# Interval Type-2 Fuzzy Envelope of Proportional Hesitant Fuzzy Linguistic Term Set: Application to Large-Scale Group Decision Making

^{1}

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## Abstract

**:**

## 1. Introduction

- (1)
- In order to decrease the information loss in the preferences collection process, a novel LS-GDM model is proposed, in which PHFLTS is applied to capture sub-group hesitation. Both the cohesion and the degree of reliability of the sub-groups can be reflected and measured based on PHFLTS. A sub-group weight determination scheme is introduced, taking into account relevant factors such as size, cohesion, and reliability of sub-groups, which are synthesized when their weights are determined.
- (2)
- To facilitate the CWW process with PHFLTS in LS-GDM under the framework of the fuzzy linguistic approach, novel fuzzy semantic representation models such as type-1 and interval type-2 fuzzy envelopes for PHFLTS will be initially studied. The current research extends the fuzzy encoding approaches from single words to PHFLTS, which increases the flexibility of linguistic preference expression in LS-GDM and further meets the demands of CWW with less information loss.
- (3)
- Entropy measures of PHFLTS are studied in a comprehensive way in order to evaluate linguistic uncertainties during the construction of the interval type-2 fuzzy envelope for PHFLTS. These explorations help to extend the CWW scheme with interval type-2 fuzzy sets from single words to more complex linguistic expressions.

## 2. Preliminary

#### 2.1. PHFLTS

**Definition**

**1**

**Definition**

**2**

**Definition**

**3**

**Definition**

**4**

#### 2.2. Interval Type-2 Fuzzy Set

**Definition**

**5**

#### 2.3. Fuzzy Linguistic Approach

#### 2.4. Large-Scale Group Decision Making under a Linguistic Environment

- A problem to be solved;
- A set of alternatives or a set of possible solutions $X=\{{x}_{1},\cdots ,{x}_{n}\}\phantom{\rule{4pt}{0ex}}(n\ge 2)$ to the problem;
- A set of decision makers $E=\{{e}_{1},\cdots ,{e}_{m}\}\phantom{\rule{4pt}{0ex}}(m\ge 20)$ who express their preferences with regard to alternatives and try to obtain a common solution to the problem;
- A linguistic domain from which decision makers could build linguistic variables to express their preferences among alternatives/solutions.

## 3. Entropy Measures of PHFLTS

**Definition**

**6**

- (E1)
- $E\left({s}_{i}\right)=0$, if and only if ${s}_{i}={s}_{0}$ or ${s}_{i}={s}_{g}$.
- (E2)
- $E\left({s}_{i}\right)=1$, if and only if ${s}_{i}={s}_{\frac{g}{2}}$.
- (E3)
- $E\left({s}_{i}\right)\le E\left({s}_{j}\right)$, if ${s}_{i}\le {s}_{j}\le {s}_{\frac{g}{2}}$ or ${s}_{i}\ge {s}_{j}\ge {s}_{\frac{g}{2}}$.
- (E4)
- $E\left({s}_{i}\right)=E\left(Neg\left({s}_{i}\right)\right)$.

- (1)
- $f(1-x)=f\left(x\right)$, moreover, $f\left(0\right)=f\left(1\right)=0,f\left(\frac{1}{2}\right)=1$.
- (2)
- $f\left(x\right)$ is strictly monotone increasing when $x\in (0,0.5]$ and strictly monotone decreasing when $x\in [0.5,1)$.

**Definition**

**7.**

**Remark**

**1.**

**Definition**

**8.**

- (F1)
- ${E}_{f}\left(P{H}_{S}\right)=0$, if and only if $P{H}_{S}=\{({s}_{0},{p}_{{s}_{0}}),({s}_{g},{p}_{{s}_{g}})\}$ where ${p}_{{s}_{0}}+{p}_{{s}_{g}}=1$.
- (F2)
- ${E}_{f}\left(P{H}_{S}\right)=1$, if and only if $P{H}_{S}=\left\{({s}_{\frac{g}{2}},1)\right\}$.
- (F3)
- Let $P{H}_{{S}_{1}}=\left\{({s}_{t},{p}_{t}^{1})\right|{s}_{t}\in S,t=0,\cdots ,g\}=\{({s}_{{\alpha}_{1}},{p}_{{\alpha}_{1}}^{1}),({s}_{{\alpha}_{2}},{p}_{{\alpha}_{2}}^{1}),\cdots ,({s}_{{\alpha}_{l}},{p}_{{\alpha}_{l}}^{1})|$${p}_{{\alpha}_{1,\cdots ,l}}^{1}>0\}$ as well as $P{H}_{{S}_{2}}=\left\{({s}_{t},{p}_{t}^{2})\right|{s}_{t}\in S,t=0,\cdots ,g\}=\{({s}_{{\alpha}_{1}},{p}_{{\alpha}_{1}}^{2}),$$({s}_{{\alpha}_{2}},{p}_{{\alpha}_{2}}^{2}),\cdots ,$$({s}_{{\alpha}_{l}},{p}_{{\alpha}_{l}}^{2})|{p}_{{\alpha}_{1,\cdots ,l}}^{2}$$>0\}$ be two PHFLTSs on S; if
- ${p}_{{\alpha}_{i}}^{1}>{p}_{{\alpha}_{i}}^{2}$ and ${p}_{{\alpha}_{j}}^{1}<{p}_{{\alpha}_{j}}^{2}$ (for ${\alpha}_{i},{\alpha}_{j}\in \{{\alpha}_{1}\cdots ,{\alpha}_{l}\}$, which satisfy $|{\alpha}_{i}-\frac{g}{2}|>|{\alpha}_{j}-\frac{g}{2}|$),
- ${p}_{{\alpha}_{k}}^{1}={p}_{{\alpha}_{k}}^{2}$ (for ${\alpha}_{k}\in \{{\alpha}_{1}\cdots ,{\alpha}_{l}\}$ and $k\ne i,j$),

then ${E}_{f}\left(P{H}_{{S}_{1}}\right)<{E}_{f}\left(P{H}_{{S}_{2}}\right)$. - (F4)
- If $P{H}_{S}=\left\{({s}_{t},{p}_{t})\right|{s}_{t}\in S,t=0,\cdots ,g\}=\{({s}_{{\alpha}_{1}},{p}_{{\alpha}_{1}}),\cdots ,({s}_{{\alpha}_{i}},{p}_{{\alpha}_{i}}),\cdots ,({s}_{{\alpha}_{l}},{p}_{{\alpha}_{l}})|$${p}_{{\alpha}_{1,\cdots ,l}}>0\}$ is a PHFLTS on S, change $({s}_{{\alpha}_{i}},{p}_{{\alpha}_{i}})$ to $({s}_{{\alpha}_{i}}^{\prime},{p}_{{\alpha}_{i}})$ to get a new PHFLTS $P{H}_{{S}_{2}}$, where $|{\alpha}_{i}-\frac{g}{2}|>|{\alpha}_{i}^{\prime}-\frac{g}{2}|$, then ${E}_{f}\left(P{H}_{{S}_{1}}\right)<{E}_{f}\left(P{H}_{{S}_{2}}\right)$.
- (F5)
- ${E}_{f}\left(P{H}_{S}\right)={E}_{f}\left(Neg\left(P{H}_{S}\right)\right)$,where $Neg\left(P{H}_{S}\right)=\{({s}_{g-{\alpha}_{l}},{p}_{{\alpha}_{l}}),({s}_{g-{\alpha}_{l}+1},{p}_{{\alpha}_{l}-1}),\cdots ,({s}_{g-{\alpha}_{1}},{p}_{{\alpha}_{1}})\}$.

- (H1)
- ${E}_{h}\left(P{H}_{S}\right)=0$, if and only if $P{H}_{S}=\left\{({s}_{{\alpha}_{1}},1)\right\}$.
- (H2)
- ${E}_{h}\left(P{H}_{S}\right)=1$, if $P{H}_{S}=\{({s}_{0},0.5),({s}_{g},0.5)\}$.
- (H3)
- ${E}_{h}\left(P{H}_{S}^{1}\right)\le {E}_{h}\left(P{H}_{S}^{2}\right)$, if $\eta \left(P{H}_{S}^{1}\right)\le \eta \left(P{H}_{S}^{2}\right)$.
- (H4)
- ${E}_{h}\left(P{H}_{S}\right)={E}_{h}\left(Neg\left(P{H}_{S}\right)\right)$,where $Neg\left(P{H}_{S}\right)=\{({s}_{g-{\alpha}_{l}},{p}_{{\alpha}_{l}}),({s}_{g-{\alpha}_{l}+1},{p}_{{\alpha}_{l}-1}),\cdots ,({s}_{g-{\alpha}_{1}},{p}_{{\alpha}_{1}})\}$.

**Theorem**

**1.**

**Proof.**

- (F1)
- If ${E}_{f}\left(P{H}_{S}\right)={\sum}_{i=1}^{l}{p}_{{\alpha}_{i}}E\left({s}_{{\alpha}_{i}}\right)=0$, then $E\left({s}_{{\alpha}_{i}}\right)=0$, where ${s}_{{\alpha}_{i}}\in P{H}_{S}$. According to the axiomatic definition of the fuzzy entropy of linguistic terms, we know that ${s}_{{\alpha}_{i}}={s}_{0}$ or ${s}_{{\alpha}_{i}}={s}_{g}$, where ${s}_{{\alpha}_{i}}\in P{H}_{S}$. Therefore, $P{H}_{S}=\{({s}_{0},{p}_{{s}_{0}}),({s}_{g},{p}_{{s}_{g}})\}$, where ${p}_{{s}_{0}}+{p}_{{s}_{g}}=1$. On the contrary, if $P{H}_{S}=\{({s}_{0},{p}_{{s}_{0}}),({s}_{g},{p}_{{s}_{g}})\}$, where ${p}_{{s}_{0}}+{p}_{{s}_{g}}=1$, then ${\sum}_{i=1}^{l}{p}_{{\alpha}_{i}}E\left({s}_{{\alpha}_{i}}\right)=0$; that is, ${E}_{f}\left(P{H}_{S}\right)=0$.
- (F2)
- Since ${\sum}_{i=1}^{l}{p}_{{\alpha}_{i}}=1$, we know that ${E}_{f}\left(P{H}_{S}\right)={\sum}_{i=1}^{l}{p}_{{\alpha}_{i}}E\left({s}_{{\alpha}_{i}}\right)=1$ if and only if $E\left({s}_{{\alpha}_{i}}\right)=1$. From the axiomatic definition of the fuzzy entropy of linguistic terms, $E\left({s}_{i}\right)=1$ if and only if ${s}_{i}={s}_{\frac{g}{2}}$, then we know that ${E}_{f}\left(P{H}_{S}\right)={\sum}_{i=1}^{l}{p}_{{\alpha}_{i}}E\left({s}_{{\alpha}_{i}}\right)=1$ if and only if $P{H}_{S}=\left\{({s}_{\frac{g}{2}},1)\right\}$.
- (F3)
- If $P{H}_{{S}_{1}}=\left\{({s}_{t},{p}_{t}^{1})\right|{s}_{t}\in S,t=0,\cdots ,g\}=\{({s}_{{\alpha}_{1}},{p}_{{\alpha}_{1}}^{1}),({s}_{{\alpha}_{2}},{p}_{{\alpha}_{2}}^{1}),\cdots ,({s}_{{\alpha}_{l}},{p}_{{\alpha}_{l}}^{1})|$${p}_{{\alpha}_{1,\cdots ,l}}^{1}>0\}$, and $P{H}_{{S}_{2}}=\left\{({s}_{t},{p}_{t}^{2})\right|{s}_{t}\in S,t=0,\cdots ,g\}=\{({s}_{{\alpha}_{1}},{p}_{{\alpha}_{1}}^{2}),({s}_{{\alpha}_{2}},{p}_{{\alpha}_{2}}^{2}),\cdots ,$$({s}_{{\alpha}_{l}},{p}_{{\alpha}_{l}}^{2})|{p}_{{\alpha}_{1,\cdots ,l}}^{2}$$>0\}$ are two PHFLTSs on S, $|{\alpha}_{i}-\frac{g}{2}|>|{\alpha}_{j}-\frac{g}{2}|$, then $E\left({\alpha}_{i}\right)<E\left({\alpha}_{j}\right)$. If ${p}_{{\alpha}_{i}}^{1}>{p}_{{\alpha}_{i}}^{2}$, ${p}_{{\alpha}_{j}}^{1}<{p}_{{\alpha}_{j}}^{2}$, then ${p}_{{\alpha}_{i}}^{1}E\left({\alpha}_{i}\right)+{p}_{{\alpha}_{j}}^{1}E\left({\alpha}_{j}\right)<{p}_{{\alpha}_{i}}^{2}E\left({\alpha}_{i}\right)+{p}_{{\alpha}_{j}}^{2}E\left({\alpha}_{j}\right)$. If ${p}_{{\alpha}_{k}}^{1}={p}_{{\alpha}_{k}}^{2}(k\ne i,j)$, then ${\sum}_{i=1}^{l}{p}_{{\alpha}_{i}}^{1}E\left({s}_{{\alpha}_{i}}\right)<{\sum}_{i=1}^{l}{p}_{{\alpha}_{i}}^{2}E\left({s}_{{\alpha}_{i}}\right)$, and ${E}_{f}\left(P{H}_{{S}_{1}}\right)<{E}_{f}\left(P{H}_{{S}_{2}}\right)$.
- (F4)
- If $({s}_{{\alpha}_{t}},{p}_{{\alpha}_{t}}^{1})$ in $P{H}_{{S}_{1}}$ is changed to $({s}_{{\alpha}_{t}}^{\prime},{p}_{{\alpha}_{t}}^{1})$ to get a new PHFLTS $P{H}_{{S}_{2}}$, where $|{\alpha}_{t}-\frac{g}{2}|>|{\alpha}_{t}^{\prime}-\frac{g}{2}|$, then $E\left({s}_{{\alpha}_{t}}\right)<E\left({s}_{{\alpha}_{t}^{\prime}}\right)$; therefore, ${\sum}_{i=1}^{l}{p}_{{\alpha}_{i}}^{1}E\left({s}_{{\alpha}_{i}}\right)<{\sum}_{i=1,i\ne t}^{l}{p}_{{\alpha}_{i}}^{1}E\left({s}_{{\alpha}_{i}}\right)+{p}_{{\alpha}_{t}}^{1}E\left({s}_{{\alpha}_{t}^{\prime}}\right)$, that is, ${E}_{f}\left(P{H}_{{S}_{1}}\right)<{E}_{f}\left(P{H}_{{S}_{2}}\right)$.
- (F5)
- From the axiomatic definition of the fuzzy entropy of linguistic terms, $E\left({s}_{i}\right)=E\left(Neg\left({s}_{i}\right)\right)$; therefore, ${\sum}_{i=1}^{l}{p}_{{\alpha}_{i}}E\left({s}_{{\alpha}_{i}}\right)={\sum}_{i=1}^{l}{p}_{{\alpha}_{i}}E\left(Neg\left({s}_{{\alpha}_{i}}\right)\right)={\sum}_{i=1}^{l}{p}_{{\alpha}_{i}}E\left({s}_{g-{\alpha}_{i}}\right)$. That is, ${E}_{f}\left(P{H}_{S}\right)={E}_{f}\left(Neg\left(P{H}_{S}\right)\right)$, where $Neg\left(P{H}_{S}\right)=\{({s}_{g-{\alpha}_{l}},{p}_{{\alpha}_{l}}),({s}_{g-{\alpha}_{l}+1},{p}_{{\alpha}_{l}-1}),\cdots ,$$({s}_{g-{\alpha}_{1}},{p}_{{\alpha}_{1}})\}$.

**Theorem**

**2.**

**Proof.**

- (H1)
- If ${E}_{h}\left(P{H}_{S}\right)=0$, then ${\sum}_{j=i+1}{\sum}_{i=1}^{l-1}{p}_{{\alpha}_{i}}{p}_{{\alpha}_{j}}(I\left({s}_{{\alpha}_{j}}\right)-I\left({s}_{{\alpha}_{i}}\right))=0$, that is, $I\left({s}_{{\alpha}_{j}}\right)=I\left({s}_{{\alpha}_{i}}\right)$ for all ${s}_{{\alpha}_{i}},{s}_{{\alpha}_{j}}\in P{H}_{S}$, so there is only one term contained in $P{H}_{S}$, $P{H}_{S}=\left\{({s}_{{\alpha}_{1}},1)\right\}$; On the contrary, if $P{H}_{S}=\left\{({s}_{{\alpha}_{1}},1)\right\}$, then ${\sum}_{j=i+1}{\sum}_{i=1}^{l-1}{p}_{{\alpha}_{i}}{p}_{{\alpha}_{j}}(I\left({s}_{{\alpha}_{j}}\right)-I\left({s}_{{\alpha}_{i}}\right))=0$, therefore, ${E}_{h}\left(P{H}_{S}\right)=0$.
- (H2)
- If $P{H}_{S}=\{({s}_{0},0.5),({s}_{g},0.5)\}$, then ${\sum}_{j=i+1}{\sum}_{i=1}^{l-1}{p}_{{\alpha}_{i}}{p}_{{\alpha}_{j}}(I\left({s}_{{\alpha}_{j}}\right)-I\left({s}_{{\alpha}_{i}}\right))=\frac{g}{4}$, and ${E}_{h}\left(P{H}_{S}\right)=\frac{4}{g}\times {\sum}_{j=i+1}{\sum}_{i=1}^{l-1}{p}_{{\alpha}_{i}}{p}_{{\alpha}_{j}}(I\left({s}_{{\alpha}_{j}}\right)-I\left({s}_{{\alpha}_{i}}\right))=1$.
- (H3)
- From the definition of $\eta \left(P{H}_{S}\right)$ in Equation (7), it is easy to obtain that ${E}_{h}\left(P{H}_{S}^{1}\right)\le {E}_{h}\left(P{H}_{S}^{2}\right)$ if $\eta \left(P{H}_{S}^{1}\right)\le \eta \left(P{H}_{S}^{2}\right)$.
- (H4)
- Since $I\left({s}_{{\alpha}_{j}}\right)-I\left({s}_{{\alpha}_{i}}\right)=I\left({s}_{g-{\alpha}_{i}}\right)-I\left({s}_{g-{\alpha}_{j}}\right)$ for all ${s}_{{\alpha}_{j}},{s}_{{\alpha}_{i}}\in P{H}_{S}$ and ${s}_{g-{\alpha}_{i}},{s}_{g-{\alpha}_{j}}\in Neg\left(P{H}_{S}\right)$, we have ${E}_{h}\left(P{H}_{S}\right)=\frac{4}{g}\times {\sum}_{j=i+1}{\sum}_{i=1}^{l-1}{p}_{{\alpha}_{i}}{p}_{{\alpha}_{j}}(I\left({s}_{{\alpha}_{j}}\right)-I\left({s}_{{\alpha}_{i}}\right))=\frac{4}{g}\times {\sum}_{j=i+1}{\sum}_{i=1}^{l-1}{p}_{{\alpha}_{i}}{p}_{{\alpha}_{j}}(I\left({s}_{g-{\alpha}_{i}}\right)-I\left({s}_{g-{\alpha}_{j}}\right))={E}_{h}\left(Neg\left(P{H}_{S}\right)\right).$

**Definition**

**9.**

- (C1)
- ${E}_{c}\left(P{H}_{S}\right)=0$, if and only if $P{H}_{S}=\left\{({s}_{0},1)\right\}$ or $P{H}_{S}=\left\{({s}_{g},1)\right\}$.
- (C2)
- ${E}_{c}\left(P{H}_{S}\right)=1$, if and only if $P{H}_{S}=\left\{({s}_{\frac{g}{2}},1)\right\}$.
- (C3)
- If ${E}_{f}\left(P{H}_{{S}_{1}}\right)\le {E}_{f}\left(P{H}_{{S}_{2}}\right)$ and ${E}_{h}\left(P{H}_{{S}_{1}}\right)\le {E}_{h}\left(P{H}_{{S}_{2}}\right)$, then ${E}_{c}\left(P{H}_{{S}_{1}}\right)\le {E}_{c}\left(P{H}_{{S}_{2}}\right)$.
- (C4)
- ${E}_{c}\left(P{H}_{S}\right)={E}_{c}\left(Neg\left(P{H}_{S}\right)\right)$, where$Neg\left(P{H}_{S}\right)=\{({s}_{g-{\alpha}_{l}},{p}_{{\alpha}_{l}}),({s}_{g-{\alpha}_{l}+1},{p}_{{\alpha}_{l}-1}),\cdots ,({s}_{g-{\alpha}_{1}},{p}_{{\alpha}_{1}})\}$.

**Theorem**

**3.**

- (1)
- $g(0,0)=0$, $g(1,0)=1$.
- (2)
- $g(x,y)$ is strictly monotone increasing with respect to x and y, respectively,

**Example**

**1.**

**Definition**

**10.**

- (T1)
- ${E}_{t}\left(P{H}_{S}\right)=0$, if and only if $P{H}_{S}=\left\{({s}_{0},1)\right\}$ or $P{H}_{S}=\left\{({s}_{g},1)\right\}$.
- (T2)
- ${E}_{t}\left(P{H}_{S}\right)=1$, if and only if $P{H}_{S}=\left\{({s}_{\frac{g}{2}},1)\right\}$ or $P{H}_{S}=\{({s}_{0},0.5),({s}_{g},0.5)\}$.
- (T3)
- If ${E}_{f}\left(P{H}_{{S}_{1}}\right)\le {E}_{f}\left(P{H}_{{S}_{2}}\right)$ and ${E}_{h}\left(P{H}_{{S}_{1}}\right)\le {E}_{h}\left(P{H}_{{S}_{2}}\right)$, then ${E}_{t}\left(P{H}_{{S}_{1}}\right)\le {E}_{t}\left(P{H}_{{S}_{2}}\right)$.
- (T4)
- ${E}_{t}\left(P{H}_{S}\right)={E}_{t}\left(Neg\left(P{H}_{S}\right)\right)$, where $Neg\left(P{H}_{S}\right)=\{({s}_{g-{\alpha}_{l}},{p}_{{\alpha}_{l}}),({s}_{g-{\alpha}_{l}+1},{p}_{{\alpha}_{l}-1}),\cdots ,$$({s}_{g-{\alpha}_{1}},{p}_{{\alpha}_{1}})\}$.

**Theorem**

**4.**

- (1)
- $f(0,0)=0$, $f(1,0)=1$, $f(x,y)=f(y,x)$.
- (2)
- $f(x,y)$ is monotone increasing with respect to x and y, respectively,

**Example**

**2.**

## 4. Type-1 Fuzzy Envelope of PHFLTS

#### 4.1. General Process

**Definition**

**11**

#### 4.2. Type-1 Fuzzy Envelope of PHFLTS Corresponding to Linguistic Expression “between ${s}_{i}$ and ${s}_{j}$"

- (1)
- If l is odd, then$$b=OW{A}_{{W}^{2}}({\xi}_{\frac{l+1}{2}},{\xi}_{\frac{l+3}{2}},\cdots ,{\xi}_{l})/{p}_{max}$$$$c=OW{A}_{{W}^{1}}({\xi}_{1},{\xi}_{2},\cdots ,{\xi}_{\frac{l+1}{2}})/{p}_{max}$$${w}_{1}^{2}={\beta}_{1}^{\frac{l-1}{2}},{w}_{2}^{2}=(1-{\beta}_{1}){\beta}_{1}^{\frac{l-3}{2}},\cdots ,{w}_{\frac{l-1}{2}}^{2}=(1-{\beta}_{1}){\beta}_{1},{w}_{\frac{l+1}{2}}^{2}=1-{\beta}_{1}$, (${\beta}_{1}\in [0,1]$),and ${W}^{1}={({w}_{1}^{1},{w}_{2}^{1},\cdots ,{w}_{\frac{l+1}{2}}^{1})}^{T}$ in Equation (15) is${w}_{1}^{1}={\beta}_{2}$, ${w}_{2}^{1}={\beta}_{2}(1-{\beta}_{2})$,…,${w}_{\frac{l-1}{2}}^{1}={\beta}_{2}{(1-{\beta}_{2})}^{\frac{l-3}{2}}$, ${w}_{\frac{l+1}{2}}^{1}={(1-{\beta}_{2})}^{\frac{l-1}{2}}$, (${\beta}_{2}\in [0,1]$).
- (2)
- If l is even, then$$b=OW{A}_{{W}^{2}}({\xi}_{\frac{l+2}{2}},{\xi}_{\frac{l+4}{2}},\cdots ,{\xi}_{l})/{p}_{max}$$$$c=OW{A}_{{W}^{1}}({\xi}_{1},{\xi}_{2},\cdots ,{\xi}_{\frac{l}{2}})/{p}_{max}$$${w}_{1}^{2}={\beta}_{1}^{\frac{l-2}{2}},{w}_{2}^{2}=(1-{\beta}_{1}){\beta}_{1}^{\frac{l-4}{2}},\cdots ,{w}_{\frac{l-2}{2}}^{2}=(1-{\beta}_{1}){\beta}_{1},{w}_{\frac{l}{2}}^{2}=1-{\beta}_{1}$, (${\beta}_{1}\in [0,1]$);where ${p}_{max}=max\{{p}_{{\alpha}_{1}},{p}_{{\alpha}_{2}},\cdots ,{p}_{{\alpha}_{l}}\}$ and ${W}^{1}={({w}_{1}^{1},{w}_{2}^{1},\cdots ,{w}_{\frac{l}{2}}^{1})}^{T}$ in Equation (17) is${w}_{1}^{1}={\beta}_{2}$, ${w}_{2}^{1}={\beta}_{2}(1-{\beta}_{2})$, ⋯,${w}_{\frac{l-2}{2}}^{1}={\beta}_{2}{(1-{\beta}_{2})}^{\frac{l-4}{2}}$, ${w}_{\frac{l}{2}}^{1}={(1-{\beta}_{2})}^{\frac{l-2}{2}}$, (${\beta}_{2}\in [0,1]$).

#### 4.3. Type-1 Fuzzy Envelope of PHFLTS Corresponding to Linguistic Expression “at Least ${s}_{i}$"

#### 4.4. Type-1 Fuzzy Envelope of PHFLTS Corresponding to Linguistic Expression “ at Most ${s}_{i}$"

#### 4.5. A Strategy for Determining Parameter $\beta $ in Uncertainty Evaluation for PHFLTS

- For the PHFLTS corresponding to the expression “at least $s{}_{i}$”:If $\varphi =0$, then $\beta =\frac{{\alpha}_{1}}{g}$, which is obtained from $\beta =\frac{i}{g}$ in [29], and the orness reaches its maximum, which can be computed by Equations (20) and (21). If $\varphi =g-1$, then the orness reaches its minimum, and orness $=0$ and $\beta =0$, which is also computed by Equations (20) and (21). The value of $\varphi $ increases from 0 to $g-1$, while $\beta $ decreases from $\beta =\frac{{\alpha}_{1}}{g}$ to 0. A function f can be defined as follows:$$f:[0,g-1]\to [0,\frac{{\alpha}_{1}}{g}]$$$$f\left(0\right)=\frac{{\alpha}_{1}}{g},f(g-1)=0$$Moreover, it is easy to find a linear function that satisfies such conditions:$$\beta =\frac{{\alpha}_{1}}{g(g-1)}(g-1-\varphi )$$
- For the PHFLTS corresponding to the expression “between $s{}_{i}$ and $s{}_{j}$”:If $\varphi =0$, then ${\beta}_{1}=\frac{g-({\alpha}_{l}-{\alpha}_{1})}{g-1}$, which is obtained from ${\beta}_{1}=\frac{g-(j-i)}{g-1}$ in [29], and the orness reaches its maximum, which can be computed by Equations (20) and (21). If $\varphi =g-1$, then the orness reaches its minimum, orness $=0$ and ${\beta}_{1}=0$, which is also computed by Equations (20) and (21). The value of ${\varphi}_{1}$ increases from 0 to $g-1$, while ${\beta}_{1}$ decreases from ${\beta}_{1}=\frac{g-({\alpha}_{l}-{\alpha}_{1})}{g-1}$ to 0. A function f can be defined as follows:$${f}_{1}:[0,g-1]\to [0,\frac{g-({\alpha}_{l}-{\alpha}_{1})}{g-1}],$$$${f}_{1}\left(0\right)=\frac{g-({\alpha}_{l}-{\alpha}_{1})}{g-1},{f}_{1}(g-1)=0$$Moreover, it is easy to find a linear function that satisfies such conditions:$${\beta}_{1}=\frac{g-({\alpha}_{l}-{\alpha}_{1})}{{(g-1)}^{2}}(g-1-\varphi )$$If $\varphi =0$, then ${\beta}_{2}=\frac{({\alpha}_{l}-{\alpha}_{1})-1}{g-1}$, which is obtained from ${\beta}_{2}=\frac{(j-i)-1}{g-1}$ in [29], and the orness reaches its maximum, which can be computed by Equations (20) and (21). If $\varphi =g-1$, then the orness reaches its minimum, orness $=0$ and ${\beta}_{2}=0$, which is also computed by Equations (20) and (21). The value of $\varphi $ increases from 0 to $g-1$, while ${\beta}_{2}=0$ decreases from ${\beta}_{2}=\frac{({\alpha}_{l}-{\alpha}_{1})-1}{g-1}$ to 0. A function ${f}_{2}$ can be defined as follows:$${f}_{2}:[0,g-1]\to [0,\frac{({\alpha}_{l}-{\alpha}_{1})-1}{g-1}],$$$${f}_{2}\left(0\right)=\frac{({\alpha}_{l}-{\alpha}_{1})-1}{g-1},{f}_{2}(g-1)=0$$Moreover, it is easy to find a linear function that satisfies such conditions:$${\beta}_{2}=\frac{({\alpha}_{l}-{\alpha}_{1})-1}{{(g-1)}^{2}}(g-1-\varphi )$$
**Lemma****1.**If $\varphi =0$, then ${\beta}_{1}+{\beta}_{2}=1$.**Proof.**${\beta}_{1}+{\beta}_{2}=\frac{g-({\alpha}_{l}-{\alpha}_{1})}{{(g-1)}^{2}}(g-1-\varphi )+\frac{({\alpha}_{l}-{\alpha}_{1})-1}{{(g-1)}^{2}}(g-1-\varphi )=\frac{1}{g-1}(g-1-\varphi )$. Then, if $\varphi =0$, then ${\beta}_{1}+{\beta}_{2}=1$. □ - For the PHFLTS corresponding to the expression “at most $s{}_{i}$”:If $\varphi =0$, then $\beta =\frac{{\alpha}_{l}}{g}$, which is obtained from $\beta =\frac{{\alpha}_{l}}{g}$ in [29], and the orness reaches its maximum, which can be computed by Equations (20) and (21). If $\varphi =g-1$, then the orness reaches its minimum, orness $=0$ and $\beta =0$, which is also computed by Equations (20) and (21). The value of $\varphi $ increases from 0 to $g-1$, while $\beta $ decreases from $\beta =\frac{{\alpha}_{l}}{g}$ to 0. A function f can be defined as follows:$$f:[0,g-1]\to [0,\frac{{\alpha}_{l}}{g}],$$$$f\left(0\right)=\frac{{\alpha}_{l}}{g},f(g-1)=0$$Moreover, it is easy to find a linear function that satisfies such conditions:$$\beta =\frac{{\alpha}_{l}}{g(g-1)}(g-1-\varphi )$$

## 5. Type-2 Fuzzy Envelope of PHFLTS

- (P1)
- $\gamma \left(P{H}_{S}\right)=1$ if $P{H}_{S}=\{({s}_{0},{p}_{1}),\cdots ,({s}_{g},{p}_{g})|{p}_{1,\cdots ,g}>0\}$.This principle indicates that the importance degree of hesitancy reaches its highest when all terms appear in the PHFLTS.
- (P2)
- $\gamma \left(P{H}_{S}\right)=0$ if $P{H}_{S}=\left\{({s}_{{\alpha}_{1}},1)\right\}$.This principle indicates that the importance degree of hesitancy reaches its lowest when there is no hesitancy in a PHFLTS.
- (P3)
- $\gamma \left(P{H}_{S}\right)<\gamma \left(P{H}_{S}^{\prime}\right)$ if $P{H}_{S}=\{({s}_{{\alpha}_{1}},{p}_{{\alpha}_{1}}),\cdots ,({s}_{t},{p}_{t}),\cdots ,({s}_{{\alpha}_{l}},{p}_{{\alpha}_{l}})|{p}_{{\alpha}_{1,\cdots ,l}}>0\}$,and $P{H}_{S}^{\prime}=\{({s}_{{\alpha}_{1}},{p}_{{\alpha}_{1}}),\cdots ,({s}_{t},{p}_{{t}_{1}}),({s}_{f},{p}_{{t}_{2}}),\cdots ,({s}_{{\alpha}_{l}},{p}_{{\alpha}_{l}})|{p}_{{\alpha}_{1,\cdots ,l}}>0\}$, where ${p}_{t}={p}_{{t}_{1}}+{p}_{{t}_{2}}$.This principle indicates that when a new term is added in a PHFLTS, the importance degree of hesitancy should be increased.
- (P4)
- If $P{H}_{S}=\{({s}_{{\alpha}_{1}},{p}_{{\alpha}_{1}}),\cdots ,({s}_{t},{p}_{t}),\cdots ,({s}_{{\alpha}_{l}},{p}_{{\alpha}_{l}})|{p}_{{\alpha}_{1,\cdots ,l}}>0\}$, and$P{H}_{{S}_{1}}=\{({s}_{{\alpha}_{1}},{p}_{{\alpha}_{1}}),\cdots ,({s}_{t},{p}_{{t}_{1}}),({s}_{{f}_{1}},{p}_{{t}_{2}}),\cdots ,({s}_{{\alpha}_{l}},{p}_{{\alpha}_{l}})|{p}_{{\alpha}_{1,\cdots ,l}}>0\}$,$P{H}_{{S}_{2}}=\{({s}_{{\alpha}_{1}},{p}_{{\alpha}_{1}}),\cdots ,({s}_{t},{p}_{{t}_{1}}),({s}_{{f}_{2}},{p}_{{t}_{2}}),\cdots ,({s}_{{\alpha}_{l}},{p}_{{\alpha}_{l}})|{p}_{{\alpha}_{1,\cdots ,l}}>0\}$,where ${p}_{t}={p}_{{t}_{1}}+{p}_{{t}_{2}}$, and $|{t}_{1}-\frac{g}{2}|<|{t}_{2}-\frac{g}{2}|$, then $\gamma \left(P{H}_{{S}_{1}}\right)>\gamma \left(P{H}_{{S}_{2}}\right)$.This principle indicates that when two different terms are added in a PHFLTS ${H}_{S}$, the change of $\gamma \left(P{H}_{S}\right)$ is positively related to the fuzzy degree of the added linguistic term.
- (P5)
- $\gamma \left(P{H}_{S}\right)=\gamma \left(Neg\left(P{H}_{S}\right)\right)$.This principle indicates the importance degree of hesitancy should be the same for a PHFLTS and its negative because the hesitancy degree is the same.

**Theorem**

**5.**

**Proof.**

#### 5.1. Comparative Analysis: Type-1 and Type-2 Fuzzy Envelopes of PHFLTS

- (1)
- From the point of view of computation complexity: The main difference is reflected by the parameter determination process. The complexity of type-1 fuzzy envelopes is a bit lower since only four parameters (i.e., the four parameters $a,b,c,d$ in $T(a,b,c,d)$) need to be computed to achieve a trapezoidal fuzzy number as the type-1 fuzzy envelope for PHFLTS, while five parameters (i.e., the five parameters $a,b,c,d,\underline{h}$ in $T(a,b,c,d,{a}^{\prime},{b}^{\prime},{c}^{\prime},{d}^{\prime},\underline{h}$)) need to be determined to achieve the corresponding type-2 fuzzy envelope for PHFLTS. The extra parameter $\underline{h}$ is determined on the basis of the linguistic uncertainty contained in the PHFLTS, which can be evaluated by using the proposed entropy measures.
- (2)
- From the point of view of information lost: During the past few decades, IT2FS has been a widely accepted tool in the presentation of linguistic information. The current work is an extension study of the fuzzy encoding approach for linguistic information; we extend the IT2FS encoding technique from single words to more complex linguistic expressions. Compared with the type-1 fuzzy envelope, when the type-2 fuzzy envelope is applied, the uncertainties contained in linguistic information can be evaluated by using the proposed comprehensive entropy. Therefore, more information can be reflected and restored during the process of CWW. In this way, the information lost could be decreased, which is the main reason why we recommend the use of the type-2 fuzzy envelope.
- (3)
- From the point of view of feasibility: There are plenty of research that pay attention to decision making based on either type-1 fuzzy sets or type-2 fuzzy sets. Therefore, as long as the fuzzy envelopes of PHFLTSs can be computed, most decision-making problems based on PHFLTS can be transformed into decision-making problems based on type-1 or type-2 fuzzy sets; then, the decision results can be obtained. The proposed fuzzy envelopes allow the CWW process that is based on PHFLTS in decision making follow the framework of the fuzzy linguistic approach. Hence, these two fuzzy representation models can make the CWW process with PHFLTS feasible.

## 6. An LS-GDM Approach Based on PHFLTS

#### 6.1. Problem Formulation

#### 6.2. Framework Overview

- (1)
- Information collection.At this stage, every decision maker is requested to express their preferences from the alternatives by using single terms or HFLTS or EHFLTS. In this way, the t preference matrix is obtained.
- (2)
- Cluster and information fusion.
- Cluster principle.In order to make a group decision based on the assessments provided from a large group of decision makers, first of all, we should select a suitable cluster approach to classify the large-scale group into several small-scale sub-groups. At this stage, our expectation is to “classify the decision makers who hold similar opinions into one class”.
- Cluster objective.For each alternative with respect to each criteria, group evaluations gathered from all decision makers in a sub-group will form a PHFLTS. This process is called an information-fusion process. We expect the linguistic terms contained in each EHFLT to be consecutive rather than discrete. From the view point of position in the linguistic-term set, linguistic terms contained in each EHFLT should be “the closer, the better”. Meanwhile, we expect the number of linguistic terms in an EHFLT to be “the less, the better". To illustrate this, the sub-group evaluation $P{H}_{{S}_{1}}=\{{s}_{1},{s}_{3},{s}_{4}\}$ is more expected than $P{H}_{{S}_{2}}=\{{s}_{1},{s}_{3},{s}_{6}\}$, and $P{H}_{{S}_{3}}=\{{s}_{3},{s}_{4}\}$ is more expected than $P{H}_{{S}_{4}}=\{{s}_{1},{s}_{3},{s}_{4}\}$.Our objective is just to put adjacent linguistic terms together after the classification. Therefore, the similarity of two linguistic terms ${s}_{i}$ and ${s}_{j}$ could be briefly evaluated by using the euclidean distance between their subscripts i and j.
- Cluster method.In the current research, we suggest the use of the cluster method based on the fuzzy equivalence relationship when the cluster starts with linguistic terms/HFLTSs and results with PHFLTSs. In this way, (1) the fuzzy property of linguistic values could be considered; (2) compared with the fuzzy-c means approach, it can avoid a possible unreasonable cluster result caused by the inappropriate selection of the initial cluster center; (3) the computation process is simple.

- (3)
- Best alternative selection.
- Computation tool.The proposed fuzzy representation models, i.e., fuzzy envelopes of PHFLTS, will be adopted as the computation tool during the CWW process.
- Alternative selection.We will compute the utility values for alternatives and then select the one with the largest utility value as the best selection.

#### 6.3. Method Description

- Cluster and information fusion.To classify decision makers in an LS-GDM problem into several sub-groups, the cluster scheme based on a fuzzy equivalence relation [50] is extended in order to deal with linguistic decision matrices formed by terms in $S=\{{s}_{0},{s}_{1},\cdots ,{s}_{g}\}$.
- (1)
- The construction of similarity matrix.Firstly, we should select an approach to compute the similarity between two linguistic decision matrices. Suppose that there are two linguistic decision matrices,$$A=\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ {a}_{21}& {a}_{22}& \cdots & {a}_{2n}\\ \cdots & \cdots & \cdots & \cdots \\ {a}_{n1}& {a}_{n2}& \cdots & {a}_{nn}\end{array}\right)$$$$B=\left(\begin{array}{cccc}{b}_{11}& {b}_{12}& \cdots & {b}_{1n}\\ {b}_{21}& {b}_{22}& \cdots & {b}_{2n}\\ \cdots & \cdots & \cdots & \cdots \\ {b}_{n1}& {b}_{n2}& \cdots & {b}_{nn}\end{array}\right)$$The similarity between these two matrices is defined as follows:$$1-\frac{{\sum}_{i=1}^{n}{\sum}_{j=1}^{n}{|p\left({a}_{ij}\right)-p\left({b}_{ij}\right)|}^{k}}{{g}^{k}\xb7{n}^{2}}$$
**Remark****2.**Suppose that ${a}_{ij}=\{{s}_{{\alpha}_{1}},\cdots ,{s}_{{\alpha}_{q}}\}$, we can define $p\left({a}_{ij}\right)=\frac{{\alpha}_{1}+\cdots +{\alpha}_{q}}{q}$. More specifically, when only one term exists in ${a}_{ij}$, $p\left({a}_{ij}\right)$ will be the subscript of the linguistic term. For instance, for ${a}_{ij}={s}_{6}$, we have $p\left({a}_{ij}\right)=6$.There are t decision makers, therefore the similarity matrix is constructed as a $t\times t$ matrix:$$R=\left(\begin{array}{cccc}{r}_{11}& {r}_{12}& \cdots & {r}_{1t}\\ {r}_{21}& {r}_{22}& \cdots & {r}_{2t}\\ \cdots & \cdots & \cdots & \cdots \\ {r}_{t1}& {r}_{t2}& \cdots & {r}_{tt}\end{array}\right)$$ - (2)
- Compute the transitive closure and the fuzzy equivalence relationship. If ${R}^{h}\xb7{R}^{h}={R}^{h}$, then ${R}^{h}$ is the fuzzy equivalence relationship. Denote ${R}^{h}=\left({r}_{ij}^{h}\right)$.
- (3)
- Choose a threshold value $\gamma $ and use the $\alpha -cut$ theory to obtain a dynamic cluster result.From the transitive closure ${R}^{h}=\left({r}_{ij}^{h}\right)$, construct a matrix ${R}^{*}=\left({r}_{ij}^{*}\right)$ by ${r}_{ij}^{*}=\left\{\begin{array}{cc}1\hfill & {r}_{ij}^{*}\ge \gamma ;\hfill \\ 0,\hfill & {r}_{ij}^{*}<\gamma .\hfill \end{array}\right.$The initial selection of the value $\gamma $ depends on the expectations from the similarity of the evaluations of decision makers within the sub-groups. It could be adjusted until the expected number of clustered sub-groups is obtained.
- (4)
- Adjust the threshold value and obtain several clusters.If ${r}_{ij}^{*}=1$, this means that decision makers ${d}_{i}$ and ${d}_{j}$ should be clustered into one class. In this way, we obtain different clusters. The cluster result could be easily adjusted by controlling $\gamma $. If the number of clusters is too large, we can decrease the threshold value $\gamma $; conversely, we can increase the threshold value $\gamma $.

- 2.
- Obtain the sub-group preference matrix.Suppose that the group G is classified into ${G}_{1}$, ${G}_{2}$, ⋯, ${G}_{f}$$(f<t)$ after the cluster process. There are h decision makers in a sub-group ${G}_{k}$$(1\le k\le f)$, which are denoted as $({d}_{1}^{k},{d}_{2}^{k},\cdots ,{d}_{h}^{k})$. The preference provided by the decision maker ${d}_{l}^{k}(\forall l\in \{1,2,\cdots ,h\})$ on ${x}_{i}$ over ${x}_{j}$ is denoted by ${a}_{ij,kl}$.Then, the group preference of ${G}_{k}$ on ${x}_{i}$ over ${x}_{j}$ forms a PHFLTS,$$P{H}_{S}^{({G}_{k},i,j)}=\left\{({s}_{i},{p}_{i})\right|{s}_{i}\in {a}_{ij,kl},l\in \{1,2,\cdots ,h\},{p}_{i}=\frac{\u25b5\left({s}_{i}\right)}{\square \left({a}_{ij,kl}\right)}\}$$
- 3.
- Compute the interval type-2 fuzzy envelopes of PHFLTSs.To avoid confusion, we set rules to compute the fuzzy envelope for $PHFLTSs$ according to the corresponding CLEs.
- If ${s}_{0}\in P{H}_{S},{s}_{g}\notin P{H}_{S}$, then the interval type-2 fuzzy envelope of the $PHFLTS$, $P{H}_{S}$ is computed according to the CLE “ at most ${s}_{i}$”;
- If ${s}_{0}\notin P{H}_{S},{s}_{g}\in P{H}_{S}$, then the interval type-2 fuzzy envelope of the $PHFLTS$, $P{H}_{S}$ is computed according to the CLE “ at least ${s}_{i}$”;
- If ${s}_{0}\in P{H}_{S},{s}_{g}\in P{H}_{S}$, or ${s}_{0}\notin P{H}_{S},{s}_{g}\notin P{H}_{S}$, then the interval type-2 fuzzy envelope of the $PHFLTS$, $P{H}_{S}$ is computed according to the CLE “ between ${s}_{i}$ and ${s}_{j}$”.

- 4.
- Compute the utility values for alternatives according to the sub-group preference.Suppose that the group decision matrix of a sub-group ${G}_{l}$ could be presented as follows:$${G}_{l}=\left(\begin{array}{cccc}P{H}_{S}^{({G}_{l},1,1)}& P{H}_{S}^{({G}_{l},1,2)}& \cdots & P{H}_{S}^{({G}_{l},1,n)}\\ P{H}_{S}^{({G}_{l},2,1)}& P{H}_{S}^{({G}_{l},2,2)}& \cdots & P{H}_{S}^{({G}_{l},2,n)}\\ \cdots & \cdots & \cdots & \cdots \\ P{H}_{S}^{({G}_{l},n,1)}& P{H}_{S}^{({G}_{l},n,2)}& \cdots & P{H}_{S}^{({G}_{l},n,n)}\end{array}\right)$$The utility value of ${x}_{i}$ (according to evaluation of sub-group ${G}_{l}$) is computed by the following equation:$${u}_{{G}_{l}}\left({x}_{i}\right)=\frac{1}{n}\sum _{j=1}^{n}R\left({\tilde{F}}_{P{H}_{S}^{({G}_{l},i,j)}}\right)$$
- 5.
- Determine the weights for sub-groups.Three different factors will be considered when we compute the weights for sub-groups in the proposed decision scheme.
- Size: The more members there are in a sub-group, the larger the weight of this sub-group should be;
- Cohesion[52]: The higher the level of togetherness that the preferences of a sub-group has, the larger the weight of this sub-group should be;
- Reliability: The higher the level of reliability that a sub-group has, the larger the weight of this sub-group should be.

Since size is a commonly used index in sub-group weight determination in LS-GDM, and cohesion has also been proved a useful index in [52] during the sub-group weight determination progress, here, we only explain “the reliability of the assessments of a sub-group”, which is first postulated in the current work in a detailed way. After the cluster process, we hope that the opinions of decision makers within a sub-group are close to each other. Suppose that the gathered preference of this sub-group for one alternative over another alternative is $\{{s}_{i},{s}_{i+1},{s}_{i+2}\}$, then we think that the degree of “decision makers in a sub-group are close to each other” is higher than the situation when the gathered preference is $\{{s}_{i},{s}_{i+2}\}$. This is because in the former situation, for each decision maker in this sub-group, there is another decision maker who provides a preference that is close to him/her. That is, assessments among the sub-groups are closer, and no obvious inharmonious mechanisms exist. The sub-group is more reliable when the gathered preference is $\{{s}_{i},{s}_{i+1},{s}_{i+2}\}$ rather than $\{{s}_{i},{s}_{i+2}\}$. In summary, the more continuous the linguistic terms in a PHFLTS are, the more reliable the corresponding preference of a sub-group is. On the contrary, the more discrete, the less reliable.Subsequently, we will introduce specific measures to compute the “Size”, “Cohesion”, and “Reliability” of a sub-group when PHFLTSs are applied in LS-MALGDM problems.- Size:The size of sub-group $\left({G}_{l}\right)$ is computed by:$$Size\left({G}_{l}\right)=\frac{|{G}_{l}|}{\left|G\right|}$$
- Cohesion:The preference of sub-group ${G}_{l}$ for alternative ${x}_{i}$ over ${x}_{j}(i,j\in \{1,\cdots ,n\})$ is a PHFLTS, denoted by the following equation:$$P{H}_{S}^{({G}_{l},i,j)}=\{({s}_{{\alpha}_{1}}^{({G}_{l},i,j)},{p}_{{\alpha}_{1}}^{({G}_{l},i,j)}),({s}_{{\alpha}_{2}}^{({G}_{l},i,j)},{p}_{{\alpha}_{2}}^{({G}_{l},i,j)}),\cdots ,({s}_{{\alpha}_{l({G}_{l},i,j)}}^{({G}_{l},i,j)},{p}_{{\alpha}_{2}}^{({G}_{l},i,j)})\}$$Let $I\left({s}_{{\alpha}_{k}}\right)$ be the index of linguistic term ${s}_{{\alpha}_{k}}$. The cohesion of sub-group $\left({G}_{l}\right)$ on ${x}_{i}$ over ${x}_{j}(i,j\in \{1,\cdots ,n\})$ is computed by$$Cohesion({G}_{l},i,j)=1-\frac{\frac{{\sum}_{{\delta}_{1}<{\delta}_{2},{\delta}_{1},{\delta}_{2}\in \{{\alpha}_{1},\cdots ,{\alpha}_{l({G}_{l},i,j)}\}}{p}_{{\delta}_{2}}^{({G}_{l},i,j)}{p}_{{\delta}_{1}}^{({G}_{l},i,j)}(I\left({s}_{{\delta}_{2}}^{({G}_{l},i,j)}\right)-I\left({s}_{{\delta}_{1}}^{({G}_{l},i,j)}\right))}{{C}_{l({G}_{l},i,j)}^{2}}+1}{g+1}$$The cohesion of sub-group $\left({G}_{l}\right)$ is computed by$$Cohesion\left({G}_{l}\right)=1-\frac{{\sum}_{1\le i,j\le n}(\frac{{\sum}_{{\delta}_{1}<{\delta}_{2},{\delta}_{1},{\delta}_{2}\in \{{\alpha}_{1},\cdots ,{\alpha}_{l({G}_{l},i,j)}\}}{p}_{{\delta}_{2}}^{({G}_{l},i,j)}{p}_{{\delta}_{1}}^{({G}_{l},i,j)}(I\left({s}_{{\delta}_{2}}^{({G}_{l},i,j)}\right)-I\left({s}_{{\delta}_{1}}^{({G}_{l},i,j)}\right))}{{C}_{l({G}_{l},i,j)}^{2}}+1)}{(g+1)\xb7{n}^{2}}$$
- Reliability:Let $\#\left(P{H}_{S}^{({G}_{l},i,j)}\right)$ be the number of linguistic terms in the PHFLTS, $P{H}_{S}^{({G}_{l},i,j)}$. The preference reliability of sub-group $\left({G}_{l}\right)$ for ${x}_{i}$ over ${x}_{j}$ is computed by$$Reliability({G}_{l},i,j)=\frac{\#\left(P{H}_{S}^{({G}_{l},i,j)}\right)}{I\left({s}_{{\alpha}_{l({G}_{l},i,j)}}^{({G}_{l},i,j)}\right)-I\left({s}_{{\alpha}_{1}}^{({G}_{l},i,j)}\right)+1}$$The reliability of sub-group $\left({G}_{l}\right)$ is computed by$$Reliability\left({G}_{l}\right)=\frac{{\sum}_{1\le i,j\le n}\frac{\#\left(P{H}_{S}^{({G}_{l},i,j)}\right)}{I\left({s}_{{\alpha}_{l({G}_{l},i,j)}}^{({G}_{l},i,j)}\right)-I\left({s}_{{\alpha}_{1}}^{({G}_{l},i,j)}\right)+1}}{{n}^{2}}$$The three indices above are synthesized by using the function below:$$\alpha \left({G}_{l}\right)={(1+Size\left({G}_{l}\right))}^{{\gamma}_{1}\xb7Cohesion\left({G}_{l}\right)+{\gamma}_{2}\xb7Reliability\left({G}_{l}\right)}$$Finally, the weight for sub-group ${G}_{l}$ is computed by the following equation:$$w\left({G}_{l}\right)=\frac{\alpha \left({G}_{l}\right)}{{\sum}_{h=1}^{f}\left(\alpha \left({G}_{h}\right)\right)},\forall l\in \{1\cdots ,f\}$$

- 6.
- Compute the utility value of alternatives according to the group preferences.The utility value of ${x}_{i}$ is computed by$$u\left({x}_{i}\right)=\sum _{l=1}^{f}w\left({G}_{l}\right)\xb7{u}_{{G}_{l}}\left({x}_{i}\right)$$

## 7. Case Study: Application to Urban Renewal Plan Selection

#### 7.1. Problem Description

#### 7.2. Solution to the Sample Problem

- Cluster and information fusion.There are 30 decision makers who provide evaluations by using the linguistic term set $S=\{{s}_{0},{s}_{1},\cdots ,{s}_{6}\}$, therefore $t=30$ and $g=6$. Without loss of generality, we set $k=2$, ${r}_{1}={r}_{2}=0.5$, and $r=0.99$. According to the classification strategy introduced, the group is classified into five sub-groups, i.e.,${G}_{1}=\{{t}_{1},{t}_{2},{t}_{3},{t}_{5},{t}_{7}\}$, ${G}_{2}=\{{t}_{4},{t}_{6},{t}_{8},{t}_{20},{t}_{21}\}$, ${G}_{3}=\{{t}_{9},{t}_{10},{t}_{17},{t}_{18},{t}_{19},{t}_{22},{t}_{28},{t}_{29},{t}_{30}\}$,${G}_{4}=\{{t}_{11},{t}_{12},{t}_{13},{t}_{14},{t}_{15},{t}_{16}\}$, ${G}_{5}=\{{t}_{23},{t}_{24},{t}_{25},{t}_{26},{t}_{27}\}$.
- Obtain sub-group preference matrix.From each sub-group, the evaluations on alternative ${x}_{i}$ with parameter ${e}_{j}$ are gathered to form a PHFLTS. In this way, the evaluation of the five sub-groups could be obtained as follows.${G}_{1}=\left(\begin{array}{cccc}\left\{({s}_{3},1.0)\right\}& \{({s}_{0},0.455),({s}_{1},0.455),({s}_{2},0.091)\}& \{({s}_{3},0.5),({s}_{4},0.5)\}& \{({s}_{1},0.5),({s}_{3},0.4),({s}_{4},0.1))\}\\ \{({s}_{5},0.455),({s}_{6},0.455),({s}_{4},0.091)\}& \left\{({s}_{3},1.0)\right\}& \{({s}_{3},0.5),({s}_{4},0.5)\}& \{({s}_{5},0.4),({s}_{6},0.4),({s}_{4},0.2)\}\\ \{({s}_{2},0.5),({s}_{3},0.5)\}& \{({s}_{2},0.5),({s}_{3},0.5)\}& \left\{({s}_{3},1.0)\right\}& \{({s}_{2},0.5),({s}_{3},0.5)\}\\ \{({s}_{3},0.4),({s}_{5},0.5),({s}_{2},0.1)\}& \{({s}_{0},0.4),({s}_{1},0.4),({s}_{2},0.2)\}& \{({s}_{3},0.5),({s}_{4},0.5)\}& \left\{({s}_{3},1.0)\right\}\end{array}\right)$${G}_{2}=\left(\begin{array}{cccc}\left\{({s}_{3},1.0)\right\}& \{({s}_{3},0.222),({s}_{5},0.556),({s}_{4},0.111),({s}_{6},0.111)\}& \left\{({s}_{2},1.0)\right\}& \{({s}_{4},0.357),({s}_{5},0.286),({s}_{6},0.357)\}\\ \{({s}_{1},0.625),({s}_{3},0.25),({s}_{2},0.125)\}& \left\{({s}_{3},1.0)\right\}& \{({s}_{5},0.5),({s}_{6},0.5)\}& \{({s}_{3},0.667),({s}_{2},0.333)\}\\ \left\{({s}_{4},1.0)\right\}& \{({s}_{0},0.5),({s}_{1},0.5)\}& \left\{({s}_{3},1.0)\right\}& \left\{({s}_{5},1.0)\right\}\\ \{({s}_{0},0.357),({s}_{1},0.286),({s}_{2},0.357)\}& \{({s}_{3},0.667),({s}_{4},0.333)\}& \left\{({s}_{1},1.0)\right\}& \left\{({s}_{3},1.0)\right\}\end{array}\right)$${G}_{3}=\left(\begin{array}{cccc}\left\{({s}_{3},1.0)\right\}& \{({s}_{4},0.263),({s}_{5},0.474),({s}_{3},0.263)\}& \{({s}_{2},0.692),({s}_{0},0.077),({s}_{3},0.154),({s}_{1},0.077)\}& \{({s}_{4},0.32),({s}_{5},0.36),({s}_{6},0.32)\}\\ \{({s}_{1},0.474),({s}_{2},0.263),({s}_{3},0.263)\}& \left\{({s}_{3},1.0)\right\}& \{({s}_{5},0.471),({s}_{6},0.353),({s}_{3},0.118),({s}_{4},0.059)\}& \{({s}_{5},0.5),({s}_{6},0.333),({s}_{4},0.167)\}\\ \{({s}_{4},0.692),({s}_{6},0.077),({s}_{3},0.154),({s}_{5},0.077)\}& \{({s}_{0},0.353),({s}_{1},0.471),({s}_{3},0.118),({s}_{2},0.059)\}& \left\{({s}_{3},1.0)\right\}& \{({s}_{5},0.4),({s}_{6},0.333),({s}_{4},0.2),({s}_{3},0.067)\}\\ \{({s}_{0},0.32),({s}_{1},0.36),({s}_{2},0.32)\}& \{({s}_{1},0.5),({s}_{0},0.333),({s}_{2},0.167)\}& \{({s}_{0},0.333),({s}_{1},0.4),({s}_{2},0.2),({s}_{3},0.067)\}& \left\{({s}_{3},1.0)\right\}\end{array}\right)$${G}_{4}=\left(\begin{array}{cccc}\left\{({s}_{3},1.0)\right\}& \{({s}_{4},0.5),({s}_{6},0.5)\}& \left\{({s}_{4},1.0)\right\}& \left\{({s}_{4},1.0)\right\}\\ \{({s}_{0},0.5),({s}_{2},0.5)\}& \left\{({s}_{3},1.0)\right\}& \left\{({s}_{5},1.0)\right\}& \left\{({s}_{2},1.0)\right\}\\ \left\{({s}_{2},1.0)\right\}& \left\{({s}_{1},1.0)\right\}& \left\{({s}_{3},1.0)\right\}& \left\{({s}_{2},1.0)\right\}\\ \left\{({s}_{2},1.0)\right\}& \left\{({s}_{4},1.0)\right\}& \left\{({s}_{4},1.0)\right\}& \left\{({s}_{3},1.0)\right\}\end{array}\right)$${G}_{5}=\left(\begin{array}{cccc}\left\{({s}_{3},1.0)\right\}& \{({s}_{3},0.5),({s}_{5},0.5)\}& \{({s}_{4},0.833),({s}_{3},0.167)\}& \{({s}_{3},0.833),({s}_{5},0.167)\}\\ \{({s}_{1},0.5),({s}_{3},0.5)\}& \left\{({s}_{3},1.0)\right\}& \{({s}_{3},0.8),({s}_{4},0.2)\}& \left\{({s}_{6},1.0)\right\}\\ \{({s}_{2},0.833),({s}_{3},0.167)\}& \{({s}_{3},0.8),({s}_{2},0.2)\}& \left\{({s}_{3},1.0)\right\}& \{({s}_{4},0.5),({s}_{6},0.4),({s}_{5},0.1)\}\\ \{({s}_{3},0.833),({s}_{1},0.167)\}& \left\{({s}_{0},1.0)\right\}& \{({s}_{0},0.4),({s}_{2},0.5),({s}_{1},0.1)\}& \left\{({s}_{3},1.0)\right\}\end{array}\right)$
- Compute the interval type-2 fuzzy envelopes of PHFLTSs.The context-free grammar ${G}_{H}=({v}_{N},{V}_{T},I,P)$ in [24] will be applied during the process of computing the interval type-2 fuzzy envelopes of PHFLTSs. Let $T(a,b,c,d,{a}^{\prime},{b}^{\prime},{c}^{\prime},{d}^{\prime};\underline{h})$ be the type-2 fuzzy envelope of a PHFLTS, $P{H}_{S}$. Following the scheme introduced in Section 4, to compute parameters $a,b,c,d$ for the PHFLTS corresponding to the CLE “at least ${s}_{i}$", the parameter $\beta $ is calculated by using Equation (22); for the PHFLTS corresponding to the CLE “between ${s}_{i}$ and ${s}_{j}$", the parameters ${\beta}_{1}$ and ${\beta}_{2}$ are calculated by using Equation (23) and Equation (24), respectively; for the PHFLTS corresponding to the CLE “at most ${s}_{i}$", the parameter $\beta $ is calculated by using Equation (25). Following the scheme introduced in Section 5, to compute parameter $\underline{h}$, it is necessary to evaluate the uncertainty contained in PHFLTS; we also adopt the comprehensive entropy ${E}_{c}\left(P{H}_{S}\right)=\frac{{E}_{f}\left(P{H}_{S}\right)+\gamma \left(P{H}_{S}\right){E}_{h}\left(P{H}_{S}\right)}{1+\gamma \left(P{H}_{S}\right){E}_{h}\left(P{H}_{S}\right)}$ to calculate the uncertainties, where $\gamma \left(P{H}_{S}\right)$ is computed by Equations (28) and (29). In this way, we can compute the interval type-2 fuzzy envelope of each PHFLTS; the evaluations obtained from the five sub-groups are fuzzy-encoded and presented below.${G}_{1}=\left(\begin{array}{cccc}(0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)& (0,0,0.070,0.5,0,0,0.070,0.333;0.611)& (0.333,0.5,0.667,0.833,0.491,0.5,0.667,0.675;0.053)& (0.0,0.149,0.241,0.833,0.120,0.149,0.241,0.358;0.197)\\ (0.5,0.674,1,1,0.568,0.674,1,1;0.611)& (0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)& (0.333,0.5,0.667,0.833,0.491,0.5,0.667,0.675;0.053)& (0.5,0.741,1,1,0.611,0.741,1,1;0.541)\\ (0.167,0.333,0.5,0.667,0.324,0.333,0.5,0.509;0.053)& (0.167,0.333,0.5,0.667,0.324,0.333,0.5,0.509;0.053)& (0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)\\ (0.167,0.333,0.5,0.667,0.325,0.333,0.5,0.509;0.053)& (0.167,0.227,0.539,1.0,0.215,0.227,0.539,0.629;0.197)& (0,0,0.093,0.5,0,0,0.093,0.313;0.541)& (0.333,0.5,0.667,0.833,0.491,0.5,0.667,0.675;0.053)\end{array}\right)$${G}_{2}=\left(\begin{array}{cccc}(0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)& (0.333,0.246,1,1,0.273,0.246,1,1;0.310)& (0.167,0.333,0.333,0.5,0.315,0.333,0.333,0.352;0.111)& (0.5,0.815,1,1,0.668,0.815,1,1;0.465)\\ (0.0,0.147,0.173,0.667,0.109,0.147,0.173,0.300;0.258)& (0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)& (0.667,0.972,1,1,0.761,0.972,1,1;0.693)& (0.167,0.167,0.5,0.667,0.167,0.167,0.5,0.506;0.035)\\ (0.5,0.667,0.667,0.833,0.648,0.667,0.667,0.685;0.111)& (0,0,0.028,0.333,0,0,0.028,0.239;0.693)& (0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)& (0.667,0.833,0.833,1.0,0.759,0.833,0.833,0.907;0.444)\\ (0,0,0.141,0.5,0,0,0.141,0.308;0.465)& (0.333,0.333,0.5,0.833,0.333,0.333,0.5,0.512;0.035)& (0.0,0.167,0.167,0.333,0.093,0.167,0.167,0.241;0.444)& (0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)\end{array}\right)$${G}_{3}=\left(\begin{array}{cccc}(0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)& (0.333,0.352,0.463,1.0,0.348,0.352,0.463,0.576;0.210)& (0,0,0.197,0.667,0,0,0.197,0.275;0.166)& (0.5,0.778,1,1,0.650,0.778,1,1;0.461)\\ (0.0,0.181,0.204,0.667,0.143,0.181,0.204,0.301;0.210)& (0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)& (0.333,0.271,1,1,0.301,0.271,1,1;0.479)& (0.5,0.593,1,1,0.544,0.593,1,1;0.522)\\ (0.333,0.171,1,1,0.198,0.171,1,1;0.166)& (0,0,0.120,0.667,0,0,0.120,0.382;0.479)& (0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)& (0.333,0.333,1,1,0.333,0.333,1,1;0.449)\\ (0,0,0.136,0.5,0,0,0.136,0.304;0.461)& (0,0,0.080,0.5,0,0,0.080,0.299;0.522)& (0,0,0.135,0.667,0,0,0.135,0.374;0.449)& (0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)\end{array}\right)$${G}_{4}=\left(\begin{array}{cccc}(0.333,0.5,0.5,0.6676,0.5,0.5,0.5,0.5;0.0)& (0.5,0.844,1,1,0.669,0.844,1,1;0.509)& (0.5,0.667,0.667,0.833,0.648,0.667,0.667,0.685;0.111)& (0.5,0.667,0.667,0.833,0.648,0.667,0.667,0.685;0.111)\\ (0,0,0.089,0.5,0,0,0.089,0.298;0.509)& (0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)& (0.667,0.833,0.833,1.0,0.759,0.833,0.833,0.907;0.444)& (0.167,0.333,0.333,0.5,0.315,0.333,0.333,0.352;0.111)\\ (0.167,0.333,0.333,0.5,0.315,0.333,0.333,0.352;0.111)& (0.0,0.167,0.167,0.333,0.093,0.167,0.167,0.241;0.444)& (0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)& (0.167,0.333,0.333,0.5,0.315,0.333,0.333,0.352;0.111)\\ (0.167,0.333,0.333,0.5,0.315,0.333,0.333,0.352;0.111)& (0.5,0.667,0.667,0.833,0.648,0.667,0.667,0.685;0.111)& (0.5,0.667,0.667,0.833,0.648,0.667,0.667,0.685;0.111)& (0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)\end{array}\right)$${G}_{5}=\left(\begin{array}{cccc}(0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)& (0.333,0.5,0.833,1.0,0.466,0.5,0.833,0.867;0.201)& (0.333,0.100,0.667,0.833,0.121,0.100,0.667,0.682;0.090)& (0.333,0.167,0.5,1.0,0.178,0.167,0.5,0.535;0.070)\\ (0.0,0.167,0.5,0.667,0.133,0.167,0.5,0.534;0.201)& (0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)& (0.333,0.167,0.5,0.833,0.170,0.167,0.5,0.507;0.021)& (0.833,1.0,1,1,0.833,1.0,1,1;1.0)\\ (0.167,0.100,0.333,0.667,0.106,0.100,0.333,0.363;0.090)& (0.167,0.083,0.5,0.667,0.085,0.083,0.5,0.504;0.021)& (0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)& (0.5,0.559,1,1,0.533,0.559,1,1;0.441)\\ (0.0,0.033,0.5,0.667,0.031,0.033,0.5,0.512;0.070)& (0,0,0.0,0.167,0,0,0.0,0.167;1.0)& (0,0,0.119,0.5,0,0,0.119,0.287;0.441)& (0.333,0.5,0.5,0.667,0.5,0.5,0.5,0.5;0.0)\end{array}\right)$
- Compute the utility values of alternatives according to the sub-group preference.By applying Equation (36), we obtain the following:${u}_{{G}_{1}}\left({x}_{1}\right)=5.206,{u}_{{G}_{2}}\left({x}_{1}\right)=6.641,{u}_{{G}_{3}}\left({x}_{1}\right)=6.043,{u}_{{G}_{4}}\left({x}_{1}\right)=6.852,{u}_{{G}_{5}}\left({x}_{1}\right)=5.989$;${u}_{{G}_{1}}\left({x}_{2}\right)=7.356,{u}_{{G}_{2}}\left({x}_{2}\right)=6.066,{u}_{{G}_{3}}\left({x}_{2}\right)=6.658,{u}_{{G}_{4}}\left({x}_{2}\right)=5.611,{u}_{{G}_{5}}\left({x}_{2}\right)=6.486$;${u}_{{G}_{1}}\left({x}_{3}\right)=5.237,{u}_{{G}_{2}}\left({x}_{3}\right)=6.094,{u}_{{G}_{3}}\left({x}_{3}\right)=6.414,{u}_{{G}_{4}}\left({x}_{3}\right)=4.665,{u}_{{G}_{5}}\left({x}_{3}\right)=5.748$;${u}_{{G}_{1}}\left({x}_{4}\right)=5.497,{u}_{{G}_{2}}\left({x}_{4}\right)=4.827,{u}_{{G}_{3}}\left({x}_{4}\right)=4.645,{u}_{{G}_{4}}\left({x}_{4}\right)=5.725,{u}_{{G}_{5}}\left({x}_{4}\right)=4.725$.
- Determine the weights for sub-groups.Adopt Equations (37), (40), and (42) to compute the sub-group size, cohesion, and reliability, respectively, and then adopt Equation (44) to compute the sub-group weights; it is easy to obtain $w\left({G}_{1}\right)=0.194$, $w\left({G}_{2}\right)=0.195$, $w\left({G}_{3}\right)=0.216$, $w\left({G}_{4}\right)=0.201$, and $w\left({G}_{5}\right)=0.194$.
- Decision result calculation.By applying Equation (45), we obtain the following: $u\left({x}_{1}\right)=6.149$, $u\left({x}_{2}\right)=6.434$, $u\left({x}_{3}\right)=5.643$, and $u\left({x}_{4}\right)=5.079$; then, alternative ${x}_{2}$ is the decision result.
- Numerical ComparisonFollowing the same weight-determination scheme and decision-making strategy, if the type-1 fuzzy envelope of PHFLTS is applied in this LS-GDM problem, by applying the magnitude [53] of the trapezoidal fuzzy number as the rank value, the utility values can be computed as $u\left({x}_{1}\right)=0.518$, $u\left({x}_{2}\right)=0.539$, $u\left({x}_{3}\right)=0.440$, and $u\left({x}_{4}\right)=0.329$. It is obvious that $u\left({x}_{2}\right)>u\left({x}_{1}\right)>u\left({x}_{3}\right)>u\left({x}_{4}\right)$, and therefore the best alternative is also ${x}_{2}$, which is consistent with the result when the type-2 fuzzy envelope is applied. However, since IT2FS can restore more linguistic uncertainty than the type-1 fuzzy set, the application of the proposed type-2 fuzzy envelope will contribute to a more precise result in specific decision-making situations (see more detailed discussions in Section 5.1).

## 8. Conclusions and Future Works

- Interval type-2 fuzzy encoding techniques need to be further developed, from single words to more flexible linguistic expressions.
- The fuzzy encoding technology for linguistic expressions in various forms still need to be further developed in the near future.
- More strategies adopting PHFLTS to solve LS-GDM problems need to be explored in order to pursue a suitable decision result in a flexible way.
- Consensus models need to be studied under the framework of LS-LSGDM with PHFLTS.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Decision Matrices

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