Dual Hesitant q-Rung Orthopair Fuzzy Interaction Partitioned Bonferroni Mean Operators and Their Applications

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Introduction
In real life, many problems, from the purchase of commodities to the formulation of national policies, all refect the widespread application of decision making (DM) [1] ideas, making the fnal decision results satisfactory, which involves the importance of correct DM approach.Multicriteria group decision making (MCGDM) is one of the DM methods which can address various uncertain problems.Due to the uncertainty factors and increase of complexity gradually in the actual decision-making process, it has been extensively studied.Intuitionistic fuzzy sets (IFSs) presented by Atanassov [2] are better than fuzzy sets (FSs) introduced by Zadeh [3], which can portray the uncertainty information more completely and accurately from the aspects of hesitancy degree (HD), nonmembership degree (NMD), and membership degree (MD).After that, considering that the evaluation value may exceed the application scope of IFSs, with the limits of the sum of the squares of MD and NMD to 1, Yager [4] proposed the Pythagorean fuzzy sets (PFSs), which are more efective than IFSs in dealing with MCGDM problems.For instance, the MD and NMD are 0.9 and 0.5, and we can fnd that only PFSs can express such data information.In addition, to meet the requirements of increasingly complex data, with the limits of the sum of the q− th power of MD and NMD to 1, Yager [5] presented q-rung orthopair fuzzy set (q-ROFS).Tus, q-ROFS is more efective and comprehensive in processing and expressing uncertainty data compared with IFSs and PFSs.For more research on hesitant fuzzy sets, see [6].
However, considering the hesitation from decisionmakers in providing specifc evaluation values in certain situations.For this, Torra [7] proposed hesitant fuzzy sets (HFSs) to more reasonably express the epistemic uncertainty of DMs by giving the possible MD (PMD).In 2012, Zhu et al. [8] incorporated the idea of IFS into HFS and proposed a new dual hesitant fuzzy set (DHFS) to express the mental cognitive state of DMs more clearly by increasing the possible NMD (PNMD).So far, many uncertainty theories have been studied, such as TOPSIS [9] and fuzzy rough set [10], making it more fexible and comprehensive in describing the cognitive uncertainty aspects of DMs.For more research on decision-making problems, see [11].
In this era of big data, information fusion plays an indispensable role, which extracts required information by integrating various kinds of information in a certain way.As a method of information fusion, information aggregation operators (AOs) have attracted more and more attention and have become a general tool of modern information processing.In general, the AOs are studied from two aspects, namely, functions and operations, represented as below: (i) From the perspective of the AO functions: Some traditional AOs have been proposed for data aggregation such as the Heronian mean (HM) [12] operator and the Bonferroni mean (BM) [13] operator, which consider from the perspective of the interrelationships between aggregating arguments.Xu et al. [14] presented dual hesitant q-rung orthopair fuzzy sets (DHq-ROFSs) and proposed some HM operators for dual hesitant q-rung orthopair fuzzy numbers (DHq-ROFNs).For all that, the BM operator is more widely used.Terefore, many studies with respect to the BM operator have emerged and remarkable results have been achieved in dealing with fuzzy problems.Te Hesitant fuzzy geometric BM (GBM) operators were proposed by Zhu et al. [15].Further, Jamil and Rashid [16] proposed dual hesitant fuzzy GBM (DHFGBM) operator for DHFNs.Furthermore, in certain situations, it should be taken into account that all attributes may not be interconnected, or certain attributes may exhibit a correlation with each other.For this, Dutta and Guha [17] frst presented the partitioned Bonferroni mean (PBM) operator.Moreover, Saha et al. [18] proposed q-rung orthopair fuzzy weighted fairly aggregation operators in 2021.
(ii) From the perspective of the AO operations: Te above AOs do not take into account the interaction between the NMD and MD of the evaluated value but merely use traditional operations to independently aggregate the MD and NMD, respectively.Terefore, these theories cannot deal with the MD or NMD with zero values.To solve this issue, He et al. [19] proposed the interaction operation and some related intuitionistic fuzzy geometric interaction averaging (IFGIA) operators.Xing et al. [20] proposed some interaction Hamy mean operators in 2017.Xu et al. [21] defned interaction AOs under dual hesitant fuzzy environment.In recent years, there are increasing research studies regarding AO operations.Lin et al. [22,23] proposed linguistic q-rung orthopair fuzzy interactional partitioned Heronian mean aggregation operators and picture fuzzy interactional partitioned Heronian mean aggregation operators to further address hesitant problems.
Te conclusion we can draw from the above is that the combination of the functions and operations of the AOs can better solve practical problems.Meanwhile, it has great advantages to combine the interaction operational laws with the PBM operator.Yang et al. [24] proposed Pythagorean fuzzy interaction PBM (PFIPBM) operators, and Liu et al. [25] proposed intuitionistic fuzzy interaction PBM (IFIPBM) operator.However, it is obvious that these existing operators have the following disadvantages: (i) Although the interaction operation has been considered in [19,23], the DHq-ROFSs and interaction operator of q-ROFSs have not been combined yet.So, it is not sufcient to process more problems for MCGDM.
(ii) Although the PBM operator has been combined with the interaction operations of IFSs and PFSs to process MCGDM problems, the PBM operator cannot get reasonable values by using interaction operational for DHq-ROFNs, which limits its application scope.
(iii) In view of the complexity of group decision making when handling MCGDM, a matrix should frst be integrated by using the correlation between attributes of BM operator, and then the correlation between partitioned attributes of PBM operator should be used for integrated calculation to obtain the optimal result.
For handling these shortcomings and improving the efectiveness of existing methods in multicriteria decisionmaking problems, we will simultaneously use the following tools.
Te DHq-ROFSs can reasonably describe hesitation attitude of DMs when giving the evaluation value.Te interaction operations of q-ROFSs can more efectively depict uncertain problems by adjusting the parameter q and considering the interaction between NMD and MD.In addition, by considering the relationship between partial attributes, the PBM operator can reduce the loss of fuzzy information.
According to the aforementioned analysis, motivated by the characteristic idea of PBM and q-rung orthopair fuzzy interaction operations, this paper should achieve the following supreme goals: (i) To propose a MCGDM method relying on the denoted interaction BM and PBM operators according to the information aggregation situation in the actual case.
(ii) To better solve the problem of extreme situation when the DMs give the evaluation value.

Preliminaries
In the following, we briefy introduce the theories of the BM, GBM, and PBM operators, q-ROFSs, DHq-ROFSs, and the interaction operational laws of q-ROFSs.
Defnition 4 (see [14]).Te score function S of DHq-ROFN d � (h, g) is where #h and #g are the numbers of the elements in h and g, respectively.
where |P h | denotes the cardinality of P h , d is the number of the partitioned sorts, and

Some Dual Hesitant q-Rung Orthopair Fuzzy Interaction PBM Operators for DHq-ROFNs
In this part, we present some basic operational rules among the DHq-ROFSs considering the interaction.

and thus
Journal of Mathematics Terefore, we can get Hence, we have completed the proof.

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Next, we provide some basic properties of the DHq-ROFIPBM operator.
Proof.Based on equation ( 15), we have Since Further, we give several special cases of the DHq − ROFIPBM s,t operator by adjusting the parameters s and t.
(1) When t ⟶ 0, we can get ζ �  j∈P h ,j≠i (3) When s ⟶ 0, we can get If all the DHq-ROFNS are partitioned into one sort, the DHq-ROFIPBM operator reduces to the dual hesitant qrung orthopair fuzzy interaction Bonferroni mean (DHq-ROFIBM) operator as follows: where s, t ≥ 0, |P h | denotes the cardinality of P h , x is the number of the partitioned sorts, and Proof.By equation (12), we have 10

Journal of Mathematics
If all the DHq-ROFNs are partitioned into one sort, the DHq-ROFWIPBM operator reduces to the dual hesitant qrung orthopair fuzzy weighted interaction Bonferroni mean (DHq-ROFWIBM) operator as follows: where s, t ≥ 0, |P h | denotes the cardinality of P h , x is the number of the partitioned sorts, and ) be a collection of DHq-ROFNs and s, t ≥ 0. Ten, the aggregated value of d i obtained by DHq-ROFIPGBM operator is DHq-ROFN, shown as follows: Te proof is similar to Teorem 12, so we omit it.Next, we can derive some basic properties for the DHq-ROFIPGBM operator.
Moreover, we give some certain situations of the DHq − ROFIPGBM s,t operator by adjusting the parameters s and t.
(1) When t ⟶ 0, we can get ζ � 1, c � 1. Tus, we have (2) When s � 1, t ⟶ 0, we can get ζ � 1, c � 1. Tus, we have (3) When s ⟶ 0, we can get If all the DHq-ROFNs are partitioned into one sort, the DHq-ROFIPGB-M operator reduces to the dual hesitant qrung orthopair fuzzy interaction geometric Bonferroni mean (DHq-ROFIGBM) operator as follows: where s, t ≥ 0, |P h | denotes the cardinality of P h , d is the number of the partitioned sorts, and Te proof is similar to Teorem 16, so we omit it.

MCGDM Approach Based on the Novel Proposed BM and PBM Operators
If all the DHq-ROFNs are partitioned into one sort, the DHq-ROFWIP-GBM operator reduces to the dual hesitant q-rung orthopair fuzzy weighted interaction Bonferroni mean (DHq-ROFWIGBM) operator as follows: In the approach, the complexity of MCGDM problem is frst considered, so the BM operator is used to aggregate several decision matrices to obtain a group matrix, and the correlation of some attributes is considered, so we obtain the decision result by PBM operator.

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m×n , which contains h p ij that indicates the MD set and g p ij that indicates the NMD set.In the following, the proposed MCGDM algorithm based on the novel BM and PBM operators is given.
Step 1. Standardizing all the decision matrices: In general, we construct the standard decision matrix by converting the cost criteria values to the beneft standard values.If there are no cost criteria values, this step can be ignored.

Numerical Example
In this part, we ofer a concrete example to purchase strategic missiles with DHq-ROFNs.Pakistan plans to purchase strategic missiles from China.After primary evaluation, it has decided to choose one of the four types of missiles X � x 1 , x 2 , x 3 , x 4   for purchase.Currently, three military weapon experts (E � e 1 , e 2 , e 3  ) with weight vector ω � (0.3, 0.4, 0.3) conduct comprehensive evaluation on the missile from four attributes (the attribute weight vector is w � (0.3, 0.3, 0.2, 0.2)), including price (A 1 ), accuracy (A 2 ), range (A 3 ), and speed (A 4 ).Further, all the attributes are partitioned into two sets P 1 � A 1 , A 2   and P 2 � A 3 , A 4   based on the interrelationship.Assume that three DMs give the decision matrices D 1 , D 2 , D 3 which are shown in Tables 1-3.Ten, based on the DHq-ROFWIBM and DHq-ROFWIPBM operators, we make use of the proposed MCGDM method to solve the "Purchase Strategic Missiles" problem, shown as follows: Step 1.It does not need to be normalized with regard to the decision matrices since all attributes are the beneft type.
Step 4. Compare and rank the results by using equation (3), and we can derive that the best strategic missile is A 3 since (51) 5.1.Infuence of Diferent Parameters q on the Results.In the following, we discuss the infuence of the parameter q on the ranking result of MCGDM problem based on the DHq-ROFWIBM and DHq-ROFWIPBM operators.For this, we calculate the ranking results of diferent q values in Table 6 (when s � 2, t � 2).From Table 6, we have observed that the score functions S(d i ) decrease with the increase of parameter value q.Simultaneously, the ranking result has not changed, and the best alternative is A 3 .Te DMs can choose the proper parameter q according to their personal preference.
For the selection of s, t parameters, we can choose the corresponding value according to the attitude of decision makers' preference for risk.Te smaller the value of the parameter, the greater the decision makers' preference for risk avoidance.

Comparison with the Existing Approach.
In the following, we aim to show the superiority and rationality of our method.To accomplish this, we compare our approach to three existing methods which include situation when q � 1, 2, 3 found in [26,27], and the result is shown in Table 7.
From [26], we can fnd that this approach cannot calculate the example data.From [27], the result is A 3 > A 2 > A 4 > A 1 , which is the same as the proposed approach.So, this shows the rationality of the proposed approach.In addition, we can observe that the ranking result is A 3 > A 2 > A 1 > A 4 based on the DHq − ROFWIGBM and DHq − ROFWIPGBM operators.By comparison, the ranking results of the two approaches are slightly diferent.However, the best alternative is A 3 .In the following, we analyze the merits of our method over the two methods from three diferent aspects.
(i) From the perspective of information data, the method in [26] is based on IFSs, and we can fnd that it cannot process such data; the method in [27] is based on Pythagorean fuzzy numbers, so it has few specifc applications.Te proposed method is based on DHq-ROFNs, and it can adjust q values according to the actual data, so it is more fexible when solving MCGDM problems.(ii) From the perspective of operational laws, the method in [27] does not consider interaction operational, so it cannot solve the case where the MD or NMD is zero, whereas the common advantage of the method in [26] and our method is that they Journal of Mathematics approach to solve MCGDM problems.Finally, by comparing diferent characteristics with existing methods in purchasing strategic missiles problems, we demonstrated the rationality and proved the superiority of the proposed method.Te next research is as follows: (1) To fnd the specifc application of the method proposed in this paper.(2) To fnd the improvement and extension of the method proposed in this paper, such as using the novel AOs to solve the practical MCGDM problems, and investigate more AOs to fuse dual hesitant qrung orthopair fuzzy interaction operational, such as Heronian mean operators [14] and TOPSIS method [9].