ,  A 2 , . . ., A m } be a finite set of m alternatives, C � C 1 ,  C 2 , . . ., C n } be a set of n attributes, and λ � (λ 1 , λ 2 , . . ., λ n ) T be the weight vector of attributes C j (j � 1, 2, . . ., n) with ) m×n is the decision matrix given by decision maker e k , where denotes the evaluating value represented by an IVDHFE of the alternative A i with respect to the attribute C j (i � 1, 2, . . ., m, j � 1, 2, . . ., n, k � 1, 2, . . ., l).
In general, there are both benefit attributes (the larger the attribute value the better) and cost attributes (the smaller the attribute value the better) in MAGDM problems.We transform the interval-valued dual hesitant fuzzy decision matrix D (k) � (d (k)  ij ) m×n into normalized interval-valued dual hesitant fuzzy decision matrix ij ) m×n , using the method in [25]: where Ω B and Ω C are benefit attributes and cost attributes, respectively.Depending on actual decision situations that weight information may be given in advance or not, in the following, we propose MAGDM approaches with known weight information or not.

Approach to MAGDM with Known Weight Information.
For some decision-making problems that weights of decision makers and attributes are determined in advance, we utilize the IVDHFWPHM or IVDHFWPGHM operator to develop the following Approach I for MAGDM problems.

Approach 1
Step 1: transform decision matrix ij are arranged in the increasing order.
Step 3: aggregate all individual interval-valued dual hesitant fuzzy decision matrices ij ) m×n into the collective interval-valued dual hesitant fuzzy decision matrix  D � (  d ij ) m×n by the IVDHFWPHM or IVDHFWPGHM operator: Step 4: calculate weights ω ij associated with IVDHFEs  d ij based on weight vector λ � (λ 1 , λ 2 , . . ., λ n ) T by the following formula: where Step 6: calculate scores of the above overall IVDHFEs  d i by equation (2).If any two scores of alternatives are the same, calculate their IVDHFCP 2 functions according to equation (5).
Step 7: rank all alternatives A i by eorem 2 and select the best one(s).

Approaches to MAGDM with Unknown Weight
Information.If the information regarding weights of decision makers and attributes are unknown, we apply the IVDHFPHM or IVDHFPGHM operator to construct an approach for MAGDM problems, which is described as follows.

Approach 2.
Step 1 is the same as the step in Approach I.
Step 2: calculate weights ω (k) ij associated with IVDHFEs  d (k) ij by the following formula: where T( Step 3: aggregate all individual interval-valued dual hesitant fuzzy decision matrices Step 4: calculate weights ω ij associated with IVDHFEs  d ij by the following formula: where T( Step 5: utilize the IVDHFPHM or IVDHFPGHM operator to aggregate all evaluation IVDHFEs  d ij in the ith line of  D and derive the overall IVDHFEs  d i : e next steps are the same as Approach I. Approach I is suitable for the known weight information, we utilize the IVDHFWPHM or IVDHFWPGHM operator to aggregate all individual decision matrices and to derive the overall preference value of each alternative.Approach II is designed for the unknown weight information, we aggregate the individual decision matrices and derive the overall preference values by the IVDHFPHM or IVDHFPGHM operator.e primary characteristics of these approaches are that they can comprehensively accommodate input values in the form of IVDHFEs, regard the interrelationship of input arguments, and relieve the influence of some unreasonable data.

Illustrative Example
Suppose that a bid inviting process through which the employer or investor is trying to find out the optimal bidding scheme.
Example 3. As the development of the internet technology, more and more people are tending to use smartphone to get information rather than reading the paper.Newspapers, as a traditional industry, must expand their business by new media to keep pace with the times.As a government procurement function department, Public Resource Trading Center decided to purchase WeChat live broadcasting system for Haimen Daily newspaper.e aim of our example is to help government decision makers to select a proper supplier according to the following three attributes: (1) C 1 is the price; (2) C 2 is the quality; (3) C 3 is the technology.Obviously, C 1 is the cost-type attribute, C 2 and C 3 are the benefit-type attributes.It is assumed that four suppliers A i (i � 1, 2, 3, 4) are participating in the tender according to the tender request.ree expert teams e k (k � 1, 2, 3) are formed from junior managers of the Government Procurement Center.en, three interval-valued dual hesitant fuzzy decision matrices are constructed, as shown in Tables 1-3, where d (k)  ij is an IVDHFE that denotes all possible interval-valued membership degrees and nonmembership degrees of the alternative A i to the attribute C j by the expert team e k .

Rank Alternatives by the Proposed Method.
For this case, we use Approach I (choose the IVDHFWPGHM operator, for example) to select the best supplier.
Step 2: using equations ( 40)-( 42) to obtain the weights ω (k) (k � 1, 2, 3) associated with IVDHFEs  d (k) ij .For example, ω (1) is calculated as follows: ω (1) � ω (1) Step 3: without generality, take p � q � 2, we aggregate all individual interval-valued dual hesitant fuzzy decision matrices Step 5: again take p � q � 2, aggregate the collective IVDHFEs  d ij (i �  Step 7: according to the score values s(  d i )(i � 1, 2, 3, 4), we obtain the ranking of alternatives A i (i � 1, 2, 3, 4):    Mathematical Problems in Engineering depends on decision makers' risk preferences.Pessimists anticipate desirable outcomes and may choose small values of parameters p and q, while optimistic experts may choose big values.(2) For the computational simplicity of MAGDM problems, the decision makers can select p � q � 1 (or 0.5 or 2), which is simple and straightforward, and also take the interrelationship of input arguments into account.

Mathematical Problems in Engineering
erefore, our proposed Approach I is a very flexible and reasonable method for MAGDM with known weight information.
Case 2. Suppose that the information about weights of decision makers and attributes is unknown, Approach II can be used to select the best supplier.We also have the same ranking results as Case 1. e detailed steps and sensitivity analysis of p and q are omitted due to the calculation process is similar to that of Case 1.

Comparative Studies.
In Section 5.1, we utilize the proposed method to solve Example 3 successfully, which has proven the availability of our methods.In addition, we also analyze the impacts of parameters p and q on ranking results in Section 5.2.e sensitivity analysis illustrates the high flexibility of the proposed methods.In order to further demonstrate the advantages of our proposed methods, we use four other existing MAGDM techniques to solve Example 3 (Case 2).ese four methods are based on IVDHFA operator [25], IVDHFG operator [25], the interval-valued dual hesitant fuzzy grey relational projection (IVDHF-GRP) method [26], and IVDHF-TOPSIS [31].e ranking results of the alternatives obtained by these methods are presented in Table 9.
As can be seen from Table 9, the ranking results derived by our proposed method and those obtained by others are the same, which verifies the effectiveness and validity of our proposed approaches.In the following, we summarize and clarify the advantages of our proposed MAGDM approaches: (1) In [25], the IVDHFA and IVDHFG operators are proposed to aggregate the interval-valued dual hesitant fuzzy information.It is assumed that attributes are independent of one another.As mentioned in Example 3, there are interrelationships between attributes.For example, the attribute price C 1 is related to other attributes quality C 2 and technology C 3 , etc.Our proposed IVDHFPHM and IVDHFPGHM operators can reflect the interrelationships between attributes.Additionally, the IVDHFA and IVDHFG operators [25] cannot reduce the influence of decision makers' unreasonable evaluations on the final ranking orders.In other words, if decision makers give unreasonable evaluations, the ranking results are also unreasonable by the method in [25].In a word, our proposed operators can simultaneously regard the interrelationship of input arguments and relieve the influence Mathematical Problems in Engineering of some unreasonable data.e approaches based on power Heronian aggregation operators utilize decision information more adequately in supporting MAGDM.
(2) e GRP method combines grey system theory and vector projection principle, which can comprehensively analyze the relationships among the attributes, reflect the influence of the whole index space, and avoid the unidirectional deviation.e basic principle of the TOPSIS method is that the optimal alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution simultaneously.e ranking results obtained by IVDHF-GRP and IVDHF-TOPSIS methods are identical to those by the IVDHFPHM and IVDHFPGHM operators, which state the validity of our proposed methods.
is can explain that the proposed interval-valued dual hesitant fuzzy power Heronian aggregation operators are good complement to existing MAGDM methods.
(3) It is known that IFS, IVIFS, HFS, IVHFS, and DHFS are subsets of IVDHFS.Our interval-valued dual hesitant fuzzy power Heronian aggregation operators can also be used to solve MAGDM with IFSs, IVIFSs, HFSs, IVHFSs, and DHFSs.Moreover, our proposed power Heronian aggregation operators are defined by incorporating PA into HM (or GHM), while Liu's PHM operators [9] are introduced by combining PA with HM. us, our operators can be considered as a generalization of [9].

Conclusions
In this paper, we introduce the IVDHFCP 2 function to compare different IVDHFEs in which some IVDHFEs have the same score values, and propose some interval-valued dual hesitant fuzzy power Heronian aggregation operators, such as IVDHFPHM, IVDHFPGHM, IVDHFWPHM, and IVDHFWPGHM.Obviously, these operators can simultaneously take advantages of PA and HM (or GHM), and accommodate input arguments in the form of IVDHFSs.In addition, we utilize these operators to solve MAGDM problems under interval-valued dual hesitant fuzzy environment, and provide a numerical example to illustrate the validity and advantages of the proposed approaches.
In future researches, the application of these operators with different interval-valued dual hesitant fuzzy group decision making methods will be developed, such as TOPSIS, VIKOR, ELECTRE, and PROMETHEE.In addition, we can also extend the power Heronian aggregation operators to interval-valued dual hesitant fuzzy linguistic set, Pythagorean hesitant fuzzy set, and so on.

□
Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Figure 2 :
Figure 2: Trends of scores for four alternatives by IVDHFWPGHM operator.
are the ith smallest h 2 has bigger different preferences of decision makers than that of  h 1 .Although s(  h 1 ) � s(  h 2 ),  h 1 is more stable than  h 2 .For nonmembership degrees  g 1 and  g 2 ,  g 1 has smaller different preferences, so  g 1 is more stable than  g 2 .erefore,  d 1 is larger than  d 2 .
) v(  h) and v( g) are similar to the variance in statistics, which indicate degrees of dispersion of interval values around s(  h) and s( g), respectively.e larger the value of v(  h) − v( g), the greater the volatility of the values in IVDHFE  d. us, IVDHFE  d is more unstable and smaller.In other words, the bigger the value of IVDHFCP 2 (  d), the more stable and larger the IVDHFE  d.Theorem 2. For two finite and nonempty IVDHFEs  d 1 and  d 2 ,  d 1 >  d 2 if and only if (a) s(  d 1 ) > s(  d 2 ) or (b) s(  d 1 ) � s(  d 2 ) and IVDHFCP 2 (  d 1 ) > IVDHFCP 2 (  d 2 ).e new ranking method is more appropriate to compare IVDHFEs, which can be used to solve Example 1.We have IVDHFCP 2 (  d 1 ) � 0.9981, IVDHFCP 2 (  d 2 ) � 0.9931 using equation (5).According to eorem 2, we obtain  d 1 >  d 2 .Moreover, the result in Example 1 can be explained reasonably by Figure 1.For membership degrees  h 1 and  h 2 , e 2 , . . ., e l   denotes a group of decision makers, with the associated weight vector ξ

Table 8 :
Scores and ranking results with different parameters p and q by IVDHFWPGHM operator.

Table 9 :
Compared with other existing methods for Case 2.