Some aggregation operators of Q-neutrosophic cubic sets for multi-attribute decision making

Abstract

A neutrosophic cubic set (NCS) can depict single-valued and interval neutrosophic information simultaneously in real life. Then, the NCS concept cannot describe neutrosophic cubic information regarding the assessment problems of two-dimensional universal sets (TDUSs), while a Q-neutrosophic set (Q-NS) can depict neutrosophic information in TDUSs but not describe neutrosophic cubic information in TDUSs. Motivated by the Q-NS and NCS concepts, we need to extend the Q-NS concept to Q-NCS for indicating neutrosophic cubic information in TDUSs. Therefore, this study first proposes a Q-NCS concept, which indicates its truth, falsity, and indeterminacy values independently in TDUSs, and then the basic operations of Q-neutrosophic cubic elements (Q-NCEs) and some weighted aggregation operators of Q-NCEs, such as a Q-NCE weighted arithmetic averaging (Q-NCEWAA) operator and a Q-NCE weighted geometric averaging (Q-NCEWGA) operator. Next, Q-neutrosophic cubic multi-attribute decision-making (MADM) methods regarding the proposed Q-NCEWAA and Q-NCEWGA operators are proposed under TDUSs and Q-NCS setting. Eventually, an illustrative example shows the applicability of the proposed MADM methods in TDUSs and Q-NCS setting.

Keywords

Q-neutrosophic cubic set two-dimensional universal sets Q-neutrosophic cubic element weighted geometric averaging (Q-NCEWAA) operator Q-neutrosophic cubic element weighted geometric averaging (Q-NCEWGA) operator decision making

1 Introduction

The theory of neutrosophic sets (NSs) [2] was introduced as the generalization of fuzzy sets (FSs), interval-valued FSs (IVFSs), intuitionistic FSs (IFSs), interval-valued IFSs, as well as (interval-valued) Pythagorean fuzzy sets [10– 12 , 15]. Then for convenient engineering applications, simplified NSs (implying single valued and interval NSs) [7] were presented as the subclasses of NSs to depict the truth, falsity, indeterminacy arguments for an objective thing, and then applied to decision making [7 , 16]. Since a cubic set introduced by Jun et al. [19] can depict partial determinate and partial uncertain information simultaneously, Kaur and Garg extended the cubic set to introduce a cubic intuitionistic fuzzy set, which consists of a IFS and an interval-valued IFS, and applied it to decision making [5] and pattern recognition and medical diagnosis [6]. To represent single-valued NSs and interval NSs simultaneously, the notion of neutrosophic cubic sets (NCSs) [9, 20] was introduced as the extension of cubic sets [19] and used for pattern recognition and decision making in NCS setting. Further, a multi-attribute decision-making (MADM) method [1] was presented by grey relational analysis in NCS setting. A multi-attribute group decision-making (MAGDM) method was proposed by using the similarity measure of NCSs [17]. Then, cosine measures of NCSs [21] were proposed for MADM in NCS setting. Further, basic operations, a neutrosophic cubic element weighted arithmetic averaging (NCEWAA) operator and a neutrosophic cubic element weighted geometric averaging (NCEWGA) operator [8] were presented and utilized for MADM problems with NCS information. However, the aforementioned NCS concept is depicted in a single universal set, and then the concepts of Q-single-valued neutrosophic soft sets and Q-single-valued NSs [18] were proposed to express the single-valued neutrosophic (soft) information in two-dimensional universal sets (TDUSs). Also Q-NSs and Q-neutrosophic soft sets [13] were introduced for Q-neutrosophic soft MADM problems in TDUSs. More recently, generalized Q-neutrosophic soft expert sets [14] were introduced for decision-making in TDUSs.

However, the Q-neutrosophic soft (expert) sets and Q-NSs in existing literature cannot describe single valued NSs and interval NSs simultaneously in TDUSs, while NCSs cannot also represent and handle neutrosophic cubic MADM problems under TDUS setting. For example, assume that a buyer wants to purchase a house in a group of four houses (a set of alternatives C = {C₁, C₂, C₃, C₄}). Then, the buyer indicates her/his attractive assessment attributes regarding the price (x₁), environment (x₂), traffic (x₃) for the four houses, which are constructed as one universal set X = {x₁, x₂, x₃}, and then choosing two cities c₁ and c₂ are implied as another universal set Y = {y₁, y₂}. Clearly, NCSs cannot express such an evaluation problem for each alternative C_j (j = 1, 2, 3, 4) under the TDUSs X = {x₁, x₂, x₃} and Y = {y₁, y₂} and NCS setting. Motivated by Q-NSs and NCSs and required by the real situation of the example, we need to extend the Q-NS concept to Q-NCS for indicating neutrosophic cubic information in TDUSs. Hence, the aims of this study are: (1) to propose the concepts of a Q-NCS and a Q-neutrosophic cubic element (Q-NCE) and some basic operations of Q-NCEs, (2) to develop a Q-NCE weighted arithmetic averaging (Q-NCEWAA) operator and a Q-NCE weighted geometric averaging (Q-NCEWGA) operator, and (3) to establish MADM methods regarding the Q-NCEWAA and Q-NCEWGA operators under TDUSs and Q-NCS setting.

Therefore, the contribution of this study is that we originally propose a Q-NCS concept and its weighted aggregation operators to express and aggregate NCS information in TDUSs in decision making problems, while existing the neutrosophic cubic decision making methods [1 , 17] or the Q-neutrosophic decision making method [13] cannot handle such a decision making problem with NCS information in TDUSs. It is obvious that the proposed Q-neutrosophic cubic decision making method is superior to existing decision making methods [1 , 17] and shows its advantage under TDUSs and Q-NCS setting.

The construction of the article is arranged as the following framework. The Section 2 proposes Q-NCS and Q-NCE concepts and some basic operations of Q-NCEs, and then introduces the ranking method of two NCEs regarding the score, accuracy and certainty functions of NCEs. In the Section 3, the Q-NCEWAA and Q-NCEWGA operators are developed to aggregate NCEs and their properties are investigated. The Section 4 develops MADM methods regarding the Q-NCEWAA and Q-NCEWGA operators and the ranking method of NCEs in TDUSs and Q-NCS setting. Next, an illustrative example indicates the applicability of the developed methods in the Section 5. Eventually, conclusions and future research are contained in the Section 6.

2 Q-neutrosophic cubic sets

This section presents the concept of Q-NCS so as to describe the neutrosophic certain and uncertain evaluation problem by the truth, indeterminacy, and falsity certain and uncertain arguments over TDUSs, and then defines the basic operations of Q-NCEs and the ranking method of NCEs.

Definition 1. Let X = {x₁, x₂, . . . , x_n} and Y = {y₁, y₂, . . . , y_m} be TDUSs, then a Q-NCS C in X and Y is defined as the mathematical expression: $C = {\begin{matrix} x_{i}, y_{j}, 〈 [T^{L} (x_{i}, y_{j}), T^{U} (x_{i}, y_{j})], [U^{L} (x_{i}, y_{j}), U^{U} (x_{i}, y_{j})], [V^{L} (x_{i}, y_{j}), V^{U} (x_{i}, y_{j})] 〉, \\ 〈 t (x_{i}, y_{j}), u (x_{i}, y_{j}), v (x_{i}, y_{j}) 〉 | x_{i} \in X, y_{j} \in Y \end{matrix}},$ where $〈 [T^{L} (x_{i}, y_{j}), T^{U} (x_{i}, y_{j})], [U^{L} (x_{i}, y_{j}), U^{U} (x_{i}, y_{j})], [V^{L} (x_{i}, y_{j}), V^{U} (x_{i}, y_{j})] 〉$ is a Q-interval NS in X, Y and $[T^{L} (x_{i}, y_{j}), T^{U} (x_{i}, y_{j})], [U^{L} (x_{i}, y_{j}), U^{U} (x_{i}, y_{j})], [V^{L} (x_{i}, y_{j}), V^{U} (x_{i}, y_{j})] \subseteq [0, 1]$ are the truth, indeterminacy, and falsity interval numbers in X and Y along with the conditions 0≤ T^U (x_i, y_j) + U^U (x_i, y_j) + V^U (x_i, y_j) ≤ 3 ; 〈 t (x_i, y_j) , u (x_i, y_j) , v (x_i, y_j) 〉 is a Q-single-valued NS in X, Y and t (x_i, y_j) , u (x_i, y_j) , v (x_i, y_j) ∈ [0, 1] are the truth, indeterminacy, and falsity degrees in TDUSs along with the condition 0 ≤ t (x_i, y_j) + u (x_i, y_j) + v (x_i, y_j) ≤ 3.

Then, each basic component (x_i, y_j, < [ T^L (x_i, y_j) , T^U (x_i, y_j)] , [ U^L (x_i, y_j) , U^U (x_i, y_j)] , [ V^L (x_i, y_j) , V^U (x_i, y_j)] > , < t (x_i, y_j) , u (x_i, y_j) , v (x_i, y_j) >) in a Q-NCS C is denoted by the simplified form $c_{ij} = (x_{i}, y_{j}, 〈 [T_{ij}^{L}, T_{ij}^{U}], [U_{ij}^{L}, U_{ij}^{U}], [V_{ij}^{L}, V_{ij}^{U}] 〉, 〈 t_{ij}, u_{ij}, v_{ij} 〉)$ , which is called a Q-NCE.

Definition 2. Let TDUSs be X = {x₁, x₂, . . . , x_n} and Y = {y₁, y₂, . . . , y_m}. A Q-NCS $C = {\begin{matrix} x_{i}, y_{j}, 〈 [T^{L} (x_{i}, y_{j}), T^{U} (x_{i}, y_{j})], [U^{L} (x_{i}, y_{j}), U^{U} (x_{i}, y_{j})], [V^{L} (x_{i}, y_{j}), V^{U} (x_{i}, y_{j})] 〉, \\ 〈 t (x_{i}, y_{j}), u (x_{i}, y_{j}), v (x_{i}, y_{j}) 〉 | x_{i} \in X, y_{j} \in Y \end{matrix}}$ in X and Y is called

(a) An internal Q-NCS C if T^L (x_i, y_j) ≤ t (x_i, y_j) ≤ T^U (x_i, y_j), U^L (x_i, y_j) ≤ u (x_i, y_j) ≤ U^U (x_i, y_j), and V^L (x_i, y_j) ≤ v (x_i, y_j) ≤ V^U (x_i, y_j) for each x_i ∈ X and each y_j ∈ Y;

(b) An external Q-NCS C if t (x_i, y_j) ∉ [ T^L (x_i, y_j) , T^U (x_i, y_j)], u (x_i, y_j) ∉ [ U^L (x_i, y_j) , U^U (x_i, y_j)], and v (x_i, y_j) ∉ [ V^L (x_i, y_j) , V^U (x_i, y_j)] for each x_i ∈ X and each y_j ∈ Y.

Definition 3. Let $c_{1, ij} = (x_{i}, y_{j}, 〈 [T_{1, ij}^{L}, T_{1, ij}^{U}], [U_{1, ij}^{L}, U_{1, ij}^{U}], [V_{1, ij}^{L}, V_{1, ij}^{U}] 〉, 〈 t_{1, ij}, u_{1, ij}, v_{1, ij} 〉)$ and $c_{2, ij} = (x_{i}, y_{j}, 〈 [T_{2, ij}^{L}, T_{2, ij}^{U}], [U_{2, ij}^{L}, U_{2, ij}^{U}], [V_{2, ij}^{L}, V_{2, ij}^{U}] 〉, 〈 t_{2, ij}, u_{2, ij}, v_{2, ij} 〉)$ be any two Q-NCEs in the TDUSs X and Y. Then there are the following operations:

$c_{1, ij}^{c} = (x_{ij}, y_{ij}, 〈 [V_{1, ij}^{L}, V_{1, ij}^{U}], [1 - U_{1, ij}^{U}, 1 - U_{1, ij}^{L}], [T_{1, ij}^{L}, T_{1, ij}^{U}] 〉, 〈 v_{1, ij}, 1 - u_{1, ij}, t_{1, ij} 〉)$ (complement of c_1,ij);

c_{1, ij} ⊆ c_{2, ij} if and only if $[T_{1, ij}^{L}, T_{1, ij}^{U}] \subseteq [T_{2, ij}^{L}, T_{2, ij}^{U}]$ , $[U_{1, ij}^{L}, U_{1, ij}^{U}] \supseteq [U_{2, ij}^{L}, U_{2, ij}^{U}]$ , $[V_{1, ij}^{L}, V_{1, ij}^{U}] \supseteq [V_{2, ij}^{L}, V_{2, ij}^{U}]$ , t_{1, ij} ≤ t_{2, ij}, u_{1, ij} ⩾ u_{2, ij}, and v_{1, ij} ⩾ v_{2, ij};

c_{1, ij} = c_{2, ij} if and only if c_{2, ij} ⊆ c_{1, ij} and c_{1, ij} ⊆ c_{2, ij}, i.e., $〈 [T_{1, ij}^{L}, T_{1, ij}^{U}], [U_{1, ij}^{L}, U_{1, ij}^{U}], [V_{1, ij}^{L}, V_{1, ij}^{U}] 〉$ = $〈 [T_{2, ij}^{L}, T_{1, ij}^{U}], [U_{2, ij}^{L}, U_{2, ij}^{U}], [V_{2, ij}^{L}, V_{2, ij}^{U}] 〉$ and 〈 t_{1, ij}, u_{1, ij}, v_{1, ij}〉 = 〈 t_{2, ij}, u_{2, ij}, v_{2, ij}〉;

$c_{1, i j} \cup c_{2, i j} = (x_{i}, y_{j}, 〈 [T_{1, i j}^{L} \lor T_{2, i j}^{L}, T_{1, i j}^{U} \lor T_{2, i j}^{U}], [U_{1, i j}^{L} \land U_{2, i j}^{L}, U_{1, i j}^{U} \land U_{2, i j}^{U}], [V_{1, i j}^{L} \land V_{2, i j}^{L}, V_{1, i j}^{U} \land V_{2, i j}^{U}] 〉, 〈 t_{1, i j} \lor t_{2, i j}, u_{1, i j} \land u_{2, i j}, v_{1, i j} \land v_{2, i j} 〉);$

$c_{1, i j} \cap c_{2, i j} = (x_{i}, y_{j}, 〈 [T_{1, i j}^{L} \land T_{2, i j}^{L}, T_{1, i j}^{U} \land T_{2, i j}^{U}], [U_{1, i j}^{L} \lor U_{2, i j}^{L}, U_{1, i j}^{U} \lor U_{2, i j}^{U}], [V_{1, i j}^{L} \lor V_{2, i j}^{L}, V_{1, i j}^{U} \lor V_{2, i j}^{U}] 〉, 〈 t_{1, i j}^{} \land t_{2, i j}^{}, u_{1, i j}^{} \lor u_{2, i j}^{}, v_{1, i j}^{} \lor v_{2, i j}^{} 〉);$

$c_{1, i j} \oplus c_{2, i j} = (x_{i}, y_{j}, 〈 [T_{1, i j}^{L} + T_{2, i j}^{L} - T_{1, i j}^{L} T_{2, i j}^{L}, T_{1, i j}^{U} + T_{2, i j}^{U} - T_{1, i j}^{U} T_{2, i j}^{U}], [U_{1, i j}^{L} U_{2, i j}^{L}, U_{1, i j}^{U} U_{2, i j}^{U}], [V_{1, i j}^{L} V_{2, i j}^{L}, V_{1, i j}^{U} V_{2, i j}^{U}] 〉, 〈 t_{1, i j} + t_{2, i j} - t_{1, i j} t_{2, i j}, u_{1, i j} u_{2, i j}, v_{1, i j} v_{2, i j} 〉);$

$c_{1, i j} \otimes c_{2, i j} = (x_{i}, y_{j}, 〈 [T_{1, i j}^{L} T_{2, i j}^{L}, T_{1, i j}^{U} T_{2, i j}^{U}], [U_{1, i j}^{L} + U_{2, i j}^{L} - U_{1, i j}^{L} U_{2, i j}^{L}, U_{1, i j}^{U} + U_{2, i j}^{U} - U_{1, i j}^{U} U_{2, i j}^{U}], [V_{1, i j}^{L} + V_{2, i j}^{L} - V_{1, i j}^{L} V_{2, i j}^{L}, V_{1, i j}^{U} + V_{2, i j}^{U} - V_{1, i j}^{U} V_{2, i j}^{U}] 〉, 〈 t_{1, i j} t_{2, i j}, u_{1, i j} + u_{2, i j} - u_{1, i j} u_{2, i j}, v_{1, i j} + v_{2, i j} - v_{1, i j} v_{2, i j} 〉);$

$ρ c_{1, i j} = (x_{i}, y_{j}, 〈 [1 - {(1 - T_{1, i j}^{L})}^{ρ}, 1 - {(1 - T_{1, i j}^{U})}^{ρ}], [{(U_{1, i j}^{L})}^{ρ}, {(U_{1, i j}^{U})}^{ρ}], [{(V_{1, i j}^{L})}^{ρ}, {(V_{1, i j}^{U})}^{ρ}] 〉, 〈 1 - {(1 - t_{1, i j})}^{ρ}, {(u_{1, i j})}^{ρ}, {(v_{1, i j})}^{ρ} 〉) f o r ρ > 0;$

${(c_{1, i j})}^{ρ} = (x_{i}, y_{j}, 〈 [{(T_{1, i j}^{L})}^{ρ}, {(T_{1, i j}^{U})}^{ρ}], [1 - {(1 - U_{1, i j}^{L})}^{ρ}, 1 - {(1 - U_{1, i j}^{U})}^{ρ}], [1 - {(1 - V_{1, i j}^{L})}^{ρ}, 1 - {(1 - V_{1, i j}^{U})}^{ρ}] 〉, 〈 {(t_{1, i j})}^{ρ}, 1 - {(1 - u_{1, i j})}^{ρ}, 1 - {(1 - v_{1, i j})}^{ρ} 〉) f o r ρ > 0.$

Definition 4 [8]. Let $c = (〈 [T^{L}, T^{U}], [U^{L}, U^{U}], [V^{L}, V^{U}] 〉, 〈 t, u, v 〉)$ be any NCE, then its score, accuracy, and certainty functions can be defined, respectively, as follows: $\begin{matrix} S (c) = [(4 + T^{L} - U^{L} - V^{L} + T^{U} - U^{U} - V^{U}) / 6 \\ + (2 + t - u - v) / 3] / 2, S (c) \in [0, 1], \end{matrix}$ (1) $\begin{matrix} H (c) = [(T^{L} - V^{L} + T^{U} - V^{U}) / 2 + (t - v)] / 2, \\ H (c) \in [- 1, 1], \end{matrix}$ (2) $D (c) = [(T^{L} + T^{U}) / 2 + t] / 2, D (c) \in [0, 1] .$ (3)

Regarding the three functions S(c), H(c), and D(c), the ranking method of two NCEs is introduced by the following definition.

Definition 5 [8]. Let two NCEs be $c_{1} = (〈 [T_{1}^{L}, T_{1}^{U}], [U_{1}^{L}, U_{1}^{U}], [V_{1}^{L}, V_{1}^{U}] 〉, 〈 t_{1}, u_{1}, v_{1} 〉)$ and $c_{2} = (〈 [T_{2}^{L}, T_{2}^{U}], [U_{2}^{L}, U_{2}^{U}], [V_{2}^{L}, V_{2}^{U}] 〉, 〈 t_{2}, u_{2}, v_{2} 〉)$ . Then the ranking method can be defined in the following:

If S(c₁)> S(c₂), then c₁ > c₂;

If S(c₂) = S(c₁) and H(c₂)> H(c₁), then c₂ > c₁;

If S (c₂) = S(c₁), H(c₂) = H(c₁) and D(c₂)> D(c₁), then c₂ > c₁;

If S (c₂) = S(c₁), H(c₂) = H(c₁) and D(c₂) = D(c₁), then c₂ = c₁.

3 Two weighted aggregation operators of Q-NCEs

To aggregate Q-NCEs, this section proposes the Q-NCEWAA and Q-NCEWGA operators, which are two basic information aggregation operations in decision-making process.

3.1 Q-NCE weighted arithmetic averaging operator

Corresponding to the operations in Definition 3, we present the Q-NCEWAA operator of Q-NCEs.

Definition 6. Set $c_{ij} = (x_{i}, y_{j}, 〈 [T_{ij}^{L}, T_{ij}^{U}], [U_{ij}^{L}, U_{1, ij}^{U}], [V_{ij}^{L}, V_{1, ij}^{U}] 〉, 〈 t_{ij}, u_{1, ij}, v_{ij} 〉)$ (j = 1, 2, . . . , m; i = 1,2, . . . , n) as a group of Q-NCEs in the TDUSs X = {x₁, x₂, . . . , x_n} and Y = {y₁, y₂, . . . , y_m}. Then, a Q-NCEWAA operator is defined below: $\begin{matrix} Q - NCEWAA (c_{11}, c_{21}, \dots, c_{n 1}, c_{12}, c_{22}, \dots, \\ c_{n 2}, . . ., c_{1 m}, c_{2 m}, \dots, c_{nm}) = \sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i} c_{ij}, \end{matrix}$ (4) where w_i (i = 1, 2, . . . , n) and ω _j (j = 1, 2, …, m) are the weights of x_i and y_j, respectively, along with w_i ∈ [0, 1], ω _j ∈ [0, 1], $\sum_{i = 1}^{n} w_{i} = 1$ , and $\sum_{j = 1}^{m} ω_{j} = 1$ .

Theorem 1. Set $c_{ij} = (x_{i}, y_{j}, 〈 [T_{ij}^{L}, T_{ij}^{U}], [U_{ij}^{L}, U_{1, ij}^{U}], [V_{ij}^{L}, V_{1, ij}^{U}] 〉, 〈 t_{ij}, u_{1, ij}, v_{ij} 〉)$ (j = 1, 2, . . . , m; i = 1, 2, . . . , n) as a group of Q-NCEs in the TDUSs X = {x₁, x₂, . . . , x_n} and Y = {y₁, y₂, . . . , y_m}, then the aggregated value of the Q-NCEWAA operator is given by $Q - NCEWAA (c_{11}, c_{21}, \dots, c_{n 1}, c_{12}, c_{22}, \dots, c_{n 2}, ., c_{1 m}, c_{2 m}, \dots, c_{n m})$ $= (〈 [1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - T_{i j}^{L})}^{w_{i} ω_{j}}, 1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - T_{i j}^{U})}^{w_{i} ω_{j}}], [\prod_{j = 1}^{m} \prod_{i = 1}^{n} {(U_{i j}^{L})}^{w_{i} ω_{j}}, \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(U_{i j}^{U})}^{w_{j}}], [\prod_{j = 1}^{m} \prod_{i = 1}^{n} {(V_{i j}^{L})}^{w_{i} ω_{j}}, \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(V_{i j}^{U})}^{w_{i} ω_{j}}] 〉, 〈 1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - t_{i j}^{})}^{w_{i} ω_{j}}, \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(u_{i j}^{})}^{w_{i} ω_{j}}, \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(v_{i j}^{})}^{w_{i} ω_{j}} 〉),$ (5)

where w_i (i = 1, 2, . . . , n) and ω _j (j = 1, 2, . . . , m) are the weights of x_i and y_j, respectively, along with w_i ∈ [0, 1], ω _j ∈ [0, 1], $\sum_{i = 1}^{n} w_{i} = 1$ , and $\sum_{j = 1}^{m} ω_{j} = 1$ .

Proof:

Corresponding to mathematical induction means, Equation (5) is verified below.

(1) Based on the NCEWAA operator of NCEs introduced in [8] for X = {x₁, x₂, . . . , x_n}, we can introduce the following Q-NCEWAA formula: $Q - N C E W A A (c_{11}, c_{21}, \dots, c_{n 1}, c_{12}, c_{22}, \dots, c_{n 2}, ., c_{1 m}, c_{2 m}, \dots, c_{n m}) = \sum_{j = 1}^{m} ω_{j} (〈 [1 - \prod_{i = 1}^{n} {(1 - T_{i j}^{L})}^{w_{j}}, 1 - \prod_{i = 1}^{n} {(1 - T_{i j}^{U})}^{w_{i}}], [\prod_{i = 1}^{n} {(U_{i j}^{L})}^{w_{i}}, \prod_{i = 1}^{n} {(U_{i j}^{U})}^{w_{i}}], [\prod_{i = 1}^{n} {(V_{i j}^{L})}^{w_{i}}, \prod_{i = 1}^{n} {(V_{i j}^{U})}^{w_{i}}] 〉, 〈 1 - \prod_{i = 1}^{n} {(1 - t_{i j}^{})}^{w_{i}}, \prod_{i = 1}^{n} {(u_{i j}^{})}^{w_{i}}, \prod_{i = 1}^{n} {(v_{i j}^{})}^{w_{i}} 〉) .$ (6)

When m = 2 in Y = {y₁, y₂, . . . , y_m}, regarding the operations of Definition 3 and Equation (6), we obtain the following result: $\begin{array}{l} Q - N C E W A A (c_{11}, c_{21}, \dots, c_{n 1}, c_{12}, c_{22}, \dots, c_{n 2}) \\ = \sum_{j = 1}^{2} ω_{j} (〈 [1 - \prod_{i = 1}^{n} {(1 - T_{i j}^{L})}^{w_{j}}, 1 - \prod_{i = 1}^{n} {(1 - T_{i j}^{U})}^{w_{i}}], [\prod_{i = 1}^{n} {(U_{i j}^{L})}^{w_{i}}, \prod_{i = 1}^{n} {(U_{i j}^{U})}^{w_{i}}], [\prod_{i = 1}^{n} {(V_{i j}^{L})}^{w_{i}}, \prod_{i = 1}^{n} {(V_{i j}^{U})}^{w_{i}}] 〉, \\ 〈 1 - \prod_{i = 1}^{n} {(1 - t_{i j}^{})}^{w_{i}}, \prod_{i = 1}^{n} {(u_{i j}^{})}^{w_{i}}, \prod_{i = 1}^{n} {(v_{i j}^{})}^{w_{i}} 〉) \end{array} = (\begin{array}{l} 〈 \begin{array}{l} [\begin{array}{l} 1 - {(\prod_{i = 1}^{n} {(1 - T_{i 1}^{L})}^{w_{j}})}^{ω_{1}} + 1 - {(\prod_{i = 1}^{n} {(1 - T_{i 2}^{L})}^{w_{i}})}^{ω_{2}} - (1 - {(\prod_{i = 1}^{n} {(1 - T_{i 1}^{L})}^{w_{j}})}^{ω_{1}} (1 - {(\prod_{i = 1}^{n} {(1 - T_{i 2}^{L})}^{w_{i}})}^{ω_{2}})), \\ 1 - {(\prod_{i = 1}^{n} {(1 - T_{i 1}^{U})}^{w_{j}})}^{ω_{1}} + 1 - {(\prod_{i = 1}^{n} {(1 - T_{i 2}^{U})}^{w_{i}})}^{ω_{2}} - (1 - {(\prod_{i = 1}^{n} {(1 - T_{i 1}^{U})}^{w_{j}})}^{ω_{1}} (1 - {(\prod_{i = 1}^{n} {(1 - T_{i 2}^{U})}^{w_{i}})}^{ω_{2}})) \end{array}], \\ [{(\prod_{i = 1}^{n} {(U_{i 1}^{L})}^{w_{i}})}^{ω_{1}} {(\prod_{i = 1}^{n} {(U_{i 2}^{L})}^{w_{i}})}^{ω_{2}}, {(\prod_{i = 1}^{n} {(U_{i 1}^{U})}^{w_{i}})}^{ω_{1}} {(\prod_{i = 1}^{n} {(U_{i 2}^{U})}^{w_{i}})}^{ω_{2}}], [{(\prod_{i = 1}^{n} {(V_{i 1}^{L})}^{w_{i}})}^{ω_{1}} {(\prod_{i = 1}^{n} {(V_{i 2}^{L})}^{w_{i}})}^{ω_{2}}, {(\prod_{i = 1}^{n} {(V_{i 1}^{U})}^{w_{i}})}^{ω_{1}} {(\prod_{i = 1}^{n} {(V_{i 2}^{U})}^{w_{i}})}^{ω_{2}}] \end{array} 〉 \\ 〈 \begin{array}{l} 1 - {(\prod_{i = 1}^{n} {(1 - t_{i 1}^{})}^{w_{j}})}^{ω_{1}} + 1 - {(\prod_{i = 1}^{n} {(1 - t_{i 2}^{})}^{w_{i}})}^{ω_{2}} - (1 - {(\prod_{i = 1}^{n} {(1 - t_{i 1}^{})}^{w_{j}})}^{ω_{1}} (1 - {(\prod_{i = 1}^{n} {(1 - t_{i 2}^{})}^{w_{i}})}^{ω_{2}})), \\ {(\prod_{i = 1}^{n} {(u_{i 1}^{})}^{w_{i}})}^{ω_{1}} {(\prod_{i = 1}^{n} {(u_{i 2}^{})}^{w_{i}})}^{ω_{2}}, {(\prod_{i = 1}^{n} {(v_{i 1}^{})}^{w_{i}})}^{ω_{1}} {(\prod_{i = 1}^{n} {(v_{i 2}^{})}^{w_{i}})}^{ω_{2}} \end{array} 〉 \end{array})$ $= (〈 [1 - \prod_{j = 1}^{2} \prod_{i = 1}^{n} {(1 - T_{i j}^{L})}^{w_{i} ω_{j}}, 1 - \prod_{j = 1}^{2} \prod_{i = 1}^{n} {(1 - T_{i j}^{U})}^{w_{i} ω_{j}}], [\prod_{j = 1}^{2} \prod_{i = 1}^{n} {(U_{i j}^{L})}^{w_{i} ω_{j}}, \prod_{j = 1}^{2} \prod_{i = 1}^{n} {(U_{i j}^{U})}^{w_{j}}], [\prod_{j = 1}^{2} \prod_{i = 1}^{n} {(V_{i j}^{L})}^{w_{i} ω_{j}}, \prod_{j = 1}^{2} \prod_{i = 1}^{n} {(V_{i j}^{U})}^{w_{i} ω_{j}}] 〉, 〈 1 - \prod_{j = 1}^{2} \prod_{i = 1}^{n} {(1 - t_{i j}^{})}^{w_{i} ω_{j}}, \prod_{j = 1}^{2} \prod_{i = 1}^{n} {(u_{i j}^{})}^{w_{i} ω_{j}}, \prod_{j = 1}^{2} \prod_{i = 1}^{n} {(v_{i j}^{})}^{w_{i} ω_{j}} 〉) .$ (7)

(2) When m = k in Y = {y₁, y₂, . . . , y_m}, corresponding to Equation (5) we have the following form: $\begin{array}{l} Q - N C E W A A (c_{11}, c_{21}, \dots, c_{n 1}, c_{12}, c_{22}, \dots, c_{n 2}, ., c_{1 k}, c_{2 k}, \dots, c_{n k}) \\ = (\begin{array}{l} 〈 [1 - \prod_{j = 1}^{k} \prod_{i = 1}^{n} {(1 - T_{i j}^{L})}^{w_{i} ω_{j}}, 1 - \prod_{j = 1}^{k} \prod_{i = 1}^{n} {(1 - T_{i j}^{U})}^{w_{i} ω_{j}}], [\prod_{j = 1}^{k} \prod_{i = 1}^{n} {(U_{i j}^{L})}^{w_{i} ω_{j}}, \prod_{j = 1}^{k} \prod_{i = 1}^{n} {(U_{i j}^{U})}^{w_{j}}], [\prod_{j = 1}^{k} \prod_{i = 1}^{n} {(V_{i j}^{L})}^{w_{i} ω_{j}}, \prod_{j = 1}^{k} \prod_{i = 1}^{n} {(V_{i j}^{U})}^{w_{i} ω_{j}}] 〉, \\ 〈 1 - \prod_{j = 1}^{k} \prod_{i = 1}^{n} {(1 - t_{i j}^{})}^{w_{i} ω_{j}}, \prod_{j = 1}^{k} \prod_{i = 1}^{n} {(u_{i j}^{})}^{w_{i} ω_{j}}, \prod_{j = 1}^{k} \prod_{i = 1}^{n} {(v_{i j}^{})}^{w_{i} ω_{j}} 〉 \end{array}), \end{array}$ (8)

(3) When m = k+1 in Y = {y₁, y₂, . . . , y_m}, based on combining Equation (7) and Equation (8), the result is given as follows: $Q - N C E W A A (c_{11}, c_{21}, \dots, c_{n 1}, c_{12}, c_{22}, \dots, c_{n 2}, ., c_{1 k}, c_{2 k}, \dots, c_{n k}, c_{1 (k + 1)}, c_{2 (k + 1)}, \dots, c_{n (k + 1)})$ $\begin{array}{l} = \sum_{j = 1}^{k + 1} ω_{j} (〈 [1 - \prod_{i = 1}^{n} {(1 - T_{i j}^{L})}^{w_{j}}, 1 - \prod_{i = 1}^{n} {(1 - T_{i j}^{U})}^{w_{i}}], [\prod_{i = 1}^{n} {(U_{i j}^{L})}^{w_{i}}, \prod_{i = 1}^{n} {(U_{i j}^{U})}^{w_{i}}], [\prod_{i = 1}^{n} {(V_{i j}^{L})}^{w_{i}}, \prod_{i = 1}^{n} {(V_{i j}^{U})}^{w_{i}}] 〉, 〈 1 - \prod_{i = 1}^{n} {(1 - t_{i j}^{})}^{w_{i}}, \prod_{i = 1}^{n} {(u_{i j}^{})}^{w_{i}}, \prod_{i = 1}^{n} {(v_{i j}^{})}^{w_{i}} 〉) \\ = (\begin{array}{l} 〈 \begin{array}{l} [\begin{array}{l} 1 - \prod_{j = 1}^{k} {(\prod_{i = 1}^{n} {(1 - T_{i j}^{L})}^{w_{j}})}^{ω_{j}} + 1 - {(\prod_{i = 1}^{n} {(1 - T_{i (k + 1)}^{L})}^{w_{i}})}^{ω_{k + 1}} - (1 - \prod_{j = 1}^{k} {(\prod_{i = 1}^{n} {(1 - T_{i j}^{L})}^{w_{j}})}^{ω_{j}} (1 - {(\prod_{i = 1}^{n} {(1 - T_{i (k + 1)}^{L})}^{w_{i}})}^{ω_{k + 1}})), \\ 1 - \prod_{j = 1}^{k} {(\prod_{i = 1}^{n} {(1 - T_{i j}^{U})}^{w_{j}})}^{ω_{j}} + 1 - {(\prod_{i = 1}^{n} {(1 - T_{i (k + 1)}^{U})}^{w_{i}})}^{ω_{k + 1}} - (1 - \prod_{j = 1}^{k} {(\prod_{i = 1}^{n} {(1 - T_{i j}^{U})}^{w_{j}})}^{ω_{j}} (1 - {(\prod_{i = 1}^{n} {(1 - T_{i (k + 1)}^{U})}^{w_{i}})}^{ω_{k + 1}})) \end{array}] \\ [\prod_{j = 1}^{k} {(\prod_{i = 1}^{n} {(U_{i j}^{L})}^{w_{i}})}^{ω_{j}} {(\prod_{i = 1}^{n} {(U_{i (k + 1)}^{L})}^{w_{i}})}^{ω_{k + 1}}, \prod_{j = 1}^{k} {(\prod_{i = 1}^{n} {(U_{i j}^{U})}^{w_{i}})}^{ω_{j}} {(\prod_{i = 1}^{n} {(U_{i (k + 1)}^{U})}^{w_{i}})}^{ω_{k + 1}}], \\ [\prod_{j = 1}^{k} {(\prod_{i = 1}^{n} {(V_{i j}^{L})}^{w_{i}})}^{ω_{j}} {(\prod_{i = 1}^{n} {(V_{i (k + 1)}^{L})}^{w_{i}})}^{ω_{k + 1}}, \prod_{j = 1}^{k} {(\prod_{i = 1}^{n} {(V_{i j}^{U})}^{w_{i}})}^{ω_{j}} {(\prod_{i = 1}^{n} {(V_{i (k + 1)}^{U})}^{w_{i}})}^{ω_{k + 1}}] \end{array} 〉, \\ 〈 \begin{array}{l} 1 - \prod_{j = 1}^{n} {(\prod_{i = 1}^{n} {(1 - t_{i j}^{})}^{w_{j}})}^{ω_{j}} + 1 - {(\prod_{i = 1}^{n} {(1 - t_{i (k + 1)}^{})}^{w_{i}})}^{ω_{k + 1}} - (1 - \prod_{j = 1}^{k} {(\prod_{i = 1}^{n} {(1 - t_{i j}^{})}^{w_{j}})}^{ω_{j}} (1 - {(\prod_{i = 1}^{n} {(1 - t_{i (k + 1)}^{})}^{w_{i}})}^{ω_{k + 1}})), \\ \prod_{j = 1}^{k} {(\prod_{i = 1}^{n} {(u_{i j}^{})}^{w_{i}})}^{ω_{j}} {(\prod_{i = 1}^{n} {(u_{i (k + 1)}^{})}^{w_{i}})}^{ω_{k + 1}}, \prod_{j = 1}^{k} {(\prod_{i = 1}^{n} {(v_{i j}^{})}^{w_{i}})}^{ω_{j}} {(\prod_{i = 1}^{n} {(v_{i (k + 1)}^{})}^{w_{i}})}^{ω_{k + 1}} \end{array} 〉 \end{array}) \end{array}$ $= (\begin{array}{l} 〈 [1 - \prod_{j = 1}^{k + 1} \prod_{i = 1}^{n} {(1 - T_{i j}^{L})}^{w_{i} ω_{j}}, 1 - \prod_{j = 1}^{k + 1} \prod_{i = 1}^{n} {(1 - T_{i j}^{U})}^{w_{i} ω_{j}}], [\prod_{j = 1}^{k + 1} \prod_{i = 1}^{n} {(U_{i j}^{L})}^{w_{i} ω_{j}}, \prod_{j = 1}^{k + 1} \prod_{i = 1}^{n} {(U_{i j}^{U})}^{w_{j}}], [\prod_{j = 1}^{k + 1} \prod_{i = 1}^{n} {(V_{i j}^{L})}^{w_{i} ω_{j}}, \prod_{j = 1}^{k + 1} \prod_{i = 1}^{n} {(V_{i j}^{U})}^{w_{i} ω_{j}}] 〉, \\ 〈 1 - \prod_{j = 1}^{k + 1} \prod_{i = 1}^{n} {(1 - t_{i j}^{})}^{w_{i} ω_{j}}, \prod_{j = 1}^{k + 1} \prod_{i = 1}^{n} {(u_{i j}^{})}^{w_{i} ω_{j}}, \prod_{j = 1}^{k + 1} \prod_{i = 1}^{n} {(v_{i j}^{})}^{w_{i} ω_{j}} 〉 \end{array}) .$

Based on the above results, Equation (5) holds for any m in Y = {y₁, y₂, . . . , y_m}.

Hence, this completes the proof.

Obviously, the Q-NCEWAA operator contains the following properties:

(1) Idempotency: Set c_ij = (x_i, y_j,<T_ij, U_ij, V_ij>,<t_i_j, u_i_j, v_i_j>) for x_i ∈ X, y_j ∈ Y, T_ij, U_ij, V_ij ⊆ [0, 1] and t_i_j, u_i_j, v_i_j ∈ [0, 1] (j = 1, 2,..., m;i = 1, 2, . . . , n) as a group of Q-NCEs. If c_ij is equal, i.e. c_ij = c = (<T, U, V>,<t, u, v>) for j = 1, 2,..., m and i = 1, 2, . . . , n, then Q - NCEWAA (c₁₁, c₂₁, ⋯ , c_n1, c₁₂, c₂₂, ⋯ , c_n2, . . . , c_{1 m}, c_{2 m}, ⋯ , c_nm) = c.

(2) Boundedness: Set c_ij = (x_i, y_j,<T_ij, U_ij, V_ij>,<t_i_j, u_i_j, v_i_j>) for x_i ∈ X, y_j ∈ Y, T_ij, U_ij, V_ij ⊆ [0, 1] and t_i_j, u_i_j, v_i_j ∈ [0, 1] (j = 1, 2,..., m;i = 1, 2, . . . , n) as a group of Q-NCEs and let $\begin{array}{l} c_{\min} = (\begin{array}{l} 〈 [\min_{_{^{i, j}}} (T_{i j}^{L}), \min_{_{^{i, j}}} (T_{i j}^{U})], [\max_{_{^{i, j}}} (U_{i j}^{L}), \max_{_{^{i, j}}} (U_{i j}^{U})], [\max_{_{^{i, j}}} (V_{i j}^{L}), \max_{_{^{i, j}}} (V_{i j}^{U})] 〉, \\ 〈 \min_{_{^{i, j}}} (t_{i j}^{}), \max_{_{^{i, j}}} (u_{i, j}^{}), \max_{_{^{i, j}}} (v_{i j}^{}) 〉 \end{array}), \\ c_{\max} = (\begin{array}{l} 〈 [\max_{_{^{i, j}}} (T_{i j}^{L}), \max_{_{^{i, j}}} (T_{i j}^{U})], [\min_{_{^{i, j}}} (U_{i j}^{L}), \min_{_{^{i, j}}} (U_{i j}^{U})], [\min_{_{^{i, j}}} (V_{i j}^{L}), \min_{_{^{i, j}}} (V_{i j}^{U})] 〉, \\ 〈 \max_{_{^{i, j}}} (t_{i j}^{}), \min_{_{^{i, j}}} (u_{i, j}^{}), \min_{_{^{i, j}}} (v_{i j}) 〉 \end{array}), \end{array}$

Then, there exists c_min ≤ Q - NCEWAA (c₁₁, c₂₁, ⋯ , c_n1, c₁₂, c₂₂, ⋯ , c_n2, . . . , c_{1 m}, c_{2 m}, ⋯ , c_nm) ≤ c_max.

(3) Monotonicity: Set c_ij = (x_i, y_j,<T_ij, U_ij, V_ij>,<t_i_j, u_i_j, v_i_j>) for x_i ∈ X, y_j ∈ Y, T_ij, U_ij, V_ij ⊆ [0, 1] and t_i_j, u_i_j, v_i_j ∈ [0, 1] and $c_{ij}^{*} = (x_{i}, y_{j}, 〈 T_{ij}^{*}, U_{ij}^{*}, V_{ij}^{*} 〉, 〈 t_{ij}^{*}, u_{ij}^{*}, v_{ij}^{*} 〉)$ for x_i ∈ X, y_j ∈ Y, $T_{ij}^{*}, U_{ij}^{*}, V_{ij}^{*} \subseteq [0, 1]$ and $t_{ij}^{*}, u_{ij}^{*}, v_{ij}^{*} \in [0, 1]$ (j = 1, 2,..., m;i = 1, 2, . . . , n) as two groups of Q-NCEs. If $c_{ij} \leq c_{ij}^{*}$ , then there is $\begin{matrix} Q - NCEWAA (c_{11}, c_{21}, \dots, c_{n 1}, c_{12}, c_{22}, \dots, c_{n 2}, . . ., c_{1 m}, c_{2 m}, \dots, c_{nm}) \\ \leq Q - NCEWAA (c_{11}^{*}, c_{21}^{*}, \dots, c_{n 1}^{*}, c_{12}^{*}, c_{22}^{*}, \dots, c_{n 2}^{*}, . . ., c_{1 m}^{*}, c_{2 m}^{*}, \dots, c_{nm}^{*}) \end{matrix}$ .

Proof:

(1) If c_ij = c for j = 1, 2, . . . , m and i = 1, 2,..., n, there is the following result: $\begin{array}{l} Q - N C E W A A (c_{11}, c_{21}, \dots, c_{n 1}, c_{12}, c_{22}, \dots, c_{n 2}, ., c_{1 m}, c_{2 m}, \dots, c_{n m}) \\ = (\begin{array}{l} 〈 [1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - T_{i j}^{L})}^{w_{i} ω_{j}}, 1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - T_{i j}^{U})}^{w_{i} ω_{j}}], [\prod_{j = 1}^{m} \prod_{i = 1}^{n} {(U_{i j}^{L})}^{w_{i} ω_{j}}, \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(U_{i j}^{U})}^{w_{j}}], [\prod_{j = 1}^{m} \prod_{i = 1}^{n} {(V_{i j}^{L})}^{w_{i} ω_{j}}, \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(V_{i j}^{U})}^{w_{i} ω_{j}}] 〉, \\ 〈 1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - t_{i j}^{})}^{w_{i} ω_{j}}, \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(u_{i j}^{})}^{w_{i} ω_{j}}, \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(v_{i j}^{})}^{w_{i} ω_{j}} 〉 \end{array}), \\ = (\begin{array}{l} 〈 [1 - {(1 - T_{}^{L})}^{\sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i}}, 1 - {(1 - T_{}^{U})}^{\sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i}}], [{(U_{}^{L})}^{\sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i}}, {(U_{}^{U})}^{\sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i}}], [{(V_{}^{L})}^{\sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i}}, {(V_{}^{U})}^{\sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i}}] 〉, \\ 〈 1 - {(1 - t)}^{\sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i}}, {(u)}^{\sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i}}, {(v)}^{\sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i}} 〉 \end{array}), \\ = (〈 [T^{L}, T^{U}], [U^{L}, U^{U}], [V^{L}, V^{U}] 〉, 〈 t, u, v 〉) = c . \end{array}$

(2) There is c_min ≤ c_ij ≤ c_max because c_min is the minimum NCE and c_max is the maximum NCE. Thus, there is $\sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i} c_{min} \leq \sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i} c_{ij} \leq \sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i} c_{max}$ . Regarding the above property (1), there exists c_min $\leq \sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i} c_{ij} \leq$ c_max, i.e., c_min ≤ Q - NCEWAA (c₁₁, c₂₁, ⋯ , c_n1, c₁₂, c₂₂, ⋯ , c_n2, . . . , c_{1 m}, c_{2 m}, ⋯ , c_nm) ≤ c_max.

(3) If $c_{ij} \leq c_{ij}^{*}$ , there is $\sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i} c_{ij} \leq \sum_{j = 1}^{m} ω_{j} \sum_{i = 1}^{n} w_{i} c_{ij}^{*}$ , i.e., $Q - NCEWAA (c_{11}, c_{21}, \dots, c_{n 1}, c_{12}, c_{22}, \dots, c_{n 2}, . . ., c_{1 m}, c_{2 m}, \dots, c_{nm}) \leq$ $Q - NCEWAA (c_{11}^{*}, c_{21}^{*}, \dots, c_{n 1}^{*}, c_{12}^{*}, c_{22}^{*}, \dots, c_{n 2}^{*}, . . ., c_{1 m}^{*}, c_{2 m}^{*}, \dots, c_{nm}^{*}) .$

Therefore, the above proofs are completed.□

3.2 Q-NCE weighted geometric averaging operator

Definition 7. Set c_ij = (x_i, y_j,<T_ij, U_ij, V_ij>,<t_i_j, u_i_j, v_i_j>) (j = 1, 2,..., m; i = 1, 2, . . . , n) as a group of Q-NCEs in the TDUSs X = {x₁, x₂, . . . , x_n} and Y = {y₁, y₂, . . . , y_m}. Then the Q-NCEWGA operator is defined as $Q - NCEWGA (c_{11}, c_{21}, \dots, c_{n 1}, c_{12}, c_{22}, \dots, c_{n 2}, . . ., c_{1 m}, c_{2 m}, \dots, c_{nm}) = \prod_{j = 1}^{m} {(\prod_{i = 1}^{n} c_{ij}^{w_{i}})}^{ω_{j}},$ (9) where w_i (i = 1, 2, . . . , n) and ω _j (j = 1, 2, . . . , m) is the weights of x_i and y_j, respectively, along with w_i ∈ [0, 1], ω _j ∈ [0, 1], $\sum_{i = 1}^{n} w_{i} = 1$ , and $\sum_{j = 1}^{m} ω_{j} = 1$ .

Regarding the operations in Definitions 3 and Equation (9), there is the following theorem.

Theorem 2. Set $c_{ij} = (x_{i}, y_{j}, 〈 [T_{ij}^{L}, T_{ij}^{U}], [U_{ij}^{L}, U_{1, ij}^{U}], [V_{ij}^{L}, V_{1, ij}^{U}] 〉, 〈 t_{ij}, u_{1, ij}, v_{ij} 〉)$ (j = 1, 2, . . . , m; i = 1, 2, . . . , n) as a group of Q-NCEs in the TDUSs X = {x₁, x₂, . . . , x_n} and Y = {y₁, y₂, . . . , y_m}, then the aggregated value of the Q-NCEWGA operator is given by $\begin{array}{l} Q - N C E W G A (c_{11}, c_{21}, \dots, c_{n 1}, c_{12}, c_{22}, \dots, c_{n 2}, ., c_{1 m}, c_{2 m}, \dots, c_{n m}) \\ = (\begin{array}{l} 〈 [\prod_{j = 1}^{m} \prod_{i = 1}^{n} {(T_{i j}^{L})}^{w_{i} ω_{j}}, \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(T_{i j}^{U})}^{w_{i} ω_{j}}], [1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - U_{i j}^{L})}^{w_{i} ω_{j}}, 1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - U_{i j}^{U})}^{w_{j}}], [1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - V_{i j}^{L})}^{w_{i} ω_{j}}, 1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - V_{i j}^{U})}^{w_{i} ω_{j}}] 〉, \\ 〈 \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(t_{i j}^{})}^{w_{i} ω_{j}}, 1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - u_{i j}^{})}^{w_{i} ω_{j}}, 1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - v_{i j}^{})}^{w_{i} ω_{j}} 〉 \end{array}), \end{array}$ (10)

where w_i (i = 1, 2, . . . , n) and ω _j (j = 1, 2, . . . , m) is the weights of x_i and y_j, respectively, along with w_i ∈ [0, 1], ω _j ∈ [0, 1], $\sum_{i = 1}^{n} w_{i} = 1$ , and $\sum_{j = 1}^{m} ω_{j} = 1$ .

By the similar mathematical induction means in Theorem 1, we can also prove Theorem 2 (omitted).

Similarly, the Q-NCEWGA operator also implies the following properties:

(1) Idempotency: Set c_ij = (x_i, y_j,<T_ij, U_ij, V_ij>,<t_i_j, u_i_j, v_i_j>) for x_i ∈ X, y_j ∈ Y, T_ij, U_ij, V_ij ⊆ [0, 1] and t_i_j, u_i_j, v_i_j ∈ [0, 1] (j = 1, 2,..., m;i = 1, 2, . . . , n) as a group of Q-NCEs. If c_ij is equal, i.e. c_ij = c = (<T, U, V>,<t, u, v>) for j = 1, 2,..., m and i = 1, 2, . . . , n, then Q - NCEWGA (c₁₁, c₂₁, ⋯ , c_n1, c₁₂, c₂₂, ⋯ , c_n2, . . . , c_{1 m}, c_{2 m}, ⋯ , c_nm) = c.

(2) Boundedness: Set c_ij = (x_i, y_j,<T_ij, U_ij, V_ij>,<t_i_j, u_i_j, v_i_j>) for x_i ∈ X, y_j ∈ Y, T_ij, U_ij, V_ij ⊆ [0, 1], and t_i_j, u_i_j, v_i_j ∈ [0, 1] (j = 1, 2,..., m; i = 1, 2, . . . , n) as a group of Q-NCEs and let $\begin{array}{l} c_{\min} = (\begin{array}{l} 〈 [\min_{i, j} (T_{i j}^{L}), \min_{i, j} (T_{i j}^{U})], [\max_{i, j} (U_{i j}^{L}), \max_{i, j} (U_{i j}^{U})], [\max_{i, j} (V_{i j}^{L}), \max_{i, j} (V_{i j}^{U})] 〉, \\ 〈 \min_{i, j} (t_{i j}^{}), \max_{i, j} (u_{i, j}^{}), \max_{i, j} (v_{i j}^{}) 〉 \end{array}), \\ c_{\max} = (\begin{array}{l} 〈 [\max_{i, j} (T_{i j}^{L}), \max_{i, j} (T_{i j}^{U})], [\min_{i, j} (U_{i j}^{L}), \min_{i, j} (U_{i j}^{U})], [\min_{i, j} (V_{i j}^{L}), \min_{i, j} (V_{i j}^{U})] 〉, \\ 〈 \max_{i, j} (t_{i j}^{}), \min_{i, j} (u_{i, j}^{}), \min_{i, j} (v_{i j}^{}) 〉 \end{array}), \end{array}$

Then, there exists c_min ≤ Q - NCEWGA (c₁₁, c₂₁, ⋯ , c_n1, c₁₂, c₂₂, ⋯ , c_n2, . . . , c_{1 m}, c_{2 m}, ⋯ , c_nm) ≤ c_max.

(3) Monotonicity: Set c_ij = (x_i, y_j,<T_ij, U_ij, V_ij>,<t_i_j, u_i_j, v_i_j>) for x_i ∈ X, y_j ∈ Y, T_ij, U_ij, V_ij ⊆ [0, 1] and t_i_j, u_i_j, v_i_j ∈ [0, 1] and $c_{ij}^{*} = (x_{i}, y_{j}, 〈 T_{ij}^{*}, U_{ij}^{*}, V_{ij}^{*} 〉, 〈 t_{ij}^{*}, u_{ij}^{*}, v_{ij}^{*} 〉)$ for x_i ∈ X, y_j ∈ Y, $T_{ij}^{*}, U_{ij}^{*}, V_{ij}^{*} \subseteq [0, 1]$ and $t_{ij}^{*}, u_{ij}^{*}, v_{ij}^{*} \in [0, 1]$ (j = 1, 2,..., m; i = 1, 2, . . . , n) as two groups of Q-NCEs. If $c_{ij} \leq c_{ij}^{*}$ , then there is $\begin{matrix} Q - NCEWGA (c_{11}, c_{21}, \dots, c_{n 1}, c_{12}, c_{22}, \dots, c_{n 2}, . . ., c_{1 m}, c_{2 m}, \dots, c_{nm}) \\ \leq Q - NCEWGA (c_{11}^{*}, c_{21}^{*}, \dots, c_{n 1}^{*}, c_{12}^{*}, c_{22}^{*}, \dots, c_{n 2}^{*}, . . ., c_{1 m}^{*}, c_{2 m}^{*}, \dots, c_{nm}^{*}) . \end{matrix}$

Then, these properties of the Q-NCEWGA operator can be also verified based on the similar proofs of the above properties for the Q-NCEWAA operator (omitted).

4 MADM approaches regarding the Q-NCEWAA and Q-NCEWGA operators

This section presents Q-NCS MADM approaches regarding the Q-NCEWAA and Q-NCEWGA operators of Q-NCEs in TDUSs and Q-NCS setting.

Provided that a MADM problem contains a set of m alternatives C = {C₁, C₂, . . . , C_p} corresponding to two kinds of attribute sets specified as the TDUSs X = {x₁, x₂, . . . , x_n} and Y = {y₁, y₂, . . . , y_m}, respectively. Then the weigh vectors of X and Y are specified as w = (w₁, w₂,..., w_n) and ω=(ω₁, ω₂,..., ω _m ). Thus, the alternatives C_k (k = 1, 2, . . . , p) are evaluated over the attributes x_i (i = 1, 2, . . . , n) and y_j (j = 1, 2, . . . , m) by Q-NCEs c_k,_ij = (x_i, y_j,<T_k,_ij, U_k,_ij, V_k,_ij>,<t_k,_i_j, u_k,_i_j, v_k,_i_j>) for T_k,_ij, U_k,_ij, V_k,_ij ⊆ [0, 1] and t_k,_i _j , u_k,_i _j , v_k,_i _j ∈ [0, 1] (k = 1, 2, . . . , p; i = 1,2, . . . , n; j = 1, 2, . . . , m). Hence, all the Q-NCEs given by the decision makers can be indicated as the decision matrix of Q-NCSs C = (c_k,_ij) _p _× _nm .

Based on the aggregation operators of Eqs. (5) and (10), we give the following two MADM methods for Q-NCS MADM problems in TDUSs and Q-NCS setting.

Method I. The Q-NCS MADM method based on the Q-NCEWAA operator is depicted as the following decision steps:

Step 1: Based on Equation (5), calculate the aggregation value of Q-NCEs c_k,_ij for C_k (k = 1, 2, . . . , p) by using the following formula: $\begin{array}{l} c_{k} = Q - N C E W A A (c_{k, 11}, c_{k, 21}, \dots, c_{k, n 1}, c_{k, 12}, c_{k, 22}, \dots, c_{k, n 2}, ., c_{k, 1 m}, c_{k, 2 m}, \dots, c_{k, n m}) \\ = (\begin{array}{l} 〈 [1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - T_{k, i j}^{L})}^{w_{i} ω_{j}}, 1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - T_{k, i j}^{U})}^{w_{i} ω_{j}}], [\prod_{j = 1}^{m} \prod_{i = 1}^{n} {(U_{k, i j}^{L})}^{w_{i} ω_{j}}, \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(U_{k, i j}^{U})}^{w_{j}}], [\prod_{j = 1}^{m} \prod_{i = 1}^{n} {(V_{k, i j}^{L})}^{w_{i} ω_{j}}, \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(V_{k, i j}^{U})}^{w_{i} ω_{j}}] 〉, \\ 〈 1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - t_{k, i j}^{})}^{w_{i} ω_{j}}, \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(u_{k, i j}^{})}^{w_{i} ω_{j}}, \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(v_{k, i j}^{})}^{w_{i} ω_{j}} 〉 \end{array}) . \end{array}$ (11)

Step 2: Compute the values of S(C_k) (H(C_k)/ D(C_k) if necessary) for k = 1, 2, …, p.

Step 3: Rank the alternatives in a descending order corresponding to the score (accuracy/certainty) values and choose the best one C_k *.

Step 4: Calculate the aggregation values of Q-NCEs c_k_* ,_ij corresponding to x_i ∈ X (i = 1, 2, . . . , n) or y_i ∈ Y (j = 1, 2, . . . , m) for C_k * by the following formula: $\begin{array}{l} c_{k^{*}, j} = Q - N C E W A A (c_{k^{*}, 1 j}, c_{k^{*}, 2 j}, \dots, c_{k^{*}, n j}) \\ = (\begin{array}{l} 〈 [1 - \prod_{i = 1}^{n} {(1 - T_{k^{*}, i j}^{L})}^{1 / n}, 1 - \prod_{i = 1}^{n} {(1 - T_{k^{*}, i j}^{U})}^{1 / n}], [\prod_{i = 1}^{n} {(U_{k^{*}, i j}^{L})}^{1 / n}, \prod_{i = 1}^{n} {(U_{k^{*}, i j}^{U})}^{1 / n}], [\prod_{i = 1}^{n} {(V_{k^{*}, i j}^{L})}^{1 / n}, \prod_{i = 1}^{n} {(V_{k^{*}, i j}^{U})}^{1 / n}] 〉, \\ 〈 1 - \prod_{i = 1}^{n} {(1 - t_{k^{*}, i j}^{})}^{1 / n}, \prod_{i = 1}^{n} {(u_{k^{*}, i j}^{})}^{1 / n}, \prod_{i = 1}^{n} {(v_{k^{*}, i j}^{})}^{1 / n} 〉 \end{array}), \end{array}$ (12)

for j = 1, 2, …, m,

or $\begin{array}{l} c_{k^{*}, i} = Q - N C E W A A (c_{k^{*}, i 1}, c_{k^{*}, i 2}, \dots, c_{k^{*}, i m}) \\ = (\begin{array}{l} 〈 [1 - \prod_{j = 1}^{m} {(1 - T_{k^{*}, i j}^{L})}^{1 / m}, 1 - \prod_{j = 1}^{m} {(1 - T_{k^{*}, i j}^{U})}^{1 / m}], [\prod_{j = 1}^{m} {(U_{k^{*}, i j}^{L})}^{1 / m}, \prod_{j = 1}^{m} {(U_{k^{*}, i j}^{U})}^{1 / m}], [\prod_{j = 1}^{m} {(V_{k^{*}, i j}^{L})}^{1 / m}, \prod_{j = 1}^{n} {(V_{k^{*}, i j}^{U})}^{1 / m}] 〉, \\ 〈 1 - \prod_{j = 1}^{m} {(1 - t_{k^{*}, i j}^{})}^{1 / m}, \prod_{j = 1}^{m} {(u_{k^{*}, i j}^{})}^{1 / m}, \prod_{j = 1}^{m} {(v_{k^{*}, i j}^{})}^{1 / m} 〉 \end{array}), \end{array}$ (13)

for i = 1, 2, …, n.

Step 5: Compute the values of either S(c_{k *, j}) (H(c_{k *, j})/ D(c_{k *, j}) if necessary) for j = 1, 2, . . . , m or S(c_{k *, i}) (H(c_{k *, i})/ D(c_{k *, i}) if necessary) for i = 1, 2, . . . , n (depending on some actual situation), and then determine the best one x_i * or y_j_* corresponding to the largest value for x_i ∈ X or y_j ∈ Y.

Step 6: End.

Method II. The Q-NCS MADM method based on the Q-NCEWGA operator is depicted as the following decision steps:

Step 1: Based on Equation (10), calculate the aggregation value of Q-NCEs c_k,_ij for C_k (k = 1, 2, . . . , p) by using the following formula: $\begin{array}{l} c_{k} = Q - N C E W G A (c_{k, 11}, c_{k, 21}, \dots, c_{k, n 1}, c_{k, 12}, c_{k, 22}, \dots, c_{k, n 2}, ., c_{k, 1 m}, c_{k, 2 m}, \dots, c_{k, n m}) \\ = (\begin{array}{l} 〈 [\prod_{j = 1}^{m} \prod_{i = 1}^{n} {(T_{k, i j}^{L})}^{w_{i} ω_{j}}, \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(T_{k, i j}^{U})}^{w_{i} ω_{j}}], [1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - U_{k, i j}^{L})}^{w_{i} ω_{j}}, 1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - U_{k, i j}^{U})}^{w_{j}}], [1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - V_{k, i j}^{L})}^{w_{i} ω_{j}}, 1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - V_{k, i j}^{U})}^{w_{i} ω_{j}}] 〉, \\ 〈 \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(t_{k, i j}^{})}^{w_{i} ω_{j}}, 1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - u_{k, i j}^{})}^{w_{i} ω_{j}}, 1 - \prod_{j = 1}^{m} \prod_{i = 1}^{n} {(1 - v_{k, i j}^{})}^{w_{i} ω_{j}} 〉 \end{array}), \end{array}$ (14)

Step 2: Compute the values of S(C_k) (H(C_k)/ D(C_k) if necessary) for k = 1, 2, . . . , p.

Step 3: Rank the alternatives in a descending order corresponding to the score (accuracy/certainty) values and choose the best one C_k *.

Step 4: Calculate the aggregation values of Q-NCEs c_k_* ,_ij corresponding to x_i ∈ X (i = 1, 2, . . . , n) or y_i ∈ Y (j = 1, 2, . . . , m) for C_k * by the following formula: $\begin{array}{l} c_{k^{*}, j} = Q - N C E W G A (c_{k^{*}, 1 j}, c_{k^{*}, 2 j}, \dots, c_{k^{*}, n j}) \\ = (\begin{array}{l} 〈 [\prod_{i = 1}^{n} {(T_{k^{*}, i j}^{L})}^{1 / n}, \prod_{i = 1}^{n} {(T_{k^{*}, i j}^{U})}^{1 / n}], [1 - \prod_{i = 1}^{n} {(1 - U_{k^{*}, i j}^{L})}^{1 / n}, 1 - \prod_{i = 1}^{n} {(1 - U_{k^{*}, i j}^{U})}^{1 / n}], [1 - \prod_{i = 1}^{n} {(1 - V_{k^{*}, i j}^{L})}^{1 / n}, 1 - \prod_{i = 1}^{n} {(1 - V_{k^{*}, i j}^{U})}^{1 / n}] 〉, \\ 〈 \prod_{i = 1}^{n} {(t_{k^{*}, i j}^{})}^{1 / n}, 1 - \prod_{i = 1}^{n} {(1 - u_{k^{*}, i j}^{})}^{1 / n}, 1 - \prod_{i = 1}^{n} {(1 - v_{k^{*}, i j}^{})}^{1 / n} 〉 \end{array}), \end{array}$ (15)

for j = 1, 2, . . . , m

or $\begin{array}{l} c_{k^{*}, i} = Q - N C E W G A (c_{k^{*}, i 1}, c_{k^{*}, i 2}, \dots, c_{k^{*}, i m}) \\ = (\begin{array}{l} 〈 [\prod_{j = 1}^{m} {(T_{k^{*}, i j}^{L})}^{1 / m}, \prod_{j = 1}^{m} {(T_{k^{*}, i j}^{U})}^{1 / m}], [1 - \prod_{j = 1}^{m} {(1 - U_{k^{*}, i j}^{L})}^{1 / m}, 1 - \prod_{j = 1}^{m} {(1 - U_{k^{*}, i j}^{U})}^{1 / m}], [1 - \prod_{j = 1}^{m} {(1 - V_{k^{*}, i j}^{L})}^{1 / m}, 1 - \prod_{j = 1}^{n} {(1 - V_{k^{*}, i j}^{U})}^{1 / m}] 〉, \\ 〈 \prod_{j = 1}^{m} {(t_{k^{*}, i j}^{})}^{1 / m}, 1 - \prod_{j = 1}^{m} {(1 - u_{k^{*}, i j}^{})}^{1 / m}, 1 - \prod_{j = 1}^{m} {(1 - v_{k^{*}, i j}^{})}^{1 / m} 〉) \end{array}), \end{array}$ (16)

for i = 1, 2, . . . , n.

Step 6: End.

5 Illustrative example and comparative analysis

5.1 Illustrative example

Assume a buyer needs to buy a house in some city and to consider his/her potential houses by a set of four alternatives C = {C₁, C₂, C₃, C₄}. Then, the buyer considers the two cities as one universal set (one attribute set) Y = {y₁, y₂} and their price (x₁), environment (x₂), and traffic (x₃) as another universal set (another attribute set) X = {x₁, x₂, x₃}, which indicate his/her buying attraction of a house. Whereas, the two weigh vectors of Y and X are indicated as ω=(0.4, 0.6) and w = (0.4, 0.25, 0.35), respectively. When the alternative C_k (k = 1, 2, 3, 4) are evaluated over the TDUSs Y = {y₁, y₂} and X = {x₁, x₂, x₃}, a group of decision makers/buyers can give the truth, indeterminacy, and falsity values for x_i and y_j regarding an alternative C_k by Q-NCEs $c_{k, ij} = (x_{i}, y_{j}, 〈 [T_{k, ij}^{L}, T_{k, ij}^{U}], [U_{k, ij}^{L}, U_{k, ij}^{U}], [V_{k, ij}^{L}, V_{k, ij}^{U}] 〉, 〈 t_{k, ij}, u_{k, ij}, v_{k, ij} 〉)$ (k = 1, 2, 3, 4; i = 1,2, 3; j = 1, 2), which are constructed as the decision-making matrix of Q-NCSs: $\begin{array}{l} C = \begin{matrix} C_{1} \\ C_{2} \\ C_{3} \\ C_{4} \end{matrix} [\begin{matrix} (x_{1}, y_{1}, 〈 [0.6, 0.7], [0.1, 0.2], [0.0, 0.1] 〉, 〈 0.6, 0.2, 0.1 〉) (x_{2}, y_{1}, 〈 [0.5, 0.6], [0.2, 0.3], [0.1, 0.2] 〉, 〈 0.6, 0.2, 0.1 〉) (x_{3}, y_{1}, 〈 [0.5, 0.6], [0.2, 0.3], [0.1, 0.2] 〉, 〈 0.6, 0.2, 0.2 〉) \\ (x_{1}, y_{1}, 〈 [0.6, 0.7], [0.1, 0.2], [0.2, 0.3] 〉, 〈 0.6, 0.2, 0.3 〉) (x_{2}, y_{1}, 〈 [0.6, 0.7], [0.0, 0.1], [0.0, 0.1] 〉, 〈 0.6, 0.1, 0.1 〉) (x_{3}, y_{1}, 〈 [0.6, 0.7], [0.1, 0.2], [0.3, 0.4] 〉, 〈 0.7, 0.2, 0.3 〉) \\ (x_{1}, y_{1}, 〈 [0.5, 0.7], [0.1, 0.2], [0.2, 0.3] 〉, 〈 0.6, 0.2, 0.3 〉) (x_{2}, y_{1}, 〈 [0.4, 0.5], [0.1, 0.2], [0.1, 0.2] 〉, 〈 0.5, 0.1, 0.1 〉) (x_{3}, y_{1}, 〈 [0.4, 0.6], [0.2, 0.3], [0.2, 0.3] 〉, 〈 0.5, 0.3, 0.2 〉) \\ (x_{1}, y_{1}, 〈 [0.5, 0.6], [0.1, 0.2], [0.1, 0.2] 〉, 〈 0.6, 0.1, 0.1 〉) (x_{2}, y_{1}, 〈 [0.5, 0.6], [0.2, 0.4], [0.3, 0.4] 〉, 〈 0.6, 0.4, 0.3 〉) (x_{3}, y_{1}, 〈 [0.5, 0.7], [0.1, 0.2], [0.1, 0.2] 〉, 〈 0.6, 0.1, 0.1 〉) \end{matrix} \\ \begin{matrix} (x_{1}, y_{2}, 〈 [0.7, 0.8], [0.2, 0.3], [0.2, 0.4] 〉, 〈 0.7, 0.2, 0.3 〉) (x_{2}, y_{2}, 〈 [0.4, 0.6], [0.1, 0.2], [0.1, 0.2] 〉, 〈 0.5, 0.1, 0.1 〉) (x_{3}, y_{2}, 〈 [0.7, 0.9], [0.2, 0.3], [0.2, 0.3] 〉, 〈 0.8, 0.2, 0.2 〉) \\ (x_{1}, y_{2}, 〈 [0.7, 0.8], [0.1, 0.2], [0.1, 0.2] 〉, 〈 0.7, 0.1, 0.1 〉) (x_{2}, y_{2}, 〈 [0.5, 0.7], [0.1, 0.2], [0.2, 0.3] 〉, 〈 0.6, 0.1, 0.2 〉) (x_{3}, y_{2}, 〈 [0.7, 0.8], [0.1, 0.2], [0.1, 0.2] 〉, 〈 0.7, 0.2, 0.1 〉) \\ (x_{1}, y_{2}, 〈 [0.5, 0.7], [0.1, 0.2], [0.3, 0.5] 〉, 〈 0.6, 0.2, 0.4 〉) (x_{2}, y_{2}, 〈 [0.5, 0.6], [0.1, 0.2], [0.1, 0.2] 〉, 〈 0.5, 0.2, 0.2 〉) (x_{3}, y_{2}, 〈 [0.5, 0.6], [0.2, 0.3], [0.3, 0.4] 〉, 〈 0.5, 0.2, 0.3 〉) \\ (x_{1}, y_{2}, 〈 [0.6, 0.8], [0.2, 0.3], [0.4, 0.6] 〉, 〈 0.7, 0.3, 0.5 〉) (x_{2}, y_{2}, 〈 [0.6, 0.7], [0.2, 0.3], [0.4, 0.5] 〉, 〈 0.6, 0.2, 0.4 〉) (x_{3}, y_{2}, 〈 [0.5, 0.7], [0.1, 0.2], [0.2, 0.3] 〉, 〈 0.6, 0.2, 0.3 〉) \end{matrix}] \end{array}$

Then, the proposed two Methods can be applied to this MADM problem in TDUSs and Q-NCS setting.

Method I. The decision steps of the proposed MADM approach based on the Q-NCEWAA operator are depicted below:

Step 1: Calculate the aggregation values of Equation(11) as follows:

c₁ = (< [0.6060, 0.7582], [0.1613, 0.2646], [0, 0.2302]>,<0.6663, 0.1803, 0.1659>),

c₂ = (< [0.6366, 0.7500], [0, 0.1866], [0, 0.2332]>,<0.6624, 0.1424, 0.1543>),

c₃ = (< [0.4776, 0.6354], [0.1275, 0.2305], [0.2018, 0.3255]>,<0.5427, 0.1975, 0.2561>),

c₄ = (< [0.5417, 0.6884], [0.1404, 0.2511], [0.2217, 0.3486]>,<0.6267, 0.1919, 0.2547>).

Step 2: Compute the score values of S(c_k) by using Equation (1) for C_k (k = 1, 2, 3, 4) as follows:

S(c₁) = 0.7790, S(c₂) = 0.8082, S(c₃) = 0.7005, and S(c₄) = 0.7244.

Step 3: Rank the four alternatives as C₂ > C₁ > C₄ > C₃ based on the score values of S(c_k) for C_k (k = 1, 2, 3, 4), and then choose the best one C₂.

Step 4: Compute the aggregation values of Q-NCEs c₂ ,_ij by using Equation (12) corresponding to y_j ∈ Y (j = 1, 2) for C₂:

c_2,1 = (< [0.6000, 0.7000], [0, 0.1587], [0, 0.2289]>,<0.6366, 0.1587, 0.2080>),

c_2,2 = (< [0.6443, 0.7711], [0.1000, 0.2000], [0.1260, 0.2289]>,<0.6698, 0.1260, 0.1260>).

Step 5: Compute the score values of S(c_2,j):

S(c_2,1) = 0.7877 and S(c_2,2) = 0.7997.

Thus, select y₂ as the best city.

Method II. The decision steps of the proposed MADM approach based on the Q-NCEWGA operator are depicted below:

Step 1: Calculate the aggregation values of Equation(14) as follows:

c₁ = (< [0.5792, 0.7175], [0.1702, 0.2704], [0.1319, 0.2602]>,<0.6435, 0.1857, 0.1869>),

c₂ = (< [0.6257, 0.7434], [0.0905, 0.1905], [0.1534, 0.2540]>,<0.6571, 0.1525, 0.1800>),

c₃ = (< [0.4739, 0.6266], [0.1363, 0.2365], [0.2241, 0.3537]>,<0.5378, 0.2055, 0.2809>),

c₄ = (< [0.5368, 0.6723], [0.1505, 0.2621], [0.2690, 0.4036]>,<0.6226, 0.2201, 0.3197>).

Step 2: Compute the score values of S(c_k) by using Equation (1) for C_k (k = 1, 2, 3, 4) as follows:

S(c₁) = 0.7505, S(c₂) = 0.7775, S(c₃) = 0.6877, and S(c₄) = 0.6968.

Step 3: Rank the four alternatives as C₂ > C₁ > C₄ > C₃ based on the score values of S(c_k) for C_k (k = 1, 2, 3, 4), and then choose the best one C₂.

Step 4: Compute the aggregation values of Q-NCEs c₂ ,_ij by using Equation (15) corresponding to y_j ∈ Y (j = 1, 2) for C₂:

c_2,1 = (< [0.6000, 0.7000], [0.0678, 0.1680], [0.1757, 0.2770]>,<0.6316, 0.1680, 0.2388>),

c_2,2 = (< [0.6257, 0.7652], [0.1000, 0.2000], [0.1347, 0.2348]>,<0.6649, 0.1347, 0.1347>).

Step 5: Compute the score values of S(c_2,j):

S(c_2,1) = 0.7551 and S(c_2,2) = 0.7927.

Thus, select y₂ as the best city.

From the decision results of Method I and Method II, we see that two kinds of ranking orders and decision results are identical, which show the effectiveness of the proposed MADM approaches in TDUSs and Q-NCS setting. Then, there exist different focal points between the Q-NCEWAA operator and the Q-NCEWGA operator. The Q-NCEWAA operator emphasizes group’s major points, while the Q-NCEWGA operator emphasizes personal major points. Hence, decision makers can select one of two aggregation operators depending on their preference or actual requirements in the decision making process.

5.2 Comparative analysis

However, existing cubic neutrosophic MADM approaches [1 , 21] cannot handle the decision making problems in TDUSs and Q-NCS setting; while the proposed Q-NCS MADM methods can carry out the MADM problems both in unique universal set (a special case of TDUSs) [1 , 21] and in TDUSs. If the alternative C_k (k = 1, 2, 3, 4) is evaluated in the unique universal set X = {x₁, x₂, x₃} in this city y₁ for the comparative convenience, the Q-NCS MADM problem is reduced to the NCS MADM problem [8]. Thus, the above decision-making matrix of Q-NCSs is reduced to the following decision-making matrix of NCSs: $C = \begin{matrix} C_{1} \\ C_{2} \\ C_{3} \\ C_{4} \end{matrix} [\begin{matrix} (x_{1}, 〈 [0.6, 0.7], [0.1, 0.2], [0.0, 0.1] 〉, 〈 0.6, 0.2, 0.1 〉) (x_{2}, 〈 [0.5, 0.6], [0.2, 0.3], [0.1, 0.2] 〉, 〈 0.6, 0.2, 0.1 〉) (x_{3}, 〈 [0.5, 0.6], [0.2, 0.3], [0.1, 0.2] 〉, 〈 0.6, 0.2, 0.2 〉) \\ (x_{1}, 〈 [0.6, 0.7], [0.1, 0.2], [0.2, 0.3] 〉, 〈 0.6, 0.2, 0.3 〉) (x_{2}, 〈 [0.6, 0.7], [0.0, 0.1], [0.0, 0.1] 〉, 〈 0.6, 0.1, 0.1 〉) (x_{3}, 〈 [0.6, 0.7], [0.1, 0.2], [0.3, 0.4] 〉, 〈 0.7, 0.2, 0.3 〉) \\ (x_{1}, 〈 [0.5, 0.7], [0.1, 0.2], [0.2, 0.3] 〉, 〈 0.6, 0.2, 0.3 〉) (x_{2}, 〈 [0.4, 0.5], [0.1, 0.2], [0.1, 0.2] 〉, 〈 0.5, 0.1, 0.1 〉) (x_{3}, 〈 [0.4, 0.6], [0.2, 0.3], [0.2, 0.3] 〉, 〈 0.5, 0.3, 0.2 〉) \\ (x_{1}, 〈 [0.5, 0.6], [0.1, 0.2], [0.1, 0.2] 〉, 〈 0.6, 0.1, 0.1 〉) (x_{2}, 〈 [0.5, 0.6], [0.2, 0.4], [0.3, 0.4] 〉, 〈 0.6, 0.4, 0.3 〉) (x_{3}, 〈 [0.5, 0.7], [0.1, 0.2], [0.1, 0.2] 〉, 〈 0.6, 0.1, 0.1 〉) \end{matrix}]$

Based on the NCS MADM method in [8], we use the following NCEWAA or NCEWGA operator [8]: $\begin{array}{l} c_{k} = N C E W A A (c_{k 1}, c_{k 2}, \dots, c_{k n}) \\ = (〈 [1 - \prod_{i = 1}^{n} {(1 - T_{k i}^{L})}^{w_{i}}, 1 - \prod_{i = 1}^{n} {(1 - T_{k i}^{U})}^{w_{i}}], [\prod_{i = 1}^{n} {(U_{k i}^{L})}^{w_{i}}, \prod_{i = 1}^{n} {(U_{k i}^{U})}^{w_{i}}], [\prod_{i = 1}^{n} {(V_{k i}^{L})}^{w_{i}}, \prod_{i = 1}^{n} {(V_{k i}^{U})}^{w_{i}}] 〉, 〈 1 - \prod_{i = 1}^{n} {(1 - t_{k i}^{})}^{w_{i}}, \prod_{i = 1}^{n} {(u_{k i}^{})}^{w_{i}}, \prod_{i = 1}^{n} {(v_{k i}^{})}^{w_{i}} 〉,) . \end{array}$ (17)

or $\begin{array}{l} c_{k} = N C E W G A (c_{k 1}, c_{k 2}, \dots, c_{k n}) \\ = (\begin{array}{l} 〈 [\prod_{i = 1}^{n} {(T_{k i}^{L})}^{w_{i}}, \prod_{i = 1}^{n} {(T_{k i}^{U})}^{w_{i}}], [1 - \prod_{i = 1}^{n} {(1 - U_{k i}^{L})}^{w_{i}}, 1 - 1 - \prod_{i = 1}^{n} {(1 - U_{k i}^{U})}^{w_{i}}], [1 - \prod_{i = 1}^{n} {(1 - V_{k i}^{L})}^{w_{i}}, 1 - 1 - \prod_{i = 1}^{n} {(1 - V_{k i}^{U})}^{w_{i}}] 〉, \\ 〈 \prod_{i = 1}^{n} {(t_{k i}^{})}^{w_{i}}, 1 - \prod_{i = 1}^{n} {(1 - u_{k i}^{})}^{w_{i}}, 1 - \prod_{i = 1}^{n} {(1 - v_{k i}^{})}^{w_{i}} 〉 \end{array}) . \end{array}$ (18)

Thus, the aggregation values of Equation (17) are given as follows: c₁ = (< [0.5427, 0.6435], [0.1516, 0.2551], [0, 0.1516]>,<0.6, 0.2, 0.1275>), c₂ = (< [0.6, 0.7], [0, 0.1682], [0, 0.2521]>,<0.6383, 0.1682, 0.228>), c₃ = (< [0.4422, 0.623], [0.1275, 0.2305], [0.1682, 0.2711]>,<0.5427, 0.1938, 0.1978>), c₄ = (< [0.5, 0.6383], [0.1189, 0.2378], [0.1316, 0.2378]>,<0.6, 0.1414, 0.1316>).

Or the aggregation values of Equation (18) are given as follows:

c₁ = (< [0.5378, 0.6382], [0.1614, 0.2616], [0.0613, 0.1614]>,<0.6, 0.2, 0.1363>), c₂ = (< [0.6, 0.7], [0.076, 0.1761], [0.1927, 0.2938]>,<0.6333, 0.1761, 0.2546>), c₃ = (< [0.4373, 0.6097], [0.1363, 0.2365], [0.1761, 0.2762]>,<0.5378, 0.2137, 0.2189>), c₄ = (< [0.5, 0.6333], [0.1261, 0.2555], [0.1548, 0.2555]>,<0.6, 0.1868, 0.1548>).

For the convenient comparison, all the score values and ranking orders are shown inTable 1.

Regarding the decision results ofTable 1, both the proposed Q-NCS MADM method using the Q-NCEWAA and Q-NCEWAA operators and the existing NCS MADM method using the NCEWAA operator [8] indicate the same ranking order and best alternative, which are different from the existing NCS MADM method using the NCEWGA operator [8]. However, the existing NCS MADM method [8] cannot select the best alternative in the two cities and only is a special case of the proposed Q-NCS MADM method. Hence, the proposed Q-NCS MADM method is superior to the existing NCS MADM method [8] and shows its advantage in TDUSs.

Table 1

The decision results of the proposed Q-NCS MADM method and existing NCS MADM method [8]

Method		Score value	Ranking order	The best alternative
The proposed Q-NCS MADM method	Q-NCEWAA operator	S(c₁) = 0.7790,	C₂ > C₁ > C₄ > C₃	C₂ in the best city y₂
		S(c₂) = 0.8082,
		S(c₃) = 0.7005,
		S(c₄) = 0.7244
	Q-NCEWGA operator	S(c₁) = 0.7505,	C₂ > C₁ > C₄ > C₃	C₂ in the best city y₂
		S(c₂) = 0.7775,
		S(c₃) = 0.6877,
		S(c₄) = 0.6968
Existing NCS MADM method [8]	NCEWAA operator	S(c₁) = 0.7644,	C₂ > C₁ > C₄ > C₃	C ₂
		S(c₂) = 0.7803,
		S(c₃) = 0.7142,
		S(c₄) = 0.7613
	NCEWGA operator	S(c₁) = 0.7548,	C₁ > C₂ > C₄ > C₃	C ₁
		S(c₂) = 0.7472,
		S(c₃) = 0.7027,
		S(c₄) = 0.7448

Furthermore, the Q-neutrosophic soft (expert) MADM approach presented in [13 , 18] also cannot coup with the MADM problems with Q-NCS information. Hence, the proposed Q-NCS MADM methods indicate the advantage in Q-NCS setting and a new way for Q-neutrosophic cubic MADM problems in TDUSs and Q-NCS setting.

6 Conclusions

This study proposed the Q-NCS concept for the first time to depict the hybrid information of both an interval NS and a single-valued NS in TDUSs, and then the Q-NCEWAAand Q-NCEWGA operators to aggregate Q-NCE information. Next, Q-neutrosophic cubic MADM approaches based on the Q-NCEWAA and Q-NCEWGA operators were developed to handle Q-neutrosophic cubic MADM problems in TDUSs and Q-NCS setting. Eventually, the developed MADM approaches were applied to an illustrative example in TDUSs and Q-NCS setting. The decision results show the applicability and effectiveness of the established Q-neutrosophic cubic MADM approaches in TDUSs and Q-NCS setting. Then, the main advantages in this study are that Q-NCS can depict both single-valued and interval neutrosophic information simultaneously in TDUSs and the developed Q-neutrosophic cubic decision making method can handle such a decision making problem with single-valued and interval neutrosophic information simultaneously in TDUSs. However, the first study will be further extended to medical diagnosis, pattern recognition, and clustering analysis in TDUSs and Q-NCS setting.

Funding

This study was supported by the National Natural Science Foundation of China (Nos. 61703280, 11975157).

Conflicts of interest

The authors declare no conflict of interest.

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