**Proof.**

1 If *ω* = ( 1*n* , 1*n* , ..., 1*n* ) , then *W INLMSM*(*m*)(*<sup>a</sup>*1, *a*2, ..., *an*) =

$$\begin{split} & \left\langle s\_{1} \right| \\ & \left. \left( \left[ \left( 1 - \prod\_{i=1}^{\sum\_{k=1}^{n}} \left( 1 - \prod\_{j=1}^{n} \left( 1 - \frac{a\_{j}(\boldsymbol{x}\_{i})}{\boldsymbol{\pi}} \right)^{\Delta} \right)^{\Delta} \right)^{\Delta} \right] \right.} \right. \\ & \left. \left( \left[ \left( 1 - \prod\_{i=1}^{\sum\_{k=1}^{n}} \left( 1 - \prod\_{j=1}^{n} \left( 1 - \left( 1 - \boldsymbol{T}^{\boldsymbol{L}\_{j}(\boldsymbol{x}\_{i})} \right) \right)^{\Delta} \right)^{\Delta} \right. \\ & \left. \left. \left( 1 - \prod\_{i=1}^{\sum\_{k=1}^{n}} \left( 1 - \prod\_{j=1}^{n} \left( 1 - \left( 1 - \binom{\boldsymbol{L}\_{j}}{\boldsymbol{\mu}\_{1}(\boldsymbol{x})} \right) \right)^{\Delta} \right)^{\Delta} \right. \\ & \left. \left. \left( 1 - \prod\_{i=1}^{\sum\_{k=1}^{n}} \left( 1 - \prod\_{j=1}^{n} \left( 1 - \left( \boldsymbol{L}\_{j(\boldsymbol{x})} \right) \right) \right)^{\Delta} \right)^{\Delta} \right. \\ & \left. \left. \left. \left( 1 - \prod\_{i=1}^{\sum\_{k=1}^{n}} \left( 1 - \prod\_{j=1}^{n} \left( 1 - \left( \boldsymbol{L}\_{j(\boldsymbol{x})} \right) \right) \right)^{\Delta} \right)^{\Delta} \right. \\ & \left. \left. \left. \left$$

2 The proofs of Monotonicity and Boundedness are similar to Property 1, which are now omitted.

Furthermore, the *WINLMSM*(*m*) operator would degrade a particular form when *m* takes some special values.

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

$$\begin{aligned} \{\Pi^{\text{INL}}\_{l}\Lambda\Lambda\!M\!M\!M^{(1)}\!(a\_{1},\ldots,a\_{n}) &= \\ \{\begin{array}{l} \left(\begin{array}{l} \left(\begin{array}{l} \left(\begin{array}{l} \left(\begin{array}{l} \left(\begin{array}{l} \left(\begin{array}{l} \left(\begin{array}{l} \end{array} \end{array}\right)^{\omega\_{1}} \end{array}\right)^{\omega\_{1}} \end{array}\right), \end{ } \end{ } \end{ } \end{ } \end{aligned} \right)} = \\ \{\begin{array}{l} \left(\begin{array}{l} \left(\begin{array}{l} \left(\begin{array}{l} \left(\begin{array}{l} \left(\begin{array}{l} \end{array} \end{array}\right)^{\omega\_{1}} \end{array}\right), \left(\begin{array}{l} \left(\begin{array}{l} \left(\begin{array}{l} \left(\begin{array}{l} \left(\begin{array}{l} \left(\begin{array}{l} \end{array} \end{array}\right)^{\omega\_{1}} \end{array}\right) \end{bmatrix}\right), \end{ } \right) \end{ } \end{ } \end{ } \} \end{aligned} \tag{41}$$

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

*W INLMSM*(2)(*<sup>a</sup>*1,..., *an*) =*s <sup>l</sup>*·(<sup>1</sup>−∏*C*<sup>2</sup>*nk*=<sup>1</sup> (<sup>1</sup>−(<sup>1</sup>−(<sup>1</sup>− *<sup>θ</sup>i*1(*k*) *l* )*<sup>n</sup>*·*<sup>ω</sup>i*<sup>1</sup> )·(<sup>1</sup>−(<sup>1</sup>− *<sup>θ</sup>i*2(*k*) *l* )*<sup>n</sup>*·*<sup>ω</sup>i*<sup>2</sup> )) 1*C*2*n* ) 1 2 , ⎛⎜⎜⎜⎝⎡⎢⎢⎢⎣⎛⎜⎝1 − <sup>∏</sup>*C*<sup>2</sup>*nk*=<sup>1</sup>5<sup>1</sup> − 51 − 51 − *<sup>T</sup>Li*1(*k*)6*<sup>n</sup>*·*<sup>ω</sup>i*<sup>1</sup> 6 · 51 − 51 − *<sup>T</sup>Li*2(*k*)6*<sup>n</sup>*·*<sup>ω</sup>i*<sup>2</sup> 66 1*C*2*n* ⎞⎟⎠ 12 , ⎛⎜⎝1 − <sup>∏</sup>*C*<sup>2</sup>*nk*=<sup>1</sup>5<sup>1</sup> − 51 − 51 − *<sup>T</sup>Ui*1(*k*)6*<sup>n</sup>*·*<sup>ω</sup>i*<sup>1</sup> 6 · 51 − 51 − *<sup>T</sup>Ui*2(*k*)6*<sup>n</sup>*·*<sup>ω</sup>i*<sup>2</sup> 66 1*C*2*n* ⎞⎟⎠ 1 2 ⎤⎥⎥⎥⎦, ⎡⎢⎢⎢⎣1 − ⎛⎜⎝1 − <sup>∏</sup>*C*<sup>2</sup>*nk*=<sup>1</sup>5<sup>1</sup> − 51 − 5*ILi*1(*k*)6*<sup>n</sup>*·*<sup>ω</sup>i*<sup>1</sup> 6 · 51 − 5*ILi*2(*k*)6*<sup>n</sup>*·*<sup>ω</sup>i*<sup>2</sup> 66 1*C*2*n* ⎞⎟⎠ 12 , 1 − ⎛⎜⎝1 − <sup>∏</sup>*C*<sup>2</sup>*nk*=<sup>1</sup>5<sup>1</sup> − 51 − 5*IUi*1(*k*)6*<sup>n</sup>*·*<sup>ω</sup>i*<sup>1</sup> 6 · 51 − 5*IUi*2(*k*)6*<sup>n</sup>*·*<sup>ω</sup>i*<sup>2</sup> 66 1*C*2*n* ⎞⎟⎠ 1 2 ⎤ ⎥⎥⎥⎦, ⎡⎢⎢⎢⎣1 − ⎛⎜⎝1 − <sup>∏</sup>*C*<sup>2</sup>*nk*=<sup>1</sup>5<sup>1</sup> − 51 − 5*FLi*1(*k*)6*<sup>n</sup>*·*<sup>ω</sup>i*<sup>1</sup> 6 · 51 − 5*FLi*2(*k*)6*<sup>n</sup>*·*<sup>ω</sup>i*<sup>2</sup> 66 1*C*2*n* ⎞⎟⎠ 12 , 1 − ⎛⎜⎝1 − <sup>∏</sup>*C*<sup>2</sup>*nk*=<sup>1</sup>5<sup>1</sup> − 51 − 5*FUi*1(*k*)6*<sup>n</sup>*·*<sup>ω</sup>i*<sup>1</sup> 6 · 51 − 5*FUi*2(*k*)6*<sup>n</sup>*·*<sup>ω</sup>i*<sup>2</sup> 66 1*C*2*n* ⎞⎟⎠ 1 2 ⎤ ⎥⎥⎥⎦⎞⎟⎟⎟⎠ (42)

#### (3) When *m* = *n*, we have the formula below.

$$\begin{split} & \quad \text{WINLMSM}^{(n)}(a\_{1}, \ldots, a\_{n}) - \left\langle s \right| \\ & \quad \left( \left[ \prod\_{j=1}^{n} \left( 1 - \left( 1 - \left( 1 - \frac{\tau\_{j}}{\tau\_{j}} \right)^{n \cdot \omega\_{j}} \right) \right)^{\frac{1}{\omega\_{j}}} \right], \\ & \quad \left[ \left( 1 - \left( 1 - \left( 1 - \left( 1 - \frac{\tau\_{j}}{\tau\_{j}} \right)^{n \cdot \omega\_{j}} \right) \right)^{\frac{1}{\omega\_{j}}} \right) \right], \\ & \quad \left[ 1 - \left( \prod\_{j=1}^{n} \left( 1 - \left( 1 \right)^{n \cdot \omega\_{j}} \right) \right)^{\frac{1}{\omega\_{j}}}, 1 - \left( \prod\_{j=1}^{n} \left( 1 - \left( 1 \right)^{n \cdot \omega\_{j}} \right) \right)^{\frac{1}{\omega\_{j}}} \right], \\ & \quad \left[ 1 - \left( \prod\_{j=1}^{n} \left( 1 - \left( 1 - \left( 1 \right)^{n \cdot \omega\_{j}} \right) \right)^{\frac{1}{\omega\_{j}}}, 1 - \left( \prod\_{j=1}^{n} \left( 1 - \left( \boldsymbol{F}\_{j}^{j} \right)^{n \cdot \omega\_{j}} \right) \right)^{\frac{1}{\omega\_{j}}} \right] \right]. \end{split} \tag{43}$$


**Definition 16.** *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* (*i* = 1, 2, ..., *n*) *be a set of INLNs. Let ω* = ( *ω*1, *ω*2, ..., *<sup>ω</sup>n*) *T is the weight vector and it satisfies* ∑*n i*=1 *ωi* = 1 *with ωi* > 0 (*i* = 1, 2, ..., *<sup>n</sup>*)*. Each ωi represents the importance of ai. Then the WINLGMSM operator:* Ω*n* → Ω *is defined below*.

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

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 WINLMSM operator shown below*.

**Theorem 4.** *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* (*i* = 1, 2, ..., *n*) *be a set of INLNs and m* = 1, 2, ..., *n. Then the value aggregated from Definition 16 is still an WINLGMSM*.

$$\begin{split} \text{WINI-MSM}(m\_{i},a\_{1},a\_{2},\ldots,a\_{n}) &= \left\langle s\_{i} \right| \\ & \quad \left( \left[ \left( 1 - \prod\_{l=1}^{\text{CFL}} \left( 1 - \prod\_{j=1}^{\text{CFL}} \left( 1 - \left( 1 - \frac{a\_{j}/b\_{j}}{b\_{j}} \right)^{\omega\_{i,j}} \right)^{\frac{1}{\omega\_{i,j}}} \right)^{\frac{1}{\omega\_{i,j}} + \frac{1}{\omega\_{i,j} - \omega\_{i,j}}} \right] \right. \\ & \left. \left. \left( 1 - \prod\_{l=1}^{\text{CFL}} \left( 1 - \prod\_{j=1}^{\text{CFL}} \left( 1 - \left( 1 - \left( 1 - \prod\_{l}^{\text{H}\_{j}} \left( h\_{j} \right)^{\omega\_{i,j}} \right)^{\omega\_{j}} \right)^{\frac{1}{\omega\_{i,j}}} \right)^{\frac{1}{\omega\_{i,j} - \omega\_{i,j} - \omega\_{i,j}}} \right) \right. \\ & \left. \left. \left. \left( 1 - \prod\_{l=1}^{\text{CFL}} \left( 1 - \prod\_{j=1}^{\text{CFL}} \left( 1 - \left( 1 - \prod\_{l}^{\text{H}\_{j}} \left( h\_{l} \right)^{\omega\_{i,j}} \right)^{\omega\_{i,j}} \right)^{\frac{1}{\omega\_{i,j} - \omega\_{i,j} - \omega\_{i,j}}} \right) \right) \right. \\ & \left. \left. \left. \left. \left( 1 - \prod\_{l=1}^{\text{CFL}} \left( 1 - \prod\_{j=1}^{\text{CFL}} \left( 1 - \left( 1$$

*where k* = 1, 2, ..., *Cm n* , *aij*(*k*) *is the ijth element of kth permutation. The process of proof is similar to Theorem 1. It is now omitted*.

⎢

**Property 4.** *Let xi* = *s <sup>α</sup>i* ,([ *<sup>T</sup><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* ,([ *<sup>T</sup><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 some properties of the W INLGMSM*(*<sup>m</sup>*,*p*1,*p*2,...,*pm*) *operator shown below*.


The process of proof is similar to Property 3 and is now omitted.

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

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

$$\begin{split} & \quad \text{WINLGMSM}^{(1,p\_{1})}(a\_{1}, \ldots, a\_{n}) = \left\langle s \right| \\ & \quad \left( \left[ \left( 1 - \prod\_{k=1}^{n} \left( 1 - \left( 1 - \left( 1 - \left( 1 - \frac{\Lambda\_{j\_{k}(k)}}{I\_{j\_{k}(k)}} \right)^{n \omega\_{1}} \right)^{n} \right)^{\frac{1}{\omega\_{1}}} \right)^{\frac{1}{\omega\_{1}}} \right)^{\frac{1}{\omega\_{1}}} \right]^{\frac{1}{\omega\_{1}}} \\ & \quad \left[ \left( 1 - \prod\_{k=1}^{n} \left( 1 - \left( 1 - \left( 1 - \left( 1 - \left( \frac{I\_{j\_{k}(k)}}{I\_{j\_{k}(k)}} \right)^{n \omega\_{1}} \right)^{n} \right)^{\frac{1}{\omega\_{1}}} \right)^{\frac{1}{\omega\_{1}}} \right)^{\frac{1}{\omega\_{1}}} \right)^{\frac{1}{\omega\_{1}}} \right], \\ & \quad \left[ \left( 1 - \left( 1 - \prod\_{k=1}^{n} \left( 1 - \left( 1 - \left( 1 - \left( \frac{I\_{j\_{k}(k)}}{I\_{j\_{k}(k)}} \right)^{n \omega\_{1}} \right)^{n} \right)^{\frac{1}{\omega\_{1}}} \right)^{\frac{1}{\omega\_{1}}} \right)^{\frac{1}{\omega\_{1}}} \right], \\ & \quad \left[ \left( 1 - \left( 1 - \prod\_{k=1}^{n} \left( 1 - \left( 1 - \left( 1 - \left( \frac{I\_{j\_{k}(k)}}{I\_{j\_{k}(k)}} \right)^{n$$

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

*W INLGMSM*(2,*p*1,*p*2)(*<sup>a</sup>*1,..., *an*) = *s <sup>l</sup>*·(<sup>1</sup>−∏*<sup>C</sup>*<sup>2</sup> *n k*=1 (<sup>1</sup>−(<sup>1</sup>−(<sup>1</sup>− *θi* 1(*k*) *l* ) *<sup>n</sup>*·*<sup>ω</sup>i* 1 ) *p*1 ·(<sup>1</sup>−(<sup>1</sup>− *θi* 2(*k*) *l* ) *<sup>n</sup>*·*<sup>ω</sup>i* 2 ) *p*2 ) 1 *C*<sup>2</sup> *n* ) 1 *p*1+*p*2 , ⎛ ⎜⎝ ⎡ ⎢ ⎣ 1 − ∏*<sup>C</sup>*<sup>2</sup> *n k*=1 5 1 − 1 − 1 − *T<sup>L</sup> <sup>i</sup>*1(*k*) *<sup>n</sup>*·*ω<sup>i</sup>* 1 *p*1 · 1 − 1 − *T<sup>L</sup> <sup>i</sup>*2(*k*) *<sup>n</sup>*·*ω<sup>i</sup>* 2 *p*2 6 1 *C*<sup>2</sup> *n* 1 *p*1+*p*2 , 1 − ∏*<sup>C</sup>*<sup>2</sup> *n k*=1 5 1 − 1 − 1 − *T<sup>U</sup> <sup>i</sup>*1(*k*) *<sup>n</sup>*·*ω<sup>i</sup>* 1 *p*1 · 1 − 1 − *T<sup>U</sup> <sup>i</sup>*2(*k*) *<sup>n</sup>*·*ω<sup>i</sup>* 2 *p*2 6 1 *C*<sup>2</sup> *n* 1 *p*1+*p*2 ⎤ ⎥ ⎦, ⎡ ⎢ ⎣1 − 1 − ∏*<sup>C</sup>*<sup>2</sup> *n k*=1 5 1 − 1 − *IL <sup>i</sup>*1(*k*) *<sup>n</sup>*·*ω<sup>i</sup>* 1 *p*1 · 1 − *IL <sup>i</sup>*2(*k*) *<sup>n</sup>*·*ω<sup>i</sup>* 2 *p*2 6 1 *C*<sup>2</sup> *n* 1 *p*1+*p*2 , 1 − 1 − ∏*<sup>C</sup>*<sup>2</sup> *n k*=1 5 1 − 1 − *I<sup>U</sup> <sup>i</sup>*1(*k*) *<sup>n</sup>*·*ω<sup>i</sup>* 1 *p*1 · 1 − *I<sup>U</sup> <sup>i</sup>*2(*k*) *<sup>n</sup>*·*ω<sup>i</sup>* 2 *p*2 6 1 *C*<sup>2</sup> *n* 1 *p*1+*p*2 ⎤ ⎥ ⎦, ⎡ ⎢ ⎣1 − 1 − ∏*<sup>C</sup>*<sup>2</sup> *n k*=1 5 1 − 1 − *F<sup>L</sup> <sup>i</sup>*1(*k*) *<sup>n</sup>*·*ω<sup>i</sup>* 1 *p*1 · 1 − *F<sup>L</sup> <sup>i</sup>*2(*k*) *<sup>n</sup>*·*ω<sup>i</sup>* 2 *p*2 6 1 *C*<sup>2</sup> *n* 1 *p*1+*p*2 , 1 − 1 − ∏*<sup>C</sup>*<sup>2</sup> *n k*=1 5 1 − 1 − *F<sup>U</sup> <sup>i</sup>*1(*k*) *<sup>n</sup>*·*ω<sup>i</sup>* 1 *p*1 · 1 − *F<sup>U</sup> <sup>i</sup>*2(*k*) *<sup>n</sup>*·*ωi* 2 *p*2 6 1 *C*<sup>2</sup> *n* 1 *p*1+*p*2 ⎤ ⎥ ⎦ ⎞ ⎟⎠ (47)

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

*W INLGMSM*(*<sup>n</sup>*,*p*1,*p*2,...,*pn*)(*<sup>a</sup>*1,..., *an*) <sup>=</sup>*sl*·(∏*nj*=<sup>1</sup> (<sup>1</sup>−(<sup>1</sup>− *θjl* )*<sup>n</sup>*·*<sup>ω</sup><sup>j</sup>* )*pj*) 1 *p*1+*p*2+...+*pn* , ⎛⎝⎡⎣ *n*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − 1 − *TLj <sup>n</sup>*·*<sup>ω</sup>jpj* 1 *p*1+*p*2+...+*pn* , *n*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − 1 − *TUj <sup>n</sup>*·*<sup>ω</sup>jpj* 1 *p*1+*p*2+...+*pn* ⎤⎦, ⎡ ⎣1 − *n*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − *ILj <sup>n</sup>*·*<sup>ω</sup>jpj* 1 *p*1+*p*2+...+*pn* , 1 − *n*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − *IUj <sup>n</sup>*·*<sup>ω</sup>jpj* 1 *p*1+*p*2+...+*pn* ⎤ ⎦, ⎡⎣1 − *n*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − *FLj <sup>n</sup>*·*<sup>ω</sup>jpj* 1 *p*1+*p*2+...+*pn* , 1 − *n*∏*<sup>j</sup>*=<sup>1</sup><sup>1</sup> − *FUj <sup>n</sup>*·*<sup>ω</sup>jpj* 1 *p*1+*p*2+...+*pn* ⎤⎦⎞⎠ (48)

#### **5. MADM Method Based on INLMSM Operator**

In this section, we introduce the MADM method based on the *WINLMSM* and *WINLGMSM* operators. Let *d* = {*d*1, *d*2, ..., *dm*} be a collection of alternatives and *c* = {*<sup>c</sup>*1, *c*2, ..., *cn*} is a collection of *n* criteria. The weight vector is *ω* = (*<sup>ω</sup>*1, *ω*2, ..., *<sup>ω</sup>n*)*<sup>T</sup>* with satisfying ∑*ni*=<sup>1</sup> *ωi* = <sup>1</sup>(*<sup>ω</sup>i* ≥ 0, *i* = 1, 2, ..., *<sup>n</sup>*), and each *ωi* represents the importance of *cj*. The performance of alternative *dj* in criteria *cj* is surveyed by INLNs and the decision matrix is *A* = (*aij*)*m*×*n*, where *a*ij = *s θij* ,([*T<sup>L</sup>*(*<sup>r</sup>*ij), *<sup>T</sup><sup>U</sup>*(*<sup>r</sup>*ij)], [*I<sup>L</sup>*(*<sup>r</sup>*ij), *<sup>I</sup><sup>U</sup>*(*<sup>r</sup>*ij)], [*F<sup>L</sup>*(*<sup>r</sup>*ij), *<sup>F</sup><sup>U</sup>*(*<sup>r</sup>*ij)]). The objective is to rank the alternatives.

The detailed steps are shown below.

#### **Step 1** Normalize the decision matrix.

We should normalize the decision-making information in the matrix. The benefit (the bigger the better) and the cost (the smaller the better) are the two possible types. In order to keep the consistency of the types, it is necessary to convert the decision matrix *A* into a standardized matrix *R* = (*<sup>r</sup>*ij)*<sup>m</sup>* × *n*.

$$\begin{aligned} \text{If } c\_{\bar{\eta}} \text{ is cost type, then } r\_{\bar{\eta}} = \langle s\_{\theta\_{\bar{\eta}'}} \left( [F^L(r\_{\bar{\eta}}), F^{\mathcal{U}}(r\_{\bar{\eta}})], [1 - I^{\mathcal{U}}(r\_{\bar{\eta}}), 1 - I^L(r\_{\bar{\eta}})] [T^L(r\_{\bar{\eta}}), T^{\mathcal{U}}(r\_{\bar{\eta}})] \right) \rangle, \\ \text{else } r\_{\bar{\eta}} = \langle s\_{\theta\_{\bar{\eta}'}} \left( [T^L(r\_{\bar{\eta}}), T^{\mathcal{U}}(r\_{\bar{\eta}})], [I^L(r\_{\bar{\eta}}), I^{\mathcal{U}}(r\_{\bar{\eta}})], [F^L(r\_{\bar{\eta}}), F^{\mathcal{U}}(r\_{\bar{\eta}})] \right) \rangle. \end{aligned}$$


## **6. Illustrative Example**

There are many decision-making problems to be solved in the current society, which requires some decision-making methods.

In this section, we investigate an example (adapted from Ref [43]) about the MADM. In a MADM problem, there are four possible alternatives for an investment company including a car company (*A*1), a food company (*A*2), a computer company (*A*3), and an arms company (*A*4). The following three attributes can be used to evaluate alternatives by the investment company: the risk (*C*1), the growth (*C*2), and the environmental impact (*C*3) where *C*1 and *C*2 are benefit types and *C*3 is cost type. Then the evaluation values of alternatives are shown in Table 1 where the LTS is *S* = {*<sup>s</sup>*0 = *extremely poor*(*EP*), *s*1 = *very poor*(*VP*), *s*2 = *poor*(*P*), *s*3 = *medium*(*M*), *s*4 = *good*(*G*), *s*5 = *very good*(*VG*), *s*6 = *extremely good*(*EG*)}, and the weight vector of criteria

is *ω* = (0.35, 0.25, 0.4)*<sup>T</sup>*. Now we will use the method proposed in this paper, according to the above LTs and three criteria. Then we evaluate and sort the four options in Table 1.


**Table 1.** Evaluation values of alternatives.

*6.1. The Method Based on the WINLMSM Operator*

Generally, we can give *m* = *n*2 , so *m* = 1 and *m* = 2. Then, according to Section 5, we have the statements below.

(1) When *m* = 1, the steps are shown below.

**Step 1** Normalize the decision matrix.

From the example, the risk (*C*1) and the growth (*C*2) are benefit types while the environmental impact (*C*3) is cost type. We set up the decision matrix as shown below.


⎤⎥⎥⎥⎥⎦

**Step 2** Aggregate all attribute values of each alternative and ge<sup>t</sup> the overall value of each alternative *ai*denoted as *ri*(*<sup>i</sup>* = 1, 2, 3, <sup>4</sup>).

> *r*1 = *<sup>s</sup>*6,([0.3268, 0.4590], [0.1275, 0.2305], [0.3325, 0.4704]), *r*2 = *<sup>s</sup>*6,([0.5271, 0.7000], [0.1320, 0.2000], [0.1516, 0.2551]), *r*3 = *<sup>s</sup>*6,([0.4375, 0.5675], [0.1000, 0.2603], [0.1933, 0.3565]), *r*4 = *<sup>s</sup>*6,([0.5216, 0.6565], [0.0000, 0.1569], [0.1189, 0.2213])

**Step 3** According to Definition 11, we assume *α* = 0.7 and calculate the score values of *ri*(*<sup>i</sup>* = 1, 2, 3, 4) below.

$$S\_{(r\_1)} = \text{s0.628}, S\_{(r\_2)} = \text{s0.8306}, \ S\_{(r\_3)} = \text{s0.7462}, \ S\_{(r\_4)} = \text{s0.7778}$$

**Step 4** According to Step 3 and Definition 12, we would ge<sup>t</sup> the ranking of the alternatives, which are *A*2 *A*4 *A*3 *A*1.

(2) When *m* = 2, the steps are shown below.

**Step 1** Normalize the decision matrix.

From the example, the risk (*C*1) and the growth (*C*2) are benefit types while the environmental impact (*C*3) is cost type. We set up the decision matrix as shown below.

*R* = ⎡ ⎢⎢⎢⎢⎢⎣ *<sup>s</sup>*5,([0.4, 0.5], [0.2, 0.3], [0.3, 0.4]) *<sup>s</sup>*6,([0.4, 0.6], [0.1, 0.2], [0.2, 0.4]) *<sup>s</sup>*6,([0.5, 0.7], [0.1, 0.2], [0.2, 0.3]) *<sup>s</sup>*5,([0.6, 0.7], [0.1, 0.2], [0.2, 0.3]) *<sup>s</sup>*6,([0.3, 0.5], [0.1, 0.2], [0.3, 0.4]) *<sup>s</sup>*5,([0.5, 0.6], [0.1, 0.3], [0.3, 0.4]) *<sup>s</sup>*4,([0.7, 0.8], [0.0, 0.1], [0.1, 0.2]) *<sup>s</sup>*4,([0.5, 0.7], [0.1, 0.2], [0.2, 0.3]) *<sup>s</sup>*5,([0.2, 0.3], [0.1, 0.2], [0.5, 0.6]) *<sup>s</sup>*5,([0.5, 0.7], [0.2, 0.2], [0.1, 0.2]) *<sup>s</sup>*4([0.5, 0.6], [0.1, 0.3], [0.1, 0.3]) *<sup>s</sup>*6([0.3, 0.4], [0.1, 0.2], [0.1, 0.2]) ⎤⎥⎥⎥⎥⎥⎦ **Step 2** Aggregate all attribute values of each alternative and ge<sup>t</sup> the overall value of each alternative *ai* denoted as *ri*(*<sup>i</sup>* = 1, 2, 3, <sup>4</sup>).

> *r*1 = *<sup>s</sup>*5.4841,([0.3190, 0.4520], [0.1406, 0.2420], [0.3391, 0.4771]), *r*2 = *<sup>s</sup>*5.3016,([0.5260, 0.6922], [0.1366, 0.2083], [0.1791, 0.2772]), *r*3 = *<sup>s</sup>*4.9567,([0.4224, 0.5587], [0.1077, 0.2741], [0.2494, 0.3754]), *r*4 = *<sup>s</sup>*4.4896,([0.4794, 0.6190], [0.0711, 0.1739], [0.1415, 0.2416])

**Step 3** According to Definition 11, we assume *α* = 0.7 and calculate the score values of*ri*(*<sup>i</sup>* = 1, 2, 3, <sup>4</sup>). We ge<sup>t</sup> the values below.

$$S\_{(r\_1)} = s\_{0.5695\prime} S\_{(r\_2)} = s\_{0.7170\prime} \ S\_{(r\_3)} = s\_{0.6004\prime} \ S\_{(r\_4)} = s\_{0.5765\prime}$$

**Step 4** According to Step 3 and Definition 12, we ge<sup>t</sup> the ranking of the alternatives below.

$$A\_2 \succ A\_3 \succ A\_4 \succ A\_1$$

*6.2. The Method Based on the WINLGMSM Operator*

When *m* = 1, *p* = 1, the *W INLGMSM*(1) operator is the same as the *W INLMSM*(1) operator. The steps are omitted here. When *m* = 2, the steps are below.

**Step 1** Normalize the decision matrix.

> From the example, the risk (*C*1), the growth (*C*2) are benefit types and the environmental impact (*C*3) is cost type, so we set up the matrix as step 1 of Section 6.1.

**Step 2** Aggregate all attribute values of each alternative by the *W INLMSM*(2) operator and ge<sup>t</sup> the overall value of each alternative *ai* denoted as *ri*(*<sup>i</sup>* = 1, 2, 3, 4)

$$\begin{array}{l} r\_1 = \langle \text{s}\_{5.4988}, \langle [0.3221, 0.4549], [0.1387, 0.2401], [0.3374, 0.4752] \rangle \rangle\_t \\ r\_2 = \langle \text{s}\_{5.3735}, \langle [0.5264, 0.6938], [0.1358, 0.2070], [0.1772, 0.2745] \rangle \rangle\_t \\ r\_3 = \langle \text{s}\_{5.0083}, \langle [0.4296, 0.5610], [0.1069, 0.2702], [0.2449, 0.3721] \rangle \rangle\_t \\ r\_4 = \langle s\_{4.5371}, \langle [0.4892, 0.6244], [0.0634, 0.1698], [0.1390, 0.2381] \rangle \rangle \end{array}$$

**Step 3** According to Definition 11, we assume *α* = 0.7, calculate the score values of *ri*(*<sup>i</sup>* = 1, 2, 3, <sup>4</sup>), and ge<sup>t</sup> the values shown below.

$$S\_{(r\_1)} = \mathbf{s}\_{0.5722\prime} S\_{(r\_2)} = \mathbf{s}\_{0.7276\prime} S\_{(r\_3)} = \mathbf{s}\_{0.6087\prime} S\_{(r\_4)} = \mathbf{s}\_{0.5839}$$

**Step 4** According to Step 3 and Definition 12, we ge<sup>t</sup> the rankings of the alternatives, which are shown below.

$$A\_2 \succ A\_3 \succ A\_4 \succ A\_1$$

#### *6.3. Comparative Analysis and Discussion*


or *P*2 = 0, the interrelationship between the attributes doesn't need to be considered, so it can ge<sup>t</sup> the same ranking results as the ones when *m* = 1.


**Table 2.** Comparison of different operator.


**Table 3.** Comparisons of different values of *P*1 and *P*2 when *m* = 2.

Furthermore, in order to verify the validity of the methods proposed in this paper, we can compare them with methods from Ye [16] and the ranking results are shown in Table 4.

From Table 4, we know that the best choice is *A*2 for all methods, which is the same as the results produced above. However, the ranking results are different. Compared with the approach proposed by Ye [16], when *m* = 1, our ranking results have the same values as that of Ye [16], but when *m* = 2, our ranking results are different from the Ye method [16]. When *m* = 1, all methods don't consider the interrelationship. They produce the same results, however, when *m* = 2. Our methods in this paper can take into account the interrelationship while the method by Ye [16] doesn't consider the interrelationship. Therefore, there are different ranking results. Therefore, our methods are more suitable for the different applications.



From the above comparison results, we can obtain that the methods proposed by this paper are feasible and adaptable for the MADM problems. Additionally, they have better reliability and wider application space than other existing methods.
