**2. Preliminaries**

In this section, we will introduce some existing definitions and basic concepts in order to understand this study.

#### *2.1. The NS and INS*

**Definition 1 [7–9].** *Let X be a space of points (objects) with a generic element in X denoted by x. A NS A in X is expressed by a MD TA*(*x*)*, an IMD <sup>I</sup>*(*x*)*, and a NMD FA*(*x*).

*Then a NS A is denoted below*.

$$A = \{ \langle \mathbf{x}, T\_A(\mathbf{x}), I\_A(\mathbf{x}), F\_A(\mathbf{x}) \rangle | \mathbf{x} \in X \}\tag{1}$$

*TA*(*x*), *<sup>I</sup>*(*x*), *and FA*(*x*) *are real standard or non-standard subsets of* ]−0, <sup>1</sup>+[*. That is*

$$[T\_A:X\to]^-0, 1^+[; \ I\_A:X\to]^-0, 1^+[; \ F\_A:X\to]^-0, 1^+[$$

*With the condition* −0 ≤ *TA*(*x*) + *IA*(*x*) + *FA*(*x*) ≤ 3+.

**Definition 2 [10,11].** *Let X be a space of points (objects) with a generic element in X denoted by x. For convenience, the lower and upper ends of T, I, F are expressed as <sup>T</sup>LA*(*x*), *TUA* (*x*), *<sup>I</sup>LA*(*x*), *IUA* (*x*), *<sup>F</sup>LA*(*x*)*, and FUA* (*x*)*. An INS A in X is defined below*.

$$A = \left\{ \mathbf{x}, \left\langle \left[ T\_A^L(\mathbf{x}), T\_A^{\mathrm{LI}}(\mathbf{x}) \right], \left[ I\_A^L(\mathbf{x}), I\_A^{\mathrm{LI}}(\mathbf{x}) \right], \left[ F\_A^L(\mathbf{x}), F\_A^{\mathrm{LI}}(\mathbf{x}) \right] \right\rangle \middle| \mathbf{x} \in X \right\} \tag{2}$$

*For each point x in X, we have that* I*TLA*(*x*), *TUA* (*x*)J ⊆ [0, 1], I*ILA*(*x*), *IUA* (*x*)J ⊆ [0, 1], I*FLA*(*x*), *FUA* (*x*)J ⊆ [0, 1]*, and* 0 ≤ *TUA* (*x*) + *IUA* (*x*) + *FUA* (*x*) ≤ 3.

**Definition 3 [10,11].** *An INS A is contained in the INS B*, *A* ⊆ *B, if and only if <sup>T</sup>LA*(*x*) ≤ *TLB* (*x*), *TUA* (*x*) ≤ *TUB* (*x*), *<sup>I</sup>LA*(*x*) ≥ *<sup>I</sup>LB*(*x*), *IUA* (*x*) ≥ *IUB* (*x*), *<sup>F</sup>LA*(*x*) ≥ *FLB* (*x*)*, and FUA* (*x*) ≥ *FUB* (*x*)*. If A* ⊆ *B and A* ⊇ *B, then A* = *B*.

*2.2. LVs*

**Definition 4 [36,37].** *Let S* = { *si*|*<sup>i</sup>* = 0, 1, . . . , *l*, *l* ∈ *N*∗} *be a LT set (LTS) where N*∗ *is a set of positive integers and si represents LV*.

Because the LTS is convenient and efficient, it is widely used by DMs in decision making. For instance, when we evaluate the production quality, we can set l = 9, then *S* is given below.

*S* = {*<sup>s</sup>*0 = *extremely bad*,*s*<sup>1</sup> = *very bad*,*s*<sup>2</sup> = *bad*,*s*<sup>3</sup> = *slightly bad*,*s*<sup>4</sup> = *f air*,*s*<sup>5</sup> = *slightly good*, *s*6 = *good*,*s*<sup>7</sup> = *very good*,*s*<sup>8</sup> = *extremely good*}

To relieve the loss of linguistic information in operations, Xu [38,39] extended LTS S to continuous LTS *S* = { *sθ* |0 ≤ *θ* ≤ *l*}. About the characteristics of LTS, please refer to References [38–40].

**Definition 5 [13].** *Let sα and sβ be any two LVs in S. The related operations can be defined below.*

$$s\_{\mathfrak{A}} \oplus s\_{\mathfrak{F}} = s\_{a + \mathfrak{F} - \frac{a \cdot \mathfrak{F}}{T}} \tag{3}$$

$$
\lambda s\_{\mathfrak{A}} = s\_{1 - \mathfrak{I} \cdot (1 - \mathfrak{I})^{\lambda}}, \lambda > 0 \tag{4}
$$

$$s\_a \odot s\_\beta = s\_{\frac{a\cdot\beta}{\dagger}} \tag{5}$$

$$s\_{\left(\mathbf{s}\_{\mathbf{d}}\right)}{}^{\lambda} = s\_{\mathbf{l}\cdot\left(\frac{\mathbf{s}}{\mathbf{T}}\right)^{\lambda}} \lambda > 0 \tag{6}$$

## *2.3. MSM Operator*

**Definition 6 [15,32].** *Let xi*(*<sup>i</sup>* = 1, 2, ... , *n*) *be the set of the non-negative real number. An MSM operator of dimension n is a mapping MSM*(*m*) : (*R*+)*<sup>n</sup>* → *R*<sup>+</sup> *and it can be defined below*.

$$MSM^{(m)}(\mathbf{x}\_1, \dots, \mathbf{x}\_n) = \left(\frac{\sum\_{1 \le i\_1 < \dots < i\_m \le n} \prod\_{j=1}^m \mathbf{x}\_{i\_j}}{\mathbb{C}\_n^m}\right)^{\frac{1}{m}} \tag{7}$$

*where* (*<sup>i</sup>*1, *i*2, ... , *im*) *traverses all the m-tuple combination of* (1, 2, ... , *n*) *and Cmn* = *n*! *<sup>m</sup>*!(*<sup>n</sup>*−*<sup>m</sup>*)! *is the binomial coefficient. In addition, xijrefers to ijth element in a particular arrangement.*

There are some properties of the *MSM*(*m*) operator, which are defined below.

(1) Idempotency. If *xi* = *x* for each *i*, and then *MSM*(*m*)(*<sup>x</sup>*, *x*,..., *x*) = *x*;


Furthermore, the *MSM*(*m*) operator would degrade some particular forms when *m* takes some special values, which are shown as follows.

1. When *m* = 1, the *MSM*(*m*) operator would become the average operator.

$$MSM^{(1)}(\mathbf{x}\_1, \mathbf{x}\_2, \dots \mathbf{x}\_n) = \left(\frac{\sum\_{1 \le i\_1 \le n} \mathbf{x} i\_1}{C^1\_n}\right) = \frac{\sum^n \mathbf{x} i}{n} \tag{8}$$

2. When *m* = 2, the *MSM*(*m*) operator would become the following BM operator (*p* = *q* = 1).

$$\begin{split} MSM^{(2)}(\mathbf{x}\_{1},\ldots,\mathbf{x}\_{n}) &= \left(\frac{\sum\_{1\le i\_{1}$$

3. When *m* = *n*, the *MSM*(*m*) operator would become the geometric mean.

$$MSM^{(n)}(\mathbf{x}\_1, \dots, \mathbf{x}\_n) = \left(\prod\_{j=1}^n \mathbf{x}\_j\right)^{\frac{1}{n}} \tag{10}$$

**Definition 7 [15].** *Let xi*(*<sup>i</sup>* = 1, 2, ... , *n*) *be the set of non-negative real numbers and p*1, *p*2, ... , *pm* ≥ 0*. A generalized MSM operator of dimension n is a mapping GMSM*(*<sup>m</sup>*,*p*1,*p*2,...,*pm*) : (*R*+)*<sup>n</sup>* → *R*<sup>+</sup> *and it is defined below.*

$$GMSM^{(m, p\_1, p\_2, \dots, p\_m)}(\mathbf{x}\_1, \dots, \mathbf{x}\_n) = (\frac{\sum\_{1 \le i\_1 < \dots < i\_m \le n} \prod\_{j=1}^m x\_{i\_j}^{p\_j}}{\mathbf{C}\_n^m})^{\frac{1}{p\_1 + p\_2 + \dots + p\_m}} \tag{11}$$

*where* (*<sup>i</sup>*1, *i*2, ... , *im*) *traverses all the m-tuple combination of* (1, 2, ... , *n*) *and Cmn* = *n*! *<sup>m</sup>*!(*<sup>n</sup>*−*<sup>m</sup>*)! *is the binomial coefficient*.

There are some properties of the *GMSM*(*<sup>m</sup>*,*P*1,*P*2,...,*Pm*) operator below.


In addition, the *GMSM*(*<sup>m</sup>*,*P*1,*P*2,...,*Pm*) operator would degrade to some particular forms when *m* takes some special values, which are shown below.

1. When *m* = 1, we have the formula below.

$$GMSM^{(1, P\_1)}(\mathbf{x}\_1, \mathbf{x}\_2, \dots \mathbf{x}\_n) = \left(\frac{\sum\_{1 \le i\_1 \le n} x\_{i\_1}^{p\_1}}{\mathbf{C}\_n^1}\right)^{\frac{1}{p\_1}} = \left(\frac{\sum\_{i=1}^n x\_{i}^{p\_1}}{n}\right) \tag{12}$$

1

2. When *m* = 2, the *GMSM*(*<sup>m</sup>*,*P*1,*P*2,...,*Pm*) operator would become the following BM operator.

$$\begin{split}GMSM(^{2,p\_1,p\_2}(\mathbf{x}\_1,\ldots,\mathbf{x}\_n) &= \left(\frac{^{\sum\_{1\le i\_1$$

3. When *m* = *n*, the *MSM*(*m*) operator would become the following formula.

$$GMSM^{(n,p\_1,p\_2,\ldots,p\_n)}(\mathbf{x}\_1,\ldots,\mathbf{x}\_n) = \left(\prod\_{j=1}^n \mathbf{x}\_j^{p\_j}\right)^{\frac{1}{p\_1+p\_2+\ldots+p\_n}}\tag{14}$$

4. When *p*1 = *p*2 = ... = *pm* = 1, the *GMSM*(*<sup>m</sup>*,*P*1,*P*2,...,*Pm*) operator would degenerate to the *MSM* operator and the parameter is *m* below.

$$GMSM^{(m,1,1,\ldots,1)}(\mathbf{x}\_1,\ldots,\mathbf{x}\_n) = \left(\frac{\sum\_{1 \le i\_1 < \ldots < i\_m \le n} \prod\_{j=1}^m \mathbf{x}\_{ij}^1}{\mathbb{C}\_n^m}\right)^{\frac{1}{m}} = MSM^{(m)}(\mathbf{x}\_1,\ldots,\mathbf{x}\_n). \tag{15}$$

#### **3. INLNs and Operations**

**Definition 8 [16,41].** *Let X be a finite universal set. An INLS in X is defined by the equation below.*

$$A = \left\{ \mathbf{x}, \left< \mathbf{s}\_{\theta(\mathbf{x})\prime} \left[ T\_A(\mathbf{x}), I\_A(\mathbf{x}), F\_A(\mathbf{x}) \right] \right> \middle| \mathbf{x} \in X \right\} \tag{16}$$

*where <sup>s</sup>θ*(*x*) ∈ *S*, *TA*(*x*) = I*TLA*(*x*), *TUA* (*x*)J ⊆ [0, 1], *IA*(*x*) = I*ILA*(*x*), *IUA* (*x*)J ⊆ [0, 1], *FA*(*x*) = I*FLA*(*x*), *FUA* (*x*)J ⊆ [0, 1] *represent the MD, the IMD, and the NMD of the element x in X to the LV <sup>s</sup>θ*(*x*)*, respectively, with the condition* 0 ≤ *TUA* (*x*) + *IUA* (*x*) + *FUA* (*x*) ≤ 3 *for any x* ∈ *X*.

Then the seven tuple *<sup>s</sup>θ*(*x*),(I*TLA*(*x*), *TUA* (*x*)J, I*ILA*(*x*), *IUA* (*x*)J, I*FLA*(*x*), *FUA* (*x*)J)in *A* is called an INLN. For convenience, an INLN can be represented as *a* = *<sup>s</sup>θ*(*a*),(I*T<sup>L</sup>*(*a*), *<sup>T</sup><sup>U</sup>*(*a*)J, I*I<sup>L</sup>*(*a*), *<sup>I</sup><sup>U</sup>*(*a*)J, I*F<sup>L</sup>*(*a*), *<sup>F</sup><sup>U</sup>*(*a*)J).

Then we introduced the operational rules of operators of INLNs. **Definition 9 [16,37,42].** *Let a*1 = *<sup>s</sup>θ*(*<sup>a</sup>*1),(I*T<sup>L</sup>*(*<sup>a</sup>*1), *<sup>T</sup><sup>U</sup>*(*<sup>a</sup>*1)J, I*I<sup>L</sup>*(*<sup>a</sup>*1), *<sup>I</sup><sup>U</sup>*(*<sup>a</sup>*1)J, I*F<sup>L</sup>*(*<sup>a</sup>*1), *<sup>F</sup><sup>U</sup>*(*<sup>a</sup>*1)J) *and a*2 = *<sup>s</sup>θ*(*<sup>a</sup>*2),(I*T<sup>L</sup>*(*<sup>a</sup>*2), *<sup>T</sup><sup>U</sup>*(*<sup>a</sup>*2)J, I*I<sup>L</sup>*(*<sup>a</sup>*2), *<sup>I</sup><sup>U</sup>*(*<sup>a</sup>*2)J, I*F<sup>L</sup>*(*<sup>a</sup>*2), *<sup>F</sup><sup>U</sup>*(*<sup>a</sup>*2)J) *be two INLNs and λ* ≥ 0*. Then the operation of the INLNs can be expressed by the equation below*.

$$\begin{aligned} a\_1 \oplus a\_2 &= \left< s\_{\theta(a\_1) + \theta(a\_2)}, \left( \left[ T^L(a\_1) + T^L(a\_2) - T^L(a\_1) \times T^L(a\_2), T^{\mathcal{U}}(a\_1) + T^{\mathcal{U}}(a\_2) - T^{\mathcal{U}}(a\_1) \times T^{\mathcal{U}}(a\_2) \right], \\ \left[ \left[ I^L(a\_1) \times I^L(a\_2), I^{\mathcal{U}}(a\_1) \times I^{\mathcal{U}}(a\_2) \right], \left[ F^L(a\_1) \times F^L(a\_2), F^{\mathcal{U}}(a\_1) \times F^{\mathcal{U}}(a\_2) \right] \right) \right\rangle \end{aligned} \tag{17}$$

$$\begin{aligned} &I^{\mathrm{L}}(a\_{1}) + I^{\mathrm{L}}(a\_{2}) - I^{\mathrm{L}}(a\_{1}) \times I^{\mathrm{L}}(a\_{2}) \Big|\_{}, \Big[ \boldsymbol{F}^{\mathrm{L}}(a\_{1}) + \boldsymbol{F}^{\mathrm{L}}(a\_{2}) - \boldsymbol{F}^{\mathrm{L}}(a\_{1}) \times \boldsymbol{F}^{\mathrm{L}}(a\_{2}),\\ &a\_{1} \odot a\_{2} = \left\langle \boldsymbol{s}\_{\boldsymbol{\theta}(a\_{1}) \times \boldsymbol{\theta}(a\_{2})}, \Big( \left[ \boldsymbol{T}^{\mathrm{L}}(a\_{1}) \times \boldsymbol{T}^{\mathrm{L}}(a\_{2}), \boldsymbol{T}^{\mathrm{L}}(a\_{1}) \times \boldsymbol{T}^{\mathrm{L}}(a\_{2}) \right], \left[ \boldsymbol{I}^{\mathrm{L}}(a\_{1}) + \boldsymbol{I}^{\mathrm{L}}(a\_{2}) - \boldsymbol{I}^{\mathrm{L}}(a\_{1}) \times \boldsymbol{I}^{\mathrm{L}}(a\_{2}), \quad \left( \left[ \boldsymbol{R}^{\mathrm{L}}(a\_{1}) \times \boldsymbol{T}^{\mathrm{L}}(a\_{2}) \right] \right) \right], \\ &\boldsymbol{F}^{\mathrm{L}}(a\_{1}) + \boldsymbol{F}^{\mathrm{L}}(a\_{2}) - \boldsymbol{F}^{\mathrm{L}}(a\_{1}) \times \boldsymbol{F}^{\mathrm{L}}(a\_{2}) \Big] \Big\rangle \end{aligned} \tag{18}$$

$$
\lambda a\_1 = \left\langle s\_{\lambda \times \theta(a\_1)}, \left( \left[ 1 - \left( 1 - T^L(a\_1) \right)^{\lambda}, 1 - \left( 1 - T^{\text{II}}(a\_1) \right)^{\lambda} \right] \right) \left[ \left( I^L(a\_1) \right)^{\lambda}, \left( I^{\text{II}}(a\_1) \right)^{\lambda} \right] \right\rangle \tag{19}
$$

$$
\left[ \left( F^L(a\_1) \right)^{\lambda}, \left( F^{\text{II}}(a\_1) \right)^{\lambda} \right] \rangle \left( \lambda > 0 \right) \tag{10}
$$

$$a\_1^{\lambda} = s\_{\theta^{\lambda}(a\_1)} \left( \left[ \left( T^L(a\_1) \right)^{\lambda}, \left( T^{I\!\!I}(a\_1) \right)^{\lambda} \right] \left[ 1 - \left( 1 - I^L(a\_1) \right)^{\lambda}, 1 - \left( 1 - I^{\!\!I}(a\_1) \right)^{\lambda} \right] \right. \\ \left. \left. \left. 1 - \left( 1 - F^L(a\_1) \right)^{\lambda} \right] \right) \right. \\ \left. \left. \left. 1 - \left( 1 - F^{I\!\!I}(a\_1) \right)^{\lambda} \right] \right) \left( \lambda > 0 \right) \end{aligned} \tag{20}$$

**Example 1.** *Let a*1 = *<sup>s</sup>*3,([0.1, 0.2], [0.2, 0.3], [0.4, 0.5]) *and a*2 = *<sup>s</sup>*4,([0.3, 0.5], [0.3, 0.4], [0.5, 0.6]) *be two INLNs and S* = {*<sup>s</sup>*0 = *very bad*,*s*<sup>1</sup> = *bad*,*s*<sup>2</sup> = *slightly bad*,*s*<sup>3</sup> = *f air*,*s*<sup>4</sup> = *slightly good*, *s*5 = *good*,*s*<sup>6</sup> = *very good*}*, then we have the equations below*.

*a*1 ⊕ *a*2 = *<sup>s</sup>*3+4,([0.1 + 0.3 − 0.1 × 0.3, 0.2 + 0.5 − 0.2 × 0.5], [0.2 × 0.3, 0.3 × 0.4], [0.4 × 0.5, 0.5 × 0.6] = *<sup>s</sup>*7,([0.37, 0.6], [0.06, 0.12], [0.2, 0.3]

$$\begin{array}{l} a\_1 \otimes a\_2 = \langle s\_{3 \times 4}, ([0.1 \times 0.3, 0.2 \times 0.5], [0.2 + 0.3 - 0.2 \times 0.3, 0.3 + 0.4 - 0.3 \times 0.4] \rangle \\ \langle [0.4 + 0.5 - 0.4 \times 0.5, 0.5 + 0.6 - 0.5 \times 0.6] \rangle \rangle \\ = \langle s\_{12}, ([0.03, 0.1], [0.44, 0.58], [0.7, 0.8]) \rangle \end{array}$$

*As seen from the above examples, these results are not reasonable because they exceed the range of LTS. In order to overcome these limitations, we will improve these operations by Definition 10.*

**Definition 10.** *Let a*1 = *<sup>s</sup>θ*(*<sup>a</sup>*1),(I*T<sup>L</sup>*(*<sup>a</sup>*1), *<sup>T</sup><sup>U</sup>*(*<sup>a</sup>*1)J, I*I<sup>L</sup>*(*<sup>a</sup>*1), *<sup>I</sup><sup>U</sup>*(*<sup>a</sup>*1)J, I*F<sup>L</sup>*(*<sup>a</sup>*1), *<sup>F</sup><sup>U</sup>*(*<sup>a</sup>*1)J)*and a*2 = *<sup>s</sup>θ*(*<sup>a</sup>*2),(I*T<sup>L</sup>*(*<sup>a</sup>*2), *<sup>T</sup><sup>U</sup>*(*<sup>a</sup>*2)J, I*I<sup>L</sup>*(*<sup>a</sup>*2), *<sup>I</sup><sup>U</sup>*(*<sup>a</sup>*2)J, I*F<sup>L</sup>*(*<sup>a</sup>*2), *<sup>F</sup><sup>U</sup>*(*<sup>a</sup>*2)J) *be two INLNs and λ* ≥ 0*. Then the operations of the INLNs can be defined by the equations below*.

$$\begin{split} a\_{1} \oplus a\_{2} = \left\langle s\_{\theta(a\_{1}) + \theta(a\_{2}) - \frac{\theta(a\_{1}) \cdot \theta(a\_{2})}{2}}, \left( \left[ T^{L}(a\_{1}) + T^{L}(a\_{2}) - T^{L}(a\_{1}) \times T^{L}(a\_{2}), T^{\mathrm{UI}}(a\_{1}) + T^{\mathrm{UI}}(a\_{2}) - T^{\mathrm{UI}}(a\_{1}) \times T^{\mathrm{II}}(a\_{2}) \right] \right) \right. \\ \left. \left[ I^{L}(a\_{1}) \times I^{L}(a\_{2}), I^{\mathrm{II}}(a\_{1}) \times I^{\mathrm{II}}(a\_{2}) \right], \left[ I^{\mathrm{L}}(a\_{1}) \times I^{\mathrm{L}}(a\_{2}), I^{\mathrm{II}}(a\_{1}) \times I^{\mathrm{II}}(a\_{2}) \right] \right) \end{split} \tag{21}$$

$$a\_1 \otimes a\_2 = \left\langle s\_{\frac{d(a\_1)\times d(a\_2)}{l}}, \left( \left[ T^L(a\_1) \times T^L(a\_2), T^{\text{LI}}(a\_1) \times T^{\text{LI}}(a\_2) \right], \left[ I^L(a\_1) + I^L(a\_2) - I^L(a\_1) \times I^L(a\_2), I^L(a\_1) \right] \right) \right\rangle$$

$$\begin{split} I^{\text{LI}}(a\_1) + I^{\text{LI}}(a\_2) - I^{\text{LI}}(a\_1) \times I^{\text{LI}}(a\_2) \Big|\_{\prime} & \left[ F^L(a\_1) + F^L(a\_2) - F^L(a\_1) \times F^L(a\_2), \left[ \left[ I^L(a\_1) + I^L(a\_2) \right] \right] \right] \\ & \qquad \left[ F^{\text{LI}}(a\_1) + F^{\text{LI}}(a\_2) - F^{\text{LI}}(a\_1) \times F^{\text{LI}}(a\_2) \right] \Big| \end{split} \tag{22}$$

$$\begin{array}{c} \lambda a\_{1} = \left\langle s\_{\left[ -l \cdot \left( 1 - \frac{\vartheta(a\_{1})}{\hat{l}} \right)^{\lambda} \right)}^{} \left[ 1 - \left( 1 - T^{L}(a\_{1}) \right)^{\lambda}, 1 - \left( 1 - T^{lI}(a\_{1}) \right)^{\lambda} \right], \\\ \left[ \left( I^{L}(a\_{1}) \right)^{\lambda}, \left( I^{\hat{l}I}(a\_{1}) \right)^{\lambda} \right], \left[ \left( F^{L}(a\_{1}) \right)^{\lambda}, \left( F^{\hat{l}I}(a\_{1}) \right)^{\lambda} \right] \right) \rangle, (\lambda > 0) \end{array} \tag{23}$$

$$\begin{split} a\_{1}^{\lambda} = \mathop{\mathbf{s}}\_{l \cdot \left(\frac{\vartheta(a\_{1})}{\Lambda}\right)^{\lambda}} \cdot \left( \left[ \left( T^{L}(a\_{1}) \right)^{\lambda}, \left( T^{L}(a\_{1}) \right)^{\lambda} \right], \left[ 1 - \left( 1 - I^{L}(a\_{1}) \right)^{\lambda}, 1 - \left( 1 - I^{L}(a\_{1}) \right)^{\lambda} \right], \\ \quad \left[ 1 - \left( 1 - F^{L}(a\_{1}) \right)^{\lambda}, 1 - \left( 1 - F^{L}(a\_{1}) \right)^{\lambda} \right] \right), \left( \lambda > 0 \right). \end{split} \tag{24}$$

*Based on the operational rules above, the above example is recalculated as follow.*

**Example 2.** *Let a*1 = *<sup>s</sup>*3,([0.1, 0.2], [0.2, 0.3], [0.4, 0.5]) *and a*2 = *<sup>s</sup>*4,([0.3, 0.5], [0.3, 0.4], [0.5, 0.6]) *be two INLNs and S* = {*<sup>s</sup>*0 = *very bad*,*s*<sup>1</sup> = *bad*,*s*<sup>2</sup> = *slightly bad*,*s*<sup>3</sup> = *f air*,*s*<sup>4</sup> = *slightly good*, *s*5 = *good*,*s*<sup>6</sup> = *very good*}*, then we have the equations below*.

*a*1 ⊕ *a*2 = *<sup>s</sup>*3+4− 3×4 6 ,([0.1 + 0.3 − 0.1 × 0.3, 0.2 + 0.5 − 0.2 × 0.5], [0.2 × 0.3, 0.3 × 0.4], [0.4 × 0.5, 0.5 × 0.6] = *<sup>s</sup>*5,([0.37, 0.6], [0.06, 0.12], [0.2, 0.3]

*a*1 ⊗ *a*2 = *s* 3×4 6 ,([ 0.1 × 0.3, 0.2 × 0.5], [0.2 + 0.3 − 0.2 × 0.3, 0.3 + 0.4 − 0.3 × 0.4], [0.4 + 0.5 − 0.4 × 0.5, 0.5 + 0.6 − 0.5 × 0.6]) = *<sup>s</sup>*2,([ 0.03, 0.1], [0.44, 0.58], [0.7, 0.8])

*From the above example, the results are more reasonable than the previous ones. In the following definitions, a new scoring function and a comparison method of INLN are described*.

**Definition 11. [37].** *Let a* = *<sup>s</sup>θ*(*a*),( I *<sup>T</sup><sup>L</sup>*(*a*), *<sup>T</sup><sup>U</sup>*(*a*) J , I *<sup>I</sup><sup>L</sup>*(*a*), *<sup>I</sup><sup>U</sup>*(*a*) J , I *<sup>F</sup><sup>L</sup>*(*a*), *<sup>F</sup><sup>U</sup>*(*a*) J ) *be an INLN. Then the score function of a can be expressed by the equation below*.

$$S(a) = a \cdot \frac{\theta(a)}{6} \left[ 0.5(T^{l\bar{l}}(a) + 1 - F^{L}(a)) + aI^{l\bar{l}}(a) \right] + (1 - a) \cdot \frac{\theta(a)}{6} \left[ 0.5(T^{l}(a) + 1 - F^{l}(a)) + aI^{l}(a) \right] \tag{25}$$

*where the values of α* ∈ [0, 1] *reflect the attitudes of the decision makers*.

**Definition 12. [37].** *Let a and b be two INLNs. Then the INLN comparison method can be expressed by the statements below*.

$$\text{If } S(a) > S(b) \text{, then } a \succ b;\tag{26}$$

$$\text{If } S(a) = S(b) \text{, then } a \sim b;\tag{27}$$

$$\text{If } S(a) < (b), \text{ then } a \prec b;\tag{28}$$

#### **4. Some Interval Neutrosophic Linguistic MSM Operators**

In this section, we will propose *INLMSM* operators and *INLGMSM* operators.

#### *4.1. The INLMSM Operators*

**Definition 13.** *Let ai* = *s θi* ,([ *<sup>T</sup><sup>L</sup>*(*ai*), *<sup>T</sup><sup>U</sup>*(*ai*)], [*I<sup>L</sup>*(*ai*), *<sup>I</sup><sup>U</sup>*(*ai*)], [*F<sup>L</sup>*(*ai*), *<sup>F</sup><sup>U</sup>*(*ai*)]) (*i* = 1, 2, ..., *n*) *be a set of INLNs. Then the INLMSM operator:* Ω*n* → Ω *is shown below*.

$$INLMSM^{(m)}(a\_1, \ldots, a\_n) = \left(\frac{\stackrel{\oplus}{1 \le i\_1 < \ldots < i\_m \le n} \left(\stackrel{m}{\bigotimes} a\_{i\_j}\right)^{\frac{1}{m}}}{\mathbb{C}\_n^m}\right)^{\frac{1}{m}}\tag{29}$$

Ω *is a set of INLNs and m* = 1, 2, ..., *n*.

*According to the operational laws of INLNs in Definition 10, we can get the expression of the INLMSM operator shown below.*

**Theorem 1.** *Let ai* = *s θi* ,([*T<sup>L</sup>*(*ai*), *<sup>T</sup><sup>U</sup>*(*ai*)], [*I<sup>L</sup>*(*ai*), *<sup>I</sup><sup>U</sup>*(*ai*)], [*F<sup>L</sup>*(*ai*), *<sup>F</sup><sup>U</sup>*(*ai*)]) (*i* = 1, 2, ..., *n*) *be a set of INLNs and m* = 1, 2, ..., *n. Then the value aggregated from Definition 13 is still an INLN*.

$$\begin{aligned} & \quad \text{INL} \& \text{MSM}^{(m)} (a\_1, \dots, a\_n) = \\ & \left\langle \begin{aligned} & \quad \text{INL} \& \text{MSM}^{(m)} (a\_1, \dots, a\_n) = \\ & \quad \text{I} \left[ \left( 1 - \prod\_{k=1}^{\text{G}} \left( 1 - \prod\_{j=1}^{\text{G}} \left( 1 - \prod\_{j=1}^{\text{G}} T^{\text{L}}\_{j\_j(k)} \right)^{\frac{1}{\text{G}}} \right)^{\frac{1}{\text{G}}}, \left( 1 - \prod\_{j=1}^{\text{G}} T^{\text{L}}\_{j\_j(k)} \right)^{\frac{1}{\text{G}}} \right)^{\frac{1}{\text{G}}} \right], \\ & \left[ 1 - \left( 1 - \prod\_{k=1}^{\text{G}} \left( 1 - \prod\_{j=1}^{\text{G}} \left( 1 - I^{\text{L}}\_{j\_j(k)} \right) \right)^{\frac{1}{\text{G}}} \right)^{\frac{1}{\text{G}}}, 1 - \left( 1 - \prod\_{k=1}^{\text{G}} \left( 1 - \prod\_{j=1}^{\text{G}} \left( 1 - I^{\text{L}}\_{j\_j(k)} \right) \right)^{\frac{1}{\text{G}}} \right)^{\frac{1}{\text{G}}}, \\ & \left[ 1 - \left( 1 - \prod\_{k=1}^{\text{G}} \left( 1 - \prod\_{j=1}^{\text{G}} \left( 1 - I^{\text{L}}\_{j\_j(k)} \right) \right)^{\frac{1}{\text{G}}} \right)^{\frac{1}{\text{G}}}, 1 - \left( 1 - \prod\_{k=1}^{\text{G}} \left( 1 -$$

*where k* = 1, 2, ...*Cmn* , *aij*(*k*) *is the ijth element of k th permutation*.
