New Multigranulation Neutrosophic Rough Set with Applications

Abstract

After the neutrosophic set (NS) was proposed, NS was used in many uncertainty problems. The single-valued neutrosophic set (SVNS) is a special case of NS that can be used to solve real-word problems. This paper mainly studies multigranulation neutrosophic rough sets (MNRSs) and their applications in multi-attribute group decision-making. Firstly, the existing definition of neutrosophic rough set (we call it type-I neutrosophic rough set (NRS_I) in this paper) is analyzed, and then the definition of type-II neutrosophic rough set (NRS_II), which is similar to NRS_I, is given and its properties are studied. Secondly, a type-III neutrosophic rough set (NRS_III) is proposed and its differences from NRS_I and NRS_II are provided. Thirdly, single granulation NRSs are extended to multigranulation NRSs, and the type-I multigranulation neutrosophic rough set (MNRS_I) is studied. The type-II multigranulation neutrosophic rough set (MNRS_II) and type-III multigranulation neutrosophic rough set (MNRS_III) are proposed and their different properties are outlined. We found that the three kinds of MNRSs generate tcorresponding NRSs when all the NRs are the same. Finally, MNRS_III in two universes is proposed and an algorithm for decision-making based on MNRS_III is provided. A car ranking example is studied to explain the application of the proposed model.

1. Introduction

Many theories have been applied to solve problems with imprecision and uncertainty. Fuzzy set (FS) theories [1,2,3] use the degree of membership to solve the fuzziness. Rough set (RS) theories [4,5,6,7] deal with uncertainty by lower and upper approximation (LUA). Soft set theories [8,9,10] deal with uncertainty by using a parametrized set. However, all these theories have their own restrictions. Smarandache proposed the concept of the neutrosophic set (NS) [11], which was a generalization of the intuitionistic fuzzy set (IFS). To address real-world uncertainty problems, Wang et al. proposed the single-valued neutrosophic set (SVNS) [12]. Many theories about neutrosophic sets were studied and extended single-valued neutrosophic set [13,14,15]. Zhang et al. [16] analyzed two kinds of inclusion relations of the NS and then proposed the type-3 inclusion relation of NS. The combinations of the FS and RS are popular and produce many interesting results [17]. Broumi and Smarandache [18] combined the RS and NS, then produced a rough NS and studied its qualities. Yang et al. [19] combined the SVNS and RS, then produced the SVNRS (single-valued neutrosophic rough set) and studied its qualities.

From the view point of granular computing, the RS uses upper and lower approximations to solve uncertainty problems, shown by single granularity. However, with the complexity of real-word problems, we often encounter multiple relationship concepts. Qian and Liang [20] proposed a multigranularity rough set (MGRS). Many scholars have generalized MGRS and acquired some interesting consequences [21,22,23,24,25,26]. Zhang et al. [27] proposed non-dual MGRSs and investigated their qualities.

Few articles have been published about the combination of NSs and multigranulation rough sets. In this paper, we study three kinds of neutrosophic rough sets (NRSs) and multigranulation neutrosophic rough sets (MNRSs) that are based on three kinds of inclusion relationships of NS and corresponding union and intersection relationships [11,12,16]. Their different properties are discussed. We found that MNRSs degenerate to corresponding NRSs when the NRs are the same. Yang et al. [19] defined the NRS_I and considered its properties. Bo et al. [28] proposed MNRS_I and discussed its properties. In this paper, we study NRS_II and MNRS_II. We also study NRS_III and MNRS_III, which are based on a type-3 inclusion relationship and corresponding union and intersection relationships. Finally, we use MNRS_III on two universes to solve multi-attribute group decision-making (MAGDM) problems.

The structure of this article is as follows: In Section 2, some basic notions and operations of NRS_I and NRS_II are introduced. In Section 3, the definition of NRS_III is proposed and its qualities are investigated, and the differences between NRS_I, NRS_II, and NRS_III are illustrated using an example. In Section 4, MNRS_I and MNRS_II are discussed. In Section 5, MNRS_III is proposed and its differences from MNRS_I and MNRS_II are studied. In Section 6, MNRS_III on two universes is proposed and an application to solve the MAGDM problem is outlined. Finally, Section 7 provides our conclusions and outlook.

2. Preliminary

In this chapter, we look back at several basic concepts of type-I NRS, then propose the definition and properties of type-II NRS.

Definition 1.

[12] A single valued neutrosophic set A in X is denoted by:

$$A=\left\{\left(x,T_A\left(x\right),I_A\left(x\right),F_A\left(x\right)\right)|x\in X\right\},$$

(1)

where TA(x), IA(x), FA(x) ∈ [0, 1] for each point x in X and satisfies the condition 0 ≤ T_A(x) + I_A(x) + F_A(x) ≤ 3.

For convenience, “SVNS” is abbreviated to “NS” later. Here, NS(X) denotes the set of all SVNS in X.

Definition 2.

[29] A neutrosophic relation (NR) is a neutrosophic fuzzy subset of X × Y, that is, ∀x ∈ X, y ∈ Y,

R(x, y) = (T_R, I_R, F_R),

(2)

where T_R: X × Y → [0, 1], I_R: X × Y → [0, 1], and F_R: X × Y → [0, 1] and satisfies 0 ≤ T_R + I_R + F_R ≤ 3. NR(X × Y) denotes all the NRs in X × Y.

Definition 3.

[19] Suppose (U, R) is a neutrosophic approximation space (NAS). ∀A ∈ NS(U), the LUA of A, denoted by $\underset{\_}{R}\left(A\right)$ and $\overline{R}\left(A\right)$, is defined as: ∀x ∈ U,

$$\underset{\_}{R}\left(A\right)=\underset{y\in U}{\cap }\left(R^c\left(x,y\right)\cup A\left(y\right)\right),\overline{R}\left(A\right)=\underset{y\in U}{\cup }\left(R\left(x,y\right)\cap A\left(y\right)\right).$$

(3)

The pair$\left(\underset{\_}{R}\left(A\right),\overline{R}\left(A\right)\right)$is called the SVNRS of A. In this paper, we called it type-I neutrosophic rough set (NRS_I). Because the definition of NRS_I is based on the type-1 operator of NS, the definition can be written as:

$$\underset{\_}{NR{S}_I}\left(A\right)=\underset{y\in U}{{\cap }_1}\left(R^c\left(x,y\right){\cup }_{1}A\left(y\right)\right),\overline{NR{S}_I}\left(A\right)=\underset{y\in U}{{\cup }_1}\left(R\left(x,y\right){\cap }_{1}A\left(y\right)\right).$$

(4)

Proposition 1.

[19] Suppose (U, R) is an NAS. ∀A, B ∈ NS(U), we have:

(1) 

If A ⊆₁B, then$\underset{\_}{NR{S}_I}\left(A\right){\subseteq }_1\underset{\_}{NR{S}_I}\left(B\right)$and$\overline{NR{S}_I}\left(A\right){\subseteq }_1\overline{NR{S}_I}\left(B\right)$.

(2) 

$\underset{\_}{NR{S}_I}\left(A{\cap }_{1}B\right)=\underset{\_}{NR{S}_I}\left(A\right){\cap }_1\underset{\_}{NR{S}_I}\left(B\right)$,$\overline{NR{S}_I}\left(A{\cup }_{1}B\right)=\overline{NR{S}_I}\left(A\right){\cup }_1\overline{NR{S}_I}\left(B\right)$.

(3) 

$\underset{\_}{NR{S}_I}\left(A\right){\cup }_1\underset{\_}{NR{S}_I}\left(B\right){\subseteq }_1\underset{\_}{NR{S}_I}\left(A{\cup }_{1}B\right)$,$\overline{NR{S}_I}\left(A{\cap }_{1}B\right){\subseteq }_1\overline{NR{S}_I}\left(A\right){\cap }_1\overline{NR{S}_I}\left(B\right)$.

According to the NRS_I, we can get the definition and properties of NRS_II, which is based on the type-2 operator of NS.

Definition 4.

Suppose (U, R) is an NAS. ∀A ∈ NS(U), the type-II LUA of A, is defined as:

$$\underset{\_}{NR{S}_{II}}\left(A\right)=\underset{y\in U}{{\cap }_2}\left(R^c\left(x,y\right){\cup }_{2}A\left(y\right)\right),\overline{NR{S}_{II}}\left(A\right)=\underset{y\in U}{{\cup }_2}\left(R\left(x,y\right){\cap }_{2}A\left(y\right)\right)$$

(5)

The pair$\left(\underset{\_}{NR{S}_{II}}\left(A\right),\overline{NR{S}_{II}}\left(A\right)\right)$is called NRS_II of A.

Proposition 2.

Suppose (U, R) is an NAS. ∀A, B ∈ NS(U), we have:

(1) 

If A ⊆₂ B, then$\underset{\_}{NR{S}_{II}}\left(A\right){\subseteq }_2\underset{\_}{NR{S}_{II}}\left(B\right)$,$\overline{NR{S}_{II}}\left(A\right){\subseteq }_2\overline{NR{S}_{II}}\left(B\right)$.

(2) 

$\underset{\_}{NR{S}_{II}}\left(A{\cap }_{2}B\right)=\underset{\_}{NR{S}_{II}}\left(A\right){\cap }_2\underset{\_}{NR{S}_{II}}\left(B\right)$,$\overline{NR{S}_{II}}\left(A{\cup }_{2}B\right)=\overline{NR{S}_{II}}\left(A\right){\cup }_2\overline{NR{S}_{II}}\left(B\right)$.

(3) 

$\underset{\_}{NR{S}_{II}}\left(A\right){\cup }_2\underset{\_}{NR{S}_{II}}\left(B\right){\subseteq }_2\underset{\_}{NR{S}_{II}}\left(A{\cup }_{2}B\right)$,$\overline{NR{S}_{II}}\left(A{\cap }_{2}B\right){\subseteq }_2\overline{NR{S}_{II}}\left(A\right){\cap }_2\overline{NR{S}_{II}}\left(B\right)$.

Definition 5.

[22] Suppose A, B are two NSs, then the Hamming distance between A and B is defined as:

$$d_N\left(A,B\right)={\displaystyle \sum _{i=1}^n\left\{\left|T_A\left(x_i\right)-T_B\left(x_i\right)\right|+\left|I_A\left(x_i\right)-I_B\left(x_i\right)\right|+\left|F_A\left(x_i\right)-F_B\left(x_i\right)\right|\right\}}.$$

(6)

3. Type-III NRS

In this chapter, we introduce a new NRS, type-III NRS (NRS_III). We provide the differences between the three kinds of NRSs. The properties of NRS_III are also given.

Definition 6.

Suppose (U, R) is an NAS. ∀A ∈ NS(U), the type-III LUA of A, is defined as:

$$\underset{\_}{NR{S}_{III}}\left(A\right)=\underset{y\in U}{{\cap }_3}\left(R^c\left(x,y\right){\cup }_{3}A\left(y\right)\right),\overline{NR{S}_{III}}\left(A\right)=\underset{y\in U}{{\cup }_3}\left(R\left(x,y\right){\cap }_{3}A\left(y\right)\right).$$

The pair$\left(\underset{\_}{NR{S}_{III}}\left(A\right),\overline{NR{S}_{III}}\left(A\right)\right)$is called NRS_III of A.

Proposition 3.

Suppose (U, R) is an NAS. ∀A, B ∈ NS(U), we have:

(1) 

If A ⊆₃B, then$\underset{\_}{NR{S}_{III}}\left(A\right){\subseteq }_3\underset{\_}{NR{S}_{III}}\left(B\right)$,$\overline{NR{S}_{III}}\left(A\right){\subseteq }_3\overline{NR{S}_{III}}\left(B\right)$.

(2) 

$\underset{\_}{NR{S}_{III}}\left(A{\cap }_{3}B\right){\subseteq }_3\underset{\_}{NR{S}_{III}}\left(A\right){\cap }_3\underset{\_}{NR{S}_{III}}\left(B\right)$,$\underset{\_}{NR{S}_{III}}\left(A\right){\cup }_3\underset{\_}{NR{S}_{III}}\left(B\right){\subseteq }_3\underset{\_}{NR{S}_{III}}\left(A{\cup }_{3}B\right)$.

(3) 

$\overline{NR{S}_{III}}\left(A{\cap }_{3}B\right){\subseteq }_3\overline{NR{S}_{III}}\left(A\right){\cap }_3\overline{NR{S}_{III}}\left(B\right)$,$\overline{NR{S}_{III}}\left(A\right){\cup }_3\overline{NR{S}_{III}}\left(B\right){\subseteq }_3\overline{NR{S}_{III}}\left(A{\cup }_{3}B\right)$.

Proof. 

(1) Assume A ⊆₃ B,

Case 1: If T_A(x) < T_B(x), F_A(x) ≥ F_B(x), then:

$$T_{\underset{\_}{NR{S}_{III}}\left(A\right)}\left(x\right)=\underset{y\in U}{\wedge }\left[F_R\left(x,y\right)\vee T_A\left(y\right)\right]\le \underset{y\in U}{\wedge }\left[F_R\left(x,y\right)\vee T_B\left(y\right)\right]=T_{\underset{\_}{NR{S}_{III}}\left(B\right)}\left(x\right)$$

$$F_{\underset{\_}{NR{S}_{III}}\left(A\right)}\left(x\right)=\underset{y\in U}{\vee }\left[T_R\left(x,y\right)\wedge F_A\left(y\right)\right]\ge \underset{y\in U}{\vee }\left[T_R\left(x,y\right)\wedge F_B\left(y\right)\right]=F_{\underset{\_}{NR{S}_{III}}\left(B\right)}\left(x\right).$$

Hence,

$$\underset{\_}{NR{S}_{III}}\left(A\right){\subseteq }_3\underset{\_}{NR{S}_{III}}\left(B\right).$$

Case 2: If T_A(x) = T_B(x), F_A(x) > F_B(x), then:

$$T_{\underset{\_}{NR{S}_{III}}\left(A\right)}\left(x\right)=\underset{y\in U}{\wedge }\left[F_R\left(x,y\right)\vee T_A\left(y\right)\right]=\underset{y\in U}{\wedge }\left[F_R\left(x,y\right)\vee T_B\left(y\right)\right]=T_{\underset{\_}{NR{S}_{III}}\left(B\right)}\left(x\right)$$

$$F_{\underset{\_}{NR{S}_{III}}\left(A\right)}\left(x\right)=\underset{y\in U}{\vee }\left[T_R\left(x,y\right)\wedge F_A\left(y\right)\right]\ge \underset{y\in U}{\vee }\left[T_R\left(x,y\right)\wedge F_B\left(y\right)\right]=F_{\underset{\_}{NR{S}_{III}}\left(B\right)}\left(x\right).$$

Hence,

$$\underset{\_}{NR{S}_{III}}\left(A\right){\subseteq }_3\underset{\_}{NR{S}_{III}}\left(B\right).$$

Case 3: suppose T_A(x) = T_B(x), F_A(x) = F_B(x) and I_A(x) ≤ I_B(x), then:

$$T_{\underset{\_}{NR{S}_{III}}\left(A\right)}\left(x\right)=\underset{y\in U}{\wedge }\left[F_R\left(x,y\right)\vee T_A\left(y\right)\right]=\underset{y\in U}{\wedge }\left[F_R\left(x,y\right)\vee T_B\left(y\right)\right]=T_{\underset{\_}{NR{S}_{III}}\left(B\right)}\left(x\right)$$

$$F_{\underset{\_}{NR{S}_{III}}\left(A\right)}\left(x\right)=\underset{y\in U}{\vee }\left[T_R\left(x,y\right)\wedge F_A\left(y\right)\right]=\underset{y\in U}{\vee }\left[T_R\left(x,y\right)\wedge F_B\left(y\right)\right]=F_{\underset{\_}{NR{S}_{III}}\left(B\right)}\left(x\right)$$

$$I_{\underset{\_}{NR{S}_{III}}\left(A\right)}\left(x\right)=\{\begin{array}{l}{I}_A\left(y_j\right),R^c\left(x,y_j\right){\subseteq }_{3}A\left(y_j\right){\subseteq }_{3}A\left(y_k\right),y_k,y_j\in U\\ I_{R^c}\left(x,y_j\right),A\left(y_j\right){\subseteq }_3{R}^c\left(x,y_j\right)\\ 1,else\end{array}$$

$$I_{{\underset{\_}{MNR{S}_{III}}}^o\left(B\right)}\left(x\right)=\{\begin{array}{l}{I}_B\left(y_j\right),R_i{}^c\left(x,y_j\right){\subseteq }_{3}B\left(y_j\right){\subseteq }_{3}B\left(y_k\right),y_k,y_j\in U\\ I_{R_i{}^c}\left(x,y_j\right),B\left(y_j\right){\subseteq }_3{R}_i{}^c\left(x,y_j\right)\\ 1,else\end{array}.$$

Hence, $I_{\underset{\_}{NR{S}_{III}}\left(A\right)}\left(x\right)\le I_{\underset{\_}{NR{S}_{III}}\left(B\right)}\left(x\right)$. So $\underset{\_}{NR{S}_{III}}\left(A\right){\subseteq }_3\underset{\_}{NR{S}_{III}}\left(B\right)$.

Summing up the above, if A ⊆₃ B, then $\underset{\_}{NR{S}_{III}}\left(A\right){\subseteq }_3\underset{\_}{NR{S}_{III}}\left(B\right)$.

Similarly, we can get $\overline{NR{S}_{III}}\left(A\right){\subseteq }_3\overline{NR{S}_{III}}\left(B\right)$.

(2) According the Definition 6, we have:

$$\begin{array}{ll}\underset{\_}{NR{S}_{III}}\left(A{\cap }_{3}B\right)& =\underset{y\in U}{{\cap }_3}\left[R^c\left(x,y\right){\cup }_3\left(A{\cap }_{3}B\right)\left(y\right)\right]\\ & {\subseteq }_3\left[\underset{y\in U}{{\cap }_3}\left(R^c\left(x,y\right){\cup }_{3}A\left(y\right)\right)\right]{\cap }_3\left[\underset{y\in U}{{\cap }_3}\left(R^c\left(x,y\right){\cup }_{3}B\left(y\right)\right)\right]\\ & =\underset{\_}{NR{S}_{III}}\left(A\right){\cap }_3\underset{\_}{NR{S}_{III}}\left(B\right).\end{array}$$

Similarly,

$$\begin{array}{ll}\underset{\_}{NR{S}_{III}}\left(A\right){\cup }_3\underset{\_}{NR{S}_{III}}\left(B\right)& =\left[\underset{y\in U}{{\cap }_3}\left(R^c\left(x,y\right){\cup }_{3}A\left(y\right)\right)\right]{\cup }_3\left[\underset{y\in U}{{\cap }_3}\left(R^c\left(x,y\right){\cup }_{3}B\left(y\right)\right)\right]\\ & {\subseteq }_3\underset{y\in U}{{\cap }_3}\left[R^c\left(x,y\right){\cup }_3\left(A{\cup }_{3}B\right)\left(y\right)\right]\\ & =\underset{\_}{NR{S}_{III}}\left(A{\cup }_{3}B\right).\end{array}$$

(3) The proof is similar to that of Case 2. □

Example 1.

Define NAS (U, R), where U = {x₁, x₂} and R is given in

Table 1. A neutrosophic relation R.

R	x1	x2
x1	(0.4, 0.6, 0.7)	(0.2, 0.2, 0.9)
x2	(0.7, 0.1, 0.4)	(0.8, 0.8, 0.6)

.

Suppose A is an NS and A = {(x₁, 0.8, 0.2, 0.1), (x₂, 0.4, 0.9, 0.5). Then, by Definitions 3, 4 and 6, we can get:

$$\begin{gathered} \underset{\_}{NR{S}_I}\left(A\right)\left(x_1\right)=(0.8,0.8,0.2),\underset{\_}{NR{S}_I}\left(A\right)\left(x_2\right)=(0.6,0.2,0.5), \\ \overline{NR{S}_I}\left(A\right)\left(x_1\right)=\left(0.4,0.6,0.7\right),\overline{NR{S}_I}\left(A\right)\left(x_2\right)=\left(0.7,0.2,0.4\right), \\ \underset{\_}{NR{S}_{II}}\left(A\right)\left(x_1\right)=\left(0.8,0.4,0.2\right),\underset{\_}{NR{S}_{II}}\left(A\right)\left(x_2\right)=(0.6,0.9,0.5), \\ \overline{NR{S}_{II}}\left(A\right)\left(x_1\right)=\left(0.4,0.2,0.7\right),\overline{NR{S}_{II}}\left(A\right)\left(x_2\right)=\left(0.7,0.8,0.4\right), \\ \underset{\_}{NR{S}_{III}}\left(A\right)\left(x_1\right)=\left(0.8,1,0.2\right),\underset{\_}{NR{S}_{III}}\left(A\right)\left(x_2\right)=(0.6,0,0.5), \\ \overline{NR{S}_{III}}\left(A\right)\left(x_1\right)=\left(0.4,0.6,0.7\right),\overline{NR{S}_{III}}\left(A\right)\left(x_2\right)=\left(0.7,0.1,0.4\right). \end{gathered}$$

4. Type-I and Type-II MNRS

We have proposed a kind of multigranulation neutrosophic rough set [30] (we called it type-I multigranulation neutrosophic rough set in this paper). MNRS_I is based on a type-1 operator of NRs. In this chapter, we define the type-II multigranulation neutrosophic rough set (MNRS_II), which is based on a type-2 operator of NRs.

Definition 7.

[28] Suppose U is a non-empty finite universe, and R_i (1 ≤ i ≤ m) is a binary NR on U. We call the tuple ordered set (U, R_i) the multigranulation neutrosophic approximation space (MNAS).

Definition 8.

[28] Suppose (U, R_i) is an MNAS. ∀A ∈ NS(U), the type-I optimistic LUA of A, represented by ${\underset{\_}{MNR{S}_I}}^o\left(A\right)$ and ${\overline{MNR{S}_I}}^o\left(A\right)$, is defined as:

$${\underset{\_}{MNR{S}_I}}^o\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cup }_1}}\left(\underset{y\in U}{{\cap }_1}\left(R_i{}^c\left(x,y\right){\cup }_{1}A\left(y\right)\right)\right)$$

$${\overline{MNR{S}_I}}^o\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cap }_1}}\left(\underset{y\in U}{{\cup }_1}\left(R_i\left(x,y\right){\cap }_{1}A\left(y\right)\right)\right).$$

Then, A is named a definable NS when ${\underset{\_}{MNR{S}_I}}^o\left(A\right)={\overline{MNR{S}_I}}^o\left(A\right)$. Alternatively, we name the pair $\left({\underset{\_}{MNR{S}_I}}^o\left(A\right),{\overline{MNR{S}_I}}^o\left(A\right)\right)$ an optimistic MNRS_I.

Definition 9.

[30] Suppose (U, R_i) is an MNAS. ∀A ∈ NS(U), the type-I pessimistic LUA of A, represented by ${\underset{\_}{MNR{S}_I}}^p\left(A\right)$ and ${\overline{MNR{S}_I}}^p\left(A\right)$, is defined as:

$${\underset{\_}{MNR{S}_I}}^p\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cap }_1}}\left(\underset{y\in U}{{\cap }_1}\left(R_i{}^c\left(x,y\right){\cup }_{1}A\left(y\right)\right)\right)$$

$${\overline{MNR{S}_I}}^p\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cup }_1}}\left(\underset{y\in U}{{\cup }_1}\left(R_i\left(x,y\right){\cap }_{1}A\left(y\right)\right)\right).$$

Similarly, A is named a definable NS when ${\underset{\_}{MNR{S}_I}}^p\left(A\right)={\overline{MNR{S}_I}}^p\left(A\right)$. Alternatively, we name the pair $\left({\underset{\_}{MNR{S}_I}}^p\left(A\right),{\overline{MNR{S}_I}}^p\left(A\right)\right)$ a pessimistic MNRS_I.

Definition 10.

Suppose (U, R_i) is an MNAS. ∀A ∈ NS(U), the type-II optimistic LUA of A, represented by ${\underset{\_}{MNR{S}_{II}}}^o\left(A\right)$ and ${\overline{MNR{S}_{II}}}^o\left(A\right)$, is defined as:

$${\underset{\_}{MNR{S}_{II}}}^o\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cup }_2}}\left(\underset{y\in U}{{\cap }_2}\left(R_i{}^c\left(x,y\right){\cup }_{2}A\left(y\right)\right)\right)$$

$${\overline{MNR{S}_{II}}}^o\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cap }_2}}\left(\underset{y\in U}{{\cup }_2}\left(R_i\left(x,y\right){\cap }_{2}A\left(y\right)\right)\right).$$

Then, A is named a definable NS when ${\underset{\_}{MNR{S}_{II}}}^o\left(A\right)={\overline{MNR{S}_{II}}}^o\left(A\right)$. Alternatively, we name the pair $\left({\underset{\_}{MNR{S}_{II}}}^o\left(A\right),{\overline{MNR{S}_{II}}}^o\left(A\right)\right)$ an optimistic MNRS_II.

Definition 11.

Suppose (U, R_i) is an MNAS. ∀A ∈ NS(U), the type-II pessimistic LUA of A, represented by ${\underset{\_}{MNR{S}_{II}}}^p\left(A\right)$ and ${\overline{MNR{S}_{II}}}^p\left(A\right)$, is defined as:

$${\underset{\_}{MNR{S}_{II}}}^p\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cap }_2}}\left(\underset{y\in U}{{\cap }_2}\left(R_i{}^c\left(x,y\right){\cup }_{2}A\left(y\right)\right)\right)$$

$${\overline{MNR{S}_{II}}}^p\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cup }_2}}\left(\underset{y\in U}{{\cup }_2}\left(R_i\left(x,y\right){\cap }_{2}A\left(y\right)\right)\right).$$

Similarly, A is named a definable NS when ${\underset{\_}{MNR{S}_{II}}}^p\left(A\right)={\overline{MNR{S}_{II}}}^p\left(A\right)$. Alternatively, we name the pair $\left({\underset{\_}{MNR{S}_{II}}}^p\left(A\right),{\overline{MNR{S}_{II}}}^p\left(A\right)\right)$ a pessimistic MNRS_II.

Proposition 4.

Suppose (U, R_i) is an MNAS. ∀A, B ∈ NS(U), then:

(1) 

${\underset{\_}{MNR{S}_{II}}}^o\left(A\right)=~{\overline{MNR{S}_{II}}}^o\left(~A\right)$,${\underset{\_}{MNR{S}_{II}}}^p\left(A\right)=~{\overline{MNR{S}_{II}}}^p\left(~A\right)$.

(2) 

${\overline{MNR{S}_{II}}}^o\left(A\right)=~{\underset{\_}{MNR{S}_{II}}}^o\left(~A\right)$,${\overline{MNR{S}_{II}}}^p\left(A\right)=~{\underset{\_}{MNR{S}_{II}}}^p\left(~A\right)$.

(3) 

${\underset{\_}{MNR{S}_{II}}}^o\left(A{\cap }_{2}B\right)={\underset{\_}{MNR{S}_{II}}}^o\left(A\right){\cap }_2{\underset{\_}{MNR{S}_{II}}}^o\left(B\right)$,${\underset{\_}{MNR{S}_{II}}}^p\left(A{\cap }_{2}B\right)={\underset{\_}{MNR{S}_{II}}}^p\left(A\right){\cap }_2{\underset{\_}{MNR{S}_{II}}}^p\left(B\right)$.

(4) 

${\overline{MNR{S}_{II}}}^o\left(A{\cup }_{2}B\right)={\overline{MNR{S}_{II}}}^o\left(A\right){\cup }_2{\overline{MNR{S}_{II}}}^o\left(B\right)$,${\overline{MNR{S}_{II}}}^p\left(A{\cup }_{2}B\right)={\overline{MNR{S}_{II}}}^p\left(A\right){\cup }_2{\overline{MNR{S}_{II}}}^p\left(B\right)$.

(5) 

$A{\subseteq }_{2}B\Rightarrow {\underset{\_}{MNR{S}_{II}}}^o\left(A\right){\subseteq }_2{\underset{\_}{MNR{S}_{II}}}^o\left(B\right),{\underset{\_}{MNR{S}_{II}}}^p\left(A\right){\subseteq }_2{\underset{\_}{MNR{S}_{II}}}^p\left(B\right)$.

(6) 

$A{\subseteq }_{2}B\Rightarrow {\overline{MNR{S}_{II}}}^o\left(A\right){\subseteq }_2{\overline{MNR{S}_{II}}}^o\left(B\right),{\overline{MNR{S}_{II}}}^p\left(A\right){\subseteq }_2{\overline{MNR{S}_{II}}}^p\left(B\right)$.

(7) 

${\underset{\_}{MNR{S}_{II}}}^o\left(A\right){\cup }_2{\underset{\_}{MNR{S}_{II}}}^o\left(B\right){\subseteq }_2{\underset{\_}{MNR{S}_{II}}}^o\left(A{\cup }_{2}B\right)$,${\underset{\_}{MNR{S}_{II}}}^p\left(A\right){\cup }_2{\underset{\_}{MNR{S}_{II}}}^p\left(B\right){\subseteq }_2{\underset{\_}{MNR{S}_{II}}}^p\left(A{\cup }_{2}B\right)$.

(8) 

${\overline{MNR{S}_{II}}}^o\left(A{\cap }_{2}B\right){\subseteq }_2{\overline{MNR{S}_{II}}}^o\left(A\right){\cap }_2{\overline{MNR{S}_{II}}}^o\left(B\right)$,${\overline{MNR{S}_{II}}}^p\left(A{\cap }_{2}B\right){\subseteq }_2{\overline{MNR{S}_{II}}}^p\left(A\right){\cap }_2{\overline{MNR{S}_{II}}}^p\left(B\right)$.

Proof. 

Equations (1), (2), (5), and (6) are obviously according to Definitions 10 and 11. Next, we will prove Equations (3), (4), (7), and (8).

(3) By Definition 10,

$$\begin{array}{ll}{\underset{\_}{MNR{S}_{II}}}^o\left(A{\cap }_{2}B\right)\left(x\right)& =\underset{i=1}{\overset{m}{{\cup }_2}}\left(\underset{y\in U}{{\cap }_2}\left(R_i{}^c\left(x,y\right){\cup }_2\left(A{\cap }_{2}B\right)\left(y\right)\right)\right)\\ & =\underset{i=1}{\overset{n}{{\cup }_2}}\left(\underset{y\in U}{{\cap }_2}\left(\left(R_i{}^c\left(x,y\right){\cup }_{2}A\left(y\right)\right)\cap \left(R_i{}^c\left(x,y\right){\cup }_{2}B\left(y\right)\right)\right)\right)\\ & =\left(\underset{i=1}{\overset{n}{{\cup }_2}}\left(\underset{y\in U}{{\cap }_2}\left(R_i{}^c\left(x,y\right){\cup }_{2}A\left(y\right)\right)\right)\right){\cap }_2\left(\underset{i=1}{\overset{n}{{\cup }_2}}\left(\underset{y\in U}{{\cap }_2}\left(R_i{}^c\left(x,y\right){\cup }_{2}B\left(y\right)\right)\right)\right)\\ & ={\underset{\_}{MNR{S}_{II}}}^{o}A\left(x\right){\cap }_2{\underset{\_}{MNR{S}_{II}}}^{o}B\left(y\right).\end{array}$$

Similarly, from Definition 11, we can get the following:

$${\underset{\_}{MNR{S}_{II}}}^p\left(A{\cap }_{2}B\right)={\underset{\_}{MNR{S}_{II}}}^p\left(A\right){\cap }_2{\underset{\_}{MNR{S}_{II}}}^p\left(B\right).$$

(4) The proof is similar to that of Equation (3).

(7) By Definition 10, we can get:

$$\begin{array}{ll}{T}_{{\underset{\_}{MNR{S}_{II}}}^o\left(A{\cup }_{2}B\right)}\left(x\right)& =\underset{i=1}{\overset{m}{\max }}\underset{y\in U}{\min }\left\{\max \left[F_{R_i}\left(x,y\right),\left(\max \left(T_A\left(y\right),T_B\left(y\right)\right)\right)\right]\right\}\\ & =\underset{i=1}{\overset{m}{\max }}\underset{y\in U}{\min }\left\{\max \left[\left(\max \left(F_{R_i}\left(x,y\right),T_A\left(y\right)\right)\right),\left(\max \left(F_{R_i}\left(x,y\right),T_B\left(y\right)\right)\right)\right]\right\}\\ & \ge \max \left\{\left[\underset{i=1}{\overset{m}{\max }}\underset{y\in U}{\min }\left(\max \left(F_{R_i}\left(x,y\right),T_A\left(y\right)\right)\right)\right],\left[\underset{i=1}{\overset{m}{\max }}\underset{y\in U}{\min }\left(\max \left(F_{R_i}\left(x,y\right),T_B\left(y\right)\right)\right)\right]\right\}\\ & =\max \left(T_{{\underset{\_}{MNR{S}_{II}}}^o\left(A\right)}\left(x\right),T_{{\underset{\_}{MNR{S}_{II}}}^o\left(B\right)}\left(x\right)\right).\end{array}$$

$$\begin{gathered} \begin{array}{ll}{I}_{{\underset{\_}{MNR{S}_{II}}}^o\left(A{\cup }_{2}B\right)}\left(x\right)& =\underset{i=1}{\overset{m}{\max }}\underset{y\in U}{\min }\left\{\max \left[\left(1-I_{R_i}\left(x,y\right)\right),\left(\max \left(I_A\left(y\right),I_B\left(y\right)\right)\right)\right]\right\}\\ & =\underset{i=1}{\overset{m}{\max }}\underset{y\in U}{\min }\left\{\max \left[\left(\max \left(\left(1-I_{R_i}\left(x,y\right)\right),I_A\left(y\right)\right)\right),\left(\max \left(\left(1-I_{R_i}\left(x,y\right)\right),I_B\left(y\right)\right)\right)\right]\right\}\\ & \ge \max \left\{\left[\underset{i=1}{\overset{m}{\max }}\underset{y\in U}{\min }\left(\max \left(\left(1-I_{R_i}\left(x,y\right)\right),I_A\left(y\right)\right)\right)\right],\left[\underset{i=1}{\overset{m}{\max }}\underset{y\in U}{\min }\left(\max \left(\left(1-I_{R_i}\left(x,y\right)\right),I_B\left(y\right)\right)\right)\right]\right\}\\ & =\max \left(I_{{\underset{\_}{MNR{S}_{II}}}^o\left(A\right)}\left(x\right),I_{{\underset{\_}{MNR{S}_{II}}}^o\left(B\right)}\left(x\right)\right).\end{array} \\ \begin{array}{ll}{F}_{{\underset{\_}{MNR{S}_{II}}}^o\left(A{\cup }_{2}B\right)}\left(x\right)& =\underset{i=1}{\overset{m}{\min }}\underset{y\in U}{\max }\left\{\min \left[T_{R_i}\left(x,y\right),\left(\min \left(F_A\left(y\right),F_B\left(y\right)\right)\right)\right]\right\}\\ & =\underset{i=1}{\overset{m}{\min }}\underset{y\in U}{\max }\left\{\min \left[\min \left(T_{R_i}\left(x,y\right),F_A\left(y\right)\right)\right],\left[\min \left(T_{R_i}\left(x,y\right),F_B\left(y\right)\right)\right]\right\}\\ & \le \min \left\{\left[\underset{i=1}{\overset{m}{\min }}\underset{y\in U}{\max }\left(\min \left(T_{R_i}\left(x,y\right),F_A\left(y\right)\right)\right)\right],\left[\underset{i=1}{\overset{m}{\min }}\underset{y\in U}{\max }\left(\min \left(T_{R_i}\left(x,y\right),F_B\left(y\right)\right)\right)\right]\right\}\\ & =\min \left(F_{{\underset{\_}{MNR{S}_{II}}}^o\left(A\right)}\left(x\right),F_{{\underset{\_}{MNR{S}_{II}}}^o\left(B\right)}\left(x\right)\right).\end{array} \end{gathered}$$

Hence, ${\underset{\_}{MNR{S}_{II}}}^o\left(A\right){\cup }_2{\underset{\_}{MNR{S}_{II}}}^o\left(B\right){\subseteq }_2{\underset{\_}{MNR{S}_{II}}}^o\left(A{\cup }_{2}B\right)$.

Additionally, according to Definition 11, we can get ${\underset{\_}{MNR{S}_{II}}}^p\left(A\right){\cup }_2{\underset{\_}{MNR{S}_{II}}}^p\left(B\right){\subseteq }_2{\underset{\_}{MNR{S}_{II}}}^p\left(A{\cup }_{2}B\right)$.

(8) The proof is similar to that of Equation (7). □

Remark 1.

Note that if the NRs are the same one, then the optimistic (pessimistic) MNRS_II degenerates into NRS_II in Section 2.

5. Type-III MNRS

In this chapter, MNRS_III, which is based on a type-3 inclusion relation and corresponding union and intersection relations, is proposed and their characterizations are provided.

Definition 12.

Suppose (U, R_i) is an MNAS. ∀A ∈ NS(U), the type-III optimistic LUA of A, represented by ${\underset{\_}{MNR{S}_{III}}}^o\left(A\right)$ and ${\overline{MNR{S}_{III}}}^o\left(A\right)$, is defined as:

$${\underset{\_}{MNR{S}_{III}}}^o\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cup }_3}}\left(\underset{y\in U}{{\cap }_3}\left(R_i{}^c\left(x,y\right){\cup }_{3}A\left(y\right)\right)\right)$$

$${\overline{MNR{S}_{III}}}^o\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cap }_3}}\left(\underset{y\in U}{{\cup }_3}\left(R_i\left(x,y\right){\cap }_{3}A\left(y\right)\right)\right).$$

Then, A is named a definable NS when ${\underset{\_}{MNR{S}_{III}}}^o\left(A\right)={\overline{MNR{S}_{III}}}^o\left(A\right)$. Alternatively, we name the pair $\left({\underset{\_}{MNR{S}_{III}}}^o\left(A\right),{\overline{MNR{S}_{III}}}^o\left(A\right)\right)$ an optimistic MNRS_III.

Definition 13.

Suppose (U, R_i) is an MNAS. ∀A ∈ NS(U), the type-III pessimistic LUA of A, represented by ${\underset{\_}{MNR{S}_{III}}}^p\left(A\right)$ and ${\overline{MNR{S}_{III}}}^p\left(A\right)$, is defined as:

$${\underset{\_}{MNR{S}_{III}}}^p\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cap }_3}}\left(\underset{y\in U}{{\cap }_3}\left(R_i{}^c\left(x,y\right){\cup }_{3}A\left(y\right)\right)\right)$$

$${\overline{MNR{S}_{III}}}^p\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cup }_3}}\left(\underset{y\in U}{{\cup }_3}\left(R_i\left(x,y\right){\cap }_{3}A\left(y\right)\right)\right).$$

Similarly, A is named a definable NS when ${\underset{\_}{MNR{S}_{III}}}^p\left(A\right)={\overline{MNR{S}_{III}}}^p\left(A\right)$. Alternatively, we name the pair $\left({\underset{\_}{MNR{S}_{III}}}^p\left(A\right),{\overline{MNR{S}_{III}}}^p\left(A\right)\right)$ a pessimistic MNRS_III.

Proposition 5.

Suppose (U, R_i) is an MNAS. ∀A, B ∈ NS(U), then:

(1) 

${\underset{\_}{MNR{S}_{III}}}^o\left(A\right)=\sim {\overline{MNR{S}_{III}}}^o\left(\sim A\right)$,${\underset{\_}{MNR{S}_{III}}}^p\left(A\right)=\sim {\overline{MNR{S}_{III}}}^p\left(\sim A\right)$.

(2) 

${\overline{MNR{S}_{III}}}^o\left(A\right)=\sim {\underset{\_}{MNR{S}_{III}}}^o\left(\sim A\right)$,${\overline{MNR{S}_{III}}}^p\left(A\right)=\sim {\underset{\_}{MNR{S}_{III}}}^p\left(\sim A\right)$.

(3) 

$A{\subseteq }_{3}B\Rightarrow {\underset{\_}{MNR{S}_{III}}}^o\left(A\right){\subseteq }_3{\underset{\_}{MNR{S}_{III}}}^o\left(B\right),{\underset{\_}{MNR{S}_{III}}}^p\left(A\right){\subseteq }_3{\underset{\_}{MNR{S}_{III}}}^p\left(B\right)$.

(4) 

$A{\subseteq }_{3}B\Rightarrow {\overline{MNR{S}_{III}}}^o\left(A\right){\subseteq }_3{\overline{MNR{S}_{III}}}^o\left(B\right),{\overline{MNR{S}_{III}}}^p\left(A\right){\subseteq }_3{\overline{MNR{S}_{III}}}^p\left(B\right)$.

(5) 

${\underset{\_}{MNR{S}_{III}}}^o\left(A{\cap }_{3}B\right){\subseteq }_3{\underset{\_}{MNR{S}_{III}}}^o\left(A\right){\cap }_3{\underset{\_}{MNR{S}_{III}}}^o\left(B\right)$,${\underset{\_}{MNR{S}_{III}}}^p\left(A{\cap }_{3}B\right){\subseteq }_3{\underset{\_}{MNR{S}_{III}}}^p\left(A\right){\cap }_3{\underset{\_}{MNR{S}_{III}}}^p\left(B\right)$.

(6) 

${\overline{MNR{S}_{III}}}^o\left(A\right){\cup }_3{\overline{MNR{S}_{III}}}^o\left(B\right){\subseteq }_3{\overline{MNR{S}_{III}}}^o\left(A{\cup }_{3}B\right)$,${\overline{MNR{S}_{III}}}^p\left(A\right){\cup }_3{\overline{MNR{S}_{III}}}^p\left(B\right){\subseteq }_3{\overline{MNR{S}_{III}}}^p\left(A{\cup }_{3}B\right)$.

(7) 

${\underset{\_}{MNR{S}_{III}}}^o\left(A\right){\cup }_3{\underset{\_}{MNR{S}_{III}}}^o\left(B\right){\subseteq }_3{\underset{\_}{MNR{S}_{III}}}^o\left(A{\cup }_{3}B\right)$,${\underset{\_}{MNR{S}_{III}}}^p\left(A\right){\cup }_3{\underset{\_}{MNR{S}_{III}}}^p\left(B\right){\subseteq }_3{\underset{\_}{MNR{S}_{III}}}^p\left(A{\cup }_{3}B\right)$.

(8) 

${\overline{MNR{S}_{III}}}^o\left(A{\cap }_{3}B\right){\subseteq }_3{\overline{MNR{S}_{III}}}^o\left(A\right){\cap }_3{\overline{MNR{S}_{III}}}^o\left(B\right)$,${\overline{MNR{S}_{III}}}^p\left(A{\cap }_{3}B\right){\subseteq }_3{\overline{MNR{S}_{III}}}^p\left(A\right){\cap }_3{\overline{MNR{S}_{III}}}^p\left(B\right)$.

Proof. 

Equations (1) and (2) can be directly derived from Definitions 12 and 13. We only provide the proof of Equations (3)–(8).

(3) Suppose A ⊆₃ B, then:

Case 1: If T_A(x) < T_B(x), F_A(x) ≥ F_B(x), then:

$$T_{{\underset{\_}{MNR{S}_{III}}}^o\left(A\right)}\left(x\right)=\underset{i=1}{\overset{m}{\vee }}\underset{y\in U}{\wedge }\left[F_{R_i}\left(x,y\right)\vee T_A\left(y\right)\right]\le \underset{i=1}{\overset{m}{\vee }}\underset{y\in U}{\wedge }\left[F_{R_i}\left(x,y\right)\vee T_B\left(y\right)\right]=T_{{\underset{\_}{MNR{S}_{III}}}^o\left(B\right)}\left(x\right)$$

$$F_{{\underset{\_}{MNR{S}_{III}}}^o\left(A\right)}\left(x\right)=\underset{i=1}{\overset{m}{\wedge }}\underset{y\in U}{\vee }\left[T_{R_i}\left(x,y\right)\wedge F_A\left(y\right)\right]\ge \underset{i=1}{\overset{m}{\wedge }}\underset{y\in U}{\vee }\left[T_{R_i}\left(x,y\right)\wedge F_B\left(y\right)\right]=F_{{\underset{\_}{MNR{S}_{III}}}^o\left(B\right)}\left(x\right).$$

Hence, ${\underset{\_}{MNR{S}_{III}}}^o\left(A\right){\subseteq }_3{\underset{\_}{MNR{S}_{III}}}^o\left(B\right)$.

Case 2: If T_A(x) = T_B(x), F_A(x) > F_B(x), then:

$$T_{{\underset{\_}{MNR{S}_{III}}}^o\left(A\right)}\left(x\right)=\underset{i=1}{\overset{m}{\vee }}\underset{y\in U}{\wedge }\left[F_{R_i}\left(x,y\right)\vee T_A\left(y\right)\right]=\underset{i=1}{\overset{m}{\vee }}\underset{y\in U}{\wedge }\left[F_{R_i}\left(x,y\right)\vee T_B\left(y\right)\right]=T_{{\underset{\_}{MNR{S}_{III}}}^o\left(B\right)}\left(x\right)$$

$$F_{{\underset{\_}{MNR{S}_{III}}}^o\left(A\right)}\left(x\right)=\underset{i=1}{\overset{m}{\wedge }}\underset{y\in U}{\vee }\left[T_{R_i}\left(x,y\right)\wedge F_A\left(y\right)\right]\ge \underset{i=1}{\overset{m}{\wedge }}\underset{y\in U}{\vee }\left[T_{R_i}\left(x,y\right)\wedge F_B\left(y\right)\right]=F_{{\underset{\_}{MNR{S}_{III}}}^o\left(B\right)}\left(x\right).$$

Hence, ${\underset{\_}{MNR{S}_{III}}}^o\left(A\right){\subseteq }_3{\underset{\_}{MNR{S}_{III}}}^o\left(B\right)$.

Case 3: suppose T_A(x) = T_B(x), F_A(x) = F_B(x) and I_A(x) ≤ I_B(x), then:

$$T_{{\underset{\_}{MNR{S}_{III}}}^o\left(A\right)}\left(x\right)=\underset{i=1}{\overset{m}{\vee }}\underset{y\in U}{\wedge }\left[F_{R_i}\left(x,y\right)\vee T_A\left(y\right)\right]=\underset{i=1}{\overset{m}{\vee }}\underset{y\in U}{\wedge }\left[F_{R_i}\left(x,y\right)\vee T_B\left(y\right)\right]=T_{{\underset{\_}{MNR{S}_{III}}}^o\left(B\right)}\left(x\right)$$

$$F_{{\underset{\_}{MNR{S}_{III}}}^o\left(A\right)}\left(x\right)=\underset{i=1}{\overset{m}{\wedge }}\underset{y\in U}{\vee }\left[T_{R_i}\left(x,y\right)\wedge F_A\left(y\right)\right]\ge \underset{i=1}{\overset{m}{\wedge }}\underset{y\in U}{\vee }\left[T_{R_i}\left(x,y\right)\wedge F_B\left(y\right)\right]=F_{{\underset{\_}{MNR{S}_{III}}}^o\left(B\right)}\left(x\right)$$

$$I_{{\underset{\_}{MNR{S}_{III}}}^o\left(A\right)}\left(x\right)=\{\begin{array}{l}{I}_A\left(y_j\right),R_i{}^c\left(x,y_j\right){\subseteq }_{3}A\left(y_j\right){\subseteq }_{3}A\left(y_k\right),y_k,y_j\in U\\ I_{R_i{}^c}\left(x,y_j\right),A\left(y_j\right){\subseteq }_3{R}_i{}^c\left(x,y_j\right)\\ 0,else\end{array}$$

$$I_{{\underset{\_}{MNR{S}_{III}}}^o\left(B\right)}\left(x\right)=\{\begin{array}{l}{I}_B\left(y_j\right),R_i{}^c\left(x,y_j\right){\subseteq }_{3}B\left(y_j\right){\subseteq }_{3}B\left(y_k\right),y_k,y_j\in U\\ I_{R_i{}^c}\left(x,y_j\right),B\left(y_j\right){\subseteq }_3{R}_i{}^c\left(x,y_j\right)\\ 0,else\end{array}.$$

Hence, $I_{{\underset{\_}{MNR{S}_{III}}}^o\left(A\right)}\left(x\right)\le I_{{\underset{\_}{MNR{S}_{III}}}^o\left(B\right)}\left(x\right)$. So, ${\underset{\_}{MNR{S}_{III}}}^o\left(A\right){\subseteq }_3{\underset{\_}{MNR{S}_{III}}}^o\left(B\right)$.

Summing up the above, if A ⊆₃ B, then ${\underset{\_}{MNR{S}_{III}}}^o\left(A\right){\subseteq }_3{\underset{\_}{MNR{S}_{III}}}^o\left(B\right)$.

Similarly, we can get ${\underset{\_}{MNR{S}_{III}}}^p\left(A\right){\subseteq }_3{\underset{\_}{MNR{S}_{III}}}^p\left(B\right)$.

(4) The proof is similar to that of Equation (3).

(5) From Definition 12, we have:

$$\begin{array}{ll}{\underset{\_}{MNR{S}_{III}}}^o\left(A{\cap }_{3}B\right)& =\underset{i=1}{\overset{m}{{\cup }_3}}\left(\underset{y\in U}{{\cap }_3}\left(R_i{}^c\left(x,y\right){\cup }_3\left(A\left(y\right){\cap }_{3}B\left(y\right)\right)\right)\right)\\ & {\subseteq }_3\underset{i=1}{\overset{m}{{\cup }_3}}\left(\underset{y\in U}{{\cap }_3}\left(\left(R_i{}^c\left(x,y\right){\cup }_{3}A\left(y\right)\right){\cap }_3\left(R_i{}^c\left(x,y\right){\cup }_{3}B\left(y\right)\right)\right)\right)\\ & {\subseteq }_3\left(\underset{i=1}{\overset{m}{{\cup }_3}}\left(\underset{y\in U}{{\cap }_3}\left(R_i{}^c\left(x,y\right){\cup }_{3}A\left(y\right)\right)\right)\right){\cap }_3\left(\underset{i=1}{\overset{m}{{\cup }_3}}\left(\underset{y\in U}{{\cap }_3}\left(R_i{}^c\left(x,y\right){\cup }_{3}B\left(y\right)\right)\right)\right)\\ & ={\underset{\_}{MNR{S}_{III}}}^o\left(A\right){\cap }_3{\underset{\_}{MNR{S}_{III}}}^o\left(B\right).\end{array}$$

Similarly, from Definition 13, we can get ${\underset{\_}{MNR{S}_{III}}}^p\left(A{\cap }_{3}B\right){\subseteq }_3{\underset{\_}{MNR{S}_{III}}}^p\left(A\right){\cap }_3{\underset{\_}{MNR{S}_{III}}}^p\left(B\right).$

(6) From Definition 12, we have:

$$\begin{array}{ll}{\overline{MNR{S}_{III}}}^o\left(A\right){\cup }_3{\overline{MNR{S}_{III}}}^o\left(B\right)& =\left(\underset{i=1}{\overset{m}{{\cap }_3}}\left(\underset{y\in U}{{\cup }_3}\left(R_i\left(x,y\right){\cap }_{3}A\left(y\right)\right)\right)\right){\cup }_3\left(\underset{i=1}{\overset{m}{{\cap }_3}}\left(\underset{y\in U}{{\cup }_3}\left(R_i\left(x,y\right){\cap }_{3}B\left(y\right)\right)\right)\right)\\ & {\subseteq }_3\underset{i=1}{\overset{m}{{\cap }_3}}\left(\underset{y\in U}{{\cup }_3}\left(\left(R_i\left(x,y\right){\cap }_{3}A\left(y\right)\right){\cup }_3\left(R_i\left(x,y\right){\cap }_{3}B\left(y\right)\right)\right)\right)\\ & {\subseteq }_3\underset{i=1}{\overset{m}{{\cap }_3}}\left(\underset{y\in U}{{\cup }_3}\left(R_i\left(x,y\right){\cap }_3\left(A\left(y\right){\cup }_{3}B\left(y\right)\right)\right)\right)\\ & ={\overline{MNR{S}_{III}}}^o\left(A{\cup }_{3}B\right).\end{array}$$

Similarly, from Definition 13, we can get ${\overline{MNR{S}_{III}}}^p\left(A{\cup }_{3}B\right)={\overline{MNR{S}_{III}}}^p\left(A\right){\cup }_3{\overline{MNR{S}_{III}}}^p\left(B\right).$

(7) From Definition 12, we have:

$$\begin{array}{ll}{\underset{\_}{MNR{S}_{III}}}^o\left(A{\cup }_{3}B\right)& =\underset{i=1}{\overset{m}{{\cup }_3}}\left(\underset{y\in U}{{\cap }_3}\left(R_i{}^c\left(x,y\right){\cup }_3\left(A{\cup }_{3}B\right)\left(y\right)\right)\right)\\ & =\underset{i=1}{\overset{m}{{\cup }_3}}\left(\underset{y\in U}{{\cap }_3}\left(R_i{}^c\left(x,y\right){\cup }_3\left(A\left(y\right){\cup }_{3}B\left(y\right)\right)\right)\right)\\ & {\supseteq }_3\underset{i=1}{\overset{m}{{\cup }_3}}\left(\left(\left[\underset{y\in U}{{\cap }_3}\left(R_i{}^c\left(x,y\right){\cup }_{3}A\left(y\right)\right)\right]{\cup }_3\left[\underset{y\in U}{{\cap }_3}\left(R_i{}^c\left(x,y\right){\cup }_{3}B\left(y\right)\right)\right]\right)\right)\\ & =\left(\underset{i=1}{\overset{m}{{\cup }_3}}\left[\underset{y\in U}{{\cap }_3}\left(R_i{}^c\left(x,y\right){\cup }_{3}A\left(y\right)\right)\right]\right){\cup }_3\left(\underset{i=1}{\overset{m}{{\cup }_3}}\left[\underset{y\in U}{{\cap }_3}\left(R_i{}^c\left(x,y\right){\cup }_{3}B\left(y\right)\right)\right]\right)\\ & ={\underset{\_}{MNR{S}_{III}}}^o\left(A\right){\cup }_3{\underset{\_}{MNR{S}_{III}}}^o\left(B\right).\end{array}$$

Hence, ${\underset{\_}{MNR{S}_{III}}}^o\left(A\right){\cup }_3{\underset{\_}{MNR{S}_{III}}}^o\left(B\right){\subseteq }_3{\underset{\_}{MNR{S}_{III}}}^o\left(A{\cup }_{3}B\right)$.

Additionally, from Definition 13, we can get ${\underset{\_}{MNR{S}_{III}}}^p\left(A\right){\cup }_3{\underset{\_}{MNR{S}_{III}}}^p\left(B\right){\subseteq }_3{\underset{\_}{MNR{S}_{III}}}^p\left(A{\cup }_{3}B\right)$.

(8) From Definition 12, we have:

$$\begin{array}{ll}{\overline{MNR{S}_{III}}}^o\left(A{\cap }_{3}B\right)& =\underset{i=1}{\overset{m}{{\cap }_3}}\left(\underset{y\in U}{{\cup }_3}\left(R_i\left(x,y\right){\cap }_3\left(A{\cap }_{3}B\right)\left(y\right)\right)\right)\\ & =\underset{i=1}{\overset{m}{{\cap }_3}}\left(\underset{y\in U}{{\cup }_3}\left(R_i\left(x,y\right){\cap }_3\left(A\left(y\right){\cap }_{3}B\left(y\right)\right)\right)\right)\\ & {\subseteq }_3\underset{i=1}{\overset{m}{{\cap }_3}}\left(\left[\underset{y\in U}{{\cup }_2}\left(R_i\left(x,y\right){\cap }_{3}A\left(y\right)\right)\right]{\cap }_3\left[\underset{y\in U}{{\cup }_3}\left(R_i\left(x,y\right){\cap }_{3}B\left(y\right)\right)\right]\right)\\ & =\left(\underset{i=1}{\overset{m}{{\cap }_3}}\left[\underset{y\in U}{{\cup }_3}\left(R_i\left(x,y\right){\cap }_{3}A\left(y\right)\right)\right]\right){\cap }_3\left(\underset{i=1}{\overset{m}{{\cap }_3}}\left[\underset{y\in U}{{\cup }_3}\left(R_i\left(x,y\right){\cap }_{3}B\left(y\right)\right)\right]\right)\\ & ={\overline{MNR{S}_{III}}}^o\left(A\right){\cap }_3{\overline{MNR{S}_{III}}}^o\left(B\right).\end{array}$$

Hence, ${\overline{MNR{S}_{III}}}^o\left(A{\cap }_{3}B\right){\subseteq }_3{\overline{MNR{S}_{III}}}^o\left(A\right){\cap }_3{\overline{MNR{S}_{III}}}^o\left(B\right)$.

Similarly, from Definition 13, we can get ${\overline{MNR{S}_{III}}}^p\left(A{\cap }_{3}B\right){\subseteq }_3{\overline{MNR{S}_{III}}}^p\left(A\right){\cap }_3{\overline{MNR{S}_{III}}}^p\left(B\right)$. □

Remark 2.

Note that if the NRs are the same one, then the optimistic (pessimistic) MNRS_III degenerates into NRS_III in Section 3.

6. Type-III MNRS in Two Universes with Its Applications

In this chapter, we propose the concept of MNRS_III in two universes and use it to deal with the MAGDM problem.

Definition 14.

[28] Suppose U, V are two non-empty finite universes, and R_i ∈ NS(U × V) (1 ≤ i ≤ m) is a binary NR. We call (U, V, R_i) the MNAS in two universes.

Definition 15.

Suppose (U, V, R_i) is an MNAS in two universes. ∀A ∈ NS(V) and x ∈ U, the type-III optimistic LUA of A in (U, V, R_i), represented by${\underset{\_}{MNR{S}_{III}}}^o\left(A\right)$and${\overline{MNR{S}_{III}}}^o\left(A\right)$, is defined as:

$${\underset{\_}{MNR{S}_{III}}}^o\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cup }_3}}\left(\underset{y\in V}{{\cap }_3}\left(R_i{}^c\left(x,y\right){\cup }_{3}A\left(y\right)\right)\right)$$

$${\overline{MNR{S}_{III}}}^o\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cap }_3}}\left(\underset{y\in V}{{\cup }_3}\left(R_i\left(x,y\right){\cap }_{3}A\left(y\right)\right)\right).$$

Then, A is named a definable NS in two universes when ${\underset{\_}{MNR{S}_{III}}}^o\left(A\right)={\overline{MNR{S}_{III}}}^o\left(A\right)$. Alternatively, we name the pair $\left({\underset{\_}{MNR{S}_{III}}}^o\left(A\right),{\overline{MNR{S}_{III}}}^o\left(A\right)\right)$ an optimistic MNRS_III in two universes.

Definition 16.

Suppose (U, V, R_i) is an MNAS in two universes. ∀A ∈ NS(V) and x ∈ U, the type-III pessimistic LUA of A in (U, V, R_i), denoted by ${\underset{\_}{MNR{S}_{III}}}^p\left(A\right)$ and ${\overline{MNR{S}_{III}}}^p\left(A\right)$, is defined as follows:

$${\underset{\_}{MNR{S}_{III}}}^p\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cap }_3}}\left(\underset{y\in V}{{\cap }_3}\left(R_i{}^c\left(x,y\right){\cup }_{3}A\left(y\right)\right)\right)$$

$${\overline{MNR{S}_{III}}}^p\left(A\right)\left(x\right)=\underset{i=1}{\overset{m}{{\cup }_3}}\left(\underset{y\in V}{{\cup }_3}\left(R_i\left(x,y\right){\cap }_{3}A\left(y\right)\right)\right).$$

Similarly, A is named a definable NS when ${\underset{\_}{MNR{S}_{III}}}^p\left(A\right)={\overline{MNR{S}_{III}}}^p\left(A\right)$. Alternatively, we name the pair $\left({\underset{\_}{MNR{S}_{III}}}^p\left(A\right),{\overline{MNR{S}_{III}}}^p\left(A\right)\right)$ a pessimistic MNRS_III in two universes.

Remark 3.

Note that if the two domains are the same, then the optimistic (pessimistic) MNRS_III in two universes degenerates into the optimistic (pessimistic) MNRS_III in a single universe in Section 5.

The MAGDM problem is becoming more and more generally present in our daily life. MAGDM means to select or rank all the feasible alternatives in various criterions. There are many ways to solve the MAGDM problem, but we use MNRS to solve it in this paper. Next, we give the basic description of the considered MAGDM problem.

For the car-ranking question, suppose U = {x₁, x₂, …, x_n} is the decision set and V = {y₁, y₂, …, y_m} is the criteria set in which x₁ represents “very popular”, x₂ represents “popular”, x₃ represents “less popular”, …, x_n represents “not popular”, y₁ represents the vehicle type”, y₂ represents the size of the space, y₃ represents the ride height, y₄ represents quality, and …, y_m represents length of durability. Then, l selection experts make evaluations about the criteria sets according to their own experiences. Here, the evaluations were shown by NRs. Next, we calculate the degree of popularity for a given car. Therefore, we need to use MGNRS to solve the above problem. For the MAGDM problem under a multigranulation neutrosophic environment, the optimistic lower approximation can be regarded as an optimistic risk decision, and the optimistic upper approximation can be regarded as an optimistic conservative decision. Additionally, the pessimistic lower approximation can be regarded as a pessimistic risk decision and the pessimistic upper approximation can be regarded as a pessimistic conservative decision. According to the distance of neutrosophic sets, we define the difference function d_N(A, B)(x_i) = (1/3)(|T_A(x_i) − T_B(x_i)| + |I_A(x_i) − I_B(x_i)| + |F_A(x_i) − F_B(x_i)|). We used the difference function to represent the distance of optimistic (pessimistic) upper and lower approximation. The smaller the value of the distance is, the better the alternative x_i is, because the risk decision and the conservative decision are close. By comparing the distance value, all alternatives can be ranked and we can choose the optimal alternative. In this paper, we only used three kinds of optimistic upper and lower approximation to decision-making.

Next, we show the process of the above car-ranking question based on MGNRSs over two universes. Let R_l ∈ NR(U × V) be NRs from U to V, where ∀(x_i, y_j) ∈ U × V, R_l(x_i, y_j) denotes the degree of popularity for criteria set y_j (y_j ∈ V). R_l can be obtained according to experts’ experience. Given a car A, according to the unconventional questionnaire (suppose there are three options—“like”, “not like”, and “neutral” to choose for each of the criteria sets, and everyone can choose one or more options), then we can get the popularity of every criterion as described by an NS A in the universe V according to the questionnaire. By use of the following Algorithm 1, we can determine the degree of popularity of the given car A.

Algorithm 1 Decision algorithm

Input Multigranulation neutrosophic decision information systems (U, V, R).

Output The degree of popularity of the given car.

Step 1 Computing three kinds of optimistic multigranulation LUA ${\underset{\_}{MNR{S}_I}}^o\left(A\right)$, ${\overline{MNR{S}_I}}^o\left(A\right)$, ${\underset{\_}{MNR{S}_{II}}}^o\left(A\right)$, ${\overline{MNR{S}_{II}}}^o\left(A\right)$, ${\underset{\_}{MNR{S}_{III}}}^o\left(A\right)$, ${\overline{MNR{S}_{III}}}^o\left(A\right)$.

Step 2 Calculate $d({\underset{\_}{MNR{S}_I}}^o\left(x_i\right),{\overline{MNR{S}_I}}^o\left(x_i\right))$, $d({\underset{\_}{MNR{S}_{II}}}^o\left(x_i\right),{\overline{MNR{S}_{II}}}^o\left(x_i\right))$ and $d({\underset{\_}{MNR{S}_{III}}}^o\left(x_i\right),{\overline{MNR{S}_{III}}}^o\left(x_i\right))$.

Step 3 The best choice is to select xh (which means that the most welcome degree is xh) if $d({\underset{\_}{MNRS}}^o\left(x_h\right),{\overline{MNRS}}^o\left(x_h\right))={\min }_{i\in \left\{1,2,\cdots ,n\right\}}d({\underset{\_}{MNRS}}^o\left(x_i\right),{\overline{MNRS}}^o\left(x_i\right))$.

Step 4 If h has two or more values, then each xh will be the best choice. In this case, the car may have two or more popularities and each xk will be regarded as the most possible popularity; otherwise, we use other methods to make a decision.

Next, we use an example to explain the algorithm.

Let U = {x₁, x₂, x₃, x₄} be the decision set, in which x₁ denotes “very popular”, x₂ denotes “popular”, x₃ denotes “less popular”, and x₄ denotes “not popular”. Let V = {y₁, y₂, y₃, y₄, y₅} be criteria sets, in which y₁ denotes the vehicle type, y₂ denotes the size of the space, y₃ denotes the ride height, y₄ denotes quality, and y₅ denotes length of durability.

Suppose that R₁, R₂, and R₃ are given by three invited experts. They provide their evaluations for all criteria y_j with respect to decision set elements x_i. The evaluation R₁, R₂, and R₃ are NRs between attribute set V and decision evaluation set U., that is., there are R₁, R₂, R₃ ∈ NR(U × V).

Suppose three experts present their judgment (the neutrosophic relation R₁, R₂, and R₃) for the attribute and decision sets in

Table 2. Neutrosophic relation R₁.

R1	y1	y2	y3	y4	y5
x1	(0.8, 0.6, 0.5)	(0.2, 0.3, 0.9)	(0, 0, 1)	(0.7, 0.5, 0.6)	(0, 0, 1)
x2	(0.6, 0.4, 0.6)	(0.9, 0.3, 0.4)	(1, 0, 0)	(0, 0, 1)	(0.3, 0.6, 0.7)
x3	(0.2, 0.5, 0.9)	(0.6, 0.7, 0.5)	(0.8, 0.7, 0.8)	(0, 0, 1)	(1, 0, 0)
x4	(0.6, 0.4, 0.7)	(0, 0, 1)	(0, 0, 1)	(0.9, 0.8, 0.1)	(0, 0, 1)

,

Table 3. Neutrosophic relation R₂.

R2	y1	y2	y3	y4	y5
x1	(0.9, 0.3, 0.6)	(0, 0, 1)	(0, 0, 1)	(0.5, 0.6, 0.5)	(0.2, 0.3, 0.9)
x2	(0.3, 0.7, 0.8)	(0.7, 0.5, 0.6)	(0.9, 0.1, 0.1)	(0, 0, 1)	(0.4, 0.5, 0.8)
x3	(0.1, 0.6, 0.8)	(0.3, 0.6, 0.5)	(0.7, 0.3, 0.6)	(0, 0, 1)	(1, 0, 0)
x4	(0.7, 0.5, 0.6)	(0, 0, 1)	(0, 0, 1)	(1, 0, 0)	(0, 0, 1)

and

Table 4. Neutrosophic relation R₃.

R3	y1	y2	y3	y4	y5
x1	(0.6, 0.9, 0.4)	(0.1, 0.1, 0.8)	(0.1, 0, 0.9)	(0.8, 0.4, 0.8)	(0, 0, 1)
x2	(0.5, 0.6, 0.6)	(0.6, 0.2, 0.7)	(1, 0, 0)	(0, 0, 1)	(0, 0, 1)
x3	(0.1, 0.4, 0.7)	(0.2, 0.2, 0.7)	(0.5, 0.7, 0.6)	(0, 0, 1)	(0.9, 0.1, 0.2)
x4	(0.6, 0.3, 0.4)	(0, 0, 1)	(0, 0, 1)	(0.7, 0.5, 0.4)	(0, 0, 1)

:

Suppose A is a car and each criterion in V is as follows:

A = {(y₁, 0.9, 0.2, 0.2), (y₂, 0.2, 0.7, 0.8), (y₃, 0, 1, 0.3), (y₄, 0.7, 0.6, 0.3), (y₅, 0.1, 0.8, 0.9)}.

Then, we can calculate the three kinds of optimistic LUAs of A as follow:

$$\begin{gathered} {\underset{\_}{MNR{S}_I}}^o\left(A\right)\left(x_1\right)=\left(0.8,1,0.3\right),{\underset{\_}{MNR{S}_I}}^o\left(A\right)\left(x_2\right)=\left(0.1,0.9,0.6\right), \\ {\underset{\_}{MNR{S}_I}}^o\left(A\right)\left(x_3\right)=\left(0.2,0.8,0.9\right),{\underset{\_}{MNR{S}_I}}^o\left(A\right)\left(x_4\right)=\left(0.7,1,0.3\right), \\ {\overline{MNR{S}_I}}^o\left(A\right)\left(x_1\right)=\left(0.7,0.6,0.5\right),{\overline{MNR{S}_I}}^o\left(A\right)\left(x_2\right)=\left(0.3,0.6,0.3\right), \\ {\overline{MNR{S}_I}}^o\left(A\right)\left(x_3\right)=\left(0.2,0.6,0.8\right),{\overline{MNR{S}_I}}^o\left(A\right)\left(x_4\right)=\left(0.7,0.5,0.4\right), \\ {\underset{\_}{MNR{S}_{II}}}^o\left(A\right)\left(x_1\right)=\left(0.8,0.6,0.3\right),{\underset{\_}{MNR{S}_{II}}}^o\left(A\right)\left(x_2\right)=\left(0.1,0.6,0.6\right), \\ {\underset{\_}{MNR{S}_{II}}}^o\left(A\right)\left(x_3\right)=\left(0.2,0.6,0.9\right),{\underset{\_}{MNR{S}_{II}}}^o\left(A\right)\left(x_4\right)=\left(0.7,0.6,0.3\right), \\ {\overline{MNR{S}_{II}}}^o\left(A\right)\left(x_1\right)=\left(0.7,0.4,0.5\right),{\overline{MNR{S}_{II}}}^o\left(A\right)\left(x_2\right)=\left(0.3,0.2,0.3\right), \\ {\overline{MNR{S}_{II}}}^o\left(A\right)\left(x_3\right)=\left(0.2,0.6,0.8\right),{\overline{MNR{S}_{II}}}^o\left(A\right)\left(x_4\right)=\left(0.7,0.2,0.4\right), \\ {\underset{\_}{MNR{S}_{III}}}^o\left(A\right)\left(x_1\right)=\left(0.8,0,0.3\right),{\underset{\_}{MNR{S}_{III}}}^o\left(A\right)\left(x_2\right)=\left(0.1,0,0.6\right), \\ {\underset{\_}{MNR{S}_{III}}}^o\left(A\right)\left(x_3\right)=\left(0.2,0.9,0.9\right),{\underset{\_}{MNR{S}_{III}}}^o\left(A\right)\left(x_4\right)=\left(0.7,0.6,0.3\right), \\ {\overline{MNR{S}_{III}}}^o\left(A\right)\left(x_1\right)=\left(0.7,1,0.5\right),{\overline{MNR{S}_{III}}}^o\left(A\right)\left(x_2\right)=\left(0.3,0,0.3\right), \\ {\overline{MNR{S}_{III}}}^o\left(A\right)\left(x_3\right)=\left(0.2,0.7,0.8\right){\overline{MNR{S}_{III}}}^o\left(A\right)\left(x_4\right)=\left(0.7,0.5,0.4\right). \end{gathered}$$

Therefore, we can get:

• $d({\underset{\_}{MNR{S}_I}}^o\left(x_1\right),{\overline{MNR{S}_I}}^o\left(x_1\right))=0.7/3$, $d({\underset{\_}{MNR{S}_I}}^o\left(x_2\right),{\overline{MNR{S}_I}}^o\left(x_2\right))=0.8/3$,
• $d({\underset{\_}{MNR{S}_I}}^o\left(x_3\right),{\overline{MNR{S}_I}}^o\left(x_3\right))=0.1$, $d({\underset{\_}{MNR{S}_I}}^o\left(x_4\right),{\overline{MNR{S}_I}}^o\left(x_4\right))=0.2$,
• $d({\underset{\_}{MNR{S}_{II}}}^o\left(x_1\right),{\overline{MNR{S}_{II}}}^o\left(x_1\right))=0.5/3$, $d({\underset{\_}{MNR{S}_{II}}}^o\left(x_2\right),{\overline{MNR{S}_{II}}}^o\left(x_2\right))=0.3$,
• $d({\underset{\_}{MNR{S}_{II}}}^o\left(x_3\right),{\overline{MNR{S}_{II}}}^o\left(x_3\right))=0.1/3$, $d({\underset{\_}{MNR{S}_{II}}}^o\left(x_4\right),{\overline{MNR{S}_{II}}}^o\left(x_4\right))=0.5/3$,
• $d({\underset{\_}{MNR{S}_{III}}}^o\left(x_1\right),{\overline{MNR{S}_{III}}}^o\left(x_1\right))=1.3/3$, $d({\underset{\_}{MNR{S}_{III}}}^o\left(x_2\right),{\overline{MNR{S}_{III}}}^o\left(x_2\right))=0.5/3$,
• $d({\underset{\_}{MNR{S}_{III}}}^o\left(x_3\right),{\overline{MNR{S}_{III}}}^o\left(x_3\right))=0.1$, $d({\underset{\_}{MNR{S}_{III}}}^o\left(x_4\right),{\overline{MNR{S}_{III}}}^o\left(x_4\right))=0.2/3$.

Thus, for the type-I and type-II MNRS, the optimistic best choice is to select x₃, that is, this car is less popular; for the type-III MNRS, the optimistic best choice is to select x₄, that is, this car is not popular.

7. Conclusions

NRS and MNRS are extensions of the Pawlak rough set theory. In this paper, we analysed the NRS_I and NRS_II, we proposed model NRS_III, and used an example to outline the differences between the three kinds of NRS. We gave the definition of MNRS_III, which is based on the type-3 operator relation of NS, and considered their properties. Furthermore, we proposed MNRS_III in two universes and we presented an algorithm of the MAGDM problem based on it.

In the future, we will be researching other types of fusions of MGRSs and NSs. We will also study the applications of concepts in this paper to some algebraic systems (for example, pseudo-BCI algebras, neutrosophic triplet groups, see [30,31]).