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

Authors: Shenqing Jiang; Wei He; Fangfang Qin; Qingqing Cheng
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

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. This function can further compare different IVDHFEs. Then, 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. These 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 fuzzy MAGDM with known or unknown weight 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.


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## Body

## 1. Introduction

MAGDM plays an important part of decision-making theory, which is to select the best option from all of feasible alternatives by group decision makers. Because of the complexity environment, MAGDM has been applied successfully in several different uncertain environments, such as fuzzy sets [1–3], intuitionistic fuzzy sets (IFSs) [4–7], interval-valued intuitionistic fuzzy sets (IVIFSs) [8–10], linguistic fuzzy sets [11–13], hesitant fuzzy sets (HFSs) [14–18], Pythagorean fuzzy sets [19], q-rung orthopair sets [20], interval-valued hesitant fuzzy sets (IVHFSs) [21, 22], and dual hesitant fuzzy sets (DHFSs) [23, 24]. Taking advantages of DHFS and interval numbers, Ju et al. [25] proposed the concept of interval-valued dual hesitant fuzzy set (IVDHFS), which represents membership degrees and nonmembership degrees of an element to a set with several possible interval values. For the lack of knowledge and limited expertise about complicated MAGDM problems, decision makers are usually willing to express membership degrees and nonmembership degrees as two sets of interval values. IVDHFS is more flexible than most of the other existing fuzzy sets and has begun to attract researchers’ attentions [25–31]. Zang et al. [26] proposed the grey relational projection (GRP) method under interval-valued dual hesitant fuzzy environment. Zhang et al. [31] developed two ideal-solution-based MAGDM approaches: interval-valued dual hesitant fuzzy Technique for Order Preference by Similarity to Ideal Solution (IVDHF-TOPSIS) and interval-valued dual hesitant fuzzy VlseKriterijuska Optimizacija I Kompromisno Resenje (IVDHF-VIKOR).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 interval-valued dual hesitant fuzzy geometric (IVDHFG) operator. These 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 interval-valued 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. The main advantages of our approaches can be summarized as follows:(1)
These new approaches can accommodate input arguments in the form of IVDHFSs. IVDHFS is suitable for complex decision-making problems where experts are willing to indicate their hesitancy with interval values for membership and nonmembership degrees.(2)
Taking advantages of PA and HM (or GHM), the proposed operators can simultaneously regard the interrelationship of input arguments and relieve the influence of some unreasonable data.(3)
Our proposed MAGDM approaches can deal with the situations that the information about weights of decision makers and attributes may be given in advance or not.(4)
For two IVDHFEsd˜i=h˜i,g˜ii=1,2, score values sd˜i and accuracy values pd˜i are identical, but d˜1 and d˜2 are different. According to the comparison law in [25], d˜1 is equal to d˜2, which is a contradiction. Our new ranking method can compare different IVDHFEs by the IVDHFCP2 function.The paper is organized as follows. In Section2, we review some basic concepts of IVDHFS and IVDHFE,-give a new comparison law by introducing the IVDHFCP2 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.


## 2. Preliminaries

### 2.1. IVDHFS

Definition 1 (see [25]).
LetX be a reference set, and D0,1 be the set of all closed subintervals of 0,1. Then, an interval-valued dual hesitant fuzzy set (IVDHFS) on X is(1)D˜=x,h˜x,g˜xx∈X,where h˜x:X⟶D0,1 and g˜x:X⟶D0,1 denote all possible interval-valued membership degrees and nonmembership degrees of the element x∈X to the set D˜, respectively, with the conditions: h˜x=∪γ˜L,γ˜U∈h˜xγ˜L,γ˜U, g˜x=∪η˜L,η˜U∈g˜xη˜L,η˜U, and 0≤γ˜U++η˜U+≤1, where γ˜U+∈h˜+x=maxγ˜L,γ˜U∈h˜xγ˜U and η˜U+∈g˜+x=maxη˜L,η˜U∈g˜xη˜U for all x∈X. For convenience, we call the pair d˜x=h˜x,g˜x an interval-valued dual hesitant fuzzy element (IVDHFE), denoted by d˜=h˜,g˜.
Ifγ˜L=γ˜U=γ, for all γ˜L,γ˜U∈h˜, and η˜L=η˜U=η, for all η˜L,η˜U∈g˜, then IVDHFE d˜ reduces to dual hesitant fuzzy element (DHFE) d=∪γ∈h˜γ,∪η∈g˜η. If g˜=Φ, then IVDHFE d˜ reduces to interval-valued hesitant fuzzy element (IVHFE) h˜=∪γ˜L,γ˜U∈h˜γ˜L,γ˜U. If γ˜L=γ˜U=γ, for all γ˜L,γ˜U∈h˜, and g˜=Φ, then IVDHFE d˜ reduces to hesitant fuzzy element (HFE) h=∪γ∈h˜γ. Hence, HFE, IVHFE, and DHFE are the special cases of IVDHFE. Moreover, IVDHFS reduces to IVIFS when the membership degrees and nonmembership degrees of each element to a given set only have an interval value, respectively. Therefore, IFS, IVIFS, HFS, IVHFS, and DHFS are subsets of IVDHFS.Definition 2 (see [25]).
For three IVDHFEs,d˜=h˜,g˜, d˜1=h˜1,g˜1, and d˜2=h˜2,g˜2, and λ>0, some basic operational laws are defined as follows:(1)
d1˜⊕d2˜=∪γ˜jL,γ˜jU∈h˜j,η˜jL,η˜jU∈g˜jγ˜1L+γ˜2L−γ˜1Lγ˜2L,γ˜1U+γ˜2U−γ˜1Uγ˜2U,η˜1Lη˜2L,η˜1Uη˜2U(2)
d1˜⊗d2˜=∪γ˜jL,γ˜jU∈h˜j,η˜jL,η˜jU∈g˜jγ˜1Lγ˜2L,γ˜1Uγ˜2U,η˜1L+η˜2L−η˜1Lη˜2L,η˜1U+η˜2U−η˜1Uη˜2U(3)
λd˜=∪γ˜L,γ˜U∈h˜,η˜L,η˜U∈g˜1−1−γ˜Lλ,1−1−γ˜Uλ,η˜Lλ,η˜Uλ(4)
d˜λ=∪γ˜L,γ˜U∈h˜,η˜L,η˜U∈g˜γ˜Lλ,γ˜Uλ,1−1−η˜Lλ,1−1−η˜UλDefinition 3 (see [25]).
Letd˜=h˜,g˜ be an IVDHFE, score and accuracy functions of d˜ are described as follows:(2)sd˜=121#h˜∑γ˜L,γ˜U∈h˜​γ˜L+γ˜U−1#g˜∑η˜L,η˜U∈g˜​η˜L+η˜U,(3)pd˜=121#h˜∑γ˜L,γ˜U∈h˜​γ˜L+γ˜U+1#g˜∑η˜L,η˜U∈g˜​η˜L+η˜U,where #h˜ and #g˜ are numbers of interval values in h˜ and g˜, respectively.Theorem 1 (see [25]).
For two IVDHFEsd˜1=h˜1,g˜1 and d˜2=h˜2,g˜2, then(1)
Ifsd˜1>sd˜2, then d˜1 is superior to d˜2, denoted by d˜1>d˜2.(2)
Ifsd˜1=sd˜2, then(a)
Ifpd˜1>pd˜2, then d˜1 is superior to d˜2, denoted by d˜1>d˜2.(b)
Ifpd˜1=pd˜2, then d˜1 is equivalent to d˜2, denoted by d˜1=d˜2.For two IVDHFEsd˜1=h˜1,g˜1 and d˜2=h˜2,g˜2, #h˜1, #h˜2, #g˜1, and #g˜2 are numbers of interval values in h˜1, h˜2, g˜1, and g˜2, respectively. Let #h˜=max#h˜1,#h˜2 and #g˜=max#g˜1,#g˜2. In most cases, #h˜1 and #g˜1 are not equal to #h˜2 and #g˜2, respectively, i.e., #h˜1≠#h˜2 or #g˜1≠#g˜2. 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. The 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.Definition 4 (see [26]).
The interval-valued dual hesitant normalized Hamming distance between two IVDHFEsd˜1=h˜1,g˜1 and d˜2=h˜2,g˜2 is defined as follows:(4)dd˜1,d˜2=141#h˜∑i=1#h˜γ˜1σiL−γ˜2σiL+γ˜1σiU−γ˜2σiU+1#g˜∑i=1#g˜η˜1σiL−η˜2σiL+η˜1σiU−η˜2σiU,where intervals in h˜t, g˜tt=1,2 are arranged in the increasing order, i.e., γ˜tσiL,γ˜tσiU and η˜tσiL,η˜tσiU are the ith smallest interval values in h˜t, g˜tt=1,2, respectively.


### 2.2. Improved Comparison Laws for IVDHFEs

Score and accuracy functions in Definition3 have been successfully applied to compare alternative assessments denoted by IVDHFEs in the literatures [25–31]. In some situations, such functions cannot compare all different IVDHFEs, which can be seen by the following example.Example 1.
For two IVDHFEs,d˜1=0.2,0.4,0.3,0.4,0.4,0.5,0.5,0.6 and d˜2=0.1,0.3,0.4,0.5,0.3,0.6,0.5,0.6. By Definition 3, sd˜1=sd˜2=−0.1750 and pd˜1=pd˜2=0.8250, which means that Theorem 1 cannot be applied to rank all IVDHFEs.
In order to overcome this flaw, we define theIVDHFCP2 function as follows.Definition 5.
Letd˜=h˜,g˜=∪i=1#h˜γ˜σiL,γ˜σiU,∪i=1#g˜η˜σiL,η˜σiU be a finite and nonempty IVDHFE, where γ˜σiL,γ˜σiU and η˜σiL,η˜σiU are the ith smallest interval values in h˜ and g˜,#h˜ and #g˜ are numbers of interval values in h˜ and g˜, respectively. sh˜=∑i=1#h˜γ˜σiL+γ˜σiU/2#h˜ and sg˜=∑i=1#g˜η˜σiL+η˜σiU/2#g˜ are the central membership value and the central nonmembership value, respectively:(5)IVDHFCP2d˜=1−12#h˜∑i=1#h˜γ˜σiL−sh˜2+γ˜σiU−sh˜2−12#g˜∑i=1#g˜η˜σiL−sg˜2+η˜σiU−sg˜2,is called the interval-valued dual hesitant fuzzy 2nd-order central polymerization degree IVDHFCP2 function of d˜.
For the sake of simplicity, we letvh˜=∑i=1#h˜γ˜σiL−sh˜2+γ˜σiU−sh˜2/2#h˜ and vg˜=∑i=1#g˜η˜σiL−sg˜2+η˜σiU−sg˜2/2#g˜. Thus, equation (5) can be denoted as follows:(6)IVDHFCP2d˜=1−vh˜−vg˜.
vh˜ and vg˜ are similar to the variance in statistics, which indicate degrees of dispersion of interval values around sh˜ and sg˜, respectively. The larger the value of vh˜−vg˜, the greater the volatility of the values in IVDHFE d˜. Thus, IVDHFE d˜ is more unstable and smaller. In other words, the bigger the value of IVDHFCP2d˜, the more stable and larger the IVDHFE d˜.Theorem 2.
For two finite and nonempty IVDHFEsd˜1 and d˜2, d˜1>d˜2 if and only if (a) sd˜1>sd˜2 or (b) sd˜1=sd˜2 and IVDHFCP2d˜1>IVDHFCP2d˜2.The new ranking method is more appropriate to compare IVDHFEs, which can be used to solve Example1. We have IVDHFCP2d˜1=0.9981, IVDHFCP2d˜2=0.9931 using equation (5). According to Theorem 2, we obtain d˜1>d˜2.Moreover, the result in Example1 can be explained reasonably by Figure 1. For membership degrees h˜1 and h˜2, h˜2 has bigger different preferences of decision makers than that of h˜1. Although sh˜1=sh˜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. Therefore, d˜1 is larger than d˜2.Figure 1
Comparisons for memberships and nonmemberships ofd˜1 and d˜2.
(a)(b)


## 2.1. IVDHFS

Definition 1 (see [25]).
LetX be a reference set, and D0,1 be the set of all closed subintervals of 0,1. Then, an interval-valued dual hesitant fuzzy set (IVDHFS) on X is(1)D˜=x,h˜x,g˜xx∈X,where h˜x:X⟶D0,1 and g˜x:X⟶D0,1 denote all possible interval-valued membership degrees and nonmembership degrees of the element x∈X to the set D˜, respectively, with the conditions: h˜x=∪γ˜L,γ˜U∈h˜xγ˜L,γ˜U, g˜x=∪η˜L,η˜U∈g˜xη˜L,η˜U, and 0≤γ˜U++η˜U+≤1, where γ˜U+∈h˜+x=maxγ˜L,γ˜U∈h˜xγ˜U and η˜U+∈g˜+x=maxη˜L,η˜U∈g˜xη˜U for all x∈X. For convenience, we call the pair d˜x=h˜x,g˜x an interval-valued dual hesitant fuzzy element (IVDHFE), denoted by d˜=h˜,g˜.
Ifγ˜L=γ˜U=γ, for all γ˜L,γ˜U∈h˜, and η˜L=η˜U=η, for all η˜L,η˜U∈g˜, then IVDHFE d˜ reduces to dual hesitant fuzzy element (DHFE) d=∪γ∈h˜γ,∪η∈g˜η. If g˜=Φ, then IVDHFE d˜ reduces to interval-valued hesitant fuzzy element (IVHFE) h˜=∪γ˜L,γ˜U∈h˜γ˜L,γ˜U. If γ˜L=γ˜U=γ, for all γ˜L,γ˜U∈h˜, and g˜=Φ, then IVDHFE d˜ reduces to hesitant fuzzy element (HFE) h=∪γ∈h˜γ. Hence, HFE, IVHFE, and DHFE are the special cases of IVDHFE. Moreover, IVDHFS reduces to IVIFS when the membership degrees and nonmembership degrees of each element to a given set only have an interval value, respectively. Therefore, IFS, IVIFS, HFS, IVHFS, and DHFS are subsets of IVDHFS.Definition 2 (see [25]).
For three IVDHFEs,d˜=h˜,g˜, d˜1=h˜1,g˜1, and d˜2=h˜2,g˜2, and λ>0, some basic operational laws are defined as follows:(1)
d1˜⊕d2˜=∪γ˜jL,γ˜jU∈h˜j,η˜jL,η˜jU∈g˜jγ˜1L+γ˜2L−γ˜1Lγ˜2L,γ˜1U+γ˜2U−γ˜1Uγ˜2U,η˜1Lη˜2L,η˜1Uη˜2U(2)
d1˜⊗d2˜=∪γ˜jL,γ˜jU∈h˜j,η˜jL,η˜jU∈g˜jγ˜1Lγ˜2L,γ˜1Uγ˜2U,η˜1L+η˜2L−η˜1Lη˜2L,η˜1U+η˜2U−η˜1Uη˜2U(3)
λd˜=∪γ˜L,γ˜U∈h˜,η˜L,η˜U∈g˜1−1−γ˜Lλ,1−1−γ˜Uλ,η˜Lλ,η˜Uλ(4)
d˜λ=∪γ˜L,γ˜U∈h˜,η˜L,η˜U∈g˜γ˜Lλ,γ˜Uλ,1−1−η˜Lλ,1−1−η˜UλDefinition 3 (see [25]).
Letd˜=h˜,g˜ be an IVDHFE, score and accuracy functions of d˜ are described as follows:(2)sd˜=121#h˜∑γ˜L,γ˜U∈h˜​γ˜L+γ˜U−1#g˜∑η˜L,η˜U∈g˜​η˜L+η˜U,(3)pd˜=121#h˜∑γ˜L,γ˜U∈h˜​γ˜L+γ˜U+1#g˜∑η˜L,η˜U∈g˜​η˜L+η˜U,where #h˜ and #g˜ are numbers of interval values in h˜ and g˜, respectively.Theorem 1 (see [25]).
For two IVDHFEsd˜1=h˜1,g˜1 and d˜2=h˜2,g˜2, then(1)
Ifsd˜1>sd˜2, then d˜1 is superior to d˜2, denoted by d˜1>d˜2.(2)
Ifsd˜1=sd˜2, then(a)
Ifpd˜1>pd˜2, then d˜1 is superior to d˜2, denoted by d˜1>d˜2.(b)
Ifpd˜1=pd˜2, then d˜1 is equivalent to d˜2, denoted by d˜1=d˜2.For two IVDHFEsd˜1=h˜1,g˜1 and d˜2=h˜2,g˜2, #h˜1, #h˜2, #g˜1, and #g˜2 are numbers of interval values in h˜1, h˜2, g˜1, and g˜2, respectively. Let #h˜=max#h˜1,#h˜2 and #g˜=max#g˜1,#g˜2. In most cases, #h˜1 and #g˜1 are not equal to #h˜2 and #g˜2, respectively, i.e., #h˜1≠#h˜2 or #g˜1≠#g˜2. 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. The 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.Definition 4 (see [26]).
The interval-valued dual hesitant normalized Hamming distance between two IVDHFEsd˜1=h˜1,g˜1 and d˜2=h˜2,g˜2 is defined as follows:(4)dd˜1,d˜2=141#h˜∑i=1#h˜γ˜1σiL−γ˜2σiL+γ˜1σiU−γ˜2σiU+1#g˜∑i=1#g˜η˜1σiL−η˜2σiL+η˜1σiU−η˜2σiU,where intervals in h˜t, g˜tt=1,2 are arranged in the increasing order, i.e., γ˜tσiL,γ˜tσiU and η˜tσiL,η˜tσiU are the ith smallest interval values in h˜t, g˜tt=1,2, respectively.


## 2.2. Improved Comparison Laws for IVDHFEs

Score and accuracy functions in Definition3 have been successfully applied to compare alternative assessments denoted by IVDHFEs in the literatures [25–31]. In some situations, such functions cannot compare all different IVDHFEs, which can be seen by the following example.Example 1.
For two IVDHFEs,d˜1=0.2,0.4,0.3,0.4,0.4,0.5,0.5,0.6 and d˜2=0.1,0.3,0.4,0.5,0.3,0.6,0.5,0.6. By Definition 3, sd˜1=sd˜2=−0.1750 and pd˜1=pd˜2=0.8250, which means that Theorem 1 cannot be applied to rank all IVDHFEs.
In order to overcome this flaw, we define theIVDHFCP2 function as follows.Definition 5.
Letd˜=h˜,g˜=∪i=1#h˜γ˜σiL,γ˜σiU,∪i=1#g˜η˜σiL,η˜σiU be a finite and nonempty IVDHFE, where γ˜σiL,γ˜σiU and η˜σiL,η˜σiU are the ith smallest interval values in h˜ and g˜,#h˜ and #g˜ are numbers of interval values in h˜ and g˜, respectively. sh˜=∑i=1#h˜γ˜σiL+γ˜σiU/2#h˜ and sg˜=∑i=1#g˜η˜σiL+η˜σiU/2#g˜ are the central membership value and the central nonmembership value, respectively:(5)IVDHFCP2d˜=1−12#h˜∑i=1#h˜γ˜σiL−sh˜2+γ˜σiU−sh˜2−12#g˜∑i=1#g˜η˜σiL−sg˜2+η˜σiU−sg˜2,is called the interval-valued dual hesitant fuzzy 2nd-order central polymerization degree IVDHFCP2 function of d˜.
For the sake of simplicity, we letvh˜=∑i=1#h˜γ˜σiL−sh˜2+γ˜σiU−sh˜2/2#h˜ and vg˜=∑i=1#g˜η˜σiL−sg˜2+η˜σiU−sg˜2/2#g˜. Thus, equation (5) can be denoted as follows:(6)IVDHFCP2d˜=1−vh˜−vg˜.
vh˜ and vg˜ are similar to the variance in statistics, which indicate degrees of dispersion of interval values around sh˜ and sg˜, respectively. The larger the value of vh˜−vg˜, the greater the volatility of the values in IVDHFE d˜. Thus, IVDHFE d˜ is more unstable and smaller. In other words, the bigger the value of IVDHFCP2d˜, the more stable and larger the IVDHFE d˜.Theorem 2.
For two finite and nonempty IVDHFEsd˜1 and d˜2, d˜1>d˜2 if and only if (a) sd˜1>sd˜2 or (b) sd˜1=sd˜2 and IVDHFCP2d˜1>IVDHFCP2d˜2.The new ranking method is more appropriate to compare IVDHFEs, which can be used to solve Example1. We have IVDHFCP2d˜1=0.9981, IVDHFCP2d˜2=0.9931 using equation (5). According to Theorem 2, we obtain d˜1>d˜2.Moreover, the result in Example1 can be explained reasonably by Figure 1. For membership degrees h˜1 and h˜2, h˜2 has bigger different preferences of decision makers than that of h˜1. Although sh˜1=sh˜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. Therefore, d˜1 is larger than d˜2.Figure 1
Comparisons for memberships and nonmemberships ofd˜1 and d˜2.
(a)(b)


## 3. 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.


### 3.1. IVDHFPHM Operator

Definition 6.
Letd˜ii=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:(7)IVDHFPHMp,qd˜1,d˜2,…,d˜n=2nn+1⊕i=1n⊕j=inζij1/p+q,where ζij=ωid˜ip⊗ωjd˜jq,i=1,2,…,n,j=i,i+1,…,n, ωi=n1+Td˜i/∑t=1n1+Td˜t, Td˜i=∑s=1,s≠inSupd˜i,d˜s, and Supd˜i,d˜s is the support for d˜i from d˜s, which satisfies the following three properties:(1)
Supd˜i,d˜s∈0,1(2)
Supd˜i,d˜s=Supd˜s,d˜i(3)
Supd˜i,d˜s≥Supd˜m,d˜n, if dd˜i,d˜s<dd˜m,d˜nwhere d is a distance measure between two IVDHFEs and calculated by equation (4).Theorem 3.
Letd˜ii=1,2,…,n be a collection of IVDHFEs, p,q≥0 and p,q do not take the value 0 simultaneously. Then, the aggregated value using the IVDHFPHM operator is still an IVDHFE, and(8)IVDHFPHMp,qd˜1,d˜2,…,d˜n=∪αij∈h˜ζij1−∏i=1n∏j=in1−αij2/nn+11/p+q,∪βij∈g˜ζij1−1−∏i=1n∏j=inβij2/nn+11/p+q,where(9)αij∈h˜ζij=∪γ˜iL,γ˜iU∈h˜i,i=1,2,…,nγ˜jL,γ˜jU∈h˜j,j=i,i+1,…,n1−1−γ˜iLωip1−1−γ˜jLωjq,1−1−γ˜iUωip1−1−γ˜jUωjq,βij∈g˜ζij=∪η˜iL,η˜iU∈g˜i,i=1,2,…,nη˜jL,η˜jU∈g˜j,j=i,i+1,…,n1−1−η˜iLωip1−η˜jLωjq,1−1−η˜iUωip1−η˜jUωjq.Specially, ifSupd˜i,d˜s=cc∈0,1 for all i≠s, i.e., ωi=1, then the IVDHFPHM operator reduces to the interval-valued dual hesitant fuzzy Heronian mean (IVDHFHM) operator:(10)IVDHFHMp,qd˜1,d˜2,…,d˜n=2nn+1⊕i=1n⊕j=ind˜ip⊗d˜jq1/p+q.Theorem 4 (boundedness).
For a collection of IVDHFEsd˜i=h˜i,g˜i,i=1,2,…,n, p,q≥0 and p,q do not take the value 0 simultaneously. Let(11)γ˜L−,γ˜U−=mini=1nγ˜iL,mini=1nγ˜iU,γ˜L+,γ˜U+=maxi=1nγ˜iL,maxi=1nγ˜iU,η˜L−,η˜U−=mini=1nη˜iL,mini=1nη˜iU,η˜L+,η˜U+=maxi=1nη˜iL,maxi=1nη˜iU,we have(12)d˜−≤IVDHFPHMp,qd˜1,d˜2,…,d˜n≤d˜+,where(13)d˜−=1−∏i=1n∏j=in1−αij−2/nn+11/p+q,1−1−∏i=1n∏j=inβij+2/nn+11/p+q,in which(14)αij−=1−1−γ˜L−ωip1−1−γ˜L−ωjq,1−1−γ˜U−ωip1−1−γ˜U−ωjq,βij+=1−1−η˜L+ωip1−η˜L+ωjq,1−1−η˜U+ωip1−η˜U+ωjq,d˜+=1−∏i=1n∏j=in1−αij+2/nn+11/p+q,1−1−∏i=1n∏j=inβij−2/nn+11/p+q,in which(15)αij+=1−1−γ˜L+ωip1−1−γ˜L+ωjq,1−1−γ˜U+ωip1−1−γ˜U+ωjq,βij−=1−1−η˜L−ωip1−η˜L−ωjq,1−1−η˜U−ωip1−η˜U−ωjq.Theorem 5 (commutativity).
Letd˜i=h˜i,g˜i,i=1,2,…,n be a collection of IVDHFEs, and p=q>0. Thus,(16)IVDHFPHMp,qd˜1,d˜2,…,d˜n=IVDHFPHMp,qd˜1′,d˜2′,…,d˜n′,where d˜1′,d˜2′,…,d˜n′ is any permutation of d˜1,d˜2,…,d˜n.The proofs of Theorems3–5 are shown in Appendixes A–C, respectively. However, the IVDHFPHM operator has no properties of idempotency and monotonicity, as illustrated in the following example.Example 2.
For two IVDHFEsd˜1andd˜2 in Example 1, we have sd˜1=sd˜2=−0.1750 and d˜1>d˜2. Suppose p=q=2, the aggregated results can be obtained by equation (8) as follows:(17)IVDHFPHM2,2d˜1,d˜1=0.2000,0.4000,0.2581,0.4000,0.2581,0.4000,0.3000,0.4000,0.4000,0.5000,0.4456,0.5451,0.4456,0.5451,0.5000,0.6000,IVDHFPHM2,2d˜1,d˜2=0.1627,0.3560,0.3257,0.4549,0.2348,0.3560,0.3560,0.4549,0.3457,0.5451,0.4456,0.5451,0.3830,0.6000,0.5000,0.6000.
According to equation (2), scores of the above aggregated IVDHFEs are(18)sIVDHFPHM2,2d˜1,d˜1=−0.1707,sIVDHFPHM2,2d˜1,d˜2=−0.1579.
It is clear thatsIVDHFPHM2,2d˜1,d˜1≠sd˜1. By Theorem 2, we can know IVDHFPHM2,2d˜1,d˜1≠d˜1, which implies the IVDHFPHM operator is not idempotent.
Furthermore, sincesIVDHFPHM2,2d˜1,d˜1<sIVDHFPHM2,2d˜1,d˜2, then IVDHFPHM2,2d˜1,d˜1<IVDHFPHM2,2d˜1,d˜2, but d˜1>d˜2, which shows the IVDHFPHM operator is not monotonic.
Now, we can discuss some special cases of the IVDHFPHM operator by assigning different values of parametersp and q.Case 1: ifq⟶0, then the IVDHFPHM operator reduces to the interval-valued dual hesitant fuzzy power descending average (IVDHFPDA) operator:(19)IVDHFPHMp,0d˜1,d˜2,…,d˜n=2nn+1⊕i=1nn+1−iωid˜ip1/p.Obviously, the weight vector ofωid˜ip is n,n−1,…,1.Case 2: ifp⟶0, then the IVDHFPHM operator reduces to the interval-valued dual hesitant fuzzy power ascending average (IVDHFPAA) operator:(20)IVDHFPHM0,qd˜1,d˜2,…,d˜n=2nn+1⊕i=1niωid˜iq1/q.Obviously, the weight vector ofωid˜iq is 1,2,…,n.From equations (19) and (20), we can see that the weight vectors of ωid˜ip and ωid˜iq are different. Hence, parameters p and q of the IVDHFPHM operator are not interchangeable.


### 3.2. IVDHFPGHM Operator

In this section, we introduce the IVDHFPGHM operator by incorporating PA into GHM.Definition 7.
Letd˜ii=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 geometric Heronian mean IVDHFPGHM operator is defined as follows:(21)IVDHFPGHMp,qd˜1,d˜2,…,d˜n=1p+q⊗i=1n⊗j=inδij2/nn+1,where δij=pd˜iωi⊕qd˜jωj,i=1,2,…,n,j=i,i+1,…,n, ωi=n1+Td˜i/∑t=1n1+Td˜t, and Td˜i=∑s=1,s≠inSupd˜i,d˜s.
Based on operational laws of IVDHFEs and mathematical induction onn, we can derive the following theorem.Theorem 6.
Letd˜ii=1,2,…,n be a collection of IVDHFEs, p,q≥0 and p,q do not take the value 0 simultaneously. The aggregated result of the IVDHFPGHM operator is also an IVDHFE as follows:(22)IVDHFPGHMp,qd˜1,d˜2,…,d˜n=∪μij∈h˜δij1−1−∏i=1n∏j=inμij2/nn+11/p+q,∪νij∈g˜δij1−∏i=1n∏j=in1−νij2/nn+11/p+q,where(23)μij∈h˜δij=∪γ˜iL,γ˜iU∈h˜i,i=1,2,…,nγ˜jL,γ˜jU∈h˜j,j=i,i+1,…,n1−1−γ˜iLωip1−γ˜jLωjq,1−1−γ˜iUωip1−γ˜jUωjq,νij∈g˜δij=∪η˜iL,η˜iU∈g˜i,i=1,2,…,nη˜jL,η˜jU∈g˜j,j=i,i+1,…,n1−1−η˜iLωip1−1−η˜jLωjq,1−1−η˜iUωip1−1−η˜jUωjq.Specially, ifSupd˜i,d˜s=cc∈0,1 for all i≠s, then the IVDHFPGHM operator reduces to the interval-valued dual hesitant fuzzy geometric Heronian mean (IVDHFGHM) operator:(24)IVDHFGHMp,qd˜1,d˜2,…,d˜n=1p+q⊗i=1n⊗j=inpd˜i⊕qd˜j2/nn+1.It can be easily proved that the IVDHFPGHM operator has following properties.Theorem 7 (boundedness).
For a collection of IVDHFEsd˜i=h˜i,g˜ii=1,2,…,n, p,q≥0 and p,q do not take the value 0 simultaneously. Let(25)γ˜L−,γ˜U−=mini=1nγ˜iL,mini=1nγ˜iU,γ˜L+,γ˜U+=maxi=1nγ˜iL,maxi=1nγ˜iU,η˜L−,η˜U−=mini=1nη˜iL,mini=1nη˜iU,η˜L+,η˜U+=maxi=1nη˜iL,maxi=1nη˜iU,we have(26)a˜−≤IVDHFPGHMp,qd˜1,d˜2,…,d˜n≤a˜+,where(27)a˜−=1−1−∏i=1n∏j=inμij−2/nn+11/p+q,1−∏i=1n∏j=in1−νij+2/nn+11/p+q,in which(28)μij−=1−1−γ˜L−ωip1−γ˜L−ωjq,1−1−γ˜U−ωip1−γ˜U−ωjq,νij+=1−1−η˜L+ωip1−1−η˜L+ωjq,1−1−η˜U+ωip1−1−η˜U+ωjq,a˜+=1−1−∏i=1n∏j=inμij+2/nn+11/p+q,1−∏i=1n∏j=in1−νij−2/nn+11/p+q,in which(29)μij+=1−1−γ˜L+ωip1−γ˜L+ωjq,1−1−γ˜U+ωip1−γ˜U+ωjq,νij−=1−1−η˜L−ωip1−1−η˜L−ωjq,1−1−η˜U−ωip1−1−η˜U−ωjq.Theorem 8 (commutativity).
Letd˜i=h˜i,g˜ii=1,2,…,n be a collection of IVDHFEs, and p=q>0. Thus,(30)IVDHFPGHMp,qd˜1,d˜2,…,d˜n=IVDHFPGHMp,qd˜1′,d˜2′,…,d˜n′,where d˜1′,d˜2′,…,d˜n′ is any permutation of d˜1,d˜2,…,d˜n.The 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 ofp and q, which are shown as follows:Case 1: ifq⟶0, then the IVDHFPGHM operator reduces to the interval-valued dual hesitant fuzzy power geometric descending average (IVDHFPGDA) operator:(31)IVDHFPGHMp,0d˜1,d˜2,…,d˜n=1p⊗i=1npd˜iωi2n+1−i/nn+1.Case 2: ifp⟶0, then the IVDHFPGHM operator reduces to the interval-valued dual hesitant fuzzy power geometric ascending average (IVDHFPGAA) operator:(32)IVDHFPGHM0,qd˜1,d˜2,…,d˜n=1q⊗i=1nqd˜iωi2i/nn+1.Equations (31) and (32) also show that parameters p and q are not interchangeable, according to the differences between weight vectors of pd˜iωi and qd˜iωi.


### 3.3. 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.Definition 8.
Letd˜ii=1,2,…,n be a collection of IVDHFEs, λ=λ1,λ2,…,λnT be the associated weight vector of d˜i with λi∈0,1, ∑i=1nλi=1, p,q≥0 and p,q do not take the value 0 simultaneously.(1)
TheIVDHFWPHM operator is defined as follows:(33)IVDHFWPHMp,qd˜1,d˜2,…,d˜n=2nn+1⊕i=1n⊕j=inζ¯ij1/p+q.(2)
TheIVDHFWPGHM operator is defined as follows:(34)IVDHFWPGHMp,qd˜1,d˜2,…,d˜n=1p+q⊗i=1n⊗j=inδ¯ij2/nn+1,in which ζ¯ij=ω¯id˜ip⊗ω¯jd˜jq, δ¯ij=pd˜iω¯i⊕qd˜jω¯j,i=1,2,…,n,j=i,i+1,…,n, ω¯i=nλi1+T¯d˜i/∑t=1nλt1+T¯d˜t and T¯d˜i=∑s=1,s≠inλsSupd˜i,d˜s.
Specially, ifλ1=λ2=⋯=λn=1/n, then IVDHFWPHM and IVDHFWPGHM operators reduce to IVDHFPHM and IVDHFPGHM operators, respectively.Theorem 9.
For a collection of IVDHFEsd˜ii=1,2,…,n, p,q≥0 and p,q do not take the value 0 simultaneously; λ=λ1,λ2,…,λnT is the associated weight vector of d˜i with λi∈0,1, ∑i=1nλi=1. The aggregated value using theIVDHFWPHM or IVDHFWPGHM operator is still an IVDHFE, and(35)IVDHFWPHMp,qd˜1,d˜2,…,d˜n=∪α¯ij∈h˜ζ¯ij1−∏i=1n∏j=in1−α¯ij2/nn+11/p+q,∪β¯ij∈g˜ζ¯ij1−1−∏i=1n∏j=inβ¯ij2/nn+11/p+q,where(36)α¯ij∈h˜ζ¯ij=∪γ˜iL,γ˜iU∈h˜i,i=1,2,…,nγ˜jL,γ˜jU∈h˜j,j=i,i+1,…,n1−1−γ˜iLω¯ip1−1−γ˜jLω¯jq,1−1−γ˜iUω¯ip1−1−γ˜jUω¯jq,β¯ij∈g˜ζ¯ij=∪η˜iL,η˜iU∈g˜i,i=1,2,…,nη˜jL,η˜jU∈g˜j,j=i,i+1,…,n1−1−η˜iLω¯ip1−η˜jLω¯jq,1−1−η˜iUω¯ip1−η˜jUω¯jq,IVDHFWPGHMp,qd˜1,d˜2,…,d˜n=∪μ¯ij∈h˜δ¯ij1−1−∏i=1n∏j=inμ¯ij2/nn+11/p+q,∪ν¯ij∈g˜δ¯ij1−∏i=1n∏j=in1−ν¯ij2/nn+11/p+q,where(37)μ¯ij∈h˜δ¯ij=∪γ˜iL,γ˜iU∈h˜i,i=1,2,…,nγ˜jL,γ˜jU∈h˜j,j=i,i+1,…,n1−1−γ˜iLω¯ip1−γ˜jLω¯jq,1−1−γ˜iUω¯ip1−γ˜jUω¯jq,ν¯ij∈g˜δ¯ij=∪η˜iL,η˜iU∈g˜i,i=1,2,…,nη˜jL,η˜jU∈g˜j,j=i,i+1,…,n1−1−η˜iLω¯ip1−1−η˜jLω¯jq,1−1−η˜iUω¯ip1−1−η˜jUω¯jq.Using IVDHFE operations and mathematical induction onn, the proof of Theorem 9 is similar to that of Theorem 3.


## 3.1. IVDHFPHM Operator

Definition 6.
Letd˜ii=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:(7)IVDHFPHMp,qd˜1,d˜2,…,d˜n=2nn+1⊕i=1n⊕j=inζij1/p+q,where ζij=ωid˜ip⊗ωjd˜jq,i=1,2,…,n,j=i,i+1,…,n, ωi=n1+Td˜i/∑t=1n1+Td˜t, Td˜i=∑s=1,s≠inSupd˜i,d˜s, and Supd˜i,d˜s is the support for d˜i from d˜s, which satisfies the following three properties:(1)
Supd˜i,d˜s∈0,1(2)
Supd˜i,d˜s=Supd˜s,d˜i(3)
Supd˜i,d˜s≥Supd˜m,d˜n, if dd˜i,d˜s<dd˜m,d˜nwhere d is a distance measure between two IVDHFEs and calculated by equation (4).Theorem 3.
Letd˜ii=1,2,…,n be a collection of IVDHFEs, p,q≥0 and p,q do not take the value 0 simultaneously. Then, the aggregated value using the IVDHFPHM operator is still an IVDHFE, and(8)IVDHFPHMp,qd˜1,d˜2,…,d˜n=∪αij∈h˜ζij1−∏i=1n∏j=in1−αij2/nn+11/p+q,∪βij∈g˜ζij1−1−∏i=1n∏j=inβij2/nn+11/p+q,where(9)αij∈h˜ζij=∪γ˜iL,γ˜iU∈h˜i,i=1,2,…,nγ˜jL,γ˜jU∈h˜j,j=i,i+1,…,n1−1−γ˜iLωip1−1−γ˜jLωjq,1−1−γ˜iUωip1−1−γ˜jUωjq,βij∈g˜ζij=∪η˜iL,η˜iU∈g˜i,i=1,2,…,nη˜jL,η˜jU∈g˜j,j=i,i+1,…,n1−1−η˜iLωip1−η˜jLωjq,1−1−η˜iUωip1−η˜jUωjq.Specially, ifSupd˜i,d˜s=cc∈0,1 for all i≠s, i.e., ωi=1, then the IVDHFPHM operator reduces to the interval-valued dual hesitant fuzzy Heronian mean (IVDHFHM) operator:(10)IVDHFHMp,qd˜1,d˜2,…,d˜n=2nn+1⊕i=1n⊕j=ind˜ip⊗d˜jq1/p+q.Theorem 4 (boundedness).
For a collection of IVDHFEsd˜i=h˜i,g˜i,i=1,2,…,n, p,q≥0 and p,q do not take the value 0 simultaneously. Let(11)γ˜L−,γ˜U−=mini=1nγ˜iL,mini=1nγ˜iU,γ˜L+,γ˜U+=maxi=1nγ˜iL,maxi=1nγ˜iU,η˜L−,η˜U−=mini=1nη˜iL,mini=1nη˜iU,η˜L+,η˜U+=maxi=1nη˜iL,maxi=1nη˜iU,we have(12)d˜−≤IVDHFPHMp,qd˜1,d˜2,…,d˜n≤d˜+,where(13)d˜−=1−∏i=1n∏j=in1−αij−2/nn+11/p+q,1−1−∏i=1n∏j=inβij+2/nn+11/p+q,in which(14)αij−=1−1−γ˜L−ωip1−1−γ˜L−ωjq,1−1−γ˜U−ωip1−1−γ˜U−ωjq,βij+=1−1−η˜L+ωip1−η˜L+ωjq,1−1−η˜U+ωip1−η˜U+ωjq,d˜+=1−∏i=1n∏j=in1−αij+2/nn+11/p+q,1−1−∏i=1n∏j=inβij−2/nn+11/p+q,in which(15)αij+=1−1−γ˜L+ωip1−1−γ˜L+ωjq,1−1−γ˜U+ωip1−1−γ˜U+ωjq,βij−=1−1−η˜L−ωip1−η˜L−ωjq,1−1−η˜U−ωip1−η˜U−ωjq.Theorem 5 (commutativity).
Letd˜i=h˜i,g˜i,i=1,2,…,n be a collection of IVDHFEs, and p=q>0. Thus,(16)IVDHFPHMp,qd˜1,d˜2,…,d˜n=IVDHFPHMp,qd˜1′,d˜2′,…,d˜n′,where d˜1′,d˜2′,…,d˜n′ is any permutation of d˜1,d˜2,…,d˜n.The proofs of Theorems3–5 are shown in Appendixes A–C, respectively. However, the IVDHFPHM operator has no properties of idempotency and monotonicity, as illustrated in the following example.Example 2.
For two IVDHFEsd˜1andd˜2 in Example 1, we have sd˜1=sd˜2=−0.1750 and d˜1>d˜2. Suppose p=q=2, the aggregated results can be obtained by equation (8) as follows:(17)IVDHFPHM2,2d˜1,d˜1=0.2000,0.4000,0.2581,0.4000,0.2581,0.4000,0.3000,0.4000,0.4000,0.5000,0.4456,0.5451,0.4456,0.5451,0.5000,0.6000,IVDHFPHM2,2d˜1,d˜2=0.1627,0.3560,0.3257,0.4549,0.2348,0.3560,0.3560,0.4549,0.3457,0.5451,0.4456,0.5451,0.3830,0.6000,0.5000,0.6000.
According to equation (2), scores of the above aggregated IVDHFEs are(18)sIVDHFPHM2,2d˜1,d˜1=−0.1707,sIVDHFPHM2,2d˜1,d˜2=−0.1579.
It is clear thatsIVDHFPHM2,2d˜1,d˜1≠sd˜1. By Theorem 2, we can know IVDHFPHM2,2d˜1,d˜1≠d˜1, which implies the IVDHFPHM operator is not idempotent.
Furthermore, sincesIVDHFPHM2,2d˜1,d˜1<sIVDHFPHM2,2d˜1,d˜2, then IVDHFPHM2,2d˜1,d˜1<IVDHFPHM2,2d˜1,d˜2, but d˜1>d˜2, which shows the IVDHFPHM operator is not monotonic.
Now, we can discuss some special cases of the IVDHFPHM operator by assigning different values of parametersp and q.Case 1: ifq⟶0, then the IVDHFPHM operator reduces to the interval-valued dual hesitant fuzzy power descending average (IVDHFPDA) operator:(19)IVDHFPHMp,0d˜1,d˜2,…,d˜n=2nn+1⊕i=1nn+1−iωid˜ip1/p.Obviously, the weight vector ofωid˜ip is n,n−1,…,1.Case 2: ifp⟶0, then the IVDHFPHM operator reduces to the interval-valued dual hesitant fuzzy power ascending average (IVDHFPAA) operator:(20)IVDHFPHM0,qd˜1,d˜2,…,d˜n=2nn+1⊕i=1niωid˜iq1/q.Obviously, the weight vector ofωid˜iq is 1,2,…,n.From equations (19) and (20), we can see that the weight vectors of ωid˜ip and ωid˜iq are different. Hence, parameters p and q of the IVDHFPHM operator are not interchangeable.


## 3.2. IVDHFPGHM Operator

In this section, we introduce the IVDHFPGHM operator by incorporating PA into GHM.Definition 7.
Letd˜ii=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 geometric Heronian mean IVDHFPGHM operator is defined as follows:(21)IVDHFPGHMp,qd˜1,d˜2,…,d˜n=1p+q⊗i=1n⊗j=inδij2/nn+1,where δij=pd˜iωi⊕qd˜jωj,i=1,2,…,n,j=i,i+1,…,n, ωi=n1+Td˜i/∑t=1n1+Td˜t, and Td˜i=∑s=1,s≠inSupd˜i,d˜s.
Based on operational laws of IVDHFEs and mathematical induction onn, we can derive the following theorem.Theorem 6.
Letd˜ii=1,2,…,n be a collection of IVDHFEs, p,q≥0 and p,q do not take the value 0 simultaneously. The aggregated result of the IVDHFPGHM operator is also an IVDHFE as follows:(22)IVDHFPGHMp,qd˜1,d˜2,…,d˜n=∪μij∈h˜δij1−1−∏i=1n∏j=inμij2/nn+11/p+q,∪νij∈g˜δij1−∏i=1n∏j=in1−νij2/nn+11/p+q,where(23)μij∈h˜δij=∪γ˜iL,γ˜iU∈h˜i,i=1,2,…,nγ˜jL,γ˜jU∈h˜j,j=i,i+1,…,n1−1−γ˜iLωip1−γ˜jLωjq,1−1−γ˜iUωip1−γ˜jUωjq,νij∈g˜δij=∪η˜iL,η˜iU∈g˜i,i=1,2,…,nη˜jL,η˜jU∈g˜j,j=i,i+1,…,n1−1−η˜iLωip1−1−η˜jLωjq,1−1−η˜iUωip1−1−η˜jUωjq.Specially, ifSupd˜i,d˜s=cc∈0,1 for all i≠s, then the IVDHFPGHM operator reduces to the interval-valued dual hesitant fuzzy geometric Heronian mean (IVDHFGHM) operator:(24)IVDHFGHMp,qd˜1,d˜2,…,d˜n=1p+q⊗i=1n⊗j=inpd˜i⊕qd˜j2/nn+1.It can be easily proved that the IVDHFPGHM operator has following properties.Theorem 7 (boundedness).
For a collection of IVDHFEsd˜i=h˜i,g˜ii=1,2,…,n, p,q≥0 and p,q do not take the value 0 simultaneously. Let(25)γ˜L−,γ˜U−=mini=1nγ˜iL,mini=1nγ˜iU,γ˜L+,γ˜U+=maxi=1nγ˜iL,maxi=1nγ˜iU,η˜L−,η˜U−=mini=1nη˜iL,mini=1nη˜iU,η˜L+,η˜U+=maxi=1nη˜iL,maxi=1nη˜iU,we have(26)a˜−≤IVDHFPGHMp,qd˜1,d˜2,…,d˜n≤a˜+,where(27)a˜−=1−1−∏i=1n∏j=inμij−2/nn+11/p+q,1−∏i=1n∏j=in1−νij+2/nn+11/p+q,in which(28)μij−=1−1−γ˜L−ωip1−γ˜L−ωjq,1−1−γ˜U−ωip1−γ˜U−ωjq,νij+=1−1−η˜L+ωip1−1−η˜L+ωjq,1−1−η˜U+ωip1−1−η˜U+ωjq,a˜+=1−1−∏i=1n∏j=inμij+2/nn+11/p+q,1−∏i=1n∏j=in1−νij−2/nn+11/p+q,in which(29)μij+=1−1−γ˜L+ωip1−γ˜L+ωjq,1−1−γ˜U+ωip1−γ˜U+ωjq,νij−=1−1−η˜L−ωip1−1−η˜L−ωjq,1−1−η˜U−ωip1−1−η˜U−ωjq.Theorem 8 (commutativity).
Letd˜i=h˜i,g˜ii=1,2,…,n be a collection of IVDHFEs, and p=q>0. Thus,(30)IVDHFPGHMp,qd˜1,d˜2,…,d˜n=IVDHFPGHMp,qd˜1′,d˜2′,…,d˜n′,where d˜1′,d˜2′,…,d˜n′ is any permutation of d˜1,d˜2,…,d˜n.The 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 ofp and q, which are shown as follows:Case 1: ifq⟶0, then the IVDHFPGHM operator reduces to the interval-valued dual hesitant fuzzy power geometric descending average (IVDHFPGDA) operator:(31)IVDHFPGHMp,0d˜1,d˜2,…,d˜n=1p⊗i=1npd˜iωi2n+1−i/nn+1.Case 2: ifp⟶0, then the IVDHFPGHM operator reduces to the interval-valued dual hesitant fuzzy power geometric ascending average (IVDHFPGAA) operator:(32)IVDHFPGHM0,qd˜1,d˜2,…,d˜n=1q⊗i=1nqd˜iωi2i/nn+1.Equations (31) and (32) also show that parameters p and q are not interchangeable, according to the differences between weight vectors of pd˜iωi and qd˜iωi.


## 3.3. 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.Definition 8.
Letd˜ii=1,2,…,n be a collection of IVDHFEs, λ=λ1,λ2,…,λnT be the associated weight vector of d˜i with λi∈0,1, ∑i=1nλi=1, p,q≥0 and p,q do not take the value 0 simultaneously.(1)
TheIVDHFWPHM operator is defined as follows:(33)IVDHFWPHMp,qd˜1,d˜2,…,d˜n=2nn+1⊕i=1n⊕j=inζ¯ij1/p+q.(2)
TheIVDHFWPGHM operator is defined as follows:(34)IVDHFWPGHMp,qd˜1,d˜2,…,d˜n=1p+q⊗i=1n⊗j=inδ¯ij2/nn+1,in which ζ¯ij=ω¯id˜ip⊗ω¯jd˜jq, δ¯ij=pd˜iω¯i⊕qd˜jω¯j,i=1,2,…,n,j=i,i+1,…,n, ω¯i=nλi1+T¯d˜i/∑t=1nλt1+T¯d˜t and T¯d˜i=∑s=1,s≠inλsSupd˜i,d˜s.
Specially, ifλ1=λ2=⋯=λn=1/n, then IVDHFWPHM and IVDHFWPGHM operators reduce to IVDHFPHM and IVDHFPGHM operators, respectively.Theorem 9.
For a collection of IVDHFEsd˜ii=1,2,…,n, p,q≥0 and p,q do not take the value 0 simultaneously; λ=λ1,λ2,…,λnT is the associated weight vector of d˜i with λi∈0,1, ∑i=1nλi=1. The aggregated value using theIVDHFWPHM or IVDHFWPGHM operator is still an IVDHFE, and(35)IVDHFWPHMp,qd˜1,d˜2,…,d˜n=∪α¯ij∈h˜ζ¯ij1−∏i=1n∏j=in1−α¯ij2/nn+11/p+q,∪β¯ij∈g˜ζ¯ij1−1−∏i=1n∏j=inβ¯ij2/nn+11/p+q,where(36)α¯ij∈h˜ζ¯ij=∪γ˜iL,γ˜iU∈h˜i,i=1,2,…,nγ˜jL,γ˜jU∈h˜j,j=i,i+1,…,n1−1−γ˜iLω¯ip1−1−γ˜jLω¯jq,1−1−γ˜iUω¯ip1−1−γ˜jUω¯jq,β¯ij∈g˜ζ¯ij=∪η˜iL,η˜iU∈g˜i,i=1,2,…,nη˜jL,η˜jU∈g˜j,j=i,i+1,…,n1−1−η˜iLω¯ip1−η˜jLω¯jq,1−1−η˜iUω¯ip1−η˜jUω¯jq,IVDHFWPGHMp,qd˜1,d˜2,…,d˜n=∪μ¯ij∈h˜δ¯ij1−1−∏i=1n∏j=inμ¯ij2/nn+11/p+q,∪ν¯ij∈g˜δ¯ij1−∏i=1n∏j=in1−ν¯ij2/nn+11/p+q,where(37)μ¯ij∈h˜δ¯ij=∪γ˜iL,γ˜iU∈h˜i,i=1,2,…,nγ˜jL,γ˜jU∈h˜j,j=i,i+1,…,n1−1−γ˜iLω¯ip1−γ˜jLω¯jq,1−1−γ˜iUω¯ip1−γ˜jUω¯jq,ν¯ij∈g˜δ¯ij=∪η˜iL,η˜iU∈g˜i,i=1,2,…,nη˜jL,η˜jU∈g˜j,j=i,i+1,…,n1−1−η˜iLω¯ip1−1−η˜jLω¯jq,1−1−η˜iUω¯ip1−1−η˜jUω¯jq.Using IVDHFE operations and mathematical induction onn, the proof of Theorem 9 is similar to that of Theorem 3.


## 4. 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. LetA=A1,A2,…,Am be a finite set of m alternatives, C=C1,C2,…,Cn be a set of n attributes, and λ=λ1,λ2,…,λnT be the weight vector of attributes Cjj=1,2,…,n with λj∈0,1, ∑j=1nλj=1. E=e1,e2,…,el denotes a group of decision makers, with the associated weight vector ξ=ξ1,ξ2,…,ξlT satisfying ξk∈0,1, ∑k=1lξk=1. Assume that Dk=dijkm×n is the decision matrix given by decision maker ek, where(38)dijk=hijk,gijk=∪γijLk,γijUk∈hijkγijLk,γijUk,∪ηijLk,ηijUk∈gijkηijLk,ηijUkdenotes the evaluating value represented by an IVDHFE of the alternative Ai with respect to the attribute Cj (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 matrixDk=dijkm×n into normalized interval-valued dual hesitant fuzzy decision matrix D˜k=d˜ijkm×n, using the method in [25]:(39)d˜ijk=dijk=∪γijLk,γijUk∈hijkγijLk,γijUk,∪ηijLk,ηijUk∈gijkηijLk,ηijUk,ifj∈ΩB,dijkc=∪ηijLk,ηijUk∈gijkηijLk,ηijUk,∪γijLk,γijUk∈hijkγijLk,γijUk,ifj∈ΩC,where ΩB and ΩC 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.


### 4.1. 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 matrixDk=dijkm×n into normalized decision matrix D˜k=d˜ijkm×n using equation (39), the membership degrees h˜ijk and the nonmembership degrees g˜ijk in d˜ijk are arranged in the increasing order.Step 2: calculate weightsω¯ijk associated with IVDHFEs d˜ijk by weight vector ξ=ξ1,ξ2,…,ξlT with the following formula:(40)ω¯ijk=lξk1+T¯d˜ijk∑t=1lξt1+T¯d˜ijt,whereT¯d˜ijk is the weighted support of IVDHFE d˜ijk by other IVDHFEd˜ijk1k1=1,2,…,l,andk1≠k:(41)T¯d˜ijk=∑k1=1,k1≠klξk1Supd˜ijk,d˜ijk1,(42)Supd˜ijk,d˜ijk1=1−dd˜ijk,d˜ijk1,andSupd˜ijk,d˜ijk1 satisfies the conditions in Definition 6, dd˜ijk,d˜ijk1 is the interval-valued dual hesitant normalized Hamming distance [26] defined in equation (4).Step 3: aggregate all individual interval-valued dual hesitant fuzzy decision matricesD˜k=d˜ijkm×n into the collective interval-valued dual hesitant fuzzy decision matrix D˜=d˜ijm×n by the IVDHFWPHM or IVDHFWPGHM operator:(43)d˜ij=IVDHFWPHMp,qd˜ij1,d˜ij2,…,d˜ijl,(44)d˜ij=IVDHFWPGHMp,qd˜ij1,d˜ij2,…,d˜ijl.Step 4: calculate weightsω¯ij associated with IVDHFEs d˜ij based on weight vector λ=λ1,λ2,…,λnT by the following formula:(45)ω¯ij=nλj1+T¯d˜ij∑t=1nλt1+T¯d˜it,whereT¯d˜ij is the weighted support of IVDHFE d˜ij by other IVDHFE d˜itt=1,2,…,nandt≠j:(46)T¯d˜ij=∑t=1,t≠jnλtSupd˜ij,d˜it,(47)Supd˜ij,d˜it=1−dd˜ij,d˜it,andSupd˜ij,d˜it satisfies the conditions in Definition 6, dd˜ij,d˜it is calculated according to equation (4).Step 5: utilize the IVDHFWPHM or IVDHFWPGHM operator to aggregate all evaluation IVDHFEsd˜ij in the ith line of D˜ and derive the overall IVDHFEs d˜i:(48)d˜i=IVDHFWPHMp,qd˜i1,d˜i2,…,d˜in,(49)d˜i=IVDHFWPGHMp,qd˜i1,d˜i2,…,d˜in.Step 6: calculate scores of the above overall IVDHFEsd˜i by equation (2). If any two scores of alternatives are the same, calculate their IVDHFCP2 functions according to equation (5).Step 7: rank all alternativesAi by Theorem 2 and select the best one(s).


### 4.2. 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 weightsωijk associated with IVDHFEs d˜ijk by the following formula:(50)ωijk=l1+Td˜ijk∑t=1l1+Td˜ijt,whereTd˜ijk is the support of IVDHFE d˜ijk by other IVDHFE d˜ijk1k1=1,2,…,l,andk1≠k:(51)Td˜ijk=∑k1=1,k1≠klSupd˜ijk,d˜ijk1.Step 3: aggregate all individual interval-valued dual hesitant fuzzy decision matricesD˜k=d˜ijkm×n into the collective interval-valued dual hesitant fuzzy decision matrix D˜=d˜ijm×n by the IVDHFPHM or IVDHFPGHM operator:(52)d˜ij=IVDHFPHMp,qd˜ij1,d˜ij2,…,d˜ijl,d˜ij=IVDHFPGHMp,qd˜ij1,d˜ij2,…,d˜ijl.Step 4: calculate weightsωij associated with IVDHFEs d˜ij by the following formula:(53)ωij=n1+Td˜ij∑t=1n1+Td˜it,whereTd˜ij is the support of IVDHFE d˜ij by other IVDHFE d˜itt=1,2,…,n,andt≠j:(54)Td˜ij=∑t=1,t≠jnSupd˜ij,d˜it.Step 5: utilize the IVDHFPHM or IVDHFPGHM operator to aggregate all evaluation IVDHFEsd˜ij in the ith line of D˜ and derive the overall IVDHFEs d˜i:(55)d˜i=IVDHFPHMp,qd˜i1,d˜i2,…,d˜in,d˜i=IVDHFPGHMp,qd˜i1,d˜i2,…,d˜in.
The 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. The 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.


## 4.1. 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 matrixDk=dijkm×n into normalized decision matrix D˜k=d˜ijkm×n using equation (39), the membership degrees h˜ijk and the nonmembership degrees g˜ijk in d˜ijk are arranged in the increasing order.Step 2: calculate weightsω¯ijk associated with IVDHFEs d˜ijk by weight vector ξ=ξ1,ξ2,…,ξlT with the following formula:(40)ω¯ijk=lξk1+T¯d˜ijk∑t=1lξt1+T¯d˜ijt,whereT¯d˜ijk is the weighted support of IVDHFE d˜ijk by other IVDHFEd˜ijk1k1=1,2,…,l,andk1≠k:(41)T¯d˜ijk=∑k1=1,k1≠klξk1Supd˜ijk,d˜ijk1,(42)Supd˜ijk,d˜ijk1=1−dd˜ijk,d˜ijk1,andSupd˜ijk,d˜ijk1 satisfies the conditions in Definition 6, dd˜ijk,d˜ijk1 is the interval-valued dual hesitant normalized Hamming distance [26] defined in equation (4).Step 3: aggregate all individual interval-valued dual hesitant fuzzy decision matricesD˜k=d˜ijkm×n into the collective interval-valued dual hesitant fuzzy decision matrix D˜=d˜ijm×n by the IVDHFWPHM or IVDHFWPGHM operator:(43)d˜ij=IVDHFWPHMp,qd˜ij1,d˜ij2,…,d˜ijl,(44)d˜ij=IVDHFWPGHMp,qd˜ij1,d˜ij2,…,d˜ijl.Step 4: calculate weightsω¯ij associated with IVDHFEs d˜ij based on weight vector λ=λ1,λ2,…,λnT by the following formula:(45)ω¯ij=nλj1+T¯d˜ij∑t=1nλt1+T¯d˜it,whereT¯d˜ij is the weighted support of IVDHFE d˜ij by other IVDHFE d˜itt=1,2,…,nandt≠j:(46)T¯d˜ij=∑t=1,t≠jnλtSupd˜ij,d˜it,(47)Supd˜ij,d˜it=1−dd˜ij,d˜it,andSupd˜ij,d˜it satisfies the conditions in Definition 6, dd˜ij,d˜it is calculated according to equation (4).Step 5: utilize the IVDHFWPHM or IVDHFWPGHM operator to aggregate all evaluation IVDHFEsd˜ij in the ith line of D˜ and derive the overall IVDHFEs d˜i:(48)d˜i=IVDHFWPHMp,qd˜i1,d˜i2,…,d˜in,(49)d˜i=IVDHFWPGHMp,qd˜i1,d˜i2,…,d˜in.Step 6: calculate scores of the above overall IVDHFEsd˜i by equation (2). If any two scores of alternatives are the same, calculate their IVDHFCP2 functions according to equation (5).Step 7: rank all alternativesAi by Theorem 2 and select the best one(s).


## 4.2. 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 weightsωijk associated with IVDHFEs d˜ijk by the following formula:(50)ωijk=l1+Td˜ijk∑t=1l1+Td˜ijt,whereTd˜ijk is the support of IVDHFE d˜ijk by other IVDHFE d˜ijk1k1=1,2,…,l,andk1≠k:(51)Td˜ijk=∑k1=1,k1≠klSupd˜ijk,d˜ijk1.Step 3: aggregate all individual interval-valued dual hesitant fuzzy decision matricesD˜k=d˜ijkm×n into the collective interval-valued dual hesitant fuzzy decision matrix D˜=d˜ijm×n by the IVDHFPHM or IVDHFPGHM operator:(52)d˜ij=IVDHFPHMp,qd˜ij1,d˜ij2,…,d˜ijl,d˜ij=IVDHFPGHMp,qd˜ij1,d˜ij2,…,d˜ijl.Step 4: calculate weightsωij associated with IVDHFEs d˜ij by the following formula:(53)ωij=n1+Td˜ij∑t=1n1+Td˜it,whereTd˜ij is the support of IVDHFE d˜ij by other IVDHFE d˜itt=1,2,…,n,andt≠j:(54)Td˜ij=∑t=1,t≠jnSupd˜ij,d˜it.Step 5: utilize the IVDHFPHM or IVDHFPGHM operator to aggregate all evaluation IVDHFEsd˜ij in the ith line of D˜ and derive the overall IVDHFEs d˜i:(55)d˜i=IVDHFPHMp,qd˜i1,d˜i2,…,d˜in,d˜i=IVDHFPGHMp,qd˜i1,d˜i2,…,d˜in.
The 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. The 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.


## 5. 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. The aim of our example is to help government decision makers to select a proper supplier according to the following three attributes: (1)C1 is the price; (2) C2 is the quality; (3) C3 is the technology. Obviously, C1 is the cost-type attribute, C2 and C3 are the benefit-type attributes. It is assumed that four suppliers Aii=1,2,3,4 are participating in the tender according to the tender request. Three expert teams ekk=1,2,3 are formed from junior managers of the Government Procurement Center. Then, three interval-valued dual hesitant fuzzy decision matrices Dk=dijk4×3,i=1,2,3,4,j=1,2,3,k=1,2,3 are constructed, as shown in Tables 1–3, where dijk is an IVDHFE that denotes all possible interval-valued membership degrees and nonmembership degrees of the alternative Ai to the attribute Cj by the expert team ek.Table 1
Interval-valued dual hesitant fuzzy decision matrixD1 provided by e1.
C1C2C3A10.1,0.2,0.3,0.5,0.4,0.50.3,0.5,0.2,0.30.3,0.4,0.4,0.5,0.1,0.3A20.2,0.3,0.4,0.60.5,0.6,0.6,0.7,0.2,0.30.5,0.6,0.2,0.3A30.3,0.4,0.4,0.5,0.4,0.50.4,0.5,0.5,0.6,0.1,0.20.4,0.5,0.1,0.2,0.2,0.3A40.2,0.3,0.4,0.5,0.5,0.60.6,0.7,0.1,0.30.5,0.6,0.6,0.7,0.1,0.3Table 2
Interval-valued dual hesitant fuzzy decision matrixD2 provided by e2.
C1C2C3A10.1,0.2,0.2,0.3,0.2,0.40.2,0.3,0.4,0.5,0.2,0.40.2,0.4,0.1,0.3A20.3,0.4,0.5,0.60.5,0.6,0.2,0.30.4,0.5,0.3,0.4A30.4,0.5,0.3,0.4,0.4,0.50.3,0.5,0.1,0.3,0.2,0.40.3,0.5,0.2,0.3,0.4,0.5A40.2,0.3,0.3,0.5,0.6,0.70.5,0.6,0.2,0.3,0.3,0.40.6,0.7,0.2,0.3Table 3
Interval-valued dual hesitant fuzzy decision matrixD3 provided by e3.
C1C2C3A10.2,0.3,0.3,0.4,0.4,0.50.3,0.4,0.4,0.6,0.1,0.20.3,0.5,0.1,0.2,0.3,0.4A20.1,0.2,0.2,0.3,0.6,0.70.4,0.6,0.2,0.30.4,0.6,0.1,0.3A30.2,0.3,0.5,0.60.5,0.6,0.1,0.2,0.2,0.30.3,0.5,0.1,0.3A40.1,0.3,0.6,0.70.7,0.8,0.1,0.20.5,0.6,0.2,0.3Case 1.
Suppose that the weight information is known,λ=0.4,0.3,0.3T and ξ=0.3,0.3,0.4T are weight vectors of attributes and decision makers, respectively.


### 5.1. 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 1: by equation (39), we obtain normalized decision matrices, as shown in Tables 4–6.Step 2: using equations (40)–(42) to obtain the weights ω¯kk=1,2,3 associated with IVDHFEs d˜ijk. For example, ω¯1 is calculated as follows:(56)ω¯1=ω¯ij14×3=0.92090.92480.92230.92540.91920.92470.93190.92470.92120.92270.93300.9210.Step 3: without generality, takep=q=2, we aggregate all individual interval-valued dual hesitant fuzzy decision matrices Dk=dijk4×3,k=1,2,3 into the collective decision matrix D˜=d˜ij4×3 according to equation (44) (see Table 7).Step 4: using equations (45)–(47), we obtain the weight ω¯ associated with IVDHFEs d˜iji=1,2,3,4,j=1,2,3 as follows:(57)ω¯=ω¯ij4×3=1.15960.92080.91961.15720.92340.91941.15650.92060.92291.15780.91690.9252.Step 5: again takep=q=2, aggregate the collective IVDHFEs d˜iji=1,2,3,4andj=1,2,3 into overall IVDHFEs d˜ii=1,2,3,4 based on equation (49). For example, d˜4 is shown as follows:(58)d˜4=0.4955,0.6124,0.5046,0.6194,0.5431,0.6402,0.5515,0.6482,0.5097,0.6285,0.5172,0.6361,0.5607,0.6590,0.5697,0.6677,0.1658,0.2931,0.1792,0.3033.Step 6: calculate the scoressd˜i of d˜ii=1,2,3,4 by equation (2):(59)sd˜1=0.1397,sd˜2=0.2679,sd˜3=0.1412,sd˜4=0.3498.Step 7: according to the score valuessd˜ii=1,2,3,4, we obtain the ranking of alternatives Aii=1,2,3,4: A4≻A2≻A3≻A1. Therefore, A4 is the best alternative.Table 4
Normalized interval-valued dual hesitant fuzzy decision matrixD1 provided by e1.
C1C2C3A10.3,0.5,0.4,0.5,0.1,0.20.3,0.5,0.2,0.30.3,0.4,0.4,0.5,0.1,0.3A20.4,0.6,0.2,0.30.5,0.6,0.6,0.7,0.2,0.30.5,0.6,0.2,0.3A30.4,0.5,0.3,0.4,0.4,0.50.4,0.5,0.5,0.6,0.1,0.20.4,0.5,0.1,0.2,0.2,0.3A40.4,0.5,0.5,0.6,0.2,0.30.6,0.7,0.1,0.30.5,0.6,0.6,0.7,0.1,0.3Table 5
Normalized interval-valued dual hesitant fuzzy decision matrixD2 provided by e2.
C1C2C3A10.2,0.4,0.1,0.2,0.2,0.30.2,0.3,0.4,0.5,0.2,0.40.2,0.4,0.1,0.3A20.5,0.6,0.3,0.40.5,0.6,0.2,0.30.4,0.5,0.3,0.4A30.3,0.4,0.4,0.5,0.4,0.50.3,0.5,0.1,0.3,0.2,0.40.3,0.5,0.2,0.3,0.4,0.5A40.3,0.5,0.6,0.7,0.2,0.30.5,0.6,0.2,0.3,0.3,0.40.6,0.7,0.2,0.3Table 6
Normalized interval-valued dual hesitant fuzzy decision matrixD3 provided by e3.
C1C2C3A10.3,0.4,0.4,0.5,0.2,0.30.3,0.4,0.4,0.6,0.1,0.20.3,0.5,0.1,0.2,0.3,0.4A20.6,0.7,0.1,0.2,0.2,0.30.4,0.6,0.2,0.30.4,0.6,0.1,0.3A30.5,0.6,0.2,0.30.5,0.6,0.1,0.2,0.2,0.30.3,0.5,0.1,0.3A40.6,0.7,0.1,0.30.7,0.8,0.1,0.20.5,0.6,0.2,0.3Table 7
The collective decision matrixD˜ obtained by IVDHFWPGHM operator.
C1C2C3A10.2652,0.4261,0.2964,0.4660,0.2892,0.4261,0.3232,0.4660,0.1589,0.2534,0.1814,0.27860.2652,0.3899,0.2962,0.4535,0.3272,0.4550,0.3656,0.5363,0.1670,0.30380.2651,0.4359,0.2891,0.4659,0.1008,0.2644,0.2246,0.3497A20.4987,0.6363,0.2102,0.3039,0.2358,0.33460.4550,0.5987,0.4770,0.6232,0.2012,0.30140.4262,0.5661,0.2095,0.3343A30.3980,0.4987,0.4356,0.5359,0.3043,0.4019,0.3375,0.43520.3988,0.5365,0.4254,0.5661,0.1008,0.2355,0.1589,0.2785,0.1394,0.28070.1812,0.31460.3272,0.4990,0.1396,0.2787,0.2489,0.3603,0.1667,0.3015,0.2616,0.3766A40.4242,0.5652,0.5245,0.6227,0.4528,0.5980,0.5658,0.6653,0.1672,0.30130.5991,0.6968,0.1391,0.2643,0.1909,0.30320.5248,0.6233,0.5527,0.6501,0.1813,0.3014


### 5.2. Sensitivity Analysis

In this section, the influence of parameterspandq on the ranking results is investigated and discussed. The detailed results are shown in Table 8.Table 8
Scores and ranking results with different parameterspandq by IVDHFWPGHM operator.
Parameterssd˜1sd˜2sd˜3sd˜4Rankingp=0,q=10.15170.26920.19170.3841A4≻A2≻A3≻A1p=1,q=00.16530.27520.13440.3364A4≻A2≻A1≻A3p=0.5,q=0.50.16500.28500.18840.3797A4≻A2≻A3≻A1p=1,q=10.15730.27970.17320.3705A4≻A2≻A3≻A1p=2,q=20.13970.26790.14120.3498A4≻A2≻A3≻A1p=5,q=70.07780.22330.05500.2799A4≻A2≻A1≻A3p=8,q=50.07020.21680.04450.2672A4≻A2≻A1≻A3p=10,q=100.03920.19330.00900.2350A4≻A2≻A1≻A3From Table8, it is obvious that the ranking results obtained by the IVDHFWPGHM operator are somewhat different when pandq take different values. For more detailed investigation, Figure 2 presents the influence of parameters p and q on ranking results with one parameter fixed and another varied. As parameters p and q change simultaneously, the scores for four alternatives are illustrated in Figure 3. From Figures 2 and 3, we have the following conclusions:(1)
In [9, 36], we know that prominent interactions between aggregated arguments are more emphasized as parameters p and q are larger. If either p=0 or q=0, the proposed operator cannot capture the interactions of aggregated arguments. Therefore, the selection of values for parameters p and q mainly 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 selectp=q=1 (or 0.5 or 2), which is simple and straightforward, and also take the interrelationship of input arguments into account.Figure 2
Trends of scores for four alternatives by IVDHFWPGHM operator.
(a)(b)Figure 3
Scores by IVDHFWPGHM operator.Therefore, 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 Case1. The detailed steps and sensitivity analysis of pandq are omitted due to the calculation process is similar to that of Case 1.


### 5.3. Comparative Studies

In Section5.1, 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 pandq on ranking results in Section 5.2. The 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). These 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]. The ranking results of the alternatives obtained by these methods are presented in Table 9.Table 9
Compared with other existing methods for Case2.
MethodFinal scoresRankingIVDHFPHM(p=q=1)s1=0.1701, s2=0.2852, s3=0.2006, s4=0.3855A4≻A2≻A3≻A1IVDHFPGHM(p=q=1)s1=0.1535, s2=0.2731, s3=0.1714, s4=0.3677A4≻A2≻A3≻A1IVDHFA [25]s1=0.1743, s2=0.2894, s3=0.2097, s4=0.3917A4≻A2≻A3≻A1IVDHFG [25]s1=0.1513, s2=0.2718, s3=0.1691, s4=0.3662A4≻A2≻A3≻A1IVDHF-GRP [26]C˜1=0.2863, C˜2=0.3724, C˜3=0.2984, C˜4=0.5623A4≻A2≻A3≻A1IVDHF-TOPSIS [31]C˜1=0.5801, C˜2=0.6500, C˜3=0.5918, C˜4=0.6867A4≻A2≻A3≻A1As can be seen from Table9, 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 C1 is related to other attributes quality C2 and technology C3, 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 of some unreasonable data. The approaches based on power Heronian aggregation operators utilize decision information more adequately in supporting MAGDM.(2)
The 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. The 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. The 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. This 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. Thus, our operators can be considered as a generalization of [9].


## 5.1. 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 1: by equation (39), we obtain normalized decision matrices, as shown in Tables 4–6.Step 2: using equations (40)–(42) to obtain the weights ω¯kk=1,2,3 associated with IVDHFEs d˜ijk. For example, ω¯1 is calculated as follows:(56)ω¯1=ω¯ij14×3=0.92090.92480.92230.92540.91920.92470.93190.92470.92120.92270.93300.9210.Step 3: without generality, takep=q=2, we aggregate all individual interval-valued dual hesitant fuzzy decision matrices Dk=dijk4×3,k=1,2,3 into the collective decision matrix D˜=d˜ij4×3 according to equation (44) (see Table 7).Step 4: using equations (45)–(47), we obtain the weight ω¯ associated with IVDHFEs d˜iji=1,2,3,4,j=1,2,3 as follows:(57)ω¯=ω¯ij4×3=1.15960.92080.91961.15720.92340.91941.15650.92060.92291.15780.91690.9252.Step 5: again takep=q=2, aggregate the collective IVDHFEs d˜iji=1,2,3,4andj=1,2,3 into overall IVDHFEs d˜ii=1,2,3,4 based on equation (49). For example, d˜4 is shown as follows:(58)d˜4=0.4955,0.6124,0.5046,0.6194,0.5431,0.6402,0.5515,0.6482,0.5097,0.6285,0.5172,0.6361,0.5607,0.6590,0.5697,0.6677,0.1658,0.2931,0.1792,0.3033.Step 6: calculate the scoressd˜i of d˜ii=1,2,3,4 by equation (2):(59)sd˜1=0.1397,sd˜2=0.2679,sd˜3=0.1412,sd˜4=0.3498.Step 7: according to the score valuessd˜ii=1,2,3,4, we obtain the ranking of alternatives Aii=1,2,3,4: A4≻A2≻A3≻A1. Therefore, A4 is the best alternative.Table 4
Normalized interval-valued dual hesitant fuzzy decision matrixD1 provided by e1.
C1C2C3A10.3,0.5,0.4,0.5,0.1,0.20.3,0.5,0.2,0.30.3,0.4,0.4,0.5,0.1,0.3A20.4,0.6,0.2,0.30.5,0.6,0.6,0.7,0.2,0.30.5,0.6,0.2,0.3A30.4,0.5,0.3,0.4,0.4,0.50.4,0.5,0.5,0.6,0.1,0.20.4,0.5,0.1,0.2,0.2,0.3A40.4,0.5,0.5,0.6,0.2,0.30.6,0.7,0.1,0.30.5,0.6,0.6,0.7,0.1,0.3Table 5
Normalized interval-valued dual hesitant fuzzy decision matrixD2 provided by e2.
C1C2C3A10.2,0.4,0.1,0.2,0.2,0.30.2,0.3,0.4,0.5,0.2,0.40.2,0.4,0.1,0.3A20.5,0.6,0.3,0.40.5,0.6,0.2,0.30.4,0.5,0.3,0.4A30.3,0.4,0.4,0.5,0.4,0.50.3,0.5,0.1,0.3,0.2,0.40.3,0.5,0.2,0.3,0.4,0.5A40.3,0.5,0.6,0.7,0.2,0.30.5,0.6,0.2,0.3,0.3,0.40.6,0.7,0.2,0.3Table 6
Normalized interval-valued dual hesitant fuzzy decision matrixD3 provided by e3.
C1C2C3A10.3,0.4,0.4,0.5,0.2,0.30.3,0.4,0.4,0.6,0.1,0.20.3,0.5,0.1,0.2,0.3,0.4A20.6,0.7,0.1,0.2,0.2,0.30.4,0.6,0.2,0.30.4,0.6,0.1,0.3A30.5,0.6,0.2,0.30.5,0.6,0.1,0.2,0.2,0.30.3,0.5,0.1,0.3A40.6,0.7,0.1,0.30.7,0.8,0.1,0.20.5,0.6,0.2,0.3Table 7
The collective decision matrixD˜ obtained by IVDHFWPGHM operator.
C1C2C3A10.2652,0.4261,0.2964,0.4660,0.2892,0.4261,0.3232,0.4660,0.1589,0.2534,0.1814,0.27860.2652,0.3899,0.2962,0.4535,0.3272,0.4550,0.3656,0.5363,0.1670,0.30380.2651,0.4359,0.2891,0.4659,0.1008,0.2644,0.2246,0.3497A20.4987,0.6363,0.2102,0.3039,0.2358,0.33460.4550,0.5987,0.4770,0.6232,0.2012,0.30140.4262,0.5661,0.2095,0.3343A30.3980,0.4987,0.4356,0.5359,0.3043,0.4019,0.3375,0.43520.3988,0.5365,0.4254,0.5661,0.1008,0.2355,0.1589,0.2785,0.1394,0.28070.1812,0.31460.3272,0.4990,0.1396,0.2787,0.2489,0.3603,0.1667,0.3015,0.2616,0.3766A40.4242,0.5652,0.5245,0.6227,0.4528,0.5980,0.5658,0.6653,0.1672,0.30130.5991,0.6968,0.1391,0.2643,0.1909,0.30320.5248,0.6233,0.5527,0.6501,0.1813,0.3014


## 5.2. Sensitivity Analysis

In this section, the influence of parameterspandq on the ranking results is investigated and discussed. The detailed results are shown in Table 8.Table 8
Scores and ranking results with different parameterspandq by IVDHFWPGHM operator.
Parameterssd˜1sd˜2sd˜3sd˜4Rankingp=0,q=10.15170.26920.19170.3841A4≻A2≻A3≻A1p=1,q=00.16530.27520.13440.3364A4≻A2≻A1≻A3p=0.5,q=0.50.16500.28500.18840.3797A4≻A2≻A3≻A1p=1,q=10.15730.27970.17320.3705A4≻A2≻A3≻A1p=2,q=20.13970.26790.14120.3498A4≻A2≻A3≻A1p=5,q=70.07780.22330.05500.2799A4≻A2≻A1≻A3p=8,q=50.07020.21680.04450.2672A4≻A2≻A1≻A3p=10,q=100.03920.19330.00900.2350A4≻A2≻A1≻A3From Table8, it is obvious that the ranking results obtained by the IVDHFWPGHM operator are somewhat different when pandq take different values. For more detailed investigation, Figure 2 presents the influence of parameters p and q on ranking results with one parameter fixed and another varied. As parameters p and q change simultaneously, the scores for four alternatives are illustrated in Figure 3. From Figures 2 and 3, we have the following conclusions:(1)
In [9, 36], we know that prominent interactions between aggregated arguments are more emphasized as parameters p and q are larger. If either p=0 or q=0, the proposed operator cannot capture the interactions of aggregated arguments. Therefore, the selection of values for parameters p and q mainly 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 selectp=q=1 (or 0.5 or 2), which is simple and straightforward, and also take the interrelationship of input arguments into account.Figure 2
Trends of scores for four alternatives by IVDHFWPGHM operator.
(a)(b)Figure 3
Scores by IVDHFWPGHM operator.Therefore, 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 Case1. The detailed steps and sensitivity analysis of pandq are omitted due to the calculation process is similar to that of Case 1.


## 5.3. Comparative Studies

In Section5.1, 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 pandq on ranking results in Section 5.2. The 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). These 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]. The ranking results of the alternatives obtained by these methods are presented in Table 9.Table 9
Compared with other existing methods for Case2.
MethodFinal scoresRankingIVDHFPHM(p=q=1)s1=0.1701, s2=0.2852, s3=0.2006, s4=0.3855A4≻A2≻A3≻A1IVDHFPGHM(p=q=1)s1=0.1535, s2=0.2731, s3=0.1714, s4=0.3677A4≻A2≻A3≻A1IVDHFA [25]s1=0.1743, s2=0.2894, s3=0.2097, s4=0.3917A4≻A2≻A3≻A1IVDHFG [25]s1=0.1513, s2=0.2718, s3=0.1691, s4=0.3662A4≻A2≻A3≻A1IVDHF-GRP [26]C˜1=0.2863, C˜2=0.3724, C˜3=0.2984, C˜4=0.5623A4≻A2≻A3≻A1IVDHF-TOPSIS [31]C˜1=0.5801, C˜2=0.6500, C˜3=0.5918, C˜4=0.6867A4≻A2≻A3≻A1As can be seen from Table9, 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 C1 is related to other attributes quality C2 and technology C3, 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 of some unreasonable data. The approaches based on power Heronian aggregation operators utilize decision information more adequately in supporting MAGDM.(2)
The 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. The 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. The 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. This 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. Thus, our operators can be considered as a generalization of [9].


## 6. Conclusions

In this paper, we introduce theIVDHFCP2 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.



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*Source: 2080413-2020-06-11.xml*