Hybrid Weighted Arithmetic and Geometric Aggregation Operator of Neutrosophic Cubic Sets for MADM

: Neutrosophic cubic sets (NCSs) can express complex multi-attribute decision-making (MADM) problems with its interval and single-valued neutrosophic numbers simultaneously. The weighted arithmetic average (WAA) and geometric average (WGA) operators are common aggregation operators for handling MADM problems. However, the neutrosophic cubic weighted arithmetic average (NCWAA) and neutrosophic cubic geometric weighted average (NCWGA) operators may result in some unreasonable aggregated values in some cases. In order to overcome the drawbacks of the NCWAA and NCWGA, this paper developed a new neutrosophic cubic hybrid weighted arithmetic and geometric aggregation (NCHWAGA) operator and investigates its suitability and effectiveness. Then, we established a MADM method based on the NCHWAGA operator. Finally, a MADM problem with neutrosophic cubic information was provided to illustrate the application and effectiveness of the proposed method.

Recently, Ali et al. [18,19] also put forward the concepts of neutrosophic cubic sets (NCSs) by combining neutrosophic sets with cubic sets, and defined internal and external NCSs. NCSs are described by two parts simultaneously, where the truth, falsity, and indeterminacy membership functions can be expressed by an interval value and an exact value simultaneously. Obviously, an NCS can be combined by an INS and an SVNS, and it contains much more information than an INS or an SVNS. Thus, some researchers have applied NCSs in decision-making problems effectively. Lu et al. [20] studied cosine measures of NCSs to deal with multiple attribute decision-making (MADM) problems. Banerjee et al. [21] established a grey relational analysis (GRA) method for MADM in NCS environment. Pramanik et al. [22] investigated a multi-criteria group decision making (MCGDM) method based on the similarity measure of NCSs. Moreover, aggregation operators have been widely applied in many MADM problems [23][24][25][26][27][28][29], and some aggregation operators have been studied for MADM problems in an NCS environment. Shi et al. [30] developed Dombi aggregation operators of NCSs for MADM. Zhan et al. [31] also proposed the neutrosophic cubic weighted arithmetic average (NCWAA) and neutrosophic cubic geometric weighted average (NCWGA) operators by extending the WAA and WGA operators to NCSs. However, the aforementioned NCWAA and NCWGA operators may cause some unreasonable results in some cases. In order to overcome the shortcomings of the NCWAA and NCWGA operators, this paper developed a new neutrosophic cubic hybrid weighted arithmetic and geometric aggregation (NCHWAGA) operator and analyzed its effectiveness for MADM by numerical examples. The main advantage of the proposed NCHWAGA operator can overcome the shortcomings of the existing NCWAA and NCWGA operators in some situations and obtain the moderate aggregation values.
The rest of the paper is organized as follows. Section 2 briefly introduces some basic concepts of NCSs and analyzes the shortcomings of the NCWAA and NCWGA operators. Then, Section 3 presents the NCHWAGA operator and investigated its properties. We establish a MADM approach based on the NCHWAGA operator in Section 4. Subsequently, Section 5 provides numerical examples with neutrosophic cubic information to demonstrate the application and effectiveness of the developed approach. Finally, Section 6 presents conclusions and possible future research.

Preliminaries
Some basic concepts and ranking methods of NCSs were introduced in this section. Definition 1: [19,32] Let be a universal set. A NCS G in is denoted as follows: where < Tc(x), Vc(x), Fc(x) > is an INS [4] in , and the intervals the truth, indeterminacy, and falsity membership functions; then < tc(x), vc(x), fc(x) > is an SVNS [3,5] [20]; for convenience, we denoted it as 18,19].
be two NCNs, then there are following operational laws: ; , its score and accuracy functions [33] can be defined as follows: Based on the functions Ψ(x) and Γ(x), two NCNs can be compared and ranked by definition as follows:  , where ζi  [0, 1] (i = 1, 2, …, n), satisfying Although the above-weighted average and geometric operators were used for multi-criteria decision making [31], some unreasonable results are implied in the following two cases. Then, by Equation (3) and (4) The above aggregated results indicate that the aggregated values of NCWAA (g1, g2) operator tend to the maximum value, while the aggregated results of NCWGA (g1, g2) operator tend to the maximum weight value. It is obvious that the NCWAA and NCWGA operators may cause unreasonable results of NCNs in some cases. In order to overcome the drawbacks, it is necessary to improve the NCWAA and NCWGA operators provided in [31]. Hence, in the next section, a new NCHWAGA is proposed by extending the hybrid arithmetic and geometric aggregation operators presented in [34,35].

Hybrid Arithmetic and Geometric Aggregation Operators of NCNs
In this section, we present the NCHWAGA operator and investigated its properties.

NCHWAGA Operator
NCNs. Then, the NCHWAGA operator is defined by: Proof.
For different values of ρ  [0, 1], we can discuss the families of the NCHWAGA operator in some special cases as follows: (i) The NCHWAGA operator reduces to the NCWAA operator [31] if ρ = 1.

Numerical Example
We still consider the above two numerical examples in Section 2 to demonstrate the effectiveness of the presented NCHWAGA operator. Generally taking ρ = 0.5, we calculate aggregated values of the NCHWAGA operator. In the above two cases, we can obtain the moderate values by the NCHWAGA operator. Obviously, the NCHWAGA operator can overcome the drawbacks of the NCWAA and NCWGA provided in Reference [31].

MADM Method Using the NCHWAGA Operator
In this section, we provide a MADM method based on the NCHWAGA operator to deal with neutrosophic cubic information.
In a MADM problem, assume that G = {G1, G2, …, Gk} is a set of k alternatives and P = {P1, P2, … , Pn} is a set of attributes. Suppose that the weight vector of P is , and Then, we can construct a decision matrix G = (gij)k×n with the NCN information, and provide the following MADM procedures based on the proposed NCHWAGA operator: Step 1. Calculate the aggregated value of gi for each alternative Gi (i = 1, 2,…, k) using the NCHWAGA operator: (1 )    Step 2. Obtain the score values of Ψ(x) (the accuracy degrees of Γ(x) if necessary) of the collective NCN gi (i = 1, 2, … , k) by Equations (1) and (2).
Step 3. Rank all the alternatives corresponding to the values of Ψ(x) and Γ(x), and select the best one(s) based on the largest value.

Illustrative Example and Comparison Analysis
This section introduces an illustrative example adapted from Reference [20] to demonstrate the application of the above MADM method. A company wants to invest some money in one of the four possible alternatives Gi (i = 1, 2, 3, 4). G1, G2, G3 and G4 represent a textile company, an automobile company, a computer company, and a software company, respectively. The four alternatives need to be evaluated according to the three attributes Pj (j = 1, 2, 3). P1, P2 and P3 represent respectively the risk, the growth, and the environmental impact. Corresponding to the three attributes, the weight vector is  Table 1. Then, we apply the NCHWAGA operator to handle the MADM problem as follows: Step 1. By Equation (7) Table 2 lists the decision results based on the NCHWAGA operator and cosine similarity measures of the NCSs. Obviously, the best alternatives and the ranking orders based on the NCHWAGA operator proposed in this paper are the same as in Reference [20].
For further relative comparison, Table 3 lists the MADM results using the NCHWAGA operator proposed in this paper and the NCWAA and NCWGA operators provided in Reference [31], respectively. The results listed in Table 3 show that the aggregated values of the NCHWAGA operator tend to the moderate values between the aggregated results of the NCWAA and NCWGA operators. Then, the ranking orders based on the NCHWAGA operator have little difference with the NCWAA and NCWGA operators. However, the best alternative given in all the MADM methods is identical. Furthermore, the results in Table 3 also show that the aggregated values of the NCHWAGA operator tend to moderate values between the aggregated values of the NCWAA and NCWGA operators in [31]. Therefore, the NCHWAGA operator can overcome the drawbacks of the NCWAA and NCWGA operators, and it is more effective and more suitable than the NCWAA and NCWGA operators to handle MADM problems under a neutrosophic cubic environment in some cases.

Conclusions
This paper developed the NCHWAGA operator of NCSs and investigated its properties. The main advantage of the proposed NCHWAGA operator can overcome the drawbacks implied by the existing NCWAA and NCWGA operators [31] in some cases and reach the moderate aggregated values. Then, the MADM method based on the NCHWAGA operator was established under an NCS environment. Finally, we provided an illustrative example to demonstrate the application of the established MADM method. By comparison, we found that the developed MADM method was more effective and more suitable to solve decision-making problems with neutrosophic cubic information in some cases. In the real world, a refined neutrosophic set [14] is very suitable to express complex problems of decision-making, since it can be described by its refined types of sub-truths, subindeterminacies, and sub-falsities. Therefore, we shall further extend the NCHWAGA operator to neutrosophic refined cubic sets for MADM by using the refined neutrosophic sets. In addition, the proposed method will be also extended to neutrosophic cubic oversets/undersets/offsets using the neutrosophic overset/underset/offset [36] in the future.