Research Article | Open Access

Shenqing Jiang, Wei He, Fangfang Qin, Qingqing Cheng, "Multiple Attribute Group Decision-Making Based on Power Heronian Aggregation Operators under Interval-Valued Dual Hesitant Fuzzy Environment", *Mathematical Problems in Engineering*, vol. 2020, Article ID 2080413, 19 pages, 2020. https://doi.org/10.1155/2020/2080413

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

**Academic Editor:**Mahdi Jalili

#### 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 () 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.

#### 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 IVDHFEs , score values and accuracy values are identical, but and are different. According to the comparison law in [25], is equal to , which is a contradiction. Our new ranking method can compare different IVDHFEs by the function.

The 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 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]). *Let be a reference set, and be the set of all closed subintervals of . Then, an interval-valued dual hesitant fuzzy set (IVDHFS) on iswhere and denote all possible interval-valued membership degrees and nonmembership degrees of the element to the set , respectively, with the conditions: , , and , where and for all . For convenience, we call the pair an interval-valued dual hesitant fuzzy element (IVDHFE), denoted by .

If , for all , and , for all , then IVDHFE reduces to dual hesitant fuzzy element (DHFE) . If , then IVDHFE reduces to interval-valued hesitant fuzzy element (IVHFE) . If , for all , and , then IVDHFE reduces to hesitant fuzzy element (HFE) . 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, , , and , and , some basic operational laws are defined as follows:(1)(2)(3)(4)

*Definition 3 (see [25]). *Let be an IVDHFE, score and accuracy functions of are described as follows:where # and are numbers of interval values in and , respectively.

Theorem 1 (see [25]). *For two IVDHFEs and , then*(1)*If , then is superior to , denoted by .*(2)*If , then(a) If , then is superior to , denoted by .(b)If , then is equivalent to , denoted by .*

For two IVDHFEs and , #, , #, and # are numbers of interval values in , , , and , respectively. Let and . In most cases, and are not equal to and , respectively, i.e., or . 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 IVDHFEs and is defined as follows:where intervals in , are arranged in the increasing order, i.e., and are the th smallest interval values in , , respectively.

##### 2.2. Improved Comparison Laws for IVDHFEs

Score and accuracy functions in Definition 3 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, and . By Definition 3, and , which means that Theorem 1 cannot be applied to rank all IVDHFEs.

In order to overcome this flaw, we define the function as follows.

*Definition 5. *Let be a finite and nonempty IVDHFE, where and are the th smallest interval values in and ,# and are numbers of interval values in and , respectively. and are the central membership value and the central nonmembership value, respectively:is called the interval-valued dual hesitant fuzzy 2nd-order central polymerization degree function of .

For the sake of simplicity, we let and . Thus, equation (5) can be denoted as follows: and are similar to the variance in statistics, which indicate degrees of dispersion of interval values around and , respectively. The larger the value of , the greater the volatility of the values in IVDHFE . Thus, IVDHFE is more unstable and smaller. In other words, the bigger the value of , the more stable and larger the IVDHFE .

Theorem 2. *For two finite and nonempty IVDHFEs and , if and only if (a) or (b) and .*

The new ranking method is more appropriate to compare IVDHFEs, which can be used to solve Example 1. We have , using equation (5). According to Theorem 2, we obtain .

Moreover, the result in Example 1 can be explained reasonably by Figure 1. For membership degrees and , has bigger different preferences of decision makers than that of . Although , is more stable than . For nonmembership degrees and , has smaller different preferences, so is more stable than . Therefore, is larger than .

**(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. *Let be a collection of IVDHFEs, and do not take the value 0 simultaneously. An interval-valued dual hesitant fuzzy power Heronian mean operator is defined as follows:where , , , and is the support for from , which satisfies the following three properties:(1)(2)(3), if where is a distance measure between two IVDHFEs and calculated by equation (4).

Theorem 3. *Let be a collection of IVDHFEs, and do not take the value 0 simultaneously. Then, the aggregated value using the operator is still an IVDHFE, andwhere*

Specially, if for all , i.e., , then the IVDHFPHM operator reduces to the interval-valued dual hesitant fuzzy Heronian mean (IVDHFHM) operator:

Theorem 4 (boundedness). *For a collection of IVDHFEs , and do not take the value 0 simultaneously. Letwe havewherein whichin which*

Theorem 5 (commutativity). *Let be a collection of IVDHFEs, and . Thus,where is any permutation of .*

The proofs of Theorems 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.

*Example 2. *For two IVDHFEs in Example 1, we have and . Suppose , the aggregated results can be obtained by equation (8) as follows:According to equation (2), scores of the above aggregated IVDHFEs areIt is clear that . By Theorem 2, we can know , which implies the IVDHFPHM operator is not idempotent.

Furthermore, since , then , but , 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 and . Case 1: if , then the IVDHFPHM operator reduces to the interval-valued dual hesitant fuzzy power descending average (IVDHFPDA) operator: Obviously, the weight vector of is . Case 2: if , then the IVDHFPHM operator reduces to the interval-valued dual hesitant fuzzy power ascending average (IVDHFPAA) operator: Obviously, the weight vector of is . From equations (19) and (20), we can see that the weight vectors of and are different. Hence, parameters and of the operator are not interchangeable.

##### 3.2. IVDHFPGHM Operator

In this section, we introduce the IVDHFPGHM operator by incorporating PA into GHM.

*Definition 7. *Let be a collection of IVDHFEs, and do __n__ot take the value 0 simultaneously. An interval-valued dual hesitant fuzzy power geometric Heronian mean operator is defined as follows:where , , and .

Based on operational laws of IVDHFEs and mathematical induction on , we can derive the following theorem.

Theorem 6. *Let be a collection of IVDHFEs, and do not take the value 0 simultaneously. The aggregated result of the operator is also an IVDHFE as follows:where*

Specially, if for all , then the IVDHFPGHM operator reduces to the interval-valued dual hesitant fuzzy geometric Heronian mean (IVDHFGHM) operator:

It can be easily proved that the IVDHFPGHM operator has following properties.

Theorem 7 (boundedness). *For a collection of IVDHFEs , and do not take the value 0 simultaneously. Letwe havewherein whichin which*

Theorem 8 (commutativity). *Let be a collection of IVDHFEs, and . Thus,where is any permutation of .*

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 of and , which are shown as follows: Case 1: if , then the IVDHFPGHM operator reduces to the interval-valued dual hesitant fuzzy power geometric descending average (IVDHFPGDA) operator: Case 2: if , 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 and are not interchangeable, according to the differences between weight vectors of and .

##### 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. *Let be a collection of IVDHFEs, be the associated weight vector of with , , and do not take the value 0 simultaneously.(1)The operator is defined as follows:(2)The operator is defined as follows:in which , , and .

Specially, if , then IVDHFWPHM and IVDHFWPGHM operators reduce to IVDHFPHM and IVDHFPGHM operators, respectively.

Theorem 9. *For a collection of IVDHFEs , *