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 not 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 , and do not take the value 0 simultaneously; is the associated weight vector of with , . The aggregated value using the or operator is still an IVDHFE, andwherewhere
Using IVDHFE operations and mathematical induction on , 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. Let be a finite set of alternatives, be a set of attributes, and be the weight vector of attributes with , . denotes a group of decision makers, with the associated weight vector satisfying , . Assume that is the decision matrix given by decision maker , wheredenotes the evaluating value represented by an IVDHFE of the alternative with respect to the attribute (, , ).
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 into normalized interval-valued dual hesitant fuzzy decision matrix , using the method in [25]:where and 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 matrix into normalized decision matrix using equation (39), the membership degrees and the nonmembership degrees in are arranged in the increasing order. Step 2: calculate weights associated with IVDHFEs by weight vector with the following formula: where is the weighted support of IVDHFE by other IVDHFE: and satisfies the conditions in Definition 6, 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 matrices into the collective interval-valued dual hesitant fuzzy decision matrix by the IVDHFWPHM or IVDHFWPGHM operator: Step 4: calculate weights associated with IVDHFEs based on weight vector by the following formula: where is the weighted support of IVDHFE by other IVDHFE : and satisfies the conditions in Definition 6, is calculated according to equation (4). Step 5: utilize the IVDHFWPHM or IVDHFWPGHM operator to aggregate all evaluation IVDHFEs in the th line of and derive the overall IVDHFEs : Step 6: calculate scores of the above overall IVDHFEs by equation (2). If any two scores of alternatives are the same, calculate their functions according to equation (5). Step 7: rank all alternatives 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 weights associated with IVDHFEs by the following formula: where is the support of IVDHFE by other IVDHFE : Step 3: aggregate all individual interval-valued dual hesitant fuzzy decision matrices into the collective interval-valued dual hesitant fuzzy decision matrix by the IVDHFPHM or IVDHFPGHM operator: Step 4: calculate weights associated with IVDHFEs by the following formula: where is the support of IVDHFE by other IVDHFE : Step 5: utilize the IVDHFPHM or IVDHFPGHM operator to aggregate all evaluation IVDHFEs in the th line of and derive the overall IVDHFEs :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) is the price; (2) is the quality; (3) is the technology. Obviously, is the cost-type attribute, and are the benefit-type attributes. It is assumed that four suppliers are participating in the tender according to the tender request. Three expert teams are formed from junior managers of the Government Procurement Center. Then, three interval-valued dual hesitant fuzzy decision matrices are constructed, as shown in Tables 1–3, where is an IVDHFE that denotes all possible interval-valued membership degrees and nonmembership degrees of the alternative to the attribute by the expert team .
Case 1. Suppose that the weight information is known, and 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 associated with IVDHFEs . For example, is calculated as follows: Step 3: without generality, take , we aggregate all individual interval-valued dual hesitant fuzzy decision matrices into the collective decision matrix according to equation (44) (see Table 7). Step 4: using equations (45)–(47), we obtain the weight associated with IVDHFEs as follows: Step 5: again take , aggregate the collective IVDHFEs into overall IVDHFEs based on equation (49). For example, is shown as follows: Step 6: calculate the scores of by equation (2): Step 7: according to the score values , we obtain the ranking of alternatives : . Therefore, is the best alternative.
5.2. Sensitivity Analysis
In this section, the influence of parameters on the ranking results is investigated and discussed. The detailed results are shown in Table 8.
From Table 8, it is obvious that the ranking results obtained by the IVDHFWPGHM operator are somewhat different when take different values. For more detailed investigation, Figure 2 presents the influence of parameters and on ranking results with one parameter fixed and another varied. As parameters and 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 and are larger. If either or , the proposed operator cannot capture the interactions of aggregated arguments. Therefore, the selection of values for parameters and mainly depends on decision makers’ risk preferences. Pessimists anticipate desirable outcomes and may choose small values of parameters and , while optimistic experts may choose big values.(2)For the computational simplicity of MAGDM problems, the decision makers can select (or 0.5 or 2), which is simple and straightforward, and also take the interrelationship of input arguments into account.
(a)
(b)
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 Case 1. The detailed steps and sensitivity analysis of are omitted due to the calculation process is similar to that of Case 1.
5.3. 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 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.
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 is related to other attributes quality and technology , 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 the 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.
Appendix
A. Proof of Theorem 3
Proof. According to the operations in Definition 2, for , we haveSimilarly, for ,Thus,denoted as . With the mathematical induction on , it has
Then, by the operations in Definition 2, we obtain , i.e., equation (8) holds.
B. Proof of Theorem 4
Proof. For , we haveAccording to Theorem 2, we can get . Hence, . By the mathematical induction on , we can obtain .
Therefore,Similarly,According to Theorem 2, we have .
We can also prove that .
C. Proof of Theorem 5
Proof. Suppose that is any permutation of . Then, for , , there exist , such that , , and vice visa. We have and . Hence, and . Notice that and the operation of IVDHFEs has commutativity, therefore,i.e., . We obtainwhich completes the proof.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (nos. 11571175 and 11801281), Natural Science Foundation of Higher Education of Jiangsu Province (no. 18KJB110024), High Training Funded for Professional Leaders of Higher Vocational Colleges in Jiangsu Province (no. 2018GRFX038), and Science and Technology Research Project of Nantong Shipping College (no. HYKY/2018A03).