Neutrosophic Cubic Einstein Hybrid Geometric Aggregation Operators with Application in Prioritization Using Multiple Attribute Decision-Making Method

: Viable collection is one of the imperative instruments of decision-making hypothesis. Collection operators are not simply the operators that normalize the value; they represent progressively broad values that can underline the entire information. Geometric weighted operators weight the values only, and the ordered weighted geometric operators weight the ordering position only. Both of these operators tend to the value that relates to the biggest weight segment. Hybrid collection operators beat these impediments of weighted total and request total operators. Hybrid collection operators weight the incentive as well as the requesting position. Neutrosophic cubic sets (NCs) are a classiﬁcation of interim neutrosophic set and neutrosophic set. This distinguishing of neutrosophic cubic set empowers the decision-maker to manage ambiguous and conﬂicting data even more productively. In this paper, we characterized neutrosophic cubic hybrid geometric accumulation operator (NCHG) and neutrosophic cubic Einstein hybrid geometric collection operator (NCEHG). At that point, we outﬁtted these operators upon an everyday life issue which empoweredus to organize the key objective to develop the industry.

Not long after thisinvestigation, it became a vital tool to manage obscure and conflicting information. The neutrosophic set comprises of three segments: truth enrollment, indeterminant participation, and deception enrollment. These segments can, likewise, be alluded to as participation, aversion, andnon-membership, and these segments range from ]0 − , 1 + [. For science and designing issues, Wang et al. [8] proposed the idea of a single-valued neutrosophic set, which is a class of Definition 6. [9] An interval neutrosophic set (INs) in U is a structure N = T N (u), I N (u), F N (u) u ∈ U where T N (u), I N (u), F N (u) ∈ D[0, 1] respectively called truth, indeterminacy, and falsity function in  [0,0] 3], and 0 ≤ T N + I N + F N ≤ 3. N U denotes the collection of neutrosophic cubic sets in U. Simply denoted by N = T N , I N , F N , T N , I N , F N . Definition 8. [16] The t-operators are basically Union and Intersection operators in the theory of fuzzy sets which are denoted by t-conorm (Γ * ) and t-norm (Γ), respectively. The role of t-operators is very important in fuzzy theory and its applications. .
. Definition 12. [17] The product between two neutrosophic cubic sets, A = T A , I A , F A , T A , I A , F A , where Definition 13. [17] The scalar multiplication on a neutrosophic cubic set A = T A , I A , F A , T A , I A , F A , where and a scalar k is defined.
The exponential multiplication is followed by the following result.
is a neutrosophic cubic value, then, the exponential operation defined by where A k = A ⊗ A⊗, . . . . ⊗A(k − times), moreover, A k is a neutrosophic cubic value for every positive value of k. Definition 14. [17] The Einstein sum between two neutrosophic cubic sets A = T A , Definition 15. [17] The Einstein product between two neutrosophic cubic sets, Definition 16. [17] The scalar multiplication on a neutrosophic cubic set, The Einstein exponential multiplication is followed by the following result.
is a neutrosophic cubic value, then, the exponential operation defined by , moreover, A E k is a neutrosophic cubic value for every positive value of k.
To compare two neutrosophic cubic values the score function is defined.
is a neutrosophic cubic value, and the score function is defined as If the score function of two values are equal, the accuracy function is used.
is a neutrosophic cubic value, and theaccuracy function is defined as The following definition describes the comparison relation between two neutrosophic cubic values.
The neutrosophic cubic geometric aggregation operators weight only the neutrosophic cubic values, whereas neutrosophic cubic order geometric aggregation operators weight the orders of the values first then weight them. In the two cases, the amassed values that focused on the value relate to the biggest weight. The accompanying precedents represent the impediments of the NCWG and NCEWG.
Let W = (0.7, 0.2, 0.1) be the weight corresponding to the neutrosophic cubic values We observe that the higher the weight component, the aggregated value will tend to the corresponding neutrosophic cubic value of that vector. In NCWG, the value tendsto N 1 , as the weight that corresponds to N 1 is highest, and in NCOWG, the highest component of weight corresponds to N 2 . This situation often arises in aggregation problems. Motivated by such a situation, the idea of neutrosophic cubic hybrid geometric and neutrosophic cubic Einstein hybrid geometric operators are proposed.

Neutrosophic Cubic Hybrid Geometric and Neutrosophic Cubic Einstein Geometric Operators
This segment comprises of the following subsections. In Section 3.1 neutrosophic cubic crossbreed, the geometric operator is characterized. In Section 3.2 neutrosophic cubic Einstein crossbreed, the geometric operator is characterized. In Section 3.3, a calculation is characterized to organize the neutrosophic cubic values utilizing these tasks. In Section 3.4, a numerical model is outfitted upon Section 3.3.

Neutrosophic Cubic Hybrid Geometric Operator
NCWG operator weights only the neutrosophic cubic values, where NCOWG weights only the ordering positions. The idea of neutrosophic cubic hybrid geometric aggregation operators is developed to overcome these limitations. NCHG weights both the neutrosophic cubic values and its order positioning as well.

Definition 24. NCHG
: Ω m → Ω is a mapping from m-dimenion, which has associated weight W = (w 1 , w 2 , ..., w m ) T ,such that w j ∈ [0, 1] and m j=1 w j = 1, such that The neutrosophic cubic weighted geometric operator(NCWG) is defined as where the weight where j B is the jth largest neutrosophic cubic value, and The neutrosophic cubic geometric aggregation operators weight only the neutrosophic cubic values, whereas neutrosophic cubic order geometric aggregation operators weight the orders of the values first then weight them. In the two cases, the amassed values that focused on the value relate to the biggest weight. The accompanying precedents represent the impediments of the NCWG and NCEWG. Let and m is the balancing coefficient.
.., m) be collection of neutrosophic cubic values, then the aggregated value (NCHWG) is also a cubic value and Proof. By mathematical induction for m = 2, using Let the results hold for m.
We prove the result for m + 1, which completes the proof.

Theorem 4.
The NCWG is a special case of NCHG operator.

Neutrosophic Cubic Einstein Hybrid Geometric Operator
NCEWG operator weights only the neutrosophic cubic values, where NCEOWGA weights only the ordering positions. The idea of neutrosophic cubic Einstein hybrid aggregation operators (NCEHG) is developed to overcome these limitations, which weights both the given neutrosophic cubic value and its order position as well. = 1, such that .., m) is acollection of neutrosophic cubic values, then, their aggregated value by NCEWG operator is also a cubic value and where W = (w 1 , w 2 , ..., w m ) T is weight of N j ( j = 1, 2, 3, ..., m), with w j ∈ [0, 1] and m j=1 = 1.
Proof. We use mathematical induction to prove this result for k = 2, using definition Let the result holds for m.
We prove the result holds for m + 1.
so the result holds for all values of m.

Theorem 7.
The NCEWG is special case of the NCEHG operator.

Theorem 8.
The NCOWG is a special case of NCEHG.

An Application of Neutrosophic Cubic Hybrid Geometric and Einstein Hybrid Geometric Aggregation Operator to Group Decision-Making Problems
In this section, we develop an algorithm for group decision-making problems using the neutrosophic cubichybrid geometric and Einstein hybrid geometric aggregation (NCHWG and NCEHWG). Step 1: First of all, we construct neutrosophic cubic decision matrix D = N ij n×m .
Step 4: By using aggregation operators like (NCHG, NCEHG), the decision matrix is aggregated by the new weightsassigned to the m attributes.
Step 5: The n alternatives are ranked according to their scores and arranged in descending order to select the alternative with highest score.

NumericalApplication
A steering committee is interested in prioritizingthe set of information for improvement of the project using a multiple attribute decision-making method. The committee must prioritize the development and implementation of a set of six information technology improvement projects A j ( j = 1, 2, ..., 6). The three factors, B 1 productivity, to increase the effectiveness and efficiency, B 2 differentiation, from products and services of competitors, and B 3 management, to assist the management in improving their planning, are considered to assess the potential contribution of each project. The list of proposed information systems are A 1 Quality Assurance, (2) A 2 Budget Analysis, (3) A 3 Itemization, (4) A 4 Employee Skills Tracking, (5) A 5 Customer Returns and Complaints, and (6) A 6 Materials Acquisition. Suppose the weight W = (0.5, 0.3, 0.2) corresponds to the B j , ( j = 1, 2, 3, ) factors and characteristics of projects A i (i = 1, 2, ..., 10) by the neutrosophic cubic value N ij .
Step 3: The new weights are calculated using the normal distribution method. Let W = (0.2429, 0.5142, 0.2429) be its weighting vector derived by the normal distribution-based method [18].
Step 4: By neutrosophic cubic weighted geometric aggregation operator (NCWG), the decision matrix is aggregated by the new weights assigned to the m attributes.
Step 5: The scores are List of priorities are as follows.
Hence, the project A 1 has the highest potential contribution to the firm's strategic goal of gaining competitive advantage in the industry.

Conclusions
This paper was influenced by the impediment of neutrosophic cubic geometric and Einstein geometric collection operators as preliminarily discussed, that is, we observed that the higher the weight component, the aggregated value tended to the corresponding neutrosophic cubic value of that vector. Consequent upon such circumstances, we characterized neutrosophic cubic hybrid and neutrosophic cubic Einstein hybrid aggregation operators. At that point, these operators are outfitted upon a day-by-day life precedent structure industry to organize the potential contributions that serve to achieve the strategic objective of getting favorable circumstances in industry.
Author Contributions: All authors contributed equally.