**Proof.**

(1) Since *A*(*α*+,*β*+,*<sup>γ</sup>*+) ⊆ *<sup>H</sup>*(*<sup>α</sup>*, *β*, *γ*) ⊆ *A*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*) for all (*<sup>α</sup>*, *β*, *γ*) ∈ *L*, we have

$$A = \underset{(a,\mathfrak{E},\gamma)\in L}{\cup} (a,\mathfrak{E},\gamma) \\ A^{(a+,\mathfrak{E}+,\gamma+)} \subseteq \underset{(a,\mathfrak{E},\gamma)\in L}{\cup} (a,\mathfrak{E},\gamma) \\ H(a,\mathfrak{E},\gamma) \subseteq \underset{(a,\mathfrak{E},\gamma)\in L}{\cup} (a,\mathfrak{E},\gamma) \\ A^{(a,\mathfrak{E},\gamma)} = A.$$

Thus, *A* = ∪ (*<sup>α</sup>*,*β*,*<sup>γ</sup>*)∈*<sup>L</sup>*(*<sup>α</sup>*, *β*, *<sup>γ</sup>*)*H*(*<sup>α</sup>*, *β*, *<sup>γ</sup>*). (2) From *α*1 < *α*2, *β*1 > *β*2, *γ*1 > *γ*2, we can obtain

$$H(a\_1, \beta\_1, \gamma\_1) \supseteq A^{(a\_1 +, \beta\_1 +, \gamma\_1 +)} \supseteq A^{(a\_2, \beta\_2, \gamma\_2)} \supseteq H(a\_2, \beta\_2, \gamma\_2).$$

(3) (*I*) Suppose ∑ = {(*<sup>λ</sup>*, *μ*, *ω*)|*<sup>λ</sup>* < *α*, *μ* > *β*, *ω* > *γ*, 0 ≤ *λ* + *μ* + *ω* ≤ 3 }, then,

$$
\vee\_{(\lambda,\mu,\omega)\in\Sigma}(\lambda,\mu,\omega) = (\mathfrak{a},\mathfrak{b},\gamma).
$$

So, ∩{*H*(*<sup>λ</sup>*, *μ*, *ω*)|*<sup>λ</sup>* < *α*, *μ* > *β*, *ω* > *γ*, 0 ≤ *λ* + *μ* + *ω* ≤ 3 } ⊆ <sup>∩</sup>*A*(*<sup>λ</sup>*,*μ*,*<sup>ω</sup>*)|*<sup>λ</sup>* < *α*, *μ* > *β*, *ω* > *γ*, 0 ≤ *λ* + *μ* + *ω* ≤ 3= *A*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*). On the other hand, since *x* ∈ *<sup>A</sup>*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*), we have *TjA*(*x*) ≥ *α*, *<sup>I</sup>jA*(*x*) ≤ *β*, *<sup>F</sup>jA*(*x*) ≤ *γ*. Thus, *TjA*(*x*)<sup>≥</sup> *α* > *λ*, *<sup>I</sup>jA*(*x*) ≤ *β* < *μ*, *<sup>F</sup>jA*(*x*) ≤ *γ* < *ω*. That is, *TjA*(*x*) > *λ*, *<sup>I</sup>jA*(*x*) < *μ*, *<sup>F</sup>jA*(*x*) < *ω*. Thus, *x* ∈ *A*(*<sup>λ</sup>*+,*μ*+,*<sup>ω</sup>*+). Thus, *x* ∈ *<sup>H</sup>*(*<sup>λ</sup>*, *μ*, *<sup>ω</sup>*). Therefore, *x* ∈ ∩{*H*(*<sup>λ</sup>*, *μ*, *ω*)|*<sup>λ</sup>* < *α*, *μ* > *β*, *ω* > *γ*, 0 ≤ *λ* + *μ* + *ω* ≤ 3 }. Based on the above facts, we can obtain *A*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*) = ∩{*H*(*<sup>λ</sup>*, *μ*, *ω*)|*<sup>λ</sup>* < *α*, *μ* > *β*, *ω* > *γ*, 0 ≤ *λ* + *μ* + *ω* ≤ 3 }.

(*II*) Since *A*(*α*+,*β*+,*<sup>γ</sup>*+) ⊇ *A*(*<sup>λ</sup>*, ,*μ*,*<sup>γ</sup>*) ⊇ *<sup>H</sup>*(*<sup>λ</sup>*, *μ*, *γ*) for any *λ* > *α*, *μ* < *β*, *ω* < *γ*, 0 ≤ *λ* + *μ* + *ω* ≤ 3, we have

*A*(*α*+,*β*+,*<sup>γ</sup>*+) ⊇ ∪{*H*(*<sup>λ</sup>*, *μ*, *ω*)|*<sup>λ</sup>* > *α*, *μ* < *β*, *ω* < *γ*, 0 ≤ *λ* + *μ* + *ω* ≤ 3 }. On the other hand, from *x* ∈ *A*(*α*+,*β*+,*<sup>γ</sup>*+) we have *TjA*(*x*) > *α*, *<sup>I</sup>jA*(*x*) < *β*, *<sup>F</sup>jA*(*x*) < *γ*. It follows that there exists *λ* > *α*, *μ* < *β*, *ω* < *γ*, 0 ≤ *λ* + *μ* + *ω* ≤ 3, such that *TjA*(*x*) > *λ* > *α*, *<sup>I</sup>jA*(*x*) < *μ* < *β*, *<sup>F</sup>jA*(*x*) < *ω* < *γ*, that is, *x* ∈ *A*(*<sup>λ</sup>*+,*μ*+,*<sup>ω</sup>*+). Indeed, *A*(*<sup>λ</sup>*+,*μ*+,*<sup>ω</sup>*+) ⊆ *<sup>H</sup>*(*<sup>λ</sup>*, *μ*, *<sup>ω</sup>*), then, *x* ∈ *<sup>H</sup>*(*<sup>λ</sup>*, *μ*, *<sup>ω</sup>*). Thus, *x* ∈ ∪{ *<sup>H</sup>*(*<sup>λ</sup>*, *μ*, *ω*)|*<sup>λ</sup>* > *α*, *μ* < *β*, *ω* < *γ*, 0 ≤ *λ* + *μ* + *ω* ≤ <sup>3</sup>}. Thus,

*A*(*α*+,*β*+,*<sup>γ</sup>*+) ⊆ ∪{*H*(*<sup>λ</sup>*, *μ*, *ω*)|*<sup>λ</sup>* > *α*, *μ* < *β*, *ω* < *γ*, 0 ≤ *λ* + *μ* + *ω* ≤ 3 }. Therefore, we can obtain

$$A^{(\mathfrak{a}+,\mathfrak{b}+,\gamma+)} = \cup \{ H(\lambda,\mu,\omega) | \lambda > \mathfrak{a}, \mu < \mathfrak{b}, \omega < \gamma, 0 \le \lambda + \mu + \omega \le 3 \}.$$

(4) From *A*(*α*+,*β*+,*<sup>γ</sup>*+) ⊆ *<sup>H</sup>*(*<sup>α</sup>*, *β*, *γ*) ⊆ *<sup>A</sup>*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*), we have ∩*<sup>t</sup>*∈*T<sup>H</sup>*(*<sup>α</sup>t*, *β<sup>t</sup>*, *<sup>γ</sup>t*) ⊆ ∩*<sup>t</sup>*∈*TA*(*<sup>α</sup>t*,*β<sup>t</sup>*,*γt*) ⊆ *A*(*α* ,*β* ,*<sup>γ</sup>* ) for *α* = <sup>∨</sup>*t*∈*Tαt*,*β* = <sup>∧</sup>*t*∈*<sup>T</sup>β<sup>t</sup>*,*<sup>γ</sup>* = ∧*t*∈*Tγt*. Applying (3) (I), we ge<sup>t</sup>

$$A^{(\mathfrak{a}',\mathfrak{b}',\gamma')} = \cap \{ \, \!\!\!/ \!\!\!/ (\mathfrak{a},\mathfrak{b},\gamma) \vert \, \mathfrak{a} < \mathfrak{a}', \mathfrak{b} > \mathfrak{b}', \gamma > \gamma', 0 \le \mathfrak{a} + \mathfrak{b} + \gamma \le 3 \} + \gamma \le 3 \}.$$

Therefore,

$$\bigcap\_{t \in T} H(a\_t, \beta\_t, \gamma\_t) \subseteq \cap \{ H(a, \beta, \gamma) \mid a < a', \beta > \beta', \gamma > \gamma', 0 \le a + \beta + \gamma \le 3 \} \subset \Gamma$$

**Remark 5.** *(1) The significance of Theorem 3 (Decomposition Theorem): A SVNMS can be composed of neutrosophic nested sets which consist of self-decomposed cut sets or strong cut sets. (2) The significance of* *Theorem 4 (Generalized Decomposition Theorem): A collection of family sandwiched between cut or strong cut sets of a SVNMS must be neutrosophic nested sets, and such nested sets can also compose the original SVNMS.*

#### *4.2. Representation Theorem of SVNMS*

According to the relationship between the decomposition theorem and the representation theorem, we can obtain that each neutrosophic nested set can be combined into a single-valued neutrosophic multiset. Furthermore, its cut sets or strong cut sets can be constructed with the original neutrosophic nested set. In other words, it is theoretically explained: a family of special single-valued neutrosophic multisets can be used to completely depict and represent a single-valued neutrosophic multiset).

In this section, the representation theorem of SVNMS based on the decomposition theorem is proposed in this section.

**Theorem 5.** *Let H* ∈ *SVNL*(*X*)*, A* ∈ *SVNMS*(*X*)*, and* ∀(*<sup>α</sup>*, *β*, *γ*) ∈ *L. We have*

*(I) A*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*) = ∩{*H*(*<sup>λ</sup>*, *μ*, *ω*)|*<sup>λ</sup>* < *α*, *μ* > *β*, *ω* > *γ*, 0 ≤ *λ* + *μ* + *ω* ≤ 3 }*; (II) A*(*α*+,*β*+,*<sup>γ</sup>*+) = ∪{*H*(*<sup>λ</sup>*, *μ*, *ω*)|*<sup>λ</sup>* > *α*, *μ* < *β*, *ω* < *γ*, 0 ≤ *λ* + *μ* + *ω* ≤ 3 }.

**Proof.** Since *<sup>H</sup>*(*<sup>α</sup>*, *β*, *γ*) ∈ 2*<sup>X</sup>* for all (*<sup>α</sup>*, *β*, *γ*) ∈ *L*, and (*<sup>α</sup>*, *β*, *<sup>γ</sup>*)*H*(*<sup>α</sup>*, *β*, *γ*) ∈ *SVNMS*(*X*), we have <sup>∪</sup>(*<sup>α</sup>*,*β*,*<sup>γ</sup>*)∈*<sup>L</sup>*(*<sup>α</sup>*, *β*, *<sup>γ</sup>*)*H*(*<sup>α</sup>*, *β*, *γ*) ∈ *SVNMS*(*X*), denoted by *A*. Applying Theorem 4, we only need to prove,

$$H: L \to \mathbb{2}^X \text{ satisfies } A^{(a+,\emptyset+,\gamma+)} \subseteq H(a,\emptyset,\gamma) \subseteq A^{(a,\emptyset,\gamma)}.$$

Since *x* ∈ *<sup>A</sup>*(*α*+,*β*+,*<sup>γ</sup>*+), we have *TjA*(*x*) > *α*, *<sup>I</sup>jA*(*x*) < *β*, *<sup>F</sup>jA*(*x*) < *γ*. Thus, <sup>∨</sup>(*<sup>λ</sup>*,*μ*,*<sup>ω</sup>*)∈*<sup>L</sup>*)*<sup>λ</sup>* ∧ (*H*(*λ*))*j*(*x*)\*> *α*, <sup>∧</sup>(*<sup>λ</sup>*,*μ*,*<sup>ω</sup>*)∈*<sup>L</sup>*)*<sup>μ</sup>* ∨ (*H*(*μ*))*j*(*x*)\* < *β*, <sup>∧</sup>(*<sup>λ</sup>*,*μ*,*<sup>ω</sup>*)∈*<sup>L</sup>*)*<sup>ω</sup>* ∨ (*H*(*ω*))*j*(*x*)\* < *γ*. It follows that there exists (*<sup>λ</sup>*0, *μ*0, *<sup>ω</sup>*0) ∈*L*, such that *λ*0 ∨ (*H*(*<sup>λ</sup>*0))*j*(*x*) > *α*, *μ*0 ∨ (*H*(*μ*0))*j*(*x*) < *β*, *ω*0 ∨ (*H*(*<sup>ω</sup>*0))*j*(*x*) < *γ*, that is, *λ*0 > *α*, *μ*0 < *β*,*ω*<sup>0</sup> < *γ*. Taking (*H*(*<sup>λ</sup>*0, *μ*0, *<sup>ω</sup>*0))*j*(*x*) = (1, 1, <sup>1</sup>), we have (*<sup>λ</sup>*0, *μ*0, *<sup>ω</sup>*0) > (*<sup>α</sup>*, *β*, *<sup>γ</sup>*). Thus, *x* ∈ *<sup>H</sup>*(*<sup>λ</sup>*0, *μ*0, *<sup>ω</sup>*0) ⊆*<sup>H</sup>*(*<sup>α</sup>*, *β*, *<sup>γ</sup>*). On the other hand, from *x* ∈ *<sup>H</sup>*(*<sup>α</sup>*, *β*, *<sup>γ</sup>*), we have (*H*(*<sup>λ</sup>*, *μ*, *<sup>ω</sup>*))*j*(*x*) = (1, 1, <sup>1</sup>). Thus, <sup>∨</sup>(*<sup>λ</sup>*,*μ*,*<sup>ω</sup>*)∈*<sup>L</sup>*)*<sup>λ</sup>* ∨ (*H*(*λ*))*j*(*x*)\* ≥ *α* ∧ (*H*(*α*))*j*(*x*) = *α*, <sup>∨</sup>(*<sup>λ</sup>*,*μ*,*<sup>ω</sup>*)∈*<sup>L</sup>*)*<sup>μ</sup>* ∨ (*H*(*μ*))*j*(*x*)\* ≤ *β* ∧ (*H*(*β*))*j*(*x*) = *β* and

<sup>∨</sup>(*<sup>λ</sup>*,*μ*,*<sup>ω</sup>*)∈*<sup>L</sup>*)*<sup>ω</sup>* ∨ (*H*(*ω*))*j*(*x*)\*≤ *γ* ∧ (*H*(*γ*))*j*(*x*) = *γ*, that is, *TjA*(*x*) ≥ *α*, *<sup>I</sup>jA*(*x*) ≤ *β*, *<sup>F</sup>jA*(*x*) ≤ *γ*. Thus, *x* <sup>∈</sup>*A*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*). Therefore, *A*(*α*+,*β*+,*<sup>γ</sup>*+) ⊆ *<sup>H</sup>*(*<sup>α</sup>*, *β*, *γ*) ⊆ *A*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*) for *j* = 1, 2, ··· *l*(*x* : *A*, *<sup>B</sup>*). -

Theorem 5 (Representation Theorem) provides an effective method for constructing a SVNMS: Let *H* ∈ *SVNL*(*X*), we can construct a SVNMS with the following membership function:

$$A: X \to L, \ A(\mathbf{x}) = \vee \{ (\mathfrak{a}, \mathfrak{z}, \gamma) \in L | \mathbf{x} \in H(\mathfrak{a}, \mathfrak{z}, \gamma) \}, \ \forall \mathbf{x} \in X$$

**Example 2.** *Suppose X* = {*<sup>x</sup>*1, *x*2, *<sup>x</sup>*3}*. The neutrosophic nested sets on the given X is as follows:*

$$H(\mathbf{a}, \boldsymbol{\beta}, \gamma) = \{ \langle \mathbf{x}\_1, (1, 1), (0, 0), (0, 0) \rangle, \langle \mathbf{x}\_2, 1, 0, 0 \rangle, \langle \mathbf{x}\_3, (1, 1, 1), (0, 0, 0), (0, 0, 0) \rangle \},$$

*where α* = *β* = *γ* = 0;

$$H(\mathbf{a}, \boldsymbol{\beta}, \gamma) = \{ \langle \mathbf{x}\_1, (1, 1), (0, 0), (0, 0) \rangle, \langle \mathbf{x}\_2, 1, 0, 0 \rangle, \langle \mathbf{x}\_3, (1, 1, 1), (1, 1, 0), (1, 0, 0) \rangle \},$$

*where* 0 < *α* ≤ 0.2*,* 0 < *β* ≤ 0.2*,* 0 < *γ* ≤ 0.1;

$$H(\mathfrak{a}, \mathfrak{b}, \gamma) = \{ \langle \mathfrak{x}\_1, (1, 1), (0, 1), (1, 0) \rangle, \langle \mathfrak{x}\_2, 0, 0, 0 \rangle, \langle \mathfrak{x}\_3, (1, 1, 1), (1, 1, 1), (1, 0, 0) \rangle \},$$

*where* 0.2 < *α* ≤ 0.4*,* 0.2 < *β* ≤ 0.3*,* 0.1 < *γ* ≤ 0.2;

$$H(\mathbf{a}, \boldsymbol{\beta}, \gamma) = \{ \langle \mathbf{x}\_1, (1, 0), (0, 1), (1, 1) \rangle, \langle \mathbf{x}\_2, 0, 1, 0 \rangle, \langle \mathbf{x}\_3, (1, 1, 1), (1, 1, 1), (1, 1, 0) \rangle \},$$

*where* 0.4 < *α* ≤ 0.5*,* 0.3 < *β* ≤ 0.4*,* 0.2 < *γ* ≤ 0.3;

$$H(\mathbf{a}, \boldsymbol{\beta}, \gamma) = \{ \langle \mathbf{x}\_1, (1, 0), (1, 1), (1, 1) \rangle, \langle \mathbf{x}\_2, 0, 1, 0 \rangle, \langle \mathbf{x}\_3, (1, 1, 0), (1, 1, 1), (1, 1, 1) \rangle \},$$

*where* 0.5 < *α* ≤ 0.6*,* 0.4 < *β* ≤ 0.5*,* 0.3 < *γ* ≤ 0.4;

$$H(\mathfrak{a}, \mathfrak{b}, \gamma) = \{ \langle \mathbf{x}\_1, (0, 0), (1, 1), (1, 1) \rangle, \langle \mathbf{x}\_2, 0, 1, 1 \rangle, \langle \mathbf{x}\_{3\prime}(1, 0, 0), (1, 1, 1), (1, 1, 1) \rangle \},$$

*where* 0.6 < *α* ≤ 0.8*,* 0.5 < *β* ≤ 1*,* 0.4 < *γ* ≤ 0.7;

$$H(\mathbf{a}, \boldsymbol{\beta}, \gamma) = \{ \langle \mathbf{x}\_{1\prime}(0, 0), (1, 1), (1, 1) \rangle, \langle \mathbf{x}\_{2\prime}0, 1, 1 \rangle, \langle \mathbf{x}\_{3\prime}(0, 0, 0), (1, 1, 1), (1, 1, 1) \rangle \},$$

*where* 0.8 < *α* ≤ 1*,* 0.5 < *β* ≤ 1*,* 0.7 < *γ* ≤ 1.

> *Similarly, we can give the remaining neutrosophic nested sets. Then, the SVNMS A determined by H has the following membership function:*


*Therefore,*

$$A = \{ \langle \mathbf{x}\_1, (0.6, 0.4), (0.5, 0.3), (0.2, 0.3) \rangle, \langle \mathbf{x}\_2, 0.2, 0.4, 0.7 \rangle, \langle \mathbf{x}\_3, (0.8, 0.6, 0.5), (0.2, 0.2, 0.3), (0.1, 0.3, 0.4) \rangle \}.$$

#### **5. New Similarity Measure between SVNMSs**

On the basis of the decomposition theorem of SVNMS, this section presents a new similarity measure between SVNMSs. Then, we discuss the properties of this new similarity measure and give a concrete algorithm by example.

**Definition 14.** *Let M* = *x*, *TjM*(*x*), *<sup>I</sup>jM*(*x*), *<sup>F</sup>jM*(*x*)HHH*x* ∈ *X*, *j* = 1, 2, ··· , *l*(*x* : *M*) *and N* = *<sup>x</sup>*, *TjN*(*x*), *<sup>I</sup>jN*(*x*), *<sup>F</sup>jN*(*x*)HHH*x* ∈ *X*, *j* = 1, 2, ··· , *l*(*x* : *N*) *be two SVNMSs in X. Suppose V* = [0, 1] × [0, 1] × [0, 1]*. Then, we define a new distance measure between M and N as follows:*

$$D\_{\mathbb{C}}(\mathcal{M}, \mathcal{N}) = \bigcap\_{\mathcal{V}} \bigcap\_{\mathcal{V}} f(\mathfrak{a}\_{\prime} \mathfrak{d}\_{\prime} \gamma)dV$$

*where f*(*<sup>α</sup>*, *β*, *γ*) = *DP*(*<sup>α</sup>*, *β*, *<sup>γ</sup>*)*M*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*),(*<sup>α</sup>*, *β*, *<sup>γ</sup>*)*N*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*), *α* ∈ [0, 1]*, β* ∈ [0, 1]*, γ* ∈ [0, 1].

**Proposition 2.** *Let M, N be two SVNMSs in X*. *Then, the following properties hold (DC1-DC4): (DC1)* 0 ≤ *DC*(*<sup>M</sup>*, *N*) ≤ 1*;*

*(DC2) DC*(*<sup>M</sup>*, *N*) = 0 *if and only if M* = *N; (DC3) DC*(*<sup>M</sup>*, *N*) = *DC*(*<sup>M</sup>*, *<sup>N</sup>*)*; (DC4) If Q is a SVNMS in X and M* ⊆ *N* ⊆ *Q, then, DC*(*<sup>M</sup>*, *Q*) ≤ *DC*(*<sup>M</sup>*, *N*) + *DC*(*<sup>N</sup>*, *Q*) *for P* > 0.

According to the relationship between distance measure and similarity measures, we can introduce two distance-based similarity measures between *M* and *N*:

$$S\_{\mathbb{C}1}(M, N) = 1 - D\_{\mathbb{C}}(M, N) \tag{25}$$

$$S\_{\mathbb{C}2}(M,N) = \frac{1 - D\_{\mathbb{C}}(M,N)}{1 + D\_{\mathbb{C}}(M,N)} \tag{26}$$

**Proposition 3.** *Let M*, *N* ∈ *SVNMS*(*X*)*. The distance-based similarity measures SC <sup>f</sup>*(*<sup>M</sup>*, *<sup>N</sup>*)*,* (*f* = 1, 2) *hold the following properties (SC1-SC4):*

*(SC1)* 0 ≤ *SC <sup>f</sup>*(*<sup>M</sup>*, *N*) ≤ 1*; (SC2) SC <sup>f</sup>*(*<sup>M</sup>*, *N*) = 1 *if and only if M* = *N; (SC3) SC <sup>f</sup>*(*<sup>M</sup>*, *N*) = *SC <sup>f</sup>*(*<sup>N</sup>*, *<sup>M</sup>*)*; (SC4) If Q is a SVNMS in X and M* ⊆ *N* ⊆ *Q, then SC <sup>f</sup>*(*<sup>M</sup>*, *Q*) ≤ *SC <sup>f</sup>*(*<sup>M</sup>*, *N*) + *SC <sup>f</sup>*(*<sup>N</sup>*, *Q*).

**Proof.** The proofs of proposition 2 and 3 are straightforward. -

This method is based on the cut sets, and uses the idea of the decomposition theorem to convert the similarity measure between the two SVNMSs into the similarity measure between the corresponding special SVNMSs. Now, let us use a concrete example to illustrate the specific algorithm.

**Example 3.** *Let X* = {*<sup>x</sup>*1, *x*2, *<sup>x</sup>*3}*, M*, *N* ∈ *SVNMS*(*X*)*. That is, M* = {*<sup>x</sup>*1,(0.7, 0.8),(0.1, 0.2),(0.2, 0.3),*<sup>x</sup>*2, (0.5, 0.6),(0.2, 0.3),(0.4, 0.5)}*, N* = {*<sup>x</sup>*1,(0.5, 0.6),(0.1, 0.2),(0.4, 0.5),*<sup>x</sup>*2,(0.6, 0.7),(0.1, 0.2),(0.7, 0.8)}.

*According to the values of TjM*(*xi*)*, TjN*(*xi*)(*<sup>i</sup>* = 1, 2; *j* = 1, <sup>2</sup>)*, we divide the interval* [0, 1] *of α into 5 subintervals:* [0, 0.5]*,* (0.5, 0.6]*,* (0.6, 0.7]*,* (0.7, 0.8]*,* (0.8, 1]*. Similarly, we can obtain 4 subintervals of β:* [0, 0.1]*,* (0.1, 0.2]*,* (0.2, 0.3]*,* (0.3, 1]*, and 7 subintervals of γ:* [0, 0.2]*,* (0.2, 0.3]*,* (0.3, 0.4]*,* (0.4, 0.5]*,* (0.5, 0.7]*,* (0.7, 0.8]*,* (0.8, 1]*. Thus, we have 140 interval combinations of α, β, and γ, take* 0 ≤ *α* ≤ 0.5*,* 0.2 < *β* ≤ 0.3*,* 0.7 < *γ* ≤ 0.8 for example. In this way, for each combination of interval, we can ge<sup>t</sup> the corresponding *<sup>M</sup>*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*)*, N*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*) *and* (*<sup>α</sup>*, *β*, *<sup>γ</sup>*)*M*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*)*,* (*<sup>α</sup>*, *β*, *<sup>γ</sup>*)*N*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*). Based on the above results, the process is as follows:

*Step1, calculate f*(*<sup>α</sup>*, *β*, *γ*) = *DP*(*<sup>α</sup>*, *β*, *<sup>γ</sup>*)*M*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*),(*<sup>α</sup>*, *β*, *<sup>γ</sup>*)*N*(*<sup>α</sup>*,*β*,*<sup>γ</sup>*)in every interval combination. *Step2, use Equation (24) to perform the integral operation on f*(*<sup>α</sup>*, *β*, *γ*) *over V* = [0, 1] × [0, 1] × [0, 1]*, and get DC*(*<sup>M</sup>*, *N*) = 0.2206.

*Step3, using Equation (25) and (26), we can get SC*1(*<sup>M</sup>*, *N*) = 0.7794 *and SC*2(*<sup>M</sup>*, *N*) = 0.6385.

#### **6. Application of New Similarity Measures in Multicriteria Decision-Making Problems**

In this section, the new similarity measure is applied to a medical diagnosis problem. Next, we use the typical examples in [14] to verify the feasibility and effectiveness of the new similarity measure proposed in Section 5. Furthermore, we analyze the uniqueness of the new similarity measure by comparing the results with other similarity measures.

Assume that *I* = {*<sup>I</sup>*1, *I*2, *I*3, *<sup>I</sup>*4} represents 4 patients, set *R* = {*<sup>R</sup>*1, *R*2, *R*3, *<sup>R</sup>*4} = {viral fever, tuberculosis, typhoid, throat disease} indicates 4 diseases, and set *S* = {*<sup>S</sup>*1, *S*2, *S*3, *<sup>S</sup>*4} = {temperature, cough, sore throat, headache, body pain} indicates 5 symptoms. In medical diagnosis, in order to obtain a more accurate diagnosis, the doctor collects symptom information for the same patient at different times of the day. Therefore, we use the following SVNMSs to indicate the affiliation between the patient and the symptom:


Then, the affiliation between the symptoms and the disease is represented by the following SVNMSs:

*R*1 = {*<sup>S</sup>*1, 0.8, 0.1, 0.1,*<sup>S</sup>*2, 0.2, 0.7, 0.1,*<sup>S</sup>*3, 0.3, 0.5, 0.2,*<sup>S</sup>*4, 0.5, 0.3, 0.2,*<sup>S</sup>*5, 0.5, 0.4, 0.1}; *R*2 = {*<sup>S</sup>*1, 0.2, 0.7, 0.1,*<sup>S</sup>*2, 0.9, 0.0, 0.1,*<sup>S</sup>*3, 0.7, 0.2, 0.1,*<sup>S</sup>*4, 0.6, 0.3, 0.1,*<sup>S</sup>*4, 0.7, 0.2, 0.1}; *R*3 = {*<sup>S</sup>*1, 0.5, 0.3, 0.2,*<sup>S</sup>*2, 0.3, 0.5, 0.2,*<sup>S</sup>*3, 0.2, 0.7, 0.1,*<sup>S</sup>*4, 0.2, 0.6, 0.2,*<sup>S</sup>*5, 0.4, 0.4, 0.2}; *R*4 = {*<sup>S</sup>*1, 0.1, 0.7, 0.2,*<sup>S</sup>*2, 0.3, 0.6, 0.1,*<sup>S</sup>*3, 0.8, 0.1, 0.1,*<sup>S</sup>*4, 0.1, 0.8, 0.1,*<sup>S</sup>*5, 0.1, 0.8, 0.1}.

Then, by Definition 14, we use Equations (24) and (25) to ge<sup>t</sup> the similarity *SC*1-*Ii*, *Rj*. between each patient *Ii*(*i* = 1, 2, 3, 4) and disease *Rj*(*j* = 1, 2, 3, <sup>4</sup>), which are shown in Table 1. Similarly, we use Equations (24) and (26) to ge<sup>t</sup> the similarity *SC*2-*Ii*, *Rj*. between each patient *Ii*(*i* = 1, 2, 3, 4) and disease *Rj*(*j* = 1, 2, 3, <sup>4</sup>), which are shown in Table 2.


**Table 1.** Similarity values of *SC*1*Ii*, *Rj*.

**Table 2.** Similarity values of *SC*2*Ii*, *Rj*.


It is well known that the closeness of the relationship between two SVNMSs can be described by the similarity between the two, that is, the greater the similarity, the closer the relationship is. As can be seen from Tables 1 and 2, for these four diseases, by comparison, we can determine the most similar disease to each patient and ge<sup>t</sup> the ge<sup>t</sup> the most realistic diagnosis: patient *I*1 suffers from typhoid, patient *I*2 suffers from tuberculosis, patient *I*3 suffers from throat disease, and patient *I*4 also suffers from typhoid.

The dice similarity measures proposed in [11] are applied to the decision-making example, and the diagnosis is that patient *I*1 suffers from typhoid, patient *I*2 suffers from viral fever, patient *I*3 suffers from typhoid, and patient *I*4 suffers from tuberculosis. The distance-based similarity measures proposed in [14] also are applied in this decision-making example, and the diagnosis is that patient *I*1 suffers from viral fever, patient *I*2 suffers from tuberculosis, patient *I*3 suffers from typhoid, and patient *I*4 suffers from typhoid.

By analyzing and comparing the diagnostic results obtained by the three methods, we found that when using the new similarity to calculate, the diagnosis of disease in patient *I*1 is consistent with [11] and the diagnosis of patients *I*2 and *I*4 was consistent with [14], indicating that this method is more effective, because the results are closer to the actual situation.

According to the above comparative analysis, the method proposed in this paper has the following advantages: (1) The new similarity measure under the SVNMSs environment can deal with the indeterminacy and inconsistent information which exists in decision-making problems, that is, it can be effectively used in many practical applications. (2) The new similarity measure is based on the cut sets, with the decomposition theorem and the representation theorem as the main ideas, and the integral as the main mathematical tool. Therefore, it has a solid mathematical theoretical basis. (3) This method can make full use of all the information of SVNMSs, and use the idea of splitting and summing to simplify complex problem, provide a simple and effective method for solving practical problems.
