**1. Introduction**

In the real world, the linguistic expression is well-suited for the thinking and expressing patterns of human beings. Due to the vagueness of languages and the complexity of decision-making environments, the linguistic fuzzy theory has been well developed in the past decades and shows irreplaceable advantages in the fuzzy decision-making field. Linguistic variables (LVs) were defined for fuzzy reasoning and decision-making [1–4]. Linguistic uncertain variables [5,6] (interval-valued linguistic variables) were then defined to depict uncertain linguistic information in decision-making problems [7,8]. After that, a linguistic intuitionistic fuzzy number (LIFN) [9], which contains two independent LVs to describe the degrees of truth and falsity, respectively, was presented to handle the uncertainty and incompleteness in linguistic decision-making environments [10]. Furthermore, with the wide application of the neutrosophic theory in decision-making [11–13], Fang and Ye [14] proposed a linguistic neutrosophic number (LNN) by adding a new LV to the LIFN for representing the indeterminacy degree to do with the indeterminate and inconsistent linguistic information [15]. Although there exist some research works on LNNs [14,15], existing LNNs cannot depict the hesitancy of decision-makers (DMs) in the linguistic assessment of alternatives.

Concerning the handling of the human hesitant cognition in decision-making environments, many works have been published so far. Torra and Narukawa [16] and Torra [17] originally introduced hesitant fuzzy sets (HFSs) to express the hesitancy by allowing the membership to contain several possible values. Then, for linguistic decision-making problems, the expression of a hesitant fuzzy

linguistic set (HFLS) [18] was obtained based on combining a linguistic term (LT) set with a HFS so as to satisfy the hesitant linguistic evaluation requirements [19,20] of DMs. After that, an interval-valued HFLS [21] was presented as an extension form by combining an interval-valued LT set with a HFS. Recently, Ye [22] proposed the hesitant neutrosophic linguistic number (HNLN) to carry out hesitant decision-making problems with the neutrosophic linguistic number that contains partial determinacy and partial indeterminacy. However, there is no definition or decision-making method for the hesitant sets of LNNs in the existing literature. Additionally, in the hesitant linguistic expressions of DMs, the components between two hesitant sets generally have difference in their length sizes, and thus it is difficult to directly perform measure calculations between hesitant sets. Thus, several researchers have proposed some extension methods to extend the shorter items in the two hesitant sets by adding the minimum values, maximum values, or any values [23,24] to reach their identical length. However, these extension methods depend too much on the subjective preferences and interests of the DMs. To solve this problem, we have already introduced the least common multiple cardinality (LCMC) to extend the hesitant fuzzy elements in our previous research works [22,25], which become more objective for the decision-making calculation of HFSs.

As aforementioned, there is a gap of hesitant LNNs in existing studies. For instance, suppose that we hesitate between two single-valued LNNs, <sup>&</sup>lt;*h*7, *h*3, *h*4<sup>&</sup>gt; and <sup>&</sup>lt;*h*5, *h*3, *h*1>, from the given LT set *H* = {*hs*|*s* ∈ [0, 8]} regarding an evaluated object. However, it is difficult to express the hesitation information and the LNN information of the DMs simultaneously by a unique LNN or a unique HFS. Therefore, for the purposes of satisfying the demand of hesitant decision-making with LNNs and ensuring the objectivity of the measure calculation, this paper aims to (i) define the concept of HLNNs by combining HFSs with LNNs, (ii) present the LCMC-based generalized distance and similarity measures of HLNNs for more objective measure calculation of HLNN information, and (iii) to propose a novel multiple-attribute decision-making (MADM) method based on the proposed LCMC-based similarity measure in the HLNN setting.

In order to do so, Section 2 briefly reviews LNNs. Section 3 defines a HLNN and a HLNN set. Then, in Section 4, the LCMC-based generalized distance and similarity measures of HLNNs are presented. In Section 5, a new MADM method was developed by using the proposed similarity measure of HLNNs. In Section 6, the feasibility of the proposed approach is demonstrated by an investment case. The conclusions and future research of HLNNs are discussed in the last section.

#### **2. Linguistic Neutrosophic Numbers (LNNs)**

Fang and Ye [14] originally presented the following definition of the LNN:

**Definition 1** ([14])**.** *Let H = {h0, h1, ..., hτ} be a LT set, where τ + 1 is an odd cardinality. A LNN can be defined as ϑ =<hT, hU, hF> for hT, hU, hF* ∈ *H and T, U, F* ∈ *[0, τ], where hT, hU, hF represent the degrees of truth, indeterminacy, and falsity, respectively.*

For the comparison of LNNs, the score and accuracy functions of LNNs are defined as follows [14]:

**Definition 2** ([14])**.** *Let ϑ = <hT, hU, hF> be a LNN in H. Then its score function can be given by:*

$$S(\theta) = (2\pi + T - lI - F) / 3\pi \text{ for } S(\theta) \in [0, 1], \tag{1}$$

*and its accuracy function can be expressed as*

$$V(\theta) = (T - F) / \tau \text{ for } V(\theta) \in [-1, 1]. \tag{2}$$

**Definition 3** ([14])**.** *Let ϑα* = < *hTα* , *hUα* , *hFα* > *and ϑβ* = < *hTβ* , *hUβ* , *hFβ* > *be two LNNs in H. There exist the following relations:*


#### **3. Hesitant Linguistic Neutrosophic Numbers (HLNNs) and HLNN Set**

Torra and Narukawa [16] and Torra [17] first defined the HFS as follows:

**Definition 4** ([16,17])**.** *Assume S is a universe set, then a HFS N on S can be given by*

$$N = \{  ~~| s \in \mathcal{S} \}\_s~~$$

*where E(s) is a hesitant component of N containing a set of some values in [0, 1], which represents all possible membership degrees of s.*

By integrating HFS with LNN, we define a HLNN set as follows:

**Definition 5.** *Set a universe of discourse S = {s1, s2,* ... *, sq} and a finite LT set H = {h0, h1,* ... *, hτ}, and then a HLNN set Nl on S can be expressed as*

$$N\_l = \{  |s\_j \in S, j = 1, 2, \dots, q \}$$

*where El(sj) is a set of mj LNNs*, *denoted by a HLNN El*(*sj*) = {< *hTkj* , *hUkj* , *hFkj* > *hTkj* ∈ *H*, *hUkj* ∈ *H*, *hFkj* ∈ *H*, *k* = 1, 2, ··· , *mj*} *for sj* ∈ *S.*

#### **4. LCMC-Based Distance and Similarity Measures of HLNNs**

In most situations, the cardinal numbers (the number of LNNs) of HLNNs evaluated for the same object are usually different. Thus, it is necessary to make the cardinal numbers of the two HLNNs the same to satisfy the distance and similarity measures between them.

We assume that *p* HLNNs on *S =* {*<sup>s</sup>*1, *s*2, ... , *sq*} are *El*1 (*sj*), *El*2 (*sj*), ··· , *Elp* (*sj*) for *sj* ∈ *S* (*j* = 1, 2, ..., *q*). Then, the HLNNs *Eli*(*sj*) for *i* = 1, 2, . . . , *p* can be given by

$$\begin{split} E\_{l\_1}(s\_j) &= \{ < h\_{\mathbf{T}\_{1j}^2}^2, h\_{\mathbf{L}\_{1j}^2} h\_{\mathbf{F}\_{1j}^2} > \dots \dots < h\_{\mathbf{T}\_{1j}^{m\_1j}} h\_{\mathbf{L}\_{1j}^{m\_1j}} h\_{\mathbf{F}\_{1j}^{m\_1j}} > \}, \\ E\_{l\_2}(s\_j) &= \{ < h\_{\mathbf{T}\_{2j}^2}^2, h\_{\mathbf{L}\_{2j}^2} h\_{\mathbf{F}\_{2j}^2} > \dots < h\_{\mathbf{T}\_{2j}^{m\_2j}} h\_{\mathbf{L}\_{2j}^{m\_2j}} h\_{\mathbf{F}\_{2j}^{m\_2j}} > \}, \\ \vdots \\ E\_{l\_p}(s\_j) &= \{ < h\_{\mathbf{T}\_{pj}^2} h\_{\mathbf{L}\_{pj}^2} h\_{\mathbf{F}\_{pj}^2} > \dots < h\_{\mathbf{T}\_{pj}^{m\_jj}} h\_{\mathbf{L}\_{pj}^{m\_j}} h\_{\mathbf{L}\_{pj}^{m\_j}} > \}, \end{split}$$

where *mij* is the cardinal number of *Eli*(*sj*) (*i =* 1, 2, . . . , *p* and *j =* 1, 2, . . . , *q)*.

Provided that the LCMC of *mij (i* = 1, 2, ..., *p* and *j* = 1, 2, ..., *q)* is *cj* (*j* = 1, 2, ... , *q)*, by increasing the number of LNNs < *hTkij* , *hUkij* , *hFkij* > *(k* = 1, 2, ..., *mij)* in *Eli*(*sj*) depending on *cj* (*j* = 1, 2, ... , *q*), the extended HLNN *Eoli*(*sj*) (*i* = 1, 2, ... , *p* and *j* = 1, 2, ... , *q)* will be obtained by the extension forms:

*Eol*1 (*sj*) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ *cj* < =: ; < *hT*11*j* , *hU*11*j* , *hF*11*j* >, ··· : ;< = *<sup>R</sup>*1*j* , < *hT*21*j* , *hU*21*j* , *hF*21*j* >, ··· : ;< = *<sup>R</sup>*1*j* , ··· , < *hTm*1*j* 1*j* , *hUm*1*j* 1*j* , *hFm*1*j* 1*j* >, ··· : ;< = *<sup>R</sup>*1*j* ⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭, *Eol*2 (*sj*) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ *cj* < =: ; < *hT*12*j* , *hU*12*j* , *hF*12*j* >, ··· : ;< = *<sup>R</sup>*2*j* , < *hT*22*j* , *hU*22*j* , *hF*22*j* >, ··· : ;< = *<sup>R</sup>*2*j* , ··· , < *hTm*2*j* 2*j* , *hUm*2*j* 2*j* , *hFm*2*j* 2*j* >, ··· : ;< = *<sup>R</sup>*2*j* ⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭, ··· , *Eolp* (*xj*) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ *cj* < =: ; < *hT*1*pj* , *hU*<sup>1</sup>*pj* , *hF*1*pj* >, ··· : ;< = *Rpj* , < *hT*2*pj* , *hU*<sup>2</sup>*pj* , *hF*2*pj* >, ··· : ;< = *Rpj* , ··· , < *hTmpj pj* , *hUmpj pj* , *hFmpj pj* >, ··· : ;< = *Rpj* ⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭,

where *Rij* is the number of LNNs < *hTkij* , *hUkij* , *hFkij* > (*k* = 1, 2, ..., *mij*) in *Eoli*(*xj*) (*i* = 1, 2, ... , *p* and *j* = 1, 2, ... , *q*), calculated by:

$$R\_{ij} = \frac{c\_j}{m\_{ij}}.\tag{3}$$

Additionally, the elements *ϑ<sup>σ</sup>*(*k*) *ij* =< *hTσ*(*k*) *ij* , *hUσ*(*k*) *ij* , *hFσ*(*k*) *ij* > (*k* = 1, 2, ... , *cj*) in *Eoli*(*xj*) are arranged in an ascending order, denoted as *Eoli*(*xj*) = {*ϑ<sup>σ</sup>*(1) *ij* , *ϑ<sup>σ</sup>*(2) *ij* , ··· , *ϑ<sup>σ</sup>*(*cj*) *ij* } (*i* = 1, 2, ... , *p* and *j* = 1, 2, ... , *q*), where *σ* : (1, 2, . . . , *cj*) → (1, 2, . . . , *cj*) is a permutation satisfying *ϑ<sup>σ</sup>*(*k*) *ij* ≤ *ϑ<sup>σ</sup>*(*k*+<sup>1</sup>) *ij* (*k =* 1, 2, . . . , *cj*).

**Definition 6.** *Let Nl*1 = ,*El*1 (s1), *El*1 (*<sup>s</sup>*2), ··· , *El*1 (*sq*)/ *and Nl*2 = ,*El*2 (*<sup>s</sup>*1), *El*2 (*<sup>s</sup>*2), ··· , *El*2 (*sq*)/ *be two HLNN sets on S = {s1*, *s2*, *...*, *sq}*, *where El*1 (*sj*) *and El*2 (*sj*) *(j = 1*, *2*, ... , *q) are HLNNs in a LT set H = {h0*, *h1*, *...*, *hτ} for hj* ∈ *H. Let f(hj) = j/τ be a linguistic scale function. Then, the normalized generalized distance between Nl*1 *and Nl*2 *can be represented as:*

$$\begin{split} d(N\_{l\_{1}},N\_{l\_{2}}) &= \left\{ \frac{1}{q} \sum\_{j=1}^{q} \left[ \frac{1}{\mathcal{U}\_{l\_{1}^{\prime}}} \left( |f(h\_{T\_{1^{\prime}\_{1}}^{\prime}}) - f(h\_{T\_{2^{\prime}\_{1}}^{\prime}})|^{\rho} + |f(h\_{\mathcal{U}\_{1^{\prime}\_{1}}^{\prime}}) - f(h\_{\mathcal{U}\_{2^{\prime}\_{1}}^{\prime}})|^{\rho} + |f(h\_{\mathcal{U}\_{2^{\prime}\_{1}}^{\prime}}) - f(h\_{\mathcal{U}\_{2^{\prime}\_{1}}^{\prime}})|^{\rho} \right) \right] \right\}^{1/\rho} \\ &= \left\{ \frac{1}{q} \sum\_{j=1}^{q} \left[ \frac{1}{2\epsilon\_{j}^{\prime}\theta} \sum\_{k=1}^{\epsilon\_{j}} \left( |T^{\sigma(k)}\_{1j} - T^{\sigma(k)}\_{2j}|^{\rho} + |L^{\sigma(k)}\_{1j} - L^{\sigma(k)}\_{2j}|^{\rho} + |F^{\sigma(k)}\_{1j} - F^{\sigma(k)}\_{2j}|^{\rho}) \right) \right]^{1/\rho} \text{ for } \rho > 0. \end{split} \tag{4}$$

Obviously, *<sup>d</sup>*(*Nl*1 , *Nl*2 ) degenerates to the normalized generalized distance of Hamming for *ρ =* 1 and to the normalized generalized distance of Euclidean for *ρ =* 2.

For the generalized distance *<sup>d</sup>*(*Nl*1, *Nl*2 ), there is a proposition as follows:

**Proposition 1.** *For any two HLNN sets Nl*1 = ,*El*1 (*<sup>s</sup>*1), *El*1 (*<sup>s</sup>*2), ··· , *El*1 (*sq*)/ *and Nl*2 = ,*El*2 (*<sup>s</sup>*1), *El*2 (*<sup>s</sup>*2), ··· , *El*2 (*sq*)/*, the generalized distance <sup>d</sup>*(*Nl*1 , *Nl*2 ) *between Nl*1 *and Nl*2 *for ρ > 0 contains the following properties:*


$$\begin{array}{lll} \text{(HP4)} & \text{Let } \mathrm{N}\_{l\_3} = \left\{ E\_{l\_3}(\mathbf{s}\_1), E\_{l\_3}(\mathbf{s}\_2), \dots, E\_{l\_3}(\mathbf{s}\_q) \right\} \text{ be a HLINN set, then } d(\mathrm{N}\_{l\_1}, \mathrm{N}\_{l\_2}) \le d(\mathrm{N}\_{l\_1}, \mathrm{N}\_{l\_3}) \text{ and} \\ & d(\mathrm{N}\_{l\_2}, \mathrm{N}\_{l\_3}) \le d(\mathrm{N}\_{l\_1}, \mathrm{N}\_{l\_3}) \text{ if } \mathrm{N}\_{l\_1} \subseteq \mathrm{N}\_{l\_2} \subseteq \mathrm{N}\_{l\_3}. \end{array}$$

**Proof.** It is obvious that the properties (HP1)–(HP3) are satisfied for *<sup>d</sup>*(*Nl*1 , *Nl*2 ). Thus, we only need to prove the property (HP4).

Since there is *Nl*1 ⊆ *Nl*2 ⊆ *Nl*3 , there exists *E*0*l*1 (*sj*) ≤ *E*0*l*2 (*sj*) ≤ *E*0*l*3 (*sj*) for *sj* ∈ *S (j = 1*, *2*, *...*, *q)*, which implies *T<sup>σ</sup>*(*k*) 3*j* ≥ *T<sup>σ</sup>*(*k*) 2*j* ≥ *T<sup>σ</sup>*(*k*) 1*j* , *U<sup>σ</sup>*(*k*) 3*j* ≤ *U<sup>σ</sup>*(*k*) 2*j* ≤ *U<sup>σ</sup>*(*k*) 1*j* , *F<sup>σ</sup>*(*k*) 3*j* ≤ *F<sup>σ</sup>*(*k*) 2*j* ≤ *F<sup>σ</sup>*(*k*) 1*j* for *k* = 1, 2, ..., *cj*. It follows that

$$\begin{vmatrix} T\_{1j}^{\sigma(k)} - T\_{2j}^{\sigma(k)} \Big|\_{\rho} \leq \left| T\_{1j}^{\sigma(k)} - T\_{3j}^{\sigma(k)} \right|\_{\rho}^{\rho} & \left| T\_{2j}^{\sigma(k)} - T\_{3j}^{\sigma(k)} \right|\_{\rho} \leq \left| T\_{1j}^{\sigma(k)} - T\_{3j}^{\sigma(k)} \right|\_{\rho} \\ \left| \mathcal{U}\_{1j}^{\sigma(k)} - \mathcal{U}\_{2j}^{\sigma(k)} \right|\_{\rho} \leq \left| \mathcal{U}\_{1j}^{\sigma(k)} - \mathcal{U}\_{3j}^{\sigma(k)} \right|\_{\rho}^{\rho} & \left| \mathcal{U}\_{2j}^{\sigma(k)} - \mathcal{U}\_{3j}^{\sigma(k)} \right|\_{\rho} \leq \left| \mathcal{U}\_{1j}^{\sigma(k)} - \mathcal{U}\_{3j}^{\sigma(k)} \right|\_{\rho} \rho \\ \left| \mathcal{F}\_{1j}^{\sigma(k)} - \mathcal{F}\_{2j}^{\sigma(k)} \right|\_{\rho} \leq \left| F\_{1j}^{\sigma(k)} - F\_{3j}^{\sigma(k)} \right|\_{\rho}^{\rho} & \left| F\_{2j}^{\sigma(k)} - F\_{3j}^{\sigma(k)} \right|\_{\rho} \leq \left| F\_{1j}^{\sigma(k)} - F\_{3j}^{\sigma(k)} \right|\_{\rho} \end{vmatrix}$$

Then there are the following inequalities:

HHH*T<sup>σ</sup>*(*k*) 1*j* − *T<sup>σ</sup>*(*k*) 2*j* HHH*ρ*<sup>+</sup>HHH*U<sup>σ</sup>*(*k*) 1*j* − *U<sup>σ</sup>*(*k*) 2*j* HHH*ρ*<sup>+</sup>HHH*F<sup>σ</sup>*(*k*) 1*j* − *F<sup>σ</sup>*(*k*) 2*j* HHH*ρ* ≤HHH*T<sup>σ</sup>*(*k*) 1*j* − *T<sup>σ</sup>*(*k*) 3*j* HHH*ρ*<sup>+</sup>HHH*U<sup>σ</sup>*(*k*) 1*j* − *U<sup>σ</sup>*(*k*) 3*j* HHH*ρ*<sup>+</sup>HHH*F<sup>σ</sup>*(*k*) 1*j* − *F<sup>σ</sup>*(*k*) 3*j* HHH*ρ*, HHH*T<sup>σ</sup>*(*k*) 2*j* − *T<sup>σ</sup>*(*k*) 3*j* HHH*ρ*<sup>+</sup>HHH*U<sup>σ</sup>*(*k*) 2*j* − *U<sup>σ</sup>*(*k*) 3*j* HHH*ρ*<sup>+</sup>HHH*F<sup>σ</sup>*(*k*) 2*j* − *F<sup>σ</sup>*(*k*) 3*j* HHH*ρ* ≤HHH*T<sup>σ</sup>*(*k*) 1*j* − *T<sup>σ</sup>*(*k*) 3*j* HHH*ρ*<sup>+</sup>HHH*U<sup>σ</sup>*(*k*) 1*j* − *U<sup>σ</sup>*(*k*) 3*j* HHH*ρ*<sup>+</sup>HHH*F<sup>σ</sup>*(*k*) 1*j* − *F<sup>σ</sup>*(*k*) 3*j* HHH*ρ*.

Thus, the following relations can be further obtained:

1 <sup>3</sup>*cjτρ* M *cj* ∑*k*=1 (|(*T<sup>σ</sup>*(*k*) 1*j* ) − (*T<sup>σ</sup>*(*k*) 2*j* )|*ρ* + |(*U<sup>σ</sup>*(*k*) 1*j* ) − (*U<sup>σ</sup>*(*k*) 2*j* )|*ρ* + |(*F<sup>σ</sup>*(*k*) 1*j* ) − (*F<sup>σ</sup>*(*k*) 2*j* )|*ρ*)N ≤ 1 <sup>3</sup>*cjτρ* M *cj* ∑*k*=1 (|(*T<sup>σ</sup>*(*k*) 1*j* ) − (*T<sup>σ</sup>*(*k*) 3*j* )|*ρ* + |(*U<sup>σ</sup>*(*k*) 1*j* ) − (*U<sup>σ</sup>*(*k*) 3*j* )|*ρ* + |(*F<sup>σ</sup>*(*k*) 1*j* ) − (*F<sup>σ</sup>*(*k*) 3*j* )|*ρ*)N, 1 <sup>3</sup>*cjτρ* M *cj* ∑*k*=1 (|(*T<sup>σ</sup>*(*k*) 2*j* ) − (*T<sup>σ</sup>*(*k*) 3*j* )|*ρ* + |(*U<sup>σ</sup>*(*k*) 2*j* ) − (*U<sup>σ</sup>*(*k*) 3*j* )|*ρ* + |(*F<sup>σ</sup>*(*k*) 2*j* ) − (*F<sup>σ</sup>*(*k*) 3*j* )|*ρ*)N ≤ 1 <sup>3</sup>*cjτρ* M *cj* ∑*k*=1 (|(*T<sup>σ</sup>*(*k*) 1*j* ) − (*T<sup>σ</sup>*(*k*) 3*j* )|*ρ* + |(*U<sup>σ</sup>*(*k*) 1*j* ) − (*U<sup>σ</sup>*(*k*) 3*j* )|*ρ* + |(*F<sup>σ</sup>*(*k*) 1*j* ) − (*F<sup>σ</sup>*(*k*) 3*j* )|*ρ*)N.

By Equation (4), there are *<sup>d</sup>*(*Nl*1 , *Nl*2 ) ≤ *<sup>d</sup>*(*Nl*1 , *Nl*3 ) and *<sup>d</sup>*(*Nl*2 , *Nl*3 ) ≤ *<sup>d</sup>*(*Nl*1 , *Nl*3 ) for *ρ* > 0. Therefore, the property (HP4) can hold. -

If we consider the weight *wj* of an element *sj* ∈ *S* with *wj* ∈ *[0*, *1]* and <sup>∑</sup>*qj*=<sup>1</sup> *wj* = 1, the generalized weighted distance between *Nl*1and *Nl*2 is

$$\begin{split} d\_{\boldsymbol{w}}(\boldsymbol{N\_{l}},\boldsymbol{N\_{l\_{2}}}) &= \left\{ \sum\_{j=1}^{q} w\_{j} \left[ \frac{1}{\mathcal{X}\_{\boldsymbol{l}\_{1}}} \sum\_{k=1}^{c\_{l}} \left( |f(\boldsymbol{h}\_{\boldsymbol{T\_{1}^{q(k)}\_{1}})} - f(\boldsymbol{h}\_{\boldsymbol{T\_{2}^{q(k)}\_{1}}})|^{p} + |f(\boldsymbol{h}\_{\boldsymbol{L}^{q(k)}\_{1}}) - f(\boldsymbol{h}\_{\boldsymbol{L}^{q(k)}\_{2}})|^{p} + |f(\boldsymbol{h}\_{\boldsymbol{T\_{2}^{q(k)}\_{1}}}) - f(\boldsymbol{h}\_{\boldsymbol{T\_{2}^{q(k)}\_{2}}})|^{p} \right) \right] \right\}^{1/p} \\ &= \left\{ \sum\_{j=1}^{q} w\_{j} \left[ \frac{1}{\mathcal{X}\_{\boldsymbol{l}^{p}}^{q\top}} \sum\_{k=1}^{c\_{l}} \left( |T^{q(k)}\_{1\_{l}^{p}} - T^{q(k)}\_{2\_{l}^{p}}|^{p} + |\mathcal{U}^{r(k)}\_{1\_{l}^{p}} - \mathcal{U}^{r(k)}\_{2\_{l}^{p}}|^{p} + |F^{r(k)}\_{1\_{l}^{p}} - F^{r(k)}\_{2\_{l}^{p}}|^{p} \right) \right] \right\}^{1/p} \text{ for } \rho > 0. \end{split} \tag{5}$$

Since the measures of similarity and distance are complementary with each other, the weighted measure of similarity between *Nl*1 and *Nl*2 can be represented by

$$\begin{split} S\_{\mathbb{W}}(N\_{l\_{1}}, N\_{l\_{2}}) &= 1 - d\_{\mathbb{W}}(N\_{l\_{1}}, N\_{l\_{2}}) \\ &= 1 - \left\{ \sum\_{j=1}^{q} w\_{j} \left[ \frac{1}{\mathcal{K}\_{l}^{\mathbb{W}}} \sum\_{k=1}^{\ell\_{l}} \left( \left| T\_{1\_{l}^{\mathbb{W}}}^{\sigma(k)} - T\_{2\_{l}^{\mathbb{W}}}^{\sigma(k)} \right|\_{\rho} + \left| L\_{1\_{l}^{\mathbb{W}}}^{\sigma(k)} - L\_{2\_{l}^{\mathbb{W}}}^{\sigma(k)} \right|\_{\rho} + \left| F\_{1\_{l}^{\mathbb{W}}}^{\sigma(k)} - F\_{2\_{l}^{\mathbb{W}}}^{\sigma(k)} \right|^{\rho} \right) \right\}^{1/\rho} for \,\rho > 0. \end{split} \tag{6}$$

Similar to the properties (HP1)–(HP4) satisfied by the generalized distance measure in Proposition 1, the similarity measure *Sw*(*Nl*1, *Nl*2) also has the proposition as follows:

**Proposition 2.** *The similarity measure Sw*(*Nl*1 , *Nl*2) *for ρ > 0 contains the following properties:*

*(HP1)* 0 ≤ *Sw*(*Nl*1, *Nl*2) ≤ 1*;* 

 *(HP2) Sw*(*Nl*1, *Nl*2) = 1 *if and only if Nl*1= *Nl*2*;*

 *(HP3) Sw*(*Nl*1, *Nl*2 ) = *Sw*(*Nl*2, *Nl*1)*;*

*(HP4) Let Nl*3 *be a HLNN set*, *then there are Sw*(*Nl*1 , *Nl*2 ) ≥ *Sw*(*Nl*1 , *Nl*3 ) *and Sw*(*Nl*2 , *Nl*3 ) ≥ *Sw*(*Nl*1 , *Nl*3 ) *if Nl*1 ⊆ *Nl*2 ⊆ *Nl*3 .

**Proof.** It is clear that *Sw*(*Nl*1 , *Nl*2 ) satisfies the properties (SP1)–(SP3). Thus, we only prove the property (SP4) here.

According to the proved property (HP4) in Proposition 1, if *Nl*1 ⊆ *Nl*2 ⊆ *Nl*3 , there exists the relations of *dw*(*Nl*1 , *Nl*2 ) ≤ *dw*(*Nl*1 , *Nl*3 ) and *dw*(*Nl*2 , *Nl*3 ) ≤ *dw*(*Nl*1 , *Nl*3 ) for *ρ* > 0. Since the similarity measure is the complement of the distance measure, both *Sw*(*Nl*1 , *Nl*2 ) ≥ *Sw*(*Nl*1 , *Nl*3 ) and *Sw*(*Nl*2 , *Nl*3 ) ≥ *Sw*(*Nl*1 , *Nl*3 ) can be easily obtained. Therefore, the property (SP4) can hold. -

#### **5. MADM Method Using the Similarity Measure of HLNNs**

For a MADM problem in the HLNN setting, some DMs need to evaluate *p* alternatives (denoted by *G =* {*g*1, *g*2, ... , *gp*}) over *q* attributes (denoted by *S =* {*<sup>s</sup>*1, *s*2, ... , *sq*}) from the LT set *H =* {*h*0, *h*1, ... , *hτ}*. Then, a weight vector *W =* (*<sup>ω</sup>*1, *ω*2, ... , *<sup>ω</sup>q*), which is on the conditions of 0 ≤ *<sup>ω</sup>j* ≤ 1 (*j* = 1, 2, ..., *q*) and <sup>∑</sup>*qj*=<sup>1</sup> *<sup>ω</sup>j* = 1, represents the importance of the attributes in *S*. Thus, the HLNN decision matrix *M* can be expressed as:

$$\mathcal{M} = \left(E\_{l\_1}(s\_{\bar{\jmath}})\right)\_{p \times q} = \begin{array}{c} \mathcal{G}\_1 \\ \mathcal{G}\_2 \\ \vdots \\ \mathcal{G}\_p \end{array} \left[ \begin{array}{c} E\_{l\_1}(s\_1) \\ E\_{l\_2}(s\_1) \\ \vdots \\ E\_{l\_p}(s\_2) \end{array} \left( \begin{array}{c} \cdots & E\_{l\_1}(x\_q) \\ E\_{l\_2}(s\_2) & \cdots & E\_{l\_2}(s\_q) \\ \vdots & \vdots & \ddots & \vdots \\ E\_{l\_p}(s\_1) & E\_{l\_p}(s\_2) & \cdots & E\_{l\_p}(s\_q) \end{array} \right) \right]$$

where *Eli*(*sj*) = {< *hT*1*ij* , *hU*1*ij* , *hF*1*ij* >, < *hT*2*ij* , *hU*2*ij* , *hF*2*ij* >, ··· , < *hTmij ij* , *hUmij ij* , *hFmij ij* >} is a HLNN for *sj* ∈ *S*, and *mij*is the number of LNNs in *Eli*(*sj*) (*i* = 1, 2, . . . , *p* and *j* = 1, 2, . . . , *q*).

On the basis of the proposed similarity measure, a novel MADM method of HLNN is presented by the following steps:

**Step 1**: For any HLNN *Eli*(*sj*) (*j* = 1, 2, ... , *q*) in *M*, rank all elements *ϑ<sup>σ</sup>*(*k*) *ij (k =* 1, 2, ... , *mij*) in each HLNN *Eli*(*sj*) (*j* = 1, 2, ... , *q*) in an ascending order according to their score and accuracy functions, then yield the corresponding extended HLNN *Eoli*(*sj*) based on the LCMC *cj* and the occurrence number *Rij* of every LNN in *Eli*(*sj*) obtained by Equation (3). Hence, the extended decision matrix *M*◦ is

$$\mathcal{M}^o = \left( E\_{l\_i}^o(s\_j) \right)\_{p \times q} = \begin{array}{c} \mathcal{G}1 \\ \mathcal{G}2 \\ \vdots \\ \mathcal{G}p \end{array} \left[ \begin{array}{c} E\_{l\_1}^o(s\_1) & E\_{l\_1}^o(s\_2) & \cdots & E\_{l\_1}^o(s\_q) \\ E\_{l\_2}^o(s\_1) & E\_{l\_2}^o(s\_2) & \cdots & E\_{l\_2}^o(s\_q) \\ \vdots & \vdots & \ddots & \vdots \\ E\_{l\_p}^o(s\_1) & E\_{l\_p}^o(s\_2) & \cdots & E\_{l\_p}^o(s\_q) \end{array} \right] \end{array}$$

where *Eoli*(*sj*) = {*ϑ<sup>σ</sup>*(1) *ij* , *ϑ<sup>σ</sup>*(2) *ij* , ··· , *ϑ<sup>σ</sup>*(*cj*) *ij* } (*i* = 1, 2, ... , *p* and *j* = 1, 2, ... , *q*) satisfies *ϑ<sup>σ</sup>*(*k*) *ij* ≤ *ϑ<sup>σ</sup>*(*k*+<sup>1</sup>) *ij* (*k* = 1, 2, . . . , *cj*).

**Step 2**: Specify an ideal HLNN set as *g*∗ = {*Eol* (*<sup>s</sup>*1), *Eol* (*<sup>s</sup>*2), ... , *Eol* (*sq*)} = {{*ϑ<sup>σ</sup>*(1) 1 , *ϑ<sup>σ</sup>*(2) 1 , ... , *ϑ<sup>σ</sup>*(*<sup>c</sup>*1) 1 }, {*ϑ<sup>σ</sup>*(1) 2 , *ϑ<sup>σ</sup>*(2) 2 , ... , *ϑ<sup>σ</sup>*(*<sup>c</sup>*2) 2 }, ... , {*ϑ<sup>σ</sup>*(1) *q* , *ϑ<sup>σ</sup>*(2) *q* , ... , *ϑ<sup>σ</sup>*(*cq*) *q* }} for all *ϑ<sup>σ</sup>*(*k*) *j* =< *h<sup>τ</sup>*, *h*0, *h*0 > (*k* = 1, 2, ..., *cj* and *j* = 1, 2, ..., *q*).

Hence, the similarity measure between *gi* (*i* = 1, 2, . . . , *p*) and *g\** can be calculated by

$$\begin{split} S\_{\mathbb{W}}(g\_{i},\boldsymbol{\varrho}^{\*}) &= 1 - d\_{\mathbb{W}}(\boldsymbol{\varrho}\_{i},\boldsymbol{\varrho}^{\*}) \\ &= 1 - \left\{ \sum\_{j=1}^{q} w\_{j} \left[ \frac{1}{2\mathcal{K}\_{j}} \sum\_{k=1}^{c\_{j}} \left( |f(h\_{T\_{\boldsymbol{a}\_{\parallel}^{\*}}}) - f(h\_{\Gamma})|^{\rho} + |f(h\_{U\_{\boldsymbol{a}\_{\parallel}^{\*}}}) - f(h\_{0})|^{\rho} + |f(h\_{T\_{\boldsymbol{a}\_{\parallel}^{\*}}}) - f(h\_{0})|^{\rho} \right) \right] \right\}^{1/\rho} \\ &= 1 - \left\{ \sum\_{j=1}^{q} w\_{j} \left[ \frac{1}{2\zeta\_{j}v^{\nu}} \sum\_{k=1}^{c\_{j}} \left( |T\_{\boldsymbol{a}\_{\parallel}^{\*}}^{\boldsymbol{\sigma}^{\boldsymbol{1}}}| - \mathbf{r}|^{\rho} + |L\_{\boldsymbol{a}\_{\parallel}^{\*}}^{\boldsymbol{\sigma}^{\boldsymbol{1}}}|^{\rho} + |F\_{\boldsymbol{a}\_{\parallel}^{\*}}^{\boldsymbol{\sigma}^{\boldsymbol{1}}}|^{\rho} \right) \right] \right\}^{1/\rho} for \; \rho > 0. \end{split} \tag{7}$$

**Step 3:** According to the similarity measure results, rank the alternatives in *G =* {*g*1, *g*2, ... , *gm*} in a descending order and choose the best one.
