*2.3. HFLTS*

The HFLTS permits the membership of an element to be a set of several possible linguistic variable values. In the following, the concept of HFLTS and some related operations of HFLTS are reviewed.

**Definition 2.** *Given a fixed set X*, *let S* = { *si*|*<sup>i</sup>* = 0, 1, ··· , 2*t* } *be a LTS, then a HFLTS HS on X is expressed by [8]:*

$$H\_{\mathcal{S}} = \left\{ \left( \mathbf{x}\_{j'} h\_{\mathcal{S}} (\mathbf{x}\_{j'}) \right) \, \middle| \, \mathbf{x}\_{j} \in \mathcal{X} \right\}$$

*where hS*-*xj*. *is a subset of linguistic terms in S, it represents the membership degrees of the element xj belongs to X*. *For convenience, the element of hS*-*xj*. *is called the hesitant fuzzy linguistic element (HFLE).*

**Example 1.** *Let S* = {*<sup>s</sup>*0 : *very poor*, *s*1 : *poor*,*s*<sup>2</sup> : *slightly poor*, *s*3 : *f air*,*s*<sup>4</sup> : *slightly good*,*s*<sup>5</sup> : *good*, *s*6 : *very good*} *be a LTS. Two experts evaluate the performance of a company; one thinks the performance of a company is not less than good, the other thinks it is between fair and good. According to Definition 2, the above evaluation information can be represented as H*1*S* = {*<sup>s</sup>*5, *<sup>s</sup>*6} *and H*2*S* = {*<sup>s</sup>*3,*s*4,*s*5}*, respectively. The numbers of linguistic terms in H*1*S and H*2*S are not equal, which is not convenient for computing the similarity measure between H*1*S and <sup>H</sup>*2*S*.

In order to solve this problem, for any two HFLTSs *H*1*S* = {-*xj*, *<sup>h</sup>*1*S*-*xj*..HH*xj* ∈ *X*} and *H*2*S* = {-*xj*, *<sup>h</sup>*2*S*-*xj*..HH*xj* ∈ *X*}(*j* = 1, 2, ··· , *<sup>n</sup>*), where *<sup>h</sup>kS*-*xj*. = {*<sup>s</sup>δkl* -*xj*.|*<sup>s</sup>δkl* -*xj*. ∈ *S*, *lk* = 1, 2, ··· , *Lkj* }, if the numbers of *<sup>h</sup>kS*-*xj*. are not equal, we can let *Lj* = *max*{*L*1*j* , *L*2*j* }. Zhu et al. [35] proposed the rules of regulation: for the optimists, they extend the set with fewer numbers of elements by adding the maximum value *s*+*δkl* -*xj*. = *max l* = *L*1*j or l* = *L*2*j* {*<sup>s</sup>δkl* -*xj*.} until the two sets have the same number of

elements; while for the pessimists, they add the minimum value *s*−*δkl* -*xj*. = *min l* = *L*1*j or l* = *L*2*j* {*<sup>s</sup>δkl* -*xj*.} to the set with fewer numbers of elements. In this paper, we assume that the largest element is added to the set with fewer elements until they have the same number.

The existing score function of HFLTSs is defined as follows:

**Definition 3.** *Let S* = { *si*|*<sup>i</sup>* = 0, 1, ··· , 2*t* } *be a LTS; HS* = {*<sup>s</sup>δkl* -*xj*.HHH*<sup>s</sup>δkl* -*xj*. ∈ *S*} -*l* = 1, 2, ··· , *Lj*, *j* = 1, 2, ··· , *n*. *be a HFLTS on X, then the score function of HS is [36]:*

$$F(H\_S) = \frac{1}{n} \sum\_{j=1}^n \overline{\delta}(\mathbf{x}\_j) - \frac{\sum\_{j=1}^n \left(\frac{1}{L\_j} \sum\_{l=1}^{L\_j} \left(\delta\_l \left(\mathbf{x}\_j\right) - \overline{\delta} \left(\mathbf{x}\_j\right)\right)^2\right)}{Var(2t)}$$

*where δ* = 1*Lj* ∑*Lj l*=1 *<sup>δ</sup>l*-*xj*.*, Var*(2*t*) = ∑<sup>2</sup>*<sup>t</sup> <sup>i</sup>*=<sup>0</sup>(*<sup>i</sup>*−*<sup>t</sup>*)<sup>2</sup> 2*t*+1 .

**Lemma 1.** *For two HFLTSs H*1*Sand <sup>H</sup>*2*S, the comparison rules between them are defined as follows [36]:*

*(1) H*1*S* > *H*2*S if and only if <sup>F</sup>*-*H*1*S*. > *<sup>F</sup>*-*H*2*S*.;

*(2) H*1*S* = *H*2*S if and only if <sup>F</sup>*-*H*1*S*. = *<sup>F</sup>*-*H*2*S*..

#### *2.4. Existing Distance and Similarity Measures Between HFLTSs*

The Distance and similarity measure are effective tools for describing the deviation and closeness between different alternatives in MCDM problems; the definitions about the existing distance and similarity measures between HFLTSs are given as follows:

**Definition 4.** *Given a fixed set X, suppose that S* = { *si*|*<sup>i</sup>* = 0, 1, ··· , 2*t* } *be a LTS, <sup>h</sup>kS*-*xj*. = {*<sup>s</sup>δkl* -*xj*.HHH*<sup>s</sup>δkl* -*xj*. ∈ *S*, *l* = 1, 2, ··· , *Lj*} *, where Lj* = *max*{*l*-*h*1*S*-*xj*., *<sup>l</sup>*(*h*2*S*-*xj*..)*, <sup>l</sup>hkS*-*xj*. *represents the number of elements in <sup>h</sup>kS*-*xj*. (*k* = 1, <sup>2</sup>)*. For any two HFLTSs H*1*S* = {-*xj*, *<sup>h</sup>*1*S*-*xj*..HH*xj* ∈ *X*} *and H*2*S* = {-*xj*, *<sup>h</sup>*2*S*-*xj*..HH*xj* ∈ *X*} (*j* = 1, 2, ··· , *<sup>n</sup>*)*, assume the weight of different element xj is <sup>ω</sup>j* (*j* = 1, 2, ··· , *<sup>n</sup>*)*, then the weighted Euclidean distance measure between H*1*S and H*2*S can be defined as follows [9]:*

$$D\_{\omega HFL} \left( H\_{S\_{\mathcal{L}}}^1 H\_S^2 \right) = \left( \sum\_{j=1}^n \frac{\omega\_j}{L\_j} \sum\_{l=1}^{L\_j} \left( \frac{|\delta\_l^1 \left( x\_j \right) - \delta\_l^2 \left( x\_j \right)|}{2t+1} \right)^2 \right)^{\frac{1}{2}}.\tag{1}$$

**Remark 1.** *For all j* = 1, 2, ··· , *n, if the weight <sup>ω</sup>j* = 1*n , then the weighted Euclidean distance measure <sup>D</sup>ωHFL*-*H*1*S*, *<sup>H</sup>*2*S*. *is reduced to the Euclidean distance measure DHFL*-*H*1*S*, *<sup>H</sup>*2*S*.:

$$D\_{HFL}\left(H\_{S\prime}^1H\_S^2\right) = \left(\frac{1}{n}\left(\sum\_{j=1}^n \frac{1}{L\_j}\sum\_{l=1}^{L\_j} \left(\frac{\left|\delta\_l^1\left(\mathbf{x}\_j\right) - \delta\_l^2\left(\mathbf{x}\_j\right)\right|}{2t+1}\right)^2\right)\right)^{\frac{1}{2}}$$

Liao et al. [31] defined a cosine similarity measure between HFLTSs as follows:

**Definition 5.** *Given a fixed set X, suppose that S* = { *si*|*<sup>i</sup>* = 0, 1, ··· , 2*t* } *is a LTS, <sup>h</sup>kS*-*xj*. = {*<sup>s</sup>δkl* -*xj*.HHH*<sup>s</sup>δkl* -*xj*. ∈ *S*, *l* = 1, 2, ··· , *Lj*} *, where Lj* = *max*{*l*-*h*1*S*-*xj*., *<sup>l</sup>*(*h*2*S*-*xj*..}*, <sup>l</sup>hkS*-*xj*. *represents the number of elements in <sup>h</sup>kS*-*xj*. (*k* = 1, <sup>2</sup>)*. For any two HFLTSs H*1*S* = {-*xj*, *<sup>h</sup>*1*S*-*xj*..HH*xj* ∈ *X*} *and H*2*S* = {-*xj*, *<sup>h</sup>*2*S*-*xj*..HH*xj* ∈ *X*} (*j* = 1, 2, ··· , *<sup>n</sup>*)*, if the weight of different element xj is <sup>ω</sup>j* (*j* = 1, 2, ··· , *<sup>n</sup>*)*, then the weighted cosine similarity measure can be defined as [31]:*

$$\text{Cov}\_{\omega\text{-}H\text{FL}}\left(H\_{\text{S}}^{1},H\_{\text{S}}^{2}\right) = \frac{\sum\_{j=1}^{n}\left(\frac{\omega\_{j}}{L\_{j}}\sum\_{l=1}^{L\_{j}}\left(\frac{\delta\_{l}^{1}\left(x\_{j}\right)}{2l+1},\frac{\delta\_{l}^{2}\left(x\_{j}\right)}{2l+1}\right)\right)}{\left(\sum\_{j=1}^{n}\left(\frac{\omega\_{j}}{L\_{j}}\sum\_{l=1}^{L\_{j}}\left(\frac{\delta\_{l}^{1}\left(x\_{j}\right)}{2l+1}\right)^{2}\right)\cdot\sum\_{j=1}^{n}\left(\frac{\omega\_{j}}{L\_{j}}\sum\_{l=1}^{L\_{j}}\left(\frac{\delta\_{l}^{2}\left(x\_{j}\right)}{2l+1}\right)^{2}\right)\right)^{\frac{1}{2}}}\tag{2}$$

**Remark 2.** *For all j* = 1, 2, ··· , *n, if the weight <sup>ω</sup>j* = 1*n , then the weighted cosine similarity measure CosωHFL*-*H*1*S*, *<sup>H</sup>*2*S*. *is reduced to the cosine similarity measure CosHFL*(*H*1*S* , *<sup>H</sup>*2*S* ):

$$\text{Cost}\_{HFL}\left(H\_{S}^{1},H\_{S}^{2}\right) = \frac{\sum\_{j=1}^{n}\left(\frac{1}{L\_{j}}\sum\_{l=1}^{L\_{j}}\left(\frac{\delta\_{1}^{\ell}\left(x\_{j}\right)}{2t+1}\cdot\frac{\delta\_{1}^{\ell}\left(x\_{j}\right)}{2t+1}\right)\right)}{\left(\sum\_{j=1}^{n}\left(\frac{1}{L\_{j}}\sum\_{l=1}^{L\_{j}}\left(\frac{\delta\_{1}^{\ell}\left(x\_{j}\right)}{2t+1}\right)^{2}\right)\cdot\sum\_{j=1}^{n}\left(\frac{1}{L\_{j}}\sum\_{l=1}^{L\_{j}}\left(\frac{\delta\_{1}^{\ell}\left(x\_{j}\right)}{2t+1}\right)^{2}\right)\right)^{\frac{1}{2}}}\tag{3}$$

#### *2.5. Linguistic Scale Function*

In different semantic decision-making environments, linguistic terms have some differences in expressing alternatives. Bao et al. [37] thought that the information may be distorted when using the subscript of the linguistic term set directly in the process of operations. To solve this problem, Wang et al. [23] put forward the linguistic scale function to calculate the linguistic information. According to the decision-making environment, the decision makers choose a different linguistic scale function, which can express the linguistic information more flexibly in different semantic situations.

**Definition 6.** *Let S* = { *si*|*<sup>i</sup>* = 0, 1, ··· , 2*t* } *be a LTS; if θi* ∈ *R*+(*R*<sup>+</sup> = { *r*|*r* ≥ 0,*r* ∈ *R*}) *is a real value, then the linguistic scale function f can be defined as follows [22]:*

$$f: \mathbf{s}\_i \to \theta\_i \ (i = 0, 1, \cdot, \cdot, 2t\_\prime)$$

*where* 0 ≤ *θ*0 ≤ *θ*1 ≤ ··· ≤ *θ*2*t* ≤ 1*. The linguistic scale function f is a strictly monotonically increasing function on the subscript of si. Actually, the function value θi represents the semantics of the linguistic terms.*

Next we introduce three common linguistic scale functions as follows:

$$(1). \quad f\_1(s\_i) = \theta\_i = \frac{\dot{t}}{2t} \ (i = 0, 1, \dots, 2t)\_1).$$

$$(2). \quad f\_2(s\_i) = \theta\_i = \begin{cases} \frac{d^t - d^{t-i}}{2d^t - 2}, & (i = 0, 1, \dots, t)\_2 \\\frac{d^t + d^{i-t} - 2}{2d^t - 2}, & (i = t+1, t+2, \dots, 2t)\_t \end{cases}$$

If the LTS is a set of seven terms, then *a* ∈ [1.36, 1.4] [38]. In this paper, we assume that *a* = 1.4.

$$(3). \quad f\_3(s\_i) = \theta\_i = \begin{cases} \frac{t^a - (t-i)^a}{2t^a}, & (i = 0, 1, \cdots, t); \\\frac{t^{\beta} - (t-i)^{\beta}}{2t^{\beta}}, & (i = t+1, t+2, \cdots, 2t). \end{cases}$$

where *α*, *β* ∈ (0, 1]. If the LTS is a set of seven terms, then *α* = *β* = 0.8 [39].

**Example 2.** *Assume that S* = { *si*|*<sup>i</sup>* = 0, 1, ··· , 2*t* } *be a LTS. When t* = 3*, the corresponding linguistic scale functions are f*1(*si*)*, f*2(*si*) (*a* = 1.4)*, f*3(*si*) (*α* = *β* = 0.8) *respectively, and the characteristics of the three functions are shown in Figure 1.*

**Figure 1.** The change of the three linguistic scale functions.

**Remark 3.** *The linguistic scale function f*1(*si*) *can be explained as the decision maker's neutral attitude towards risk; the linguistic scale function f*2(*si*) *indicates that the decision maker's attitude towards risk is changing from aversion to appetite; the linguistic scale function f*3(*si*) *indicates that the decision maker's attitude towards risk is changing from appetite to aversion.*

#### **3. The Score Function, Similarity Measure, and Distance Measure Between HFLTSs Based on a Linguistic Scale Function**

In this section, we first propose the definition of a new score function of HFLTSs based on a linguistic scale function, then the new similarity measure and its properties are given. Furthermore, we construct a corresponding distance measure based on the relationship between the similarity measure and the distance measure.

*3.1. The Score Function Between HFLTSs Based on the Linguistic Scale Function*

**Definition 7.** *Let S* = { *si*|*<sup>i</sup>* = 0, 1, ··· , 2*t* } *be a LTS, HS* = {*<sup>s</sup>δkl* -*xj*.HHH*<sup>s</sup>δkl* -*xj*. ∈ *S*} -*l* = 1, 2, ··· , *Lj*, *j* = 1, 2, ··· , *n*. *be the HFLTS on X, and f be a linguistic scale function, then the score function of HS is defined as:*

$$F^\*(H\_S) = \frac{1}{n} \sum\_{j=1}^n \overline{f}(s\_{\delta\_l}(x\_j)) - \frac{\sum\_{j=1}^n \left(\frac{1}{L\_j} \sum\_{l=1}^{L\_j} \left(f\left(s\_{\delta\_l}\left(x\_j\right)\right) - \overline{f}s\_{\delta\_l}\left(x\_j\right)\right)^2\right)}{Var^\*\left(2t\right)},$$

*where f* -*<sup>s</sup>δl* -*xj*.. = 1*Lj* ∑*Lj l*=1 *f* -*<sup>s</sup>δl* -*xj*..*, Var*<sup>∗</sup>(2*t*) = ∑<sup>2</sup>*<sup>t</sup> <sup>i</sup>*=<sup>0</sup>(*f*(*si*) − *f*(*st*))2.

**Theorem 1.** *For any two HFLTSs H*1*Sand <sup>H</sup>*2*S, the comparison rules between them are defined as follows:*

*(1) If <sup>F</sup>*<sup>∗</sup>-*H*1*S*. > *<sup>F</sup>*<sup>∗</sup>-*H*2*S*.*, then H*1*S* > *<sup>H</sup>*2*S*; *(2) If <sup>F</sup>*<sup>∗</sup>-*H*1*S*. = *<sup>F</sup>*<sup>∗</sup>-*H*2*S*.*, then H*1*S*= *<sup>H</sup>*2*S*.

**Example 3.** *Let S* = {*<sup>s</sup>*0 : *very poor*, *s*1 : *poor*, *s*2 : *slightly poor*,*s*3 : *f air*,*s*<sup>4</sup> : *slightly good*,*s*<sup>5</sup> : *good*, *s*6 : *very good*} *be a LTS, three HFLTSs are given as follows: H*1*S* = {*<sup>s</sup>*0,*s*1,*s*2}*, H*2*S* = {*<sup>s</sup>*2,*s*3,*s*4} *and H*3*S* = {*<sup>s</sup>*0,*s*2}*. By Definition 7, if the linguistic scale function f* = *f*1(*si*) = *i*2*t* (*t* = <sup>3</sup>)*, we obtain <sup>F</sup>*<sup>∗</sup>-*H*1*S*. = 0.1429, *<sup>F</sup>*<sup>∗</sup>-*H*2*S*. = 0.4048, *<sup>F</sup>*<sup>∗</sup>-*H*3*S*. = 0.1310*, then the ranking of the HFLTSs is H*2*S* > *H*1*S* > *<sup>H</sup>*3*S. By Definition 3, we can obtain <sup>F</sup>*-*H*1*S*. = 0.8323, *<sup>F</sup>*-*H*2*S*. = 2.8333, *<sup>F</sup>*-*H*3*S*. = 0.75*, according to Lemma 1, and it is clearly seen that H*2*S*> *H*1*S*> *<sup>H</sup>*3*S, which is same as the proposed score function in Theorem 1.*

#### *3.2. The Similarity Measure Between HFLTSs Based on the Linguistic Scale Function*

It is already known the regular similarity measure satisfies the following Lemma 2:

**Lemma 2.** *Let S* = { *si*|*<sup>i</sup>* = 0, 1, ··· , 2*t* } *be a LTS, H*1*S and H*2*S be any two HFLTSs; if the similarity measure <sup>S</sup>*-*H*1*S*, *<sup>H</sup>*2*S*. *satisfies the following properties [9]:*


*(3) <sup>S</sup>*-*H*1*S*, *<sup>H</sup>*2*S*. = *<sup>S</sup>*-*H*2*S*, *<sup>H</sup>*1*S*..

*then the similarity measure <sup>S</sup>*-*H*1*S*, *<sup>H</sup>*2*S*. *is a regular similarity measure, and the corresponding distance measure <sup>D</sup>*-*H*1*S*, *<sup>H</sup>*2*S*. = 1 − *<sup>S</sup>*-*H*1*S*, *<sup>H</sup>*2*S*..

The cosine similarity measure proposed by Liao et al. [31] is sometimes different from human intuition in practical decision-making problems, and we can determine this from the following Example 4.

**Example 4.** *When two experts evaluate the performance of a company, they provide their opinions with hesitant fuzzy linguistic information; for the given LTS,* = {*<sup>s</sup>*0 : *very poor*, *s*1 : *poor*,*s*<sup>2</sup> : *slightly poor*,*s*3 :

*f air*,*s*<sup>4</sup> : *slightly good*,*s*<sup>5</sup> : *good*,*s*<sup>6</sup> : *very good*}*, and two experts' evaluations are represented as HFLTSs H*1*S* = {*<sup>s</sup>*1,*s*2} *and H*2*S* = {*<sup>s</sup>*2,*s*4}*, respectively.*

It is already known *H*1*S* = *<sup>H</sup>*2*S*, but from using Formula (3) to calculate the similarity measure between *H*1*S* and *<sup>H</sup>*2*S*, we have *CosHFL*-*H*1*S*, *<sup>H</sup>*2*S*. = 1. That is to say, the property (2) in Lemma 2 is not satisfied. So, the similarity measure *CosHFL* proposed by Liao et al. [31] is not a regular similarity measure. On the other hand, the similarity measure *CosHFL* as defined by Liao et al. [31] used the subscript of linguistic terms directly in the process of operations; they did not consider the semantic environment, which may cause the loss of information in the decision process. In order to overcome its disadvantages, next we will construct a new similarity measure and derive a corresponding distance measure. A scheme of this process is shown in Figure 2.

**Figure 2.** The scheme of the construction of the similarity measure.

At first, we improve the existing distance measure (1) and similarity measure (2) based on a linguistic scale function, which can be defined as follows:

**Definition 8.** *Given a fixed set X, let S* = { *si*|*<sup>i</sup>* = 0, 1, ··· , 2*t* } *be a LTS, and let f be a linguistic scale function, <sup>h</sup>kS*-*xj*. = {*<sup>s</sup>δkl* -*xj*.HHH*<sup>s</sup>δkl* -*xj*. ∈ *S*, *l* = 1, 2, ··· , *Lj*} *, where Lj* = *max*{*l*-*h*1*S*-*xj*., *<sup>l</sup>*(*h*2*S*-*xj*..}*, <sup>l</sup>hkS*-*xj*. *represents the number of elements in <sup>h</sup>kS*-*xj*. (*k* = 1, <sup>2</sup>)*. For any two HFLTSs H*1*S* = {(*xj*, *<sup>h</sup>*1*S*-*xj*.)HH*xj* ∈ *X*} *and H*2*S* = {-*xj*, *<sup>h</sup>*2*S*-*xj*..HH*xj* ∈ *X*} (*j* = 1, 2, ··· , *<sup>n</sup>*)*, if the weight of the different element xj is <sup>ω</sup>j* (*j* = 1, 2, ··· , *<sup>n</sup>*)*, then the improved weighted distance measure between HFLTSs H*1*S and H*2*S can be defined as follows:*

$$D'\_{\omega HFL} \left( H^1\_{\mathbb{S}'} H^2\_{\mathbb{S}} \right) = \left( \sum\_{j=1}^n \frac{\omega\_j}{L\_j} \sum\_{l=1}^{L\_j} \left( \left| f \left( s\_{\delta\_l^1} \left( \mathbf{x}\_j \right) \right) - f \left( s\_{\delta\_l^2} \left( \mathbf{x}\_j \right) \right) \right| \right)^2 \right)^{\frac{1}{2}}.$$

**Theorem 2.** *Let H*1*S and H*2*S be any two HFLTSs, and let f be a linguistic scale function; the distance measure D ωHFL between HFLTSs satisfies the following properties:*


**Proof.** Properties (1), (2), and (3) are obvious, and we omit the proof here. -

**Remark 4.** *For all j* = 1, 2, ··· , *n, if the weight <sup>ω</sup>j* = 1*n , then the improved weighted distance measure D ωHFL*-*H*1*S*, *<sup>H</sup>*2*S*. *is reduced to the improved Euclidean distance measure D HFL*-*H*1*S*, *<sup>H</sup>*2*S*.:

$$D'\_{H\to L}\left(H^1\_{S,\prime}H^2\_S\right) = \left(\frac{1}{n}\left(\sum\_{j=1}^n \frac{1}{L\_j} \sum\_{l=1}^{L\_j} \left(\left|f\left(s\_{\delta^1\_l}(\mathbf{x}\_{\bar{\jmath}})\right) - f\left(s\_{\delta^2\_l}(\mathbf{x}\_{\bar{\jmath}})\right)\right|\right)^2\right)^{\frac{1}{2}}\right)$$

**Definition 9.** *Given a fixed set X, let S* = { *si*|*<sup>i</sup>* = 0, 1, ··· , 2*t* } *be a LTS, and let f be a linguistic scale function, <sup>h</sup>kS*-*xj*. = {*<sup>s</sup>δkl* -*xj*.HHH*<sup>s</sup>δkl* -*xj*. ∈ *S*, *l* = 1, 2, ··· , *Lj*} *, where Lj* = *max*{*l*-*h*1*S*-*xj*., *<sup>l</sup>*(*h*2*S*-*xj*..}*, <sup>l</sup>hkS*-*xj*. *represents the number of elements in <sup>h</sup>kS*-*xj*. (*k* = 1, <sup>2</sup>)*. For any two HFLTSs H*1*S* = {(*xj*, *<sup>h</sup>*1*S*-*xj*.)HH*xj* ∈ *X*} *and H*2*S* = {-*xj*, *<sup>h</sup>*2*S*-*xj*..HH*xj* ∈ *X*} (*j* = 1, 2, ··· , *<sup>n</sup>*)*, if the weight of different element xj is <sup>ω</sup>j* (*j* = 1, 2, ··· , *<sup>n</sup>*)*, then the improved weighted cosine similarity measure between H*1*S and H*2*S can be defined as:*

$$\text{Cov}'\_{\omega\text{HFL}}\left(H^1\_{\mathcal{S}'},H^2\_{\mathcal{S}}\right) = \frac{\sum\_{j=1}^{\mathfrak{n}} \left(\frac{\omega\_{j}}{\sum\_{l}^{n}} \sum\_{l=1}^{L\_{j}} f\left(\mathbf{s}\_{\delta^1\_{l}}\left(\mathbf{x}\_{j}\right)\right) \cdot f\left(\mathbf{s}\_{\delta^2\_{l}}\left(\mathbf{x}\_{j}\right)\right)\right)}{\left(\sum\_{j=1}^{\mathfrak{n}} \left(\frac{\omega\_{j}}{\sum\_{l}^{n}} \sum\_{l=1}^{L\_{j}} \left(f\left(\mathbf{s}\_{\delta^1\_{l}}\left(\mathbf{x}\_{j}\right)\right)\right)^2\right) \cdot \sum\_{j=1}^{\mathfrak{n}} \left(\frac{\omega\_{j}}{\sum\_{l}^{n}} \sum\_{l=1}^{L\_{j}} \left(f\left(\mathbf{s}\_{\delta^2\_{l}}\left(\mathbf{x}\_{j}\right)\right)\right)^2\right)\right)^{\frac{1}{2}}}\right)^{\frac{1}{2}}$$

.

.

**Remark 5.** *For all j* = 1, 2, ··· , *n, if the weight <sup>ω</sup>j* = 1*n , then the improved weighted cosine similarity measure Cos ωHFL*-*H*1*S*, *<sup>H</sup>*2*S*. *is reduced to the similarity measure Cos HFL*-*H*1*S*, *<sup>H</sup>*2*S*.:

$$\text{Cov}'\_{\text{HFL}}\left(H^1\_{\text{S}'},H^2\_{\text{S}}\right) = \frac{\sum\_{j=1}^{\underline{n}} \left(\frac{1}{\underline{\tau}\_j} \sum\_{l=1}^{\underline{L}\_j} f\left(s\_{\delta\_l^1}(\mathbf{x}\_j)\right) \cdot f\left(s\_{\delta\_l^2}(\mathbf{x}\_j)\right)\right)}{\left(\sum\_{j=1}^{\underline{n}} \left(\frac{1}{\underline{\tau}\_j} \sum\_{l=1}^{\underline{L}\_j} \left(f\left(s\_{\delta\_l^1}(\mathbf{x}\_j)\right)\right)^2\right) \cdot \sum\_{j=1}^{\underline{n}} \left(\frac{1}{\underline{\tau}\_j} \sum\_{l=1}^{\underline{L}\_j} \left(f\left(s\_{\delta\_l^2}(\mathbf{x}\_j)\right)\right)^2\right)\right)^{\frac{1}{2}}}$$

In the following, we go on to propose a similarity measure between the HFLTSs, which combine the distance measure *D HFL* and the cosine similarity measure *Cos HFL*.

**Definition 10.** *Given a fixed set X, let S* = { *si*|*<sup>i</sup>* = 0, 1, ··· , 2*t* } *be a LTS, and let f be a linguistic scale function, <sup>h</sup>kS*-*xj*. = {*<sup>s</sup>δkl* -*xj*.HHH*<sup>s</sup>δkl* -*xj*. ∈ *S*, *l* = 1, 2, ··· , *Lj*} *, where Lj* = *max*{*l*(*h*1*S*-*xj*.*, <sup>l</sup>*(*h*2*S*-*xj*.)}*, <sup>l</sup>hkS*-*xj*. *represents the number of elements in <sup>h</sup>kS*-*xj*. (*k* = 1, <sup>2</sup>)*. For any two HFLTSs H*1*S* = {(*xj*, *<sup>h</sup>*1*S*-*xj*.)HH*xj* ∈ *X*} *and H*2*S* = {-*xj*, *<sup>h</sup>*2*S*-*xj*..HH*xj* ∈ *X*} (*j* = 1, 2, ··· , *<sup>n</sup>*)*, then the new similarity measure <sup>S</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. *can be defined as follows:*

*<sup>S</sup>*<sup>∗</sup>*HFL*,*H*1*S*, *H*2*S* = 12*Cos HFLH*1*S*, *H*2*S* + 1 − *D HFLH*1*S*, *<sup>H</sup>*2*S where Cos HFL*-*H*1*S*, *<sup>H</sup>*2*S*. = <sup>∑</sup>*nj*=<sup>1</sup>5 1*Lj* ∑*Lj l*=1 *f* 5*sδ*1*l* (*xj*)6· *f* 5*sδ*2*l* (*xj*)66 <sup>∑</sup>*nj*=<sup>1</sup> 1*Lj* ∑*Lj l*=15 *f* 5*sδ*1*l* (*xj*)66<sup>2</sup>· ∑*nj*=1 1*Lj* ∑*Lj l*=15 *f* 5*sδ*2*l* (*xj*)66<sup>2</sup> 12 , *D HFL*-*H*1*S*, *<sup>H</sup>*2*S*. = 1*<sup>n</sup>*5∑*nj*=<sup>1</sup> 1*Lj* ∑*Lj <sup>l</sup>*=<sup>1</sup>HHH*<sup>f</sup> sδ*1*l* -*xj*. − *f sδ*2*l* -*xj*.HHH<sup>2</sup>66 1 2 .

**Theorem 3.** *The similarity measure <sup>S</sup>*<sup>∗</sup>-*H*1*S*, *<sup>H</sup>*2*S*. *is a regular similarity measure.*

**Proof.** According to Lemma 2, we will prove it by three steps as follows:


From Theorem 3, we know that the proposed similarity measure *<sup>S</sup>*<sup>∗</sup>*HFL* is a regular similarity measure, which overcomes the disadvantages of the similarity measure as defined by Liao et al. [31].

**Remark 6.** *According to the relation between the distance measure and the regular similarity measure, we can obtain a new distance measure <sup>D</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*.*, which is based on the proposed similarity measure <sup>S</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. *:*

*<sup>D</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. = 1 − *<sup>S</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. = 12 -1 − *Cos HFL*-*H*1*S*, *<sup>H</sup>*2*S*. + *D HFL*-*H*1*S*, *<sup>H</sup>*2*S*.. = 1 2 ⎛⎜⎜⎜⎝1 − <sup>∑</sup>*nj*=<sup>1</sup> 5 1*Lj* ∑*Lj l*=1 *f* 5*sδ*1*l* (*xj*)6· *f* 5*sδ*2*l* (*xj*)66 <sup>∑</sup>*nj*=<sup>1</sup> 1*Lj* ∑*Lj l*=15 *f* 5*sδ*1*l* (*xj*)66<sup>2</sup>· ∑*nj*=1 1*Lj* ∑*Lj l*=15 *f* 5*sδ*2*l* (*xj*)66<sup>2</sup> 12 + 5 1*<sup>n</sup>*5∑*nj*=<sup>1</sup> 1*Lj* ∑*Lj <sup>l</sup>*=<sup>1</sup>HHH*<sup>f</sup> sδ*1*l* -*xj*. − *f*(*<sup>s</sup>δ*2*l* -*xj*.)HHH<sup>2</sup>66<sup>12</sup> ⎞⎟⎟⎟⎠.

**Theorem 4.** *The new distance measure <sup>D</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. *satisfies the following properties:*

*(1)* 0 ≤ *<sup>D</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. ≤ 1; *(2) <sup>D</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. = 0 *if and only if H*1*S* = *<sup>H</sup>*2*S*;

$$(3)\quad D\_{HFL}^\*\left(H\_{\mathbb{S'}}^1H\_{\mathbb{S}}^2\right) = D\_{HFL}^\*\left(H\_{\mathbb{S'}}^2H\_{\mathbb{S}}^1\right).$$

**Proof.** Properties (1) and (3) are obvious, here we only present the proof of property (2). If *H*1*S* = *<sup>H</sup>*2*S*, we have *<sup>S</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. = 1, then *<sup>D</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. = 1 − *<sup>S</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. = 0. On the other hand, when *<sup>D</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. = 0, we have *<sup>S</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. = 1 − *<sup>D</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. = 1. Because *<sup>S</sup>*<sup>∗</sup>*HFL* is a regular similarity measure, according to Lemma 2, we have *H*1*S* = *<sup>H</sup>*2*S*. Thus, we obtain *<sup>D</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. = 0 if and only if *H*1*S* = *<sup>H</sup>*2*S*. -

**Definition 11.** *Given a fixed set X, let S* = { *si*|*<sup>i</sup>* = 0, 1, ··· , 2*t* } *be a LTS, and let f be a linguistic scale function. <sup>h</sup>kS*-*xj*. = {*<sup>s</sup>δkl* -*xj*.HHH*<sup>s</sup>δkl* -*xj*. ∈ *S*, *l* = 1, 2, ··· , *Lj*} *, where Lj* = *max*{*l*(*h*1*S*-*xj*.*, <sup>l</sup>*(*h*2*S*-*xj*.)}*, <sup>l</sup>hkS*-*xj*. *represents the number of elements in <sup>h</sup>kS*-*xj*. (*k* = 1, <sup>2</sup>)*. For any two HFLTSs H*1*S* = {(*xj*, *<sup>h</sup>*1*S*-*xj*.)HH*xj* ∈ *X*} *and H*2*S* = {-*xj*, *<sup>h</sup>*2*S*-*xj*..HH*xj* ∈ *X*} (*j* = 1, 2, ··· , *<sup>n</sup>*)*, the associated weighting vector ω* = -*<sup>ω</sup>*1, *ω*2, ··· , *<sup>ω</sup>j*. *satisfying with* ∑*nj*=<sup>1</sup> *<sup>ω</sup>j* = 1 -0 ≤ *<sup>ω</sup>j* ≤ 1.*, then the weighted similarity measure between H*1*S and H*2*S can be defined as:*

$$S^{\*}\_{\\\\\omega\text{HFL}}\left(H^{1}\_{\text{S}\prime}H^{2}\_{\text{S}}\right) = \frac{1}{2}\left(\text{Co}^{\prime}\_{\omega\text{HFL}}\left(H^{1}\_{\text{S}\prime}H^{2}\_{\text{S}}\right) + 1 - D^{\prime}\_{\omega\text{HFL}}\left(H^{1}\_{\text{S}\prime}H^{2}\_{\text{S}}\right)\right)$$

**Theorem 5.** *The weighted similarity measure <sup>S</sup>*<sup>∗</sup>*ωHFL*-*H*1*S*, *<sup>H</sup>*2*S*. *is also a regular similarity measure.* **Proof.** The proof is similar to Theorem 3; we omit it here. -

**Remark 7.** *If the weight of the different element xj is <sup>ω</sup>j* (*j* = 1, 2, ··· , *<sup>n</sup>*)*, satisfying* ∑*nj*=<sup>1</sup> *<sup>ω</sup>j* = 1 -0 ≤ *<sup>ω</sup>j* ≤ 1.*, then the weighted distance measure between H*1*S and H*2*S can be obtained by:*

$$D^\*\_{\omega\,HFL} \left( H^1\_{\mathcal{S}\prime} H^2\_{\mathcal{S}} \right) = 1 - S^\*\_{\omega\,HFL} \left( H^1\_{\mathcal{S}\prime} H^2\_{\mathcal{S}} \right)$$

**Remark 8.** *If we take the weight <sup>ω</sup>j* = 1*n* (*j* = 1, 2, ··· , *n*) *in <sup>S</sup>*<sup>∗</sup>*ωHFL*-*H*1*S*, *<sup>H</sup>*2*S*. *and <sup>D</sup>*<sup>∗</sup>*ωHFL*-*H*1*S*, *<sup>H</sup>*2*S*.*, then <sup>S</sup>*<sup>∗</sup>*ωHFL*-*H*1*S*, *<sup>H</sup>*2*S*. *and <sup>D</sup>*<sup>∗</sup>*ωHFL*-*H*1*S*, *<sup>H</sup>*2*S*. *are reduced to <sup>S</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*. *and <sup>D</sup>*<sup>∗</sup>*HFL*-*H*1*S*, *<sup>H</sup>*2*S*.*, respectively.*

Next, we utilize the medical diagnosis example to illustrate the application of the proposed similarity measure.

**Example 5.** *In traditional Chinese medical diagnosis, doctors diagnose patients by watching, smelling, asking and touching, so the doctor always get some imprecise information about patients' symptoms. Let us consider a set of diagnoses G* = { *Viral f ever*, *Typhoid*, *Pneumonia*, *Stomach problem*} *and a set of symptoms X* = { *temperature*, *headache*, *cough*, *stomach pain}. Assume that a patient, with respect to all symptoms, can be depicted as the following LTS, respectively: S*1 = { *s*0 : *very low*, *s*1 : *low*, *s*2 : *slightly low, s*3 : *normal*,*s*<sup>4</sup> : *slightly high*, *s*5 : *high*,*s*<sup>6</sup> : *very high*}, *Sj* = {*<sup>s</sup>*0 : *none*,*s*1 : *very slight*,*s*<sup>2</sup> : *slight*, *s*3 : *normal*, *s*4 : *slightly terrible*,*s*<sup>5</sup> : *terrible*,*s*<sup>6</sup> : *very terrible*} (*j* = 2, 3, <sup>4</sup>)*. Furthermore, let <sup>ω</sup>j* = (0.25, 0.25, 0.25, 0.25) (*j* = 1, 2, 3, 4) *be the weight vector of symptoms.*

Suppose that the patient *P* = {*Richard*, *Catherine*, *Nicle*, *Kevin*} has all of the symptoms, which are represented by a HFLTS and are given in Table 1.


**Table 1.** Symptoms characteristic for the patients.

According to experience, each patient's symptoms diagnosis can be viewed as a HFLTS, and these are shown in Table 2.

**Table 2.** Symptoms characteristic for the diagnosis.


In order to diagnose what kind of symptoms that the patients belong to, we can calculate the similarity measure between each patient's symptoms and the diagnosis. If the linguistic scale function *f* = *f*1(*si*) = *i*2*t* (*t* = <sup>3</sup>), we apply the proposed similarity measure *<sup>S</sup>*<sup>∗</sup>*ωHFL* to calculate the degree of similarity between each patient's symptoms and the diagnosis; the results are shown in Table 3.


**Table 3.** Hesitant fuzzy linguistic similarity measure.

It is already known that the larger value of similarity measure, the higher the possibility of diagnosis for the patient. From the above results of Table 3, the symptoms of Richard, Catherine, Nicole, and Kevin indicate that they are suffering from typhoid, stomach problems, viral fever, and pneumonia, respectively.

#### **4. The TOPSIS Method with the Proposed Distance Measure** *D***\*** *ωHFL*

In Section 4, we will present the TOPSIS method [40] to the proposed distance measure D<sup>∗</sup> *ω*HFL for hesitant fuzzy linguistic multi-criteria decision-making problems.

Suppose that a panel of decision makers are invited to evaluate the alternatives *H* = {*<sup>H</sup>*1, *H*2, ··· , *Hm*} with respect to the criteria *C* = {*<sup>C</sup>*1,*C*2, ··· ,*Cn*}. Let *S* = {*si*|*<sup>i</sup>* = 0, 1, ··· , 2*t*} be a LTS, let *<sup>ω</sup>j* (*j* = 1, 2, ··· , *n*) be the weight of criteria *Cj*, where 0 ≤ *<sup>ω</sup>j* ≤ 1 (*j* = 1, 2, ··· , *n*) and ∑*n j*=1 *<sup>ω</sup>j* = 1; the hesitant fuzzy linguistic information decision matrix *H* are given as follows:

$$H = \begin{pmatrix} H\_S^{11} & H\_S^{12} & \cdots & H\_S^{1n} \\ H\_S^{21} & H\_S^{22} & \cdots & H\_S^{2n} \\ \vdots & \vdots & \ddots & \vdots \\ H\_S^{m1} & H\_S^{m2} & \cdots & H\_S^{mn} \end{pmatrix}'$$

where *Hij S* = {*sij δl* HH H*l* = 1, 2, ··· , *Lj*}(*i* = 1, 2, ··· , *m*; *j* = 1, 2, ··· , *n*) are HFLTSs, representing the evaluation about alternative *Hi* with respect to the criterion *Cj*.

Next, we present the TOPSIS method with the distance measure *D*∗ *ωHFL* for MCDM problems. In general, it includes the following steps:

**Step 1.** Normalize the hesitant fuzzy linguistic decision matrix *H*.

If the criteria belong to the benefit-type, we need not do anything; if the criteria belong to the cost-type, we should use *neg*(*si*) = *sj*(*<sup>i</sup>* + *j* = 2*t*) to normalize the decision matrix.

**Step 2.** For *i* = 1, 2, ··· , *m*, *j* = 1, 2, ··· , *n*, the hesitant fuzzy linguistic positive ideal solution (HFLPIS) *H*<sup>+</sup> = {*H*1<sup>+</sup> *S* , *H*2<sup>+</sup> *S* , ··· , *H<sup>n</sup>*<sup>+</sup> *S* } and hesitant fuzzy linguistic negative ideal solution (HFLNIS) H− = {H1− S , H2− S , ··· , H<sup>n</sup>− S } are given in the following:

$$H\_{\mathbb{S}}^{j+} = H\_{\mathbb{S}'}^{i\bar{j}}, \\ H\_{\mathbb{S}}^{j-} = H\_{\mathbb{S}}^{i\bar{j}}.$$

For criteria *Cj*(*j* = 1, 2, ··· , *<sup>n</sup>*), by the score function proposed in Definition 7, we can ge<sup>t</sup> the value of *F*∗(*Hij S* ) (*i* = 1, 2, ··· , *<sup>m</sup>*). According to Theorem 1, the order relationship for HFLTSs can be given as: if *F*∗- *H*<sup>1</sup> *S*. > *F*∗- *H*<sup>2</sup> *S*., then *H*<sup>1</sup> *S*> *H*<sup>2</sup> *S*, so that *Hj*<sup>+</sup> *S*and *Hj*<sup>−</sup> *S*can be obtained.

**Step 3.** Use the distance measure to calculate the separation of each alternative between the HFLPIS *H*<sup>+</sup> = {*H*1<sup>+</sup> *S* , *H*2<sup>+</sup> *S* , ··· , *H<sup>n</sup>*<sup>+</sup> *S* } and HFLNIS *H*− = {*H*1− *S* , *H*2− *S* , ··· , *H<sup>n</sup>*− *S* }, respectively.

The distance measure between *Hi*(*i* = 1, 2, ··· , *m*) and *H*<sup>+</sup> can be given as: *D*<sup>+</sup> *i* = ∑*n j*=1 *D*∗ *ωHFL*(*Hij S* , *<sup>H</sup>*+). Similarly to the distance measure *D*<sup>+</sup> *i* , the distance measure between the alternative *Hi*(*i* = 1, 2, ··· , *m*) and *H*− is obtained as: *D*− *i* = ∑*n j*=1 *D*∗ *ωHFL*(*Hij S* , *<sup>H</sup>*−).

For the given *Hi* (*i* = 1, 2, ··· , *<sup>m</sup>*), *<sup>D</sup>*+*i* = ∑*nj*=<sup>1</sup> *<sup>D</sup>*<sup>∗</sup>*ωHFL*(*HijS* , *H*+) = <sup>∑</sup>*nj*=<sup>1</sup>(<sup>1</sup> − 12 (*Cos ωHFL*(*HijS* , *H*+) + 1 − *D ωHFL*(*HijS* , *H*+))) = ∑*nj*=<sup>1</sup> 12 (<sup>1</sup>− *Cos ωHFLHijS* , *H*<sup>+</sup> + *D ωHFLHijS* , *H*<sup>+</sup> = ∑*nj*=<sup>1</sup> 12 ⎛⎜⎜⎜⎜⎝1 − <sup>∑</sup>*nj*=<sup>1</sup> 1*Lj* ∑*Lj l*=1 *f sδijl* (*xj*)· *f* 5*sδ*+*l* (*xj*)6 ⎛⎝∑*nj*=<sup>1</sup>⎛⎝ 1*Lj* ∑*Lj l*=1*f sδijl* (*xj*)<sup>2</sup>⎞⎠· ∑*nj*=1 1*Lj* ∑*Lj l*=15 *f* 5*sδ*+*l* (*xj*)66<sup>2</sup>⎞⎠ 12 + ∑*nj*=1 *<sup>ω</sup>j Lj* ∑*Lj <sup>l</sup>*=<sup>1</sup>5HHHH*<sup>f</sup>* 5*sδijl* -*xj*.6 − *f sδ*+*l* -*xj*.HHHH6<sup>2</sup> 1 2 ⎞⎠; *<sup>D</sup>*<sup>−</sup>*i* = ∑*nj*=<sup>1</sup> *<sup>D</sup>*<sup>∗</sup>*ωHFLHijS* , *<sup>H</sup>*− = <sup>∑</sup>*nj*=<sup>1</sup><sup>1</sup> − 12*Cos ωHFLHijS* , *<sup>H</sup>*− + 1 − *D ωHFLHijS* , *<sup>H</sup>*− = ∑*nj*=<sup>1</sup> 12 (<sup>1</sup>− *Cos ωHFLHijS* , *<sup>H</sup>*− + *D ωHFLHijS* , *<sup>H</sup>*− = ∑*nj*=<sup>1</sup> 12 ⎛⎜⎜⎜⎜⎝1 − <sup>∑</sup>*nj*=<sup>1</sup> 1*Lj* ∑*Lj l*=1 *f sδijl* (*xj*)· *f* 5*sδ*−*l* (*xj*)6 ⎛⎝∑*nj*=<sup>1</sup>⎛⎝ 1*Lj* ∑*Lj l*=1*f sδijl* (*xj*)<sup>2</sup>⎞⎠· ∑*nj*=1 1*Lj* ∑*Lj l*=15 *f* 5*sδ*−*l* (*xj*)66<sup>2</sup>⎞⎠ 12 + ∑*nj*=1 *<sup>ω</sup>j Lj* ∑*Lj <sup>l</sup>*=<sup>1</sup>5HHHH*<sup>f</sup>* 5*sδijl* -*xj*.6 − *f sδ*−*l* -*xj*.HHHH6<sup>2</sup> 1 2 ⎞⎠.

**Step 4.** Calculate the closeness coefficient Φ*i* of each alternative *Hi*(*i* = 1, 2, ··· , *m*):

$$\Phi\_i = \frac{D\_i^-}{D\_i^+ + D\_i^-}.$$

**Step 5.** Rank the alternatives by decreasing order of the closeness coefficient Φ*i*; the greater value Φ*i* is, the better alternative *Hi* will be.

## **5. Numerical Example**

In this section, we give a numerical example that concerns logistics outsourcing (adapted from Wang et al. [38]) to illustrate the feasibility of the TOPSIS method with the proposed distance measure *<sup>D</sup>*<sup>∗</sup>*ωHFL*.
