Multivalued neutrosophic power partitioned Hamy mean operators and their application in MAGDM

The novel multivalued neutrosophic aggregation operators are proposed in this paper to handle the complicated decision-making situations with correlation between specific information and partitioned parameters at the same time, which are based on weighted power partitioned Hamy mean (WMNPPHAM) operators for multivalued neutrosophic sets (MNS) proposed by combining the Power Average and Hamy operators. Firstly, the power partitioned Hamy mean (PPHAM) is capable of capture the correlation between aggregation parameters and the relationship among attributes dividing several parts, where the attributes are dependent definitely within the interchangeable fragment, other attributes in divergent sections are irrelevant. Secondly, because MNS can effectively represent imprecise, insufficient, and uncertain information, we proposed the multivalued neutrosophic PMHAM (WMNPHAM) operator for MNS and its partitioned variant (WMNPPHAM) with the characteristics and examples. Finally, this multiple attribute group decision making (MAGDM) technique is proven to be feasible by comparing with the existing methods to confirm this method’s usefulness and validity.


Introduction
The world is full of partial, imprecise, inconsistent, and uncertain data that can't be characterized with precise numbers [1][2][3]. In order to deal with these complex problems, the MAGDM method can sort and adopt the superlative alternative from a set of complicated options [4]. Zadeh suggested fuzzy set to solve MAGDM problems in order to decrease information loss and increase assessment accuracy [5]. However, main limitation of fuzzy set is that it can't handle complex fuzzy information adequately because its membership limit is only one value. As a result, Atanassov expanded fuzzy set to the intuitionistic fuzzy set [6][7][8]. However, in the face of conflicting, partial, and uncertain data, the foregoing has some limitations. Smarandache proposed the notion of neutrosophic sets to address this issue, which incorporated an independent indeterminacy membership function [9]. Such as the generalization of fuzzy sets, the single-valued neutrosophic set and multivalued neutrosophic sets are some of the achievements in this subject [10][11][12][13][14][15][16][17]. where U K (q), D K (q) and M K (q) denote the truth-membership, the indeterminacy-membership and the falsity-membership of the element x 2 X to the set K respectively. For each point q in Q, we have U K (q), D K (q), M K (q) 2 [0,1], 0 � U K (q) + D K (q) + M K (q) � 3. For simplicity, we may utilize the simpler form Q = (U q , D q , M q ) to represent single valued neutrosophic set, and the element q can be termed as a single valued neutrosophic number. Definition 2. K is a nonempty fixed set, the multivalued neutrosophic set (MNS) in Q could be characterized as: I ¼ q;ũðqÞ;dðqÞ;lðqÞ whereũ q ð Þ ¼ fmjm 2ũ x ð Þg,d q ð Þ ¼ fdjr 2d x ð Þg andlðqÞ ¼ fljl 2lðqÞg are three sets with some values in interval [0,1], and satisfying the limits:m;r;l 2 0; 1 ½ � and 0 � supm þ þ supr þ þ supl þ � 3.  m 1 m 2 ; r 1 þ r 2 À r 1 r 2 ; l 1 þ l 2 À l 1 l 2 f g; ð5Þ 3.

Power aggregation operators
Definition 9. The Power aggregation operator is the mapping R n ! R as: PA a 1 ; a 2 ; . . .; a n ð Þ ¼ is the support as α i from α j . And certain qualities are detailed as follows:

Partitioned Hamy mean (PHAM) operator
Definition 10. The Partitioned Hamy mean operator (PHAM) is expressed in the form as: Where (i 1 , i 2 , . . .,i p ) explores the whole p-tuple combination and C p q is the binomial coefficient,

Hamy mean operators based on multivalued neutrosophic sets
In summary, we will investigate the Hamy mean operator and Power aggregation operator to deal with MNS and build MNPPHAM operator and WMNPPHAM operator, as well as explain various attributes and specific circumstances of these new operators, with the operating regulations of MNS.

MNPHAM operator
Þ are MNS, and x = 1,2,. . .,m The MNPHAM operator is described this way: where Tñ j is support degree, which meets: tance among any two neutrosophic sets signed by the Definition 7.
The denominator C x n represents the binomial coefficient n!
x! nÀ x ð Þ! and n is the balancing coefficient in the preceding Eq (14), we could note then power weight vector is identified by (σ 1 , σ 2 ,. . .,σ n ). As a result, Eq (14) can be documented in the following simplified form: The following theorems could be derived from the operational rules of the MNS: Þ be a MNS, the result of aggregation is still MNS.

WMNPWPPHAM operator
Definition 13. Let Z ¼ñ 1 ;ñ 2 ; � � � ;ñ n ð Þ be a collection of MNS, and the elements could well be separated into l parts P = (P 1 , P 2 ,� � �,P l ), where P t ¼ñ t1 ;ñ t2 ; � � � ;ñ t jP t j n o , t = (1, 2,� � �,l), P s T P t = ∅, [ l t¼1 P t ¼ñ, the WMNPPHAM operator is described as follows: Furthermore, the foregoing operators satisfy the Theorems of Idempotency, Commutativity, and Boundedness. However, the proofs are much like the proofs of the Theorems for MNPPHAM and WMNPPHAM operators, therefore the proof procedure is omitted here.

MAGDM approach with WMNPPHAM operators
In this part, we will utilize the WMNPPHAM operators to solve the MAGDM issue. For example, a MAGDM issue concludes a set of m alternatives A = {A 1 , A 2 ,. . .,A m }, the decision makers D = {D 1 , D 2 ,. . .,D z } with their weight vector θ = (θ 1 , θ 2 ,. . .,θ z ) T and a collection of n attributes C = {C 1 , C 2 ,. . .,C n } with ω = (ω 1 , ω 2 ,. . .,ω n ) T matching θ l 2 [0,1], X z l¼1 y l ¼ 1, ω n 2 [0,1], X n j¼1 o j ¼ 1. Taking into account the relationship of the attribute set C, C can be divided into l parts P 1 , P 2 ,. . .,P l , where P t ¼ C t1 ; C t2 ; � � � ; C t P t n o , t = (1, 2,. . .,l), P s To overcome this problem, the following phases of the new MAGDM technique might be taken: Step 1. The gathered choice matrices R h must be normalized into standard matrices SR h by converting the cost-type to the benefit-type.
Step 3. Estimate the weights s k ij associated with the MNñ k ij . Since then the weights s k ij combined by theñ k ij could be collected: . . .; l: Step 4. Aggregate each expert's evaluation information.
Step 9. As shown by Definition 6, order the alternatives A i and pick the most accomplished one(s).

Illustrative example
The investment firm seeks to choose the best option from the four agricultural brands, which are A 1 , A 2 , A 3 and A 4 . Three characteristics are used to evaluate the four alternatives: (1) C 1 (the risk index), (2) C 2 (the growth index), (3) C 3 (environmental impact index), where C 1 and C 2 belong to the benefit type, C 3 is of the cost type with their weight is ω = (0. 35 Decision-making procedure Step 1. Normalize the decision matrix. The normalized MN decision matrixR k ¼ñ k ij � � 0:2 f g; 0:9 f g; 0:5  can denoted by Sup kt . According to Eq (29), the Sup kt (k,t = 1,2,3; k 6 ¼ t) can be calculated: The weights s k ij i; j ¼ 1; 2; 3; 4; k ¼ 1; 2; . . .; l ð Þ can be formed by MNñ k ij using Eq (31).
are formed by σ k (k = 1,2,3) as follows: Step 4. Combine the evaluation information of every expert.
The collective multivalued neutrosophic decision matrixR ¼ñ ij � � n�m can be computed as: Step 5. Retrieve the supports Supñ ij ;ñ ip � � :  Step 7. Evaluate the information within each partition of attributes. This step can estimate the collective evaluation values of each alternate within P t by (20), which show in Table 1.
The valueñ i of the alternative α i can be calculated by WMNPPHAM operator in Table 2: Step 8. Use the Eqs. (8) and (9) to calculate the score values. S i can be calculated by Definition 5 as: Step 9. Order all the alternatives. The results in Step 4, we can get S 1 > S 2 >S 4 >S 3 . So, the final rank of all the alternatives could be shown as

Influence of the parameter on the final result
The changing value of parameter x in the MNWHAM operator can be taken to demonstrate the effects on the ranking results in Table 3.

Comparison analysis
The efficacy and practicality of the suggested MAGDM technique by WMNPPHAM operators must be compared and verified, thus we perform a comparative analysis using the same illustrative case. The analysis might be made from the following aspects: techniques utilizing MNS with other operators and ways using the same operators with different discrete forms of neutrosophic numbers. Then these different ranking results could be shown as α 1 � α 4 � α 2 � α 3 [43]. Clearly, the ideal choice is α 1 , whereas the worst alternative is α 3 . We summarize the reasons for variances in the final rankings of all the examined approaches and the suggested methodology in Table 4. Not only can our approach consider interrelationships between any two qualities, numerous arguments, and membership and nonmembership, but it also has beneficial flexibility to represent preference and capacity to describe uncertainty. Furthermore, our technique may partition the attributes into discrete portions that include both the interdependence and the independence of the attributes. So we may infer that WMNPPHAM is more practical and efficient.

Conclusion
In this study, we propose the WMNPHAM and WMNPPHAM operators, which extend the Hamy mean and Power aggregation operator to the MNS. In addition, we describe the desirable qualities, create the score function, and use it to rank the choices. After that, based on the WMNPPHAM operator, we provide detailed procedures for solving MAGDM issues using multi-valued neutrosophic information. Furthermore, we compare the efficacy and practicality of the created technique to current methods.
Therefore, for addressing complex decision-making situations, these proposed novel multivalued neutrosophic aggregation operators can aggregate fuzzy information and partitioned parameters meantime, which can be used as a practical tool to solve the MADM challenges more efficiently and effectively. In the future, further research can expand them to other