**Proof.**

1 If each *ai* = *x*, then we ge<sup>t</sup> the equation below.

*INLMSM*(*m*)(*<sup>x</sup>*, *x*, ..., *x*) = *sl*·(<sup>1</sup>−∏*<sup>C</sup>mnk*=<sup>1</sup> (<sup>1</sup>−<sup>∏</sup>*mj*=<sup>1</sup> ( *θxl* )) 1*Cmn* ) 1*m* , ⎛⎜⎜⎝⎡⎢⎢⎣⎛⎝1 − ∏*Cmn k*=11 − *m*∏*j*=1 *TLx* 1*Cmn* ⎞⎠ 1*m* , ⎛⎝1 − ∏*Cmn k*=11 − *m*∏*j*=1 *TUx* 1*Cmn* ⎞⎠ 1*m* ⎤⎥⎥⎦ , ⎡⎢⎢⎣1 − ⎛⎝1 − ∏*Cmn k*=11 − *m*∏*<sup>j</sup>*=<sup>1</sup>-<sup>1</sup> − *ILx*. 1*Cmn* ⎞⎠ 1*m* , 1 − ⎛⎝1 − ∏*Cmn k*=11 − *m*∏*<sup>j</sup>*=<sup>1</sup>-<sup>1</sup> − *IUx*. 1*Cmn* ⎞⎠ 1*m* ⎤⎥⎥⎦, ⎡ ⎢⎢⎣1 − ⎛⎝1 − ∏*Cmn k*=11 − *m*∏*<sup>j</sup>*=<sup>1</sup>-<sup>1</sup> − *FLx*. 1*Cmn* ⎞⎠ 1 *m* , 1 − ⎛⎝1 − ∏*Cmn k*=11 − *m*∏*<sup>j</sup>*=<sup>1</sup>-<sup>1</sup> − *FUx*. 1*Cmn* ⎞⎠ 1 *m* ⎤ ⎥⎥⎦ = *<sup>s</sup>θx*,(*Tx*, *Ix*, *Fx*) = *x*.


*m* ∏ *j*=1 *αi* ≤ *m* ∏ *j*=1 *β<sup>i</sup>*, *m* ∏ *j*=1 *<sup>T</sup><sup>L</sup>*(*xi*) ≤ *m* ∏ *j*=1 *<sup>T</sup><sup>L</sup>*(*yi*), *m* ∏ *j*=1 *<sup>T</sup><sup>U</sup>*(*xi*) ≤ *m* ∏ *j*=1 *<sup>T</sup><sup>U</sup>*(*yi*), *m* ∏ *j*=1 *<sup>I</sup><sup>L</sup>*(*xi*) ≥ *m* ∏ *j*=1 *<sup>I</sup><sup>L</sup>*(*yi*),*m* ∏ *j*=1 *<sup>I</sup><sup>U</sup>*(*xi*) ≥ *m* ∏ *j*=1 *<sup>I</sup><sup>U</sup>*(*yi*), *m* ∏ *j*=1 *<sup>F</sup><sup>L</sup>*(*xi*) ≥ *m* ∏ *j*=1 *<sup>F</sup><sup>L</sup>*(*yi*), *m* ∏ *j*=1 *<sup>F</sup><sup>U</sup>*(*xi*) ≥ *m* ∏ *j*=1 *<sup>F</sup><sup>U</sup>*(*yi*) then *l* · ⎛⎜⎝1 − *Cmn* ∏*k*=11 − *m*∏*j*=1 *αil* 1*Cmn* ⎞⎟⎠ 1 *m* ≤ *l* · ⎛⎜⎝1 − *Cmn* ∏*k*=11 − *m*∏*j*=1 *βi l* 1*Cmn* ⎞⎟⎠ 1 *m* , ⎛⎜⎝1 − *Cmn* ∏*k*=11 − *m*∏*j*=1 *<sup>T</sup><sup>L</sup>*(*xi*) 1*Cmn* ⎞⎟⎠ 1 *m* ≤ ⎛⎜⎝1 − *Cmn* ∏*k*=11 − *m*∏*j*=1 *<sup>T</sup><sup>L</sup>*(*yi*) 1*Cmn* ⎞⎟⎠ 1 *m* ,

⎛⎜⎝1 − *Cmn* ∏*k*=11 − *m*∏*j*=1 *<sup>T</sup><sup>U</sup>*(*xi*) 1*Cmn* ⎞⎟⎠ 1 *m* ≤ ⎛⎜⎝1 − *Cmn* ∏*k*=11 − *m*∏*j*=1 *<sup>T</sup><sup>U</sup>*(*yi*) 1*Cmn* ⎞⎟⎠ 1 *m* , 1 − ⎛⎝1 − *Cmn* ∏*k*=11 − *m*∏*j*=11 − *<sup>I</sup><sup>L</sup>*(*xi*) 1*Cmn* ⎞⎠ 1 *m* ≥ 1 − ⎛⎝1 − *Cmn* ∏*k*=11 − *m*∏*j*=11 − *<sup>I</sup><sup>L</sup>*(*yi*) 1*Cmn* ⎞⎠ 1 *m* , 1 − ⎛⎝1 − *Cmn* ∏*k*=11 − *m*∏*j*=11 − *<sup>I</sup><sup>U</sup>*(*xi*) 1*Cmn* ⎞⎠ 1 *m* ≥ 1 − ⎛⎝1 − *Cmn* ∏*k*=11 − *m*∏*j*=11 − *<sup>I</sup><sup>U</sup>*(*yi*) 1*Cmn* ⎞⎠ 1 *m* , 1 − ⎛⎝1 − *Cmn* ∏*k*=11 − *m*∏*j*=11 − *<sup>F</sup><sup>L</sup>*(*xi*) 1*Cmn* ⎞⎠ 1 *m* ≥ 1 − ⎛⎝1 − *Cmn* ∏*k*=11 − *m*∏*j*=11 − *<sup>F</sup><sup>L</sup>*(*yi*) 1*Cmn* ⎞⎠ 1 *m* , 1 − ⎛⎝1 − *Cmn* ∏*k*=11 − *m*∏*j*=11 − *<sup>F</sup><sup>U</sup>*(*xi*) 1*Cmn* ⎞⎠ 1 *m* ≥ 1 − ⎛⎝1 − *Cmn* ∏*k*=11 − *m*∏*j*=11 − *<sup>F</sup><sup>U</sup>*(*yi*)) ) 1*Cmn* ) 1 *m* .

Therefore, we can ge<sup>t</sup> the following conclusion.

$$INLMSM^{(m)}(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) \le INLMSM^{(m)}(y\_1, y\_2, \dots, y\_n)$$

4 According to the idempotency, let min{ *x*1, *x*2, ...*xn*} = *xa* = *INLMSM*(*m*)(*xa*, *xa*, ..., *xa*) and max{ *x*1, *x*2, ...*xn*} = *xb* = *INLMSM*(*m*)(*xb*, *xb*, ..., *xb*). According to the monotonicity, if *xa* ≤ *xi* and *xb* ≥ *xi* for all i, then we have *xa* = *INLMSM*(*m*)(*xa*, *xa*, ..., *xa*) ≤ *INLMSM*(*m*)(*<sup>x</sup>*1, *x*2, ..., *xn*) and

$$INLMSM^{(m)}(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) \preceq \mathbf{x}\_b = INLMSM^{(m)}(\mathbf{x}\_b, \mathbf{x}\_b, \dots, \mathbf{x}\_b).$$

Therefore, we can ge<sup>t</sup> the conclusion below.

$$\min\{\mathbf{x}\_1, \mathbf{x}\_2, \dots \mathbf{x}\_n\} \le INLMSM^{(\mathfrak{m})} \{\mathbf{x}\_1, \mathbf{x}\_2, \dots \mathbf{x}\_n\} \le \max\{\mathbf{x}\_1, \mathbf{x}\_2, \dots \mathbf{x}\_n\}.$$

Furthermore, the *INLMSM*(*m*) operator would degrade to some particular forms when *m* takes some special values.

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

$$\begin{split} \text{INLMSM}^{(1)}(\mathbf{x}\_{1},\mathbf{x}\_{2},\ldots \mathbf{x}\_{n}) &= \left(\frac{\stackrel{\oplus\_{i=1}^{n}\mathbf{x}\_{i}}{\mathcal{C}\_{n}^{\top}}}{\mathcal{C}\_{n}^{\top}}\right) = \\ \left\langle s\_{\frac{1}{\mathcal{I}}\cdot\left(1-\prod\_{k=1}^{n}\left(1-\frac{1}{\mathcal{I}}\right)^{\frac{1}{\pi}}\right)}, \left(\left[1-\prod\_{k=1}^{n}\left(1-T^{L}\_{k}\right)^{\frac{1}{\pi}}, 1-\prod\_{k=1}^{n}\left(1-T^{\mathcal{U}\_{k}}\right)^{\frac{1}{\pi}}\right]\right. \\ \left. \left[\prod\_{k=1}^{n}\left(I^{L}\_{k}\right)^{\frac{1}{\pi}}, \prod\_{k=1}^{n}\left(I^{\mathcal{U}\_{k}}\right)^{\frac{1}{\pi}}\right], \left[\prod\_{k=1}^{n}\left(F^{L}\_{k}\right)^{\frac{1}{\pi}}, \prod\_{k=1}^{n}\left(F^{\mathcal{U}\_{k}}\right)^{\frac{1}{\pi}}\right]\right) \right\rangle \end{split} \tag{31}$$

(2) When *m* = 2, we have the formula below.

*INLMSM*(2)(*<sup>x</sup>*1, *x*2, ...*xn*) = *sl*·(<sup>1</sup>−∏*<sup>C</sup>*<sup>2</sup>*nk*=<sup>1</sup> (<sup>1</sup>−( *<sup>θ</sup>i*1 (*k*) *l* )·( *<sup>θ</sup>i*2 (*k*) *l* )) 1*C*2*n* ) 12 , ⎛⎝⎡⎣1 − <sup>∏</sup>*<sup>C</sup>*2*n <sup>k</sup>*=<sup>1</sup><sup>1</sup> − *<sup>T</sup>Li*1 (*k*) · *<sup>T</sup>Li*2 (*k*) 1*C*2*n* 12 , 1 − <sup>∏</sup>*<sup>C</sup>*2*n <sup>k</sup>*=<sup>1</sup><sup>1</sup> − *<sup>T</sup>Ui*1 (*k*) · *<sup>T</sup>Ui*2 (*k*) 1*C*2*n* 12 ⎤⎦ , M1 − 51 − <sup>∏</sup>*<sup>C</sup>*2*n <sup>k</sup>*=<sup>1</sup>-<sup>1</sup> − -1 − *<sup>I</sup>Li*1 (*k*). · -1 − *<sup>I</sup>Li*2 (*k*).. 1*C*2*n* 6 12 , 1 − 51 − <sup>∏</sup>*<sup>C</sup>*2*n <sup>k</sup>*=<sup>1</sup>-<sup>1</sup> − -1 − *<sup>I</sup>Ui*1 (*k*). · -1 − *<sup>I</sup>Ui*2 (*k*).. 1*C*2*n* 6 12 N, M1 − 51 − <sup>∏</sup>*<sup>C</sup>*2*n <sup>k</sup>*=<sup>1</sup>-<sup>1</sup> − -1 − *<sup>F</sup>Li*1 (*k*). · -1 − *<sup>F</sup>Li*2 (*k*).. 1*C*2*n* 6 12 , 1 − 51 − <sup>∏</sup>*<sup>C</sup>*2*n <sup>k</sup>*=<sup>1</sup>-<sup>1</sup> − -1 − *<sup>F</sup>Ui*1 (*k*). · -1 − *<sup>F</sup>Ui*2 (*k*).. 1*C*2*n* 6 12 N (32)

(3) When *m* = *n*, the *INLMSM*(*m*) operator would reduce to the following form.

*INLMSM*(*n*)(*<sup>x</sup>*1,..., *xn*) = *sl*·(∏*nj*=<sup>1</sup> ( *θjl* )) 1*n* , 5?∏*nj*=<sup>1</sup> *TLj* 1*n* , ∏*nj*=<sup>1</sup> *T<sup>U</sup> j* 1*n* @ ,⎡⎣1 − *n*∏*<sup>j</sup>*=<sup>1</sup>-<sup>1</sup> − *ILj*. 1*n* , 1 − *n*∏*<sup>j</sup>*=<sup>1</sup>-<sup>1</sup> − *I<sup>U</sup> j*. 1*n* ⎤⎦, ⎡⎣1 − *n*∏*<sup>j</sup>*=<sup>1</sup>-<sup>1</sup> − *FLj*. 1*n* , 1 − *n*∏*<sup>j</sup>*=<sup>1</sup>-<sup>1</sup> − *F<sup>U</sup> j*. 1*n* ⎤⎦⎞⎠ (33)


⎡

⎢⎢⎣1 − ⎛⎜⎝1

**Definition 14.** *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. Then the INLGMSM operator:* Ω*n* → Ω *is shown below*.

$$IINLGMSM^{(m,p\_1,p\_2,\ldots,p\_m)}(a\_1,\ldots,a\_n) = \left(\frac{\stackrel{\oplus}{1\leq i\_1<\ldots$$

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 2.** *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 14 is still an INLN*.

*INLGMSM*(*<sup>m</sup>*,*p*1,*p*2,...,*pm*)(*<sup>a</sup>*1,..., *an*) =*s <sup>l</sup>*·(<sup>1</sup>−∏*<sup>C</sup>mnk*=<sup>1</sup> (<sup>1</sup>−<sup>∏</sup>*mj*=<sup>1</sup> ( *θij*(*k*) *l* )*pj* ) 1 *Cmn* ) 1 *p*1+*p*2+...+*pm* , ⎛⎜⎜⎝⎡⎢⎢⎣⎛⎜⎝1 − <sup>∏</sup>*<sup>C</sup>mnk*=<sup>1</sup><sup>1</sup> − *m*∏*<sup>j</sup>*=<sup>1</sup>*TLij*(*k*)*pj* 1*Cmn* ⎞⎟⎠ 1 *p*1+*p*2+...+*pm* , ⎛⎜⎝1 − <sup>∏</sup>*<sup>C</sup>mnk*=<sup>1</sup><sup>1</sup> − *m*∏*<sup>j</sup>*=<sup>1</sup>*TUij*(*k*)*pj* 1*Cmn* ⎞⎟⎠ 1 *p*1+*p*2+...+*pm* ⎤⎥⎥⎦, ⎡⎢⎢⎣1 − ⎛⎜⎝1 − <sup>∏</sup>*<sup>C</sup>mnk*=<sup>1</sup><sup>1</sup> − *m*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − *<sup>I</sup>Lij*(*k*)*pj* 1*Cmn* ⎞⎟⎠ 1 *p*1+*p*2+...+*pm* , 1 − ⎛⎜⎝1 − <sup>∏</sup>*<sup>C</sup>mnk*=<sup>1</sup><sup>1</sup> − *m*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − *<sup>I</sup>Uij*(*k*)*pj* 1*Cmn* ⎞⎟⎠ 1 *p*1+*p*2+...+*pm* ⎤⎥⎥⎦, − <sup>∏</sup>*<sup>C</sup>mnk*=<sup>1</sup><sup>1</sup> − *m*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − *<sup>F</sup>Lij*(*k*)*pj* 1*Cmn* ⎞⎟⎠ 1 *p*1+*p*2+...+*pm* , 1 − ⎛⎜⎝1 − <sup>∏</sup>*<sup>C</sup>mnk*=<sup>1</sup><sup>1</sup> − *m*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − *<sup>F</sup>Uij*(*k*)*pj* 1*Cmn* ⎞⎟⎠ 1 *p*1+*p*2+...+*pm* ⎤ ⎥⎥⎦⎞⎟⎟⎠ (35)

*where k* = 1, 2, ...*Cmn* , *aij*(*k*) *is the ijth element of kjth permutation. Therefore, Theorem 2 is kept. The process of proof is similar to Theorem 1 and is now omitted.*

**Property 2.** *Let xi* = *s <sup>α</sup>i* ,([*T<sup>L</sup>*(*xi*), *<sup>T</sup><sup>U</sup>*(*xi*)], [*I<sup>L</sup>*(*xi*), *<sup>I</sup><sup>U</sup>*(*xi*)], [*F<sup>L</sup>*(*xi*), *<sup>F</sup><sup>U</sup>*(*xi*)]) (*i* = 1, 2, ..., *n*) *and yi* = *s βi* ,([*T<sup>L</sup>*(*yi*), *<sup>T</sup><sup>U</sup>*(*yi*)], [*I<sup>L</sup>*(*yi*), *<sup>I</sup><sup>U</sup>*(*yi*)], [*F<sup>L</sup>*(*yi*), *<sup>F</sup><sup>U</sup>*(*yi*)]) (*i* = 1, 2, ..., *n*) *be two sets of INLNs. There are four properties of INLGMSM*(*<sup>m</sup>*,*p*1,*p*2,...,*pm*) *operator shown as follows.*



The proofs are similar to Property 1, which are now omitted.

Furthermore, the *INLGMSM*(*<sup>m</sup>*,*p*1,*p*2,...,*pm*) operator would degrade to some particular forms when *m*takes some special values.

(1) When *m* = 1, we have the following formula.

*INLGMSM*(1)(*<sup>x</sup>*1, *x*2, ..., *xn*) = 5 <sup>⊕</sup>*ni*=<sup>1</sup> *xiP*1 *C*1*n* 6 1*P*1 = *sl*·(<sup>1</sup>−∏*nk*=<sup>1</sup> (<sup>1</sup>−( *kl* )*p*1 ) 1*n* ) 1*P*1 , ⎛⎝⎡⎣51 − <sup>∏</sup>*nk*=<sup>1</sup><sup>1</sup> − -*TLk*.*<sup>p</sup>*1 1*n* 6 1*P*1 , 51 − <sup>∏</sup>*nk*=<sup>1</sup><sup>1</sup> − -*TUk*.*<sup>p</sup>*1 1*n* 6 1*P*1 ⎤⎦, ⎡ ⎣1 − 51 − <sup>∏</sup>*nk*=<sup>1</sup><sup>1</sup> − -1 − *<sup>I</sup>Li*1 (*k*).*<sup>p</sup>*1 1*n* 6 1 *p*1 , 1 − 51 − <sup>∏</sup>*nk*=<sup>1</sup><sup>1</sup> − -1 − *<sup>I</sup>Ui*1 (*k*).*<sup>p</sup>*1 1*n* 6 1 *p*1 ⎤⎦, ⎡ ⎣1 − 51 − <sup>∏</sup>*nk*=<sup>1</sup><sup>1</sup> − -1 − *<sup>F</sup>Li*1 (*k*).*<sup>p</sup>*1 1*n* 6 1 *p*1 , 1 − 51 − <sup>∏</sup>*nk*=<sup>1</sup><sup>1</sup> − -1 − *<sup>F</sup>Ui*1 (*k*).*<sup>p</sup>*1 1*n* 6 1 *p*1 ⎤ ⎦⎞⎠ (36)

(2) When *m* = 2, we have the following formula.

*INLMSM*(2)(*<sup>x</sup>*1, *x*2, ..., *xn*) = *sl*·(<sup>1</sup>−∏*<sup>C</sup>*<sup>2</sup>*nk*=<sup>1</sup> (<sup>1</sup>−( *<sup>θ</sup>i*1(*k*) *l* )*p*1 ·( *<sup>θ</sup>i*2(*k*) *l* )*p*2 ) 1*C*2*n* ) 1 *p*1+*p*2 , ⎛⎜⎝⎡⎢⎣1 − <sup>∏</sup>*<sup>C</sup>*2*n <sup>k</sup>*=<sup>1</sup>5<sup>1</sup> − *TLi*1 (*k*)*<sup>P</sup>*<sup>1</sup> · *TLi*2 (*k*)*<sup>P</sup>*26 1*C*2*n* 1 *p*1+*p*2 , 1 − <sup>∏</sup>*<sup>C</sup>*2*n <sup>k</sup>*=<sup>1</sup>5<sup>1</sup> − *TUi*1 (*k*)*<sup>P</sup>*<sup>1</sup> · *TUi*2 (*k*)*<sup>P</sup>*26 1*C*2*n* 1 *p*1+*p*2 ⎤⎥⎦ ⎡ ⎣1 − 51 − <sup>∏</sup>*<sup>C</sup>*2*n <sup>k</sup>*=<sup>1</sup><sup>1</sup> − -1 − *<sup>I</sup>Li*1 (*k*).*<sup>P</sup>*<sup>1</sup> · -1 − *<sup>I</sup>Li*2 (*k*).*<sup>P</sup>*2 1*C*2*n* 6 1 *p*1+*p*2 , 1 − 51 − <sup>∏</sup>*<sup>C</sup>*2*n <sup>k</sup>*=<sup>1</sup><sup>1</sup> − -1 − *<sup>I</sup>Ui*1 (*k*).*<sup>P</sup>*<sup>1</sup> · -1 − *<sup>I</sup>Ui*2 (*k*).*<sup>P</sup>*2 1*C*2*n* 6 1 *p*1+*p*2 ⎤ ⎦, ⎡⎣1 − 51 − <sup>∏</sup>*<sup>C</sup>*2*n <sup>k</sup>*=<sup>1</sup><sup>1</sup> − -1 − *<sup>F</sup>Li*1 (*k*).*<sup>P</sup>*<sup>1</sup> · -1 − *<sup>F</sup>Li*2 (*k*).*<sup>P</sup>*2 1*C*2*n* 6 1 *p*1+*p*2 , 1 − 51 − <sup>∏</sup>*<sup>C</sup>*2*n <sup>k</sup>*=<sup>1</sup><sup>1</sup> − -1 − *<sup>F</sup>Ui*1 (*k*).*<sup>P</sup>*<sup>1</sup> · -1 − *<sup>F</sup>Ui*2 (*k*).*<sup>P</sup>*2 1*C*2*n* 6 1 *p*1+*p*2 ⎤ ⎦⎞⎠ (37)

When *m* = 2, the *INLGMSM*(*<sup>m</sup>*,*p*1,*p*2,...,*pm*) operator would reduce to the BM for INLNs (INLGBM) operator.

(3) When *m* = *n*, the *INLMSM*(*m*) operator would reduce to the form below.

$$\begin{split} \text{INL}\_{\text{L}}\text{CMSSM}^{(\text{u}\_{P},p\_{1},p\_{2},\ldots,p\_{m})}(a\_{1},\ldots,a\_{n}) &= \\ \left\langle s\bigg|\_{\text{L}\in\prod\_{j=1}^{n}\left(\prod\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i=1}^{p\_{i}}{\left(1-\sum\_{i}{\left(1-\sum\_{i}{\left(\sum\_{i}$$

#### *4.2. Some Weighted INLMSM Operators*

We will introduce two operators, which are the weighted forms of the *INLMSM* operator and *INLGMSM* operator.

**Definition 15.** *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* (*i* = 1, 2, ..., *n*) *be a set of INLNs. Let ω* = (*<sup>ω</sup>*1, *ω*2, ..., *<sup>ω</sup>n*) *T is the weight vector and satisfies* ∑*ni*=<sup>1</sup> *ωi* = 1 *with ωi* > 0 (*i* = 1, 2, ..., *<sup>n</sup>*)*. Each ωi represents the importance of ai. Then the WINLMSM operator:* Ω*n* → Ω *is defined below*.

$$WINLMSM^{(m)}(a\_1, \ldots, a\_n) = \left(\frac{\stackrel{\oplus}{1 \le i\_1 < \ldots < i\_m \le n} \left(\sum\_{j=1}^m \left(n\omega\_{i\_j}\right) a\_{i\_j}\right)}{\mathbb{C}\_n^m}\right)^{\frac{\Delta}{m}}\tag{39}$$

1

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

⎡

⎢⎢⎣1

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

**Theorem 3.** *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* (*i* = 1, 2, ..., *n*) *be a set of INLNs and m* = 1, 2, ..., *n, then the value aggregated from Definition 15 is still a WINLMSM operator*.

$$\begin{split} & \quad \text{WINLLMSM}^{(m)}(a\_{1},\ldots,a\_{k}) - \left( s\_{k} \right)^{\text{E}} \\ & \quad \left( \left[ \left( 1 - \prod\_{i=1}^{\text{CE}} \left( 1 - \prod\_{j=1}^{\text{E}} \left( 1 - \left( 1 - \prod\_{i \neq j}^{k\_{i} \text{(j)}} \right)^{\text{cov}\_{j}} \right) \right)^{\text{E}} \right]^{\text{E}} \right)^{\text{E}} \\ & \quad \left( 1 - \prod\_{k=1}^{\text{CE}} \left( 1 - \prod\_{j=1}^{\text{E}} \left( 1 - \left( 1 - \left( 1 - \prod\_{j \neq j}^{k\_{j} \text{(j)}} \right)^{\text{cov}\_{j}} \right) \right)^{\text{E}} \right)^{\text{E}} \right)^{\text{E}} \\ & \quad \left[ 1 - \prod\_{k=1}^{\text{E}} \left( 1 - \prod\_{j=1}^{\text{E}} \left( 1 - \left( 1 - \left( 1 - \left( 1 \cdot \prod\_{j \neq j}^{k\_{j} \text{(j)}} \right)^{\text{cov}\_{j}} \right) \right)^{\text{E}} \right)^{\text{E}} \right)^{\text{E}} \\ & \quad \left[ 1 - \left( 1 - \prod\_{k=1}^{\text{E}} \left( 1 - \prod\_{j=1}^{\text{E}} \left( 1 - \left( 1 \cdot \left( 1 \cdot \prod\_{j \neq j}^{k\_{j} \text{(j)}} \right)^{\text{cov}\_{j}} \right) \right)^{\text{E}} \right)^{\text{E}} \right]^{\text{E}} \right]^{\text{E}} \\ & \quad \left$$

*where k* = 1, 2, ..., *Cmn* , *aij*(*k*) *is the ijth element of kth permutation. The process of proof is similar to Theorem 1. Now it is omitted.*

**Property 3.** *Let xi* = *s <sup>α</sup>i* ,([*T<sup>L</sup>*(*xi*), *<sup>T</sup><sup>U</sup>*(*xi*)], [*I<sup>L</sup>*(*xi*), *<sup>I</sup><sup>U</sup>*(*xi*)], [*F<sup>L</sup>*(*xi*), *<sup>F</sup><sup>U</sup>*(*xi*)]) (*i* = 1, 2, ..., *n*) *and yi* = *s βi* ,([*T<sup>L</sup>*(*yi*), *<sup>T</sup><sup>U</sup>*(*yi*)], [*I<sup>L</sup>*(*yi*), *<sup>I</sup><sup>U</sup>*(*yi*)], [*F<sup>L</sup>*(*yi*), *<sup>F</sup><sup>U</sup>*(*yi*)]) (*i* = 1, 2, ..., *n*) *be sets of INLNs. There are some properties of the W INLMSM*(*m*) *operator as shown below.*

