# Dombi Aggregation Operators of Neutrosophic Cubic Sets for Multiple Attribute Decision-Making
## 1 Introduction  
Fuzzy sets were presented by [PERSON] [1] to describe fuzzy problems with the membership function. After [PERSON], 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 [PERSON] 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, [PERSON] et al. defined cubic sets by the combination of interval-valued fuzzy sets with fuzzy sets [26]. Cubic sets can describe vagueness and uncertainty using an exact value and an interval-value simultaneously. Recently, [PERSON] et al. extendedcubic sets to the neutrosophic sets and introduced the definition of neutrosophic cubic sets (NCSs), containing external NCSs and internal NCSs [27]. [PERSON] et al. discussed the P-union and P-intersection of NCSs [28]. Furthermore, several related studies have been conducted to solve decision-making problems based on NCSs. [PERSON] et al. presented the concepts of weighted geometric operator (\(\mathrm{G_{W}}\)) and the weighted average operator (\(\mathrm{A_{W}}\)) on NCSs to solve multi-criteria decision-making problem [29]. [PERSON] et al. introduced the grey relational analysis method of NCSs for MADM [30]. Some similarity measures of NCSs were introduced for decision-making problems under neutrosophic cubic set environment [31; 32]. [PERSON] et al. proposed the NC-TODIM method for solving a multiple attribute group decision-making problem [33]. Aggregation operators play an important role in decision making. Hence, many researchers have presented various aggregation operators and their extensions [34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50], such as Harmonic aggregation operators [34; 35], weighted Bonferroni mean operators [36; 37; 38; 39], Einstein prioritized weighted operators [40], generalized weighted aggregation operators [41], Choquet integral operators [42] and so on. Dombi first developed the Dombi T-norm and T-conorm operations [51]. Recently, [PERSON] et al. presented Dombi Bonferroni mean operators of intuitionistic fuzzy sets and applied them in MADM problems [52]. [PERSON] and [PERSON] also extended the Dombi operations to single valued neutrosophic sets and proposed some single-valued neutrosophic Dombi weighted aggregation operators and applied them in MADM problems [53]. From the above review, the Dombi operations have the advantage of good flexibility with the operational parameter [53] and NCSs contain much more incomplete, inconsistent and indeterminate information to express actual decision-making problems [31].