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

Zadeh [1] proposed the classic fuzzy set to describe fuzzy problems with the membership degree in the closed interval [0,1]. Atanassov [2] presented the concept of the intuitionistic fuzzy set (IFS) to express fuzzy problems by the membership function and non-membership function. Smarandache [3] defined neutrosophic logic and introduced neutrosophic sets (NSs) to describe fuzzy problems by the truth, falsity, and indeterminacy membership functions. For easy engineering applications, some subclasses of NSs are defined. Wang et al. developed interval neutrosophic sets (INSs) [4] and single-valued neutrosophic sets (SVNSs) [5]. Ye presented simplified neutrosophic sets (SNSs) [6]. Wang et al. also presented multi-valued neutrosophic sets (MVNSs) [7]. Since then, INSs, SVNSs, SNSs and MVNSs have been widely applied in decision-making [8,9], medical diagnoses [10,11], and fault diagnoses [12,13]. Furthermore, many scholars developed some extension forms of NSs by combining neutrosophic sets with other sets, such as refined single-valued neutrosophic sets [14], intuitionistic neutrosophic soft sets [15], single-valued neutrosophic hesitant fuzzy sets [16], and rough neutrosophic sets [17].

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.

Some basic concepts and ranking methods of NCSs were introduced in this section.

Then, we called a basic element (x, < T_c(x), V_c(x), F_c(x) >, < t_c(x), v_c(x), f_c(x) >) in an NCS G a neutrosophic cubic number (NCN) [20]; for convenience, we denoted it as $g=(<[T^{\mathrm{L}},T^{\mathrm{U}}],[V^{\mathrm{L}},V^{\mathrm{U}}],[F^{\mathrm{L}},F^{\mathrm{U}}]>,<t,v,f>)$, where t, v, f ∈ [0,1] and $[T^{\mathrm{L}},T^{\mathrm{U}}],[V^{\mathrm{L}},V^{\mathrm{U}}],[F^{\mathrm{L}},F^{\mathrm{U}}]\subseteq [0,1]$ satisfy the condition $0\le T^{\mathrm{U}}+V^{\mathrm{U}}+F^{\mathrm{U}}\le 3$ and 0 ≤ t + v + f ≤ 3.

Then, an NCS G = {x, < T_c(x), V_c(x), F_c(x) >, < t_c(x), v_c(x), f_c(x) > | x ∈ 𝓩} is called an internal NCS if $T_c{}^L(x)$ ≤ t_c(x) ≤ $T_c{}^U(x)$, $V_c{}^L(x)$ ≤ v_c(x) ≤ $V_c{}^U(x)$, and $F_c{}^L(x)$ ≤ f_c(x) ≤ $F_c{}^U(x)$ for x ∈ 𝓩; and an NCS G is called an external NCS if t_c(x) ∉$(T_c{}^L(x),T_c{}^U(x))$, v_c(x) ∉ $(V_c{}^L(x),V_c{}^U(x))$, and f_c(x) ∉$(F_c{}^L(x),F_c{}^U(x))$ for x ∈ 𝓩 [18,19].

Let $g_1=(<[T_1{}^{\mathrm{L}},T_1{}^{\mathrm{U}}],[V_1{}^{\mathrm{L}},V_1{}^{\mathrm{U}}],[F_1{}^{\mathrm{L}},F_1{}^{\mathrm{U}}]>,<t_1,v_1,f_1>)$ and ${\mathrm{g}}_2=(<[T_2{}^{\mathrm{L}}$, $T_2{}^{\mathrm{U}}],[V_2{}^{\mathrm{L}},V_2{}^{\mathrm{U}}],[F_2{}^{\mathrm{L}},F_2{}^{\mathrm{U}}]>,<t_2,v_2,f_2>)$ be two NCNs, then there are following operational laws:

For any NCN $g=(<[T^{\mathrm{L}},T^{\mathrm{U}}],[V^{\mathrm{L}},V^{\mathrm{U}}],[F^{\mathrm{L}},F^{\mathrm{U}}]>,<t,v,f>)$, 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:

Assume that $g_i=(<[T_i{}^{\mathrm{L}},T_i{}^{\mathrm{U}}],[V_i{}^{\mathrm{L}},V_i{}^{\mathrm{U}}],[F_i{}^{\mathrm{L}},F_i{}^{\mathrm{U}}]>,<t_i,v_i,f_i>)$ (i = 1, 2, …, n) be a collection of NCNs. Then the NCWAA and NCWGA are provided, respectively, as follows [31]:

NCWAA (g₁, g₂, …, g_n)

NCWGA (g₁, g₂, …, g_n) where ζ_i $\in$ (i = 1, 2, …, n), satisfying ${\displaystyle {\sum }_{i=1}^n{\xi }_i=1}$.

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 Equations (3) and (4), NCWAA (g₁, g₂) = (< [0.001, 1], [0, 0], [0, 0] >, <1, 0, 0 >) and NCWGA (g₁, g₂) = (< [0, 0.004], [0, 0], [0, 0] >, <0.002, 0, 0 >).

Then, by Equation (3) and (4), NCWAA (g₁, g₂) = (< [0, 1], [0, 0], [0, 0] >, <1, 0, 0 >) and NCWGA (g₁, g₂) = (< [0, 0.537], [0, 0], [0, 0] >, <0.501, 0, 0 >).

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].

In this section, we present the NCHWAGA operator and investigated its properties.