Dombi Aggregation Operators of Neutrosophic Cubic Sets for Multiple Attribute Decision-Making

: The neutrosophic cubic set can describe complex decision-making problems with its single-valued neutrosophic numbers and interval neutrosophic numbers simultaneously. The Dombi operations have the advantage of good flexibility with the operational parameter. In order to solve decision-making problems with flexible operational parameter under neutrosophic cubic environments, the paper extends the Dombi operations to neutrosophic cubic sets and proposes a neutrosophic cubic Dombi weighted arithmetic average (NCDWAA) operator and a neutrosophic cubic Dombi weighted geometric average (NCDWGA) operator. Then, we propose a multiple attribute decision-making (MADM) method based on the NCDWAA and NCDWGA operators. Finally, we provide two illustrative examples of MADM to demonstrate the application and effectiveness of the established method.


Introduction
Fuzzy sets were presented by Zadeh [1] to describe fuzzy problems with the membership function. After Zadeh, some extensions of fuzzy sets have been proposed, including interval-valued fuzzy sets [2], intuitionistic fuzzy sets [3] and interval-valued intuitionistic fuzzy sets [4]. Interval-valued fuzzy sets can be described by the membership degree in an interval value of [0, 1]. Intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets can deal with different types of uncertainties by the non-membership function and membership function. Neutrosophic sets [5] were defined by Smarandache to express fuzzy problems using the truth, indeterminacy and falsity membership functions. Based on the neutrosophic sets, some simplified forms of neutrosophic sets were introduced for engineering applications, including interval neutrosophic sets [6], single valued neutrosophic sets [7] and simplified neutrosophic sets [8] and so on. The simplified forms of neutrosophic sets have been widely applied in multiple attribute decision-making (MADM) problems [9][10][11][12][13] and fault diagnosis [14]. Some extension forms of neutrosophic sets have been proposed by combining neutrosophic sets and other sets, for instance, multi-valued neutrosophic sets [15,16], intuitionistic neutrosophic soft set [17], rough neutrosophic sets [18], single-valued neutrosophic hesitant fuzzy [19], refined single-valued neutrosophic sets [20], neutrosophic soft sets [21], linguistic neutrosophic number [22,23], normal neutrosophic sets [24] and single-valued neutrosophic hesitant fuzzy set [25].
In the real world, the membership function in some fuzzy problems cannot be described completely only by an exact value or an interval-value. Hence, Jun et al. defined cubic sets by the combination of interval-valued fuzzy sets with fuzzy sets [26]. Cubic sets can describe vagueness

Preliminaries
In this section, we firstly present some concepts of interval neutrosophic sets, single-valued neutrosophic sets, cubic sets and NCSs and then introduce some ranking methods of NCSs based on their score, accuracy and certainty functions. ,,

Definition 1 ([6]). Let Z be a non-empty set. An interval neutrosophic sets G in Z is defined as follows:
where the functions t(v), i(v), f(v)  [0, 1] with the condition 0 ≤ t(v) + i(v) + f(v) ≤ 3 for v  Z, represent respectively the degrees of the truth-membership, the indeterminacy-membership and falsity-membership. Definition 3 ([26]). Let Z be a non-empty set, then a cubic set C in Z is constructed as the following form: for v  Z. It Ali et al. [27] and Jun et al. [28] extended cubic sets to the neutrosophic sets and proposed the concept of a NCS as follows.
Definition 4 ( [27,28]). Let Z be a universe of discourse, then a neutrosophic cubic set X in Z is denoted as the following form: where <T(v), I(v), F(v)> is an interval neutrosophic set [6] in Z and the intervals ,, Then, a neutrosophic cubic sets For convenient expression, a basic element (v, <T(v), For any neutrosophic cubic number, we provide the following score, accuracy and certainty functions.
where, S(x), A(x) and C(x) represent the score, accuracy and certainty functions of the NCNs, respectively.
The score function S(x) is a useful index in ranking NCNs. For a NCN, the bigger the truth-membership is, the greater the NCN is. At the same time, the smaller the memberships of indeterminacy and falsity are, the greater the NCN is. As to the accuracy function A(x), the larger the difference between truth-membership and falsity-membership is, the more effective the statement is. For the certainty function C(x), if the truth membership is bigger, then the NCN is more certainty. Hence, the score, accuracy and certainty functions are defined as shown above.
According to the three functions S(x), A(x) and C(x), the comparison and ranking of two NCNs are defined as following definition.    According to Equations (8) and (9) T  T  T  T  I  I  T  T  T  T  I  I ( ) x x ,

Dombi Weighted Aggregation Operators of NCSs
In this section, two Dombi weighted aggregation operators of NCNs are proposed based on the Dombi operators of NCNs in Definition 8 and then their properties are investigated. Dombi weighted geometric average operators are defined, respectively, as follows: . Then, the aggregated value of the NCDWAA operator is still a NCN, which can be calculated as follows: We can prove Theorem 1 by the mathematical induction.
Then we have the following inequalities:  T  T  I   T  T  I , ,..., Theorem 2 can be proved by a similar proof process as Theorem 1. Hence, it is not repeated here.
Then,   We can prove these properties by the same way as that of Theorem 1. Thus, they are omitted here.

MADM Method Using the NCDWAA or NCDWGA Operators
In this section, a MADM method based on the NCDWAA operator or the NCDWGA operator is proposed to handle MADM problems with neutrosophic cubic information.
In  In this case, we present a MADM method based on the NCDWAA operator or the NCDWGA operator to handle MADM problems with NCN information and the decision steps can be described as following: Step 1. Derive the collective NCN xk (k = 1, 2, …, m) for the alternative Xk (k = 1, 2, …, m) by using the NCDWAA operator: or by using the NCDWGA operator: where ωYj  [0, 1] and 1 1 n Y j j     for j = 1, 2, …, n.
Step 3. Rank all the alternatives and select the best one(s) according to the values of S(xk), A(xk) and C(xk).

Illustrative Examples
In order to demonstrate the application of the proposed MADM method, in this section, we provide two illustrative examples with neutrosophic cubic information adapted from [29]. Step 3. According to the above score values, the ranking order of the alternatives is X3  X4  X1  X2 and thus X3 is the best alternative.
Or we can use the NCDWGA operator for the MADM problem as follows: Step 1'. By using Equation (23) for ρ = 1, the collective NCNs for the alternatives Xj (j = 1, 2, 3, 4) can be obtained based on the NCDWGA as follows: Step 3'. According to the above score values, the ranking order of the alternatives is X1  X3  X4  X2 and thus X1 is the best alternative.
Further, all the ranking results of alternatives are listed in Tables 1 and 2 when the parameter ρ is changed from 1 to 5 in the NCDWAA and NCWGA operators. Then, we use the NCDWAA operator or the NCDWGA operator to solve the MADM problem with NCN information. By the same steps as that of Example 2, we obtain the ranking results of the alternatives. Tables 3 and 4 list the ranking results of the NCDWAA operator and NCWGA operator, respectively, when the parameter ρ is changed from 1 to 5.

Comparison Analysis
From Tables 1-4, we see that the ranking orders corresponding to the NCDWAA and NCDWGA operators show obvious difference in the MADM problem. In Example 3, Table 1 indicates that the different parameters of ρ may not influence the ranking orders corresponding to the NCDWAA operator; while Table 2 shows the different parameters of ρ can change the ranking orders based on the NCDWGA operator. In Table 2, when ρ = 1, the best alternative is X1, while the worst alternative is X2; when ρ = 2, ρ = 3, ρ = 4 and ρ = 5, the ranking order is changed and the best alternative is X3 and the worst alternative is X4. In Example 4, Tables 3 and 4 indicate that the different values of ρ can change the ranking orders based on the NCDWGA and NCDWGA operators. In Table 3, when ρ = 1 and ρ = 2, Q1 is the worst alternative; when ρ = 3, ρ = 4 and ρ = 5, the ranking order is changed and Q3 is the worst alternative. In Table 4, when ρ = 1, Q1 is the worst alternative; when ρ = 2, ρ = 3, ρ = 4 and ρ = 5, the ranking order is changed and Q2 is the worst alternative.
From the results of Tables 1-4, we can say that the NCDWAA and NCDWGA operators are sensitive to ρ. Hence, decision makers can specify some parameter ρ according to actual requirements and/or their preference.
Compared with the existing MADM method for NCSs introduced in [29], Table 5 lists the MADM results using NCDWAA and NCDWGA operators proposed in this paper and the weighted average operator ( W) of NCSs in the relevant paper [29], respectively. From Table 5, we see that the ranking orders based on the Dombi operators proposed in this paper and the weighted average operator ( W) of NCSs have obvious difference since different aggregation operators may be result in different ranking orders. Due to no parameter selected in [29], the proposed MADM based on Dombi aggregation operators is more flexible than the approach provided in [29].
Weighted average operator ( W) [29] Q3  Q1  Q2 Q3 For further comparison, the existing related decision-making approaches [51][52][53] based on some Dombi operations cannot deal with the decision-making problem with NCSs. However, the decision-making method presented in this paper can describe attributes with interval neutrosophic sets and single valued neutrosophic sets information simultaneously. Therefore, the paper provides a new effective way for decision makers to deal with MADM problems under neutrosophic cubic environment.

Conclusions
This paper proposed the NCDWAA and NCDWGA operators and discussed their properties. Then, we presented a MADM method based on the NCDWAA and NCDWGA operators to handle MADM problems under a NCN environment, in which attribute values of the alternatives were ranked and the best one(s) was determined according to their score (accuracy) function values.
Finally, two illustrative examples were provided to illustrate the application and effectiveness of the established MADM method. The developed MADM method can effectively solve decision-making problems with flexible operational parameter under neutrosophic cubic environments. In future work, we will further develop more aggregation operators for hesitant neutrosophic cubic sets and apply them in these areas, such as decision-making problems and fault diagnosis.