# Multicriteria Decision Making Based on Generalized Maclaurin Symmetric Means with Multi-Hesitant Fuzzy Linguistic Information

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. HFLTSs

_{i}is called a LV. Then the following requirements must be satisfied [2]:

- (1)
- The set is ordered: $\alpha >\beta \iff {s}_{\alpha}>{s}_{\beta}$;
- (2)
- There is a negation operator: $neg\left({s}_{\alpha}\right)={s}_{2g-\alpha}$.

**Definition**

**1.**

**Definition**

**2.**

- (1)
- The upper bound ${h}_{{s}^{+}}:{h}_{{s}^{+}}=\mathrm{max}\left\{{h}_{\tau}|{h}_{\tau}\in {H}_{s}\right\}$; and the lower bound ${h}_{{s}^{-}}$ of H
_{s}: ${h}_{{s}^{-}}=\mathrm{min}\left\{{h}_{\tau}|{h}_{\tau}\in {H}_{s}\right\}$ and ${h}_{{s}^{-}}\le {h}_{{s}^{+}}$; - (2)
- The intersection between ${H}_{s}^{1}$ and ${H}_{s}^{2}$ : ${H}_{s}^{1}{\cap}_{s}{H}_{s}^{2}=\left\{{h}_{\tau}|{h}_{\tau}\in {H}_{s}^{1}\text{}\mathrm{and}\text{}{h}_{\tau}\in {H}_{s}^{2}\right\}$;
- (3)
- The union of ${H}_{s}^{1}$ and ${H}_{s}^{2}$ : ${H}_{s}^{1}{\cup}_{s}{H}_{s}^{2}=\left\{{h}_{\tau}|{h}_{\tau}\in {H}_{s}^{1}\text{}\mathrm{or}\text{}{h}_{\tau}\in {H}_{s}^{2}\right\}$.

**Example**

**1.**

- (1)
- ${h}_{{s}^{+}}^{1}=\left\{{s}_{6}\right\}$ and ${h}_{{s}^{-}}^{1}=\left\{{s}_{5}\right\}$;
- (2)
- ${H}_{s}^{1}{\cap}_{s}{H}_{s}^{2}=\left\{{s}_{5}\right\}$ and ${H}_{s}^{1}{\cup}_{s}{H}_{s}^{3}=\left\{{s}_{1},{s}_{2},{s}_{3},{s}_{5},{s}_{6}\right\}$.

#### 2.2. HFLSs

**Definition**

**3.**

**Definition**

**4.**

**Example**

**2.**

**Definition**

**5.**

**Example**

**3.**

#### 2.3. MHFLTS

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

#### 2.4. Basic Operations of MHFLTS

**Definition**

**9.**

_{1}and h

_{2}be any two MHFLTEs on $\overline{S}$. Then, we can give the following operations.

**Definition**

**10.**

_{1}, h

_{2}be arbitrary MHFLTEs on $\overline{S}$. Then we give the score function of h as follows [31]:

- (1)
- If $\vartheta \left({h}_{1}^{-}\right)>\vartheta \left({h}_{1}^{+}\right)$, then ${h}_{1}$ is strictly greater than ${h}_{2}$, denoted by ${h}_{1}>{h}_{2}$;
- (2)
- If $S\left({h}_{1}\right)>S\left({h}_{2}\right)$, then ${h}_{1}$ is greater than ${h}_{2}$, denoted by ${h}_{1}>{h}_{2}$; If $S\left({h}_{1}\right)=S\left({h}_{2}\right)$, and $V\left({h}_{1}\right)<V\left({h}_{2}\right)$, then ${h}_{1}>{h}_{2}$; If $S\left({h}_{1}\right)=S\left({h}_{2}\right)$, and $V\left({h}_{1}\right)=V\left({h}_{2}\right)$, then ${h}_{1}={h}_{2}$.

#### 2.5. MSM Operator

**Definition**

**11.**

- (1)
- Idempotency. $MS{M}^{\left(m\right)}\left(a,a,\cdots ,a\right)=a$;
- (2)
- Monotonicity. $MS{M}^{\left(m\right)}\left({a}_{1},{a}_{2},\cdots ,{a}_{n}\right)\le MS{M}^{\left(m\right)}\left({b}_{1},{b}_{2},\cdots ,{b}_{n}\right)$, if ${a}_{i}<{b}_{i}$ for all i;
- (3)
- Boundedness. $\mathrm{min}\left\{{a}_{1},{a}_{2},\cdots ,{a}_{n}\right\}\le MS{M}^{\left(m\right)}\left({a}_{1},{a}_{2},\cdots ,{a}_{n}\right)\le \mathrm{max}\left\{{a}_{1},{a}_{2},\cdots ,{a}_{n}\right\}$.

**Definition**

**12.**

**Property**

**1.**

- (1)
- Idempotency. $GMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left(a,a,\cdots ,a\right)=a$;
- (2)
- Monotonicity. $GMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left({a}_{1},{a}_{2},\cdots ,{a}_{n}\right)\le GMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left({b}_{1},{b}_{2},\cdots ,{b}_{n}\right)$, if ${a}_{i}\le {b}_{i}$ for all i;
- (3)
- Boundedness. $\mathrm{min}\left\{{a}_{1},{a}_{2},\cdots ,{a}_{n}\right\}\le GMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left({a}_{1},{a}_{2},\cdots ,{a}_{n}\right)\le \mathrm{max}\left\{{a}_{1},{a}_{2},\cdots ,{a}_{n}\right\}$.

**Proof.**

- (1)
- $\begin{array}{l}GMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left(a,a,\cdots ,a\right)={\left(\frac{{\displaystyle {\sum}_{1\le {i}_{1}<\cdots <{i}_{m}\le {i}_{n}}{\displaystyle {\prod}_{j=1}^{m}{a}^{{p}_{j}}}}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}={\left(\frac{{\displaystyle {\sum}_{1\le {i}_{1}<\cdots <{i}_{m}\le {i}_{n}}{a}^{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}\\ ={\left(\frac{{C}_{n}^{m}{a}^{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}=a\end{array}$
- (2)
- Assume that m-tuple $\left({i}_{1},{i}_{2},\cdots ,{i}_{m}\right)$ is given randomly, and ${p}_{1},{p}_{2},\cdots ,{p}_{m}\ge 0$. if $0\le {a}_{i}\le {b}_{i}$ for all $i$, then $\prod}_{j=1}^{m}{a}_{{i}_{j}}^{{p}_{j}}}\le {\displaystyle {\prod}_{j=1}^{m}{b}_{{i}_{j}}^{{p}_{j}$ and $\sum}_{1\le {i}_{1}<\cdots <{i}_{m}\le {i}_{n}}{\displaystyle {\prod}_{j=1}^{m}{a}_{{i}_{j}}^{{p}_{j}}}}\le {\displaystyle {\sum}_{1\le {i}_{1}<\cdots <{i}_{m}\le {i}_{n}}{\displaystyle {\prod}_{j=1}^{m}{b}_{{i}_{j}}^{{p}_{j}}$, therefore,$${\left(\frac{{\displaystyle {\sum}_{1\le {i}_{1}<\cdots <{i}_{m}\le {i}_{n}}{\displaystyle {\prod}_{j=1}^{m}{a}_{{i}_{j}}^{{p}_{j}}}}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}\le {\left(\frac{{\displaystyle {\sum}_{1\le {i}_{1}<\cdots <{i}_{m}\le {i}_{n}}{\displaystyle {\prod}_{j=1}^{m}{b}_{{i}_{j}}^{{p}_{j}}}}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}$$
- (3)
- Let ${a}^{-}=\mathrm{min}\left\{{a}_{1},{a}_{2},\cdots ,{a}_{n}\right\}$ and ${a}^{+}=\mathrm{max}\left\{{a}_{1},{a}_{2},\cdots ,{a}_{n}\right\}$. According to the property of idempotency, $\mathrm{min}\left\{{a}_{1},{a}_{2},\cdots ,{a}_{n}\right\}={a}^{-}=GMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left({a}^{-},{a}^{-},\cdots ,{a}^{-}\right)$. According to the property of monotonicity, when ${a}^{-}\le {a}_{i}$ for all i, we have $GMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left({a}^{-},{a}^{-},\cdots ,{a}^{-}\right)\le GMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left({a}_{1},{a}_{2},\cdots ,{a}_{n}\right)$. Similarly, we also have $GMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left({a}_{1},{a}_{2},\cdots ,{a}_{n}\right)\le GMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left({a}^{+},{a}^{+},\cdots ,{a}^{+}\right)$.Finally, $MS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left({a}_{1},{a}_{2},\cdots ,{a}_{n}\right)\le MS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left({b}_{1},{b}_{2},\cdots ,{b}_{n}\right)$. □

**Definition**

**13.**

**Property**

**2.**

- (1)
- Idempotency. $GGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left(a,a,\cdots ,a\right)=a$;
- (2)
- Monotonicity. $GGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left({a}_{1},{a}_{2},\cdots ,{a}_{n}\right)\le GGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left({b}_{1},{b}_{2},\cdots ,{b}_{n}\right)$, if $0\le {a}_{i}\le {b}_{i}$ for all $i$;
- (3)
- Boundedness. $\mathrm{min}\left\{{a}_{1},{a}_{2},\cdots ,{a}_{n}\right\}\le GGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{m}\right)}\left({a}_{1},{a}_{2},\cdots ,{a}_{n}\right)\le \mathrm{max}\left\{{a}_{1},{a}_{2},\cdots ,{a}_{n}\right\}$.

## 3. Some Multi-Hesitant Fuzzy Linguistic MSM Operators

#### 3.1. MHFLGMSM Operator

**Definition**

**14.**

**Theorem**

**1.**

**Property**

**3.**

- (1)
- Idempotency. If the ${h}_{i}=h$, then$$S\left(MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left(h,h,\cdots ,h\right)\right)=S\left(h\right)$$
- (2)
- Monotonicity. If $\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)$ and $\left({h}_{1}^{\prime},{h}_{2}^{\prime},\cdots ,{h}_{n}^{\prime}\right)$ are two sets of MHFLTESs on $\overline{S}$ and ${h}_{i}^{+}<{h}_{i}^{\prime -}$, then for $i=1,2,\cdots ,n$,$$MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)<MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1}^{\prime},{h}_{2}^{\prime},\cdots ,{h}_{n}^{\prime}\right)$$
- (3)
- Boundedness. If ${h}_{i}^{\mathrm{min}}$ is obtained by replacing the minimum of ${h}_{i}$ for each element of ${h}_{i}$, ${h}_{i}^{\mathrm{max}}$ is obtained by replacing the maximum of ${h}_{i}$ for each element of ${h}_{i}$, then$$\begin{array}{l}MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1}^{\mathrm{min}},{h}_{2}^{\mathrm{min}},\cdots ,{h}_{n}^{\mathrm{min}}\right)\le MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)\\ \le MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1}^{\mathrm{max}},{h}_{2}^{\mathrm{max}},\cdots ,{h}_{n}^{\mathrm{max}}\right)\end{array}$$
- (4)
- Commutativity. Let $\left({h}_{1}^{\prime},{h}_{2}^{\prime},\cdots ,{h}_{n}^{\prime}\right)$ be any permutation of $\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)$, then$$MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1}^{\prime},{h}_{2}^{\prime},\cdots ,{h}_{n}^{\prime}\right)=MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)$$

**Proof**

- (1)
- Since the MHFLTSs ${h}_{i}=h$, then$$\begin{array}{l}S\left(MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left(h,h,\cdots ,h\right)\right)=S\left({\cup}_{{h}_{{\tau}_{1}}\in {h}_{1},{h}_{{\tau}_{2}}\in {h}_{2},\cdots ,{h}_{{\tau}_{n}}\in {h}_{n}}\left({\vartheta}^{-1}{\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}\left({\displaystyle {\prod}_{j=1}^{m}{\left(\vartheta \left({h}_{{\tau}_{i}}\right)\right)}^{{p}_{j}}}\right)}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}\right)\right)\\ =S\left({\cup}_{{h}_{{\tau}_{1}}\in {h}_{1},{h}_{{\tau}_{2}}\in {h}_{2},\cdots ,{h}_{{\tau}_{n}}\in {h}_{n}}\left({\vartheta}^{-1}\left(\vartheta \left({h}_{{\tau}_{i}}\right)\right)\right)\right)=\frac{1}{{\left(\#h\right)}^{n}}{\displaystyle {\sum}_{k=1}^{{\left(\#h\right)}^{n}}\vartheta \left({h}_{{\tau}_{i}}\right)}=S\left(h\right).\end{array}$$
- (2)
- If $\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)$ and $\left({h}_{1}^{\prime},{h}_{2}^{\prime},\cdots ,{h}_{n}^{\prime}\right)$ are two sets of MHFLTESs on $\overline{S}$ and ${h}_{i}^{+}<{h}_{i}^{\prime -}$, then for $i=1,2,\cdots ,n$$$\begin{array}{l}S\left(MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)\right)\\ =S\left({\cup}_{{h}_{{\tau}_{1}}\in {h}_{1},{h}_{{\tau}_{2}}\in {h}_{2},\cdots ,{h}_{{\tau}_{n}}\in {h}_{n}}\left({\vartheta}^{-1}{\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}\left({\displaystyle {\prod}_{j=1}^{m}{\left(\vartheta \left({h}_{{\tau}_{i}}\right)\right)}^{{p}_{j}}}\right)}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}\right)\right)\\ =\frac{1}{\#{h}_{1}\times \#{h}_{2}\times \cdots \times \#{h}_{n}}{\displaystyle \sum _{{h}_{{\tau}_{1}}\in {h}_{1},{h}_{{\tau}_{2}}\in {h}_{2},\cdots ,{h}_{{\tau}_{n}}\in {h}_{n}}\vartheta \left({\vartheta}^{-1}{\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}\left({\displaystyle {\prod}_{j=1}^{m}{\left(\vartheta \left({h}_{{\tau}_{i}}\right)\right)}^{{p}_{j}}}\right)}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}\right)}\\ =\frac{1}{\#{h}_{1}\times \#{h}_{2}\times \cdots \times \#{h}_{n}}{\displaystyle \sum _{{h}_{{\tau}_{1}}\in {h}_{1},{h}_{{\tau}_{2}}\in {h}_{2},\cdots ,{h}_{{\tau}_{n}}\in {h}_{n}}{\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}\left({\displaystyle {\prod}_{j=1}^{m}{\left(\vartheta \left({h}_{{\tau}_{i}}\right)\right)}^{{p}_{j}}}\right)}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}}\end{array}$$The ${h}_{i}^{+}<{h}_{i}^{\prime -}$, and for $i=1,2,\cdots ,n$ states the minimum element of ${h}_{i}^{\prime -}$ is more than the maximum element of ${h}_{i}^{+}$, then $\vartheta \left({h}_{i}^{+}\right)<\vartheta \left({h}_{i}^{\prime -}\right)$. In addition, MHFLGMSM operator can satisfy the property of monotonicity, so$${\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}\left({\displaystyle {\prod}_{j=1}^{m}{\left(\vartheta \left({h}_{{\tau}_{i}}^{}\right)\right)}^{{p}_{j}}}\right)}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}<{\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}\left({\displaystyle {\prod}_{j=1}^{m}{\left(\vartheta \left({h}_{{\tau}_{i}}^{\prime}\right)\right)}^{{p}_{j}}}\right)}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}$$Finally,$$MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)<MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1}^{\prime},{h}_{2}^{\prime},\cdots ,{h}_{n}^{\prime}\right)$$
- (3)
- If ${h}_{i}^{\mathrm{min}}$ is obtained by replacing the minimum of ${h}_{i}$ for each element of ${h}_{i}$ and ${h}_{i}^{\mathrm{max}}$ is obtained by replacing the maximum of ${h}_{i}$ for each element of ${h}_{i}$, then ${h}_{{\tau}_{i}}^{\mathrm{min}}\le {h}_{{\tau}_{i}}\le {h}_{{\tau}_{i}}^{\mathrm{max}}$ and $\vartheta \left({h}_{{\tau}_{i}}^{\mathrm{min}}\right)\le \vartheta \left({h}_{{\tau}_{i}}\right)\le \vartheta \left({h}_{{\tau}_{i}}^{\mathrm{max}}\right)$ for ${h}_{{\tau}_{i}}^{\mathrm{min}}\in {h}_{i}^{\mathrm{min}},{h}_{{\tau}_{i}}\in {h}_{i},{h}_{{\tau}_{i}}^{\mathrm{max}}\in {h}_{i}^{\mathrm{max}}$. In addition, the MHFLGMSM operator can satisfy the property of monotonicity, so$$\begin{array}{l}{\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}\left({\displaystyle {\prod}_{j=1}^{m}{\left(\vartheta \left({h}_{{\tau}_{i}}^{\mathrm{min}}\right)\right)}^{{p}_{j}}}\right)}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}\le {\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}\left({\displaystyle {\prod}_{j=1}^{m}{\left(\vartheta \left({h}_{{\tau}_{i}}^{}\right)\right)}^{{p}_{j}}}\right)}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}\\ \le {\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}\left({\displaystyle {\prod}_{j=1}^{m}{\left(\vartheta \left({h}_{{\tau}_{i}}^{\mathrm{max}}\right)\right)}^{{p}_{j}}}\right)}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}\end{array}$$Then$$\begin{array}{l}S\left({\cup}_{{h}_{{\tau}_{1}}^{\mathrm{min}}\in {h}_{1}^{\mathrm{min}},{h}_{{\tau}_{2}}^{\mathrm{min}}\in {h}_{2}^{\mathrm{min}},\cdots ,{h}_{{\tau}_{n}}^{\mathrm{min}}\in {h}_{n}^{\mathrm{min}}}\left({\vartheta}^{-1}{\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}\left({\displaystyle {\prod}_{j=1}^{m}{\left(\vartheta \left({h}_{{\tau}_{i}}^{\mathrm{min}}\right)\right)}^{{p}_{j}}}\right)}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}\right)\right)\\ \le S\left({\cup}_{{h}_{{\tau}_{1}}\in {h}_{1},{h}_{{\tau}_{2}}\in {h}_{2},\cdots ,{h}_{{\tau}_{n}}\in {h}_{n}}\left({\vartheta}^{-1}{\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}\left({\displaystyle {\prod}_{j=1}^{m}{\left(\vartheta \left({h}_{{\tau}_{i}}\right)\right)}^{{p}_{j}}}\right)}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}\right)\right)\\ \le S\left({\cup}_{{h}_{{\tau}_{1}}^{\mathrm{max}}\in {h}_{1}^{\mathrm{max}},{h}_{{\tau}_{2}}^{\mathrm{max}}\in {h}_{2}^{\mathrm{max}},\cdots ,{h}_{{\tau}_{n}}^{\mathrm{max}}\in {h}_{n}^{\mathrm{max}}}\left({\vartheta}^{-1}{\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}\left({\displaystyle {\prod}_{j=1}^{m}{\left(\vartheta \left({h}_{{\tau}_{i}}^{\mathrm{max}}\right)\right)}^{{p}_{j}}}\right)}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}\right)\right)\end{array}$$Finally,$$\begin{array}{l}MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1}^{\mathrm{min}},{h}_{2}^{\mathrm{min}},\cdots ,{h}_{n}^{\mathrm{min}}\right)\le MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)\\ \le MHFLGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1}^{\mathrm{max}},{h}_{2}^{\mathrm{max}},\cdots ,{h}_{n}^{\mathrm{max}}\right).\end{array}$$
- (4)
- Because $\left({h}_{1}^{\prime},{h}_{2}^{\prime},\cdots ,{h}_{n}^{\prime}\right)$ is a permutation of $\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)$, so $\prod \left(\#{h}_{i},i=1,2,\cdots ,n\right)}={\displaystyle \prod \left(\#{h}_{i}^{\prime},i=1,2,\cdots ,n\right)$. Therefore$${\left(\frac{{\oplus}_{1\le {\tau}_{{i}_{1}}<\cdots <{\tau}_{{i}_{m}}<n}\left({\otimes}_{j=1}^{m}{h}_{{\tau}_{{i}_{j}}}^{{p}_{j}}\right)}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}={\left(\frac{{\oplus}_{1\le {\tau}_{{i}_{1}}<\cdots <{\tau}_{{i}_{m}}<n}\left({\otimes}_{j=1}^{m}{\left({h}_{{\tau}_{{i}_{j}}}^{\prime}\right)}^{{p}_{j}}\right)}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}$$$$\begin{array}{l}MHFLGMS{M}^{(m,{p}_{1},{p}_{2},\cdots ,{p}_{m})}({h}_{1},{h}_{2},\cdots ,{h}_{n})\\ ={\cup}_{{h}_{{\tau}_{1}}\in {h}_{1},{h}_{{\tau}_{2}}\in {h}_{2},\cdots ,{h}_{{\tau}_{n}}\in {h}_{n}}\left({\vartheta}^{-1}{\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}\left({\displaystyle {\prod}_{j=1}^{m}{\left(\vartheta \left({h}_{{\tau}_{i}{}_{{}_{j}}^{k}}\right)\right)}^{{p}_{j}}}\right)}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}\right)\\ ={\cup}_{{h}_{{\tau}_{{}_{1}}}^{\prime}\in {h}_{1}^{\prime},{h}_{{\tau}_{{}_{2}}}^{\prime}\in {h}_{2}^{\prime},\cdots ,{h}_{{\tau}_{n}}^{\prime}\in {h}_{n}^{\prime}}\left({\vartheta}^{-1}{\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}\left({\displaystyle {\prod}_{j=1}^{m}{\left(\vartheta \left({h}_{{\tau}_{i}{}_{{}_{j}}^{k}}^{\prime}\right)\right)}^{{p}_{j}}}\right)}}{{C}_{n}^{m}}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+\cdots +{p}_{m}}}\right)\\ =MHFLGMS{M}^{(m,{p}_{1},{p}_{2},\cdots ,{p}_{m})}({h}_{1}^{\prime},{h}_{2}^{\prime},\cdots ,{h}_{n}^{\prime}).\end{array}$$The proof of Theorem 2 is completed now. □

- (1)
- When m = 1, then$$MHFLGMS{M}^{(1,{p}_{1})}({h}_{1},{h}_{2},\cdots ,{h}_{n})={\left(\frac{{\oplus}_{i=1}^{n}{h}_{i}^{{p}_{1}}}{n}\right)}^{\frac{1}{{p}_{1}}}={\cup}_{{h}_{{\tau}_{1}}\in {h}_{1},{h}_{{\tau}_{2}}\in {h}_{2},\cdots ,{h}_{{\tau}_{n}}\in {h}_{n}}\left({\vartheta}^{-1}{\left(\frac{{\displaystyle {\sum}_{k=1}^{{C}_{n}^{m}}{\left(\vartheta \left({h}_{{\tau}_{i}^{k}}\right)\right)}^{{p}_{1}}}}{n}\right)}^{\frac{1}{{p}_{1}}}\right)$$
- (2)
- When m = 2, then$$\begin{array}{l}MHFLGMS{M}^{(2,{p}_{1},{p}_{2})}({h}_{1},{h}_{2},\cdots ,{h}_{n})={\left(\frac{{\oplus}_{i=1,j=1,i\ne j}^{n}{h}_{i}^{{p}_{1}}{h}_{j}^{{p}_{2}}}{n\left(n-1\right)}\right)}^{\frac{1}{{p}_{1}+{p}_{2}}}\\ ={\cup}_{{h}_{{\tau}_{1}}\in {h}_{1},{h}_{{\tau}_{2}}\in {h}_{2},\cdots ,{h}_{{\tau}_{n}}\in {h}_{n}}\left({\vartheta}^{-1}{\left(\frac{{\displaystyle {\sum}_{i=1,j=1,i\ne j}^{n}{\left(\vartheta \left({h}_{{\tau}_{i}^{k}}\right)\right)}^{{p}_{1}}{\left(\vartheta \left({h}_{{\tau}_{j}^{k}}\right)\right)}^{{p}_{2}}}}{n\left(n-1\right)}\right)}^{\frac{1}{{p}_{1}+{p}_{2}}}\right)\end{array}$$
- (3)
- When m = 3, then$$\begin{array}{l}MHFLGMS{M}^{(3,{p}_{1},{p}_{2},{p}_{3})}({h}_{1},{h}_{2},\cdots ,{h}_{n})={\left(\frac{{\oplus}_{i=1,j=1,r=1,i\ne j\ne r}^{n}{h}_{i}^{{p}_{1}}{h}_{j}^{{p}_{2}}{h}_{r}^{{p}_{3}}}{n\left(n-1\right)\left(n-2\right)}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+{p}_{3}}}\\ ={\cup}_{{h}_{{\tau}_{1}}\in {h}_{1},{h}_{{\tau}_{2}}\in {h}_{2},\cdots ,{h}_{{\tau}_{n}}\in {h}_{n}}\left({\vartheta}^{-1}{\left(\frac{{\displaystyle {\sum}_{i=1,j=1,r=1,i\ne j\ne r}^{n}{\left(\vartheta \left({h}_{{\tau}_{i}{}_{}^{k}}\right)\right)}^{{p}_{1}}{\left(\vartheta \left({h}_{{\tau}_{j}{}_{}^{k}}\right)\right)}^{{p}_{2}}{\left(\vartheta \left({h}_{{\tau}_{r}{}_{}^{k}}\right)\right)}^{{p}_{3}}}}{n\left(n-1\right)\left(n-2\right)}\right)}^{\frac{1}{{p}_{1}+{p}_{2}+{p}_{3}}}\right)\end{array}$$

#### 3.2. MHFLGGMSM Operator

**Definition**

**15.**

**Theorem**

**2.**

_{j}th element in kth union of each permutation which consists of one element from every ${h}_{i}\left(i=1,2,\cdots ,n\right)$. Because the form of MHFLGGMSM operator involves selecting an element from every ${h}_{i}\left(i=1,2,\cdots ,n\right)$, each h

_{i}must mutual calculation, and obviously the MHFLGGMSM operator will be used $\prod \left(\#{h}_{i},i=1,2,\cdots ,n\right)$ times, where $\#{h}_{i}$ indicates the count of elements in h

_{i}. The final aggregated result consists of $\prod \left(\#{h}_{i},i=1,2,\cdots ,n\right)$ elements because of each aggregated result becomes an element which is based on the operation law of MHFLTSs.

**Property**

**4.**

- (1)
- Idempotency. If the ${h}_{i}=h$, then$$S\left(MHFLGGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left(h,h,\cdots ,h\right)\right)=S\left(h\right)$$
- (2)
- Monotonicity. If $\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)$ and $\left({h}_{1}^{\prime},{h}_{2}^{\prime},\cdots ,{h}_{n}^{\prime}\right)$ are two collections of MHFLTSs on $\overline{S}$ and ${h}_{i}^{+}<{h}_{i}^{\prime -}$, then for $i=1,2,\cdots ,n$.$$MHFLGGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)<MHFLGGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1}^{\prime},{h}_{2}^{\prime},\cdots ,{h}_{n}^{\prime}\right).$$
- (3)
- Boundedness. If ${h}_{i}^{\mathrm{min}}$ is obtained by replacing the minimum of ${h}_{i}$ for each element of ${h}_{i}$, ${h}_{i}^{\mathrm{max}}$ is obtained by replacing the maximum of ${h}_{i}$ for each element of ${h}_{i}$, then$$\begin{array}{l}MHFLGGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1}^{\mathrm{min}},{h}_{2}^{\mathrm{min}},\cdots ,{h}_{n}^{\mathrm{min}}\right)\le MHFLGGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)\\ \le MHFLGGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1}^{\mathrm{max}},{h}_{2}^{\mathrm{max}},\cdots ,{h}_{n}^{\mathrm{max}}\right).\end{array}$$
- (4)
- Commutativity. Let $\left({h}_{1}^{\prime},{h}_{2}^{\prime},\cdots ,{h}_{n}^{\prime}\right)$ be any permutation of $\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)$, then $MHFLGGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1}^{\prime},{h}_{2}^{\prime},\cdots ,{h}_{n}^{\prime}\right)=MHFLGGMS{M}^{\left(m,{p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}\left({h}_{1},{h}_{2},\cdots ,{h}_{n}\right)$.

- (1)
- When m = 1, then$$\begin{array}{l}MHFLGGMS{M}^{(1,{p}_{1})}({h}_{1},{h}_{2},\cdots ,{h}_{n})=\frac{1}{{p}_{1}}{\left({\otimes}_{i=1}^{n}\left({p}_{1}{h}_{i}\right)\right)}^{\frac{1}{n}}\\ ={\cup}_{{h}_{{\tau}_{1}}\in {h}_{1},{h}_{{\tau}_{2}}\in {h}_{2},\cdots ,{h}_{{\tau}_{n}}\in {h}_{n}}\left({\vartheta}^{-1}\left(\frac{1}{{p}_{1}}{\left({\displaystyle {\prod}_{k=1}^{n}\left({p}_{1}\vartheta \left({h}_{{\tau}_{i}^{k}}\right)\right)}\right)}^{\frac{1}{n}}\right)\right)\\ ={\cup}_{{h}_{{\tau}_{1}}\in {h}_{1},{h}_{{\tau}_{2}}\in {h}_{2},\cdots ,{h}_{{\tau}_{n}}\in {h}_{n}}\left({\vartheta}^{-1}{\left({\displaystyle {\prod}_{k=1}^{n}\vartheta \left({h}_{{\tau}_{i}^{k}}\right)}\right)}^{\frac{1}{n}}\right)\end{array}$$
- (2)
- When m = 2, then$$\begin{array}{l}MHFLGGMS{M}^{(2,{p}_{1},{p}_{2})}({h}_{1},{h}_{2},\cdots ,{h}_{n})=\frac{1}{{p}_{1}+{p}_{2}}{\left({\otimes}_{i=1,j=1,i\ne j}^{n}\left({p}_{1}{h}_{i}+{p}_{2}{h}_{j}\right)\right)}^{\frac{1}{n\left(n-1\right)}}\\ ={\cup}_{{h}_{{\tau}_{1}}\in {h}_{1},{h}_{{\tau}_{2}}\in {h}_{2},\cdots ,{h}_{{\tau}_{n}}\in {h}_{n}}\left({\vartheta}^{-1}\left(\frac{1}{{p}_{1}+{p}_{2}}{\left({\displaystyle {\prod}_{i=1,j=1,i\ne j}^{n}\left({p}_{1}\vartheta \left({h}_{{\tau}_{i}}\right)+{p}_{2}\vartheta \left({h}_{{\tau}_{j}}\right)\right)}\right)}^{\frac{1}{n\left(n-1\right)}}\right)\right)\end{array}$$
- (3)
- When m = 3, then$$\begin{array}{l}MHFLGGMS{M}^{(3,{p}_{1},{p}_{2},{p}_{3})}({h}_{1},{h}_{2},\cdots ,{h}_{n})\\ =\frac{1}{{p}_{1}+{p}_{2}+{p}_{3}}{\left({\otimes}_{i=1,j=1,r=1,i\ne j\ne r}^{n}\left({p}_{1}{h}_{i}+{p}_{2}{h}_{j}+{p}_{3}{h}_{r}\right)\right)}^{\frac{1}{n\left(n-1\right)\left(n-2\right)}}\\ ={\cup}_{{h}_{{\tau}_{1}}\in {h}_{1},{h}_{{\tau}_{2}}\in {h}_{2},\cdots ,{h}_{{\tau}_{n}}\in {h}_{n}}\left({\vartheta}^{-1}\left(\frac{1}{{p}_{1}+{p}_{2}+{p}_{3}}{\left({\displaystyle {\prod}_{i=1,j=1,r=1,i\ne j\ne r}^{n}\left({p}_{1}\vartheta \left({h}_{{\tau}_{i}}\right)+{p}_{2}\vartheta \left({h}_{{\tau}_{j}}\right)+{p}_{3}\vartheta \left({h}_{{\tau}_{r}}\right)\right)}\right)}^{\frac{1}{n\left(n-1\right)\left(n-2\right)}}\right)\right)\end{array}$$

#### 3.3. WMHFLGMSM Operator

**Definition**

**16.**

**Theorem**

**3.**

**Property**

**5.**

**Proof**

#### 3.4. WMHFLGGMSM Operator

**Definition**

**17.**

**Theorem**

**4.**

**Property**

**6.**

## 4. A MCDM Approach with MHFLTs

_{i}given by e

_{k}under r

_{j}is denoted by ${a}_{ij}^{k}$ based on the LTs $S=\left\{{s}_{i}|i=0,1,2,\cdots ,2g,g\in N\right\}$. b

_{ij}is a MHFLTE that contains inconsecutive and repetitive LTs by combing with the result ${a}_{ij}^{k}$ from l experts, then we obtain the decision matrix $B={\left({b}_{ij}\right)}_{d\times n}$, and the goal is to rank the alternatives.

**Step 1.**- Normalize the MHFLTE matrix $B={\left({b}_{ij}\right)}_{d\times n}$. Only for the cost criterion r
_{j}, b_{ij}is normalized by using the negation operator. **Step 2.**- Integrate the assessment value of every alternative under n criteria and obtain the comprehensive assessment results b
_{i}for ${t}_{i}\left(i=1,2,\cdots ,d\right)$. **Step 3.**- Calculate S(b
_{i}) and V(b_{i}) for alternative t_{i}. **Step 4.**- Ranking all alternatives based on S(b
_{i}) and V(b_{i}).

## 5. An Illustrative Example

_{1}: cost (such as the total cost of logistics operation); r

_{2}: relationship (such as shared risks and cooperation rewards); r

_{3}: service (such as breadth, specialization, variety); r

_{4}: quality (such as management and improvement). The weight vector of the criteria is $w={\left(0.40,0.26,0.17,0.17\right)}^{T}$. The evaluation is carry out by three experts, denoted by $E=\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$. S = {s

_{0}= Extremely Poor (EP), s

_{1}= Very Poor (VP), s

_{2}= Poor (P), s

_{3}= Slightly Poor (SP), s

_{4}= Fair (F), s

_{5}= Slightly Good (SG), s

_{6}= Good (G), s

_{7}= Very Good (VG), s

_{8}= Extremely Good (EG)} be a LTs. The evaluation value of each criteria b

_{ij}from three experts are represented in the form of MHFLTEs as demonstrated in Table 1.

#### 5.1. Procedure of Decision Making Based on WMHFLGMSM Operator

**Step 1.**- Get the normalized decision matrix $B={\left({b}_{ij}\right)}_{n\times m}$.

**Step 2.**- Integrate the evaluation value of every alternative for four criteria and obtain the comprehensive assessment results ${b}_{i}$ for ${t}_{i}\left(i=1,2,\cdots ,5\right)$ by the WMHFGMSM operator (suppose $m=2$, ${p}_{1}={p}_{2}=1$).
**Step 3.**- Calculate $S\left({b}_{i}\right)$ for alternative ${t}_{i}\left(i=1,2,\cdots ,5\right)$, and get$$S\left({b}_{1}\right)=0.5970,S\left({b}_{2}\right)=0.5987,S\left({b}_{3}\right)=0.4820,S\left({b}_{4}\right)=0.6329,S\left({b}_{5}\right)=0.5838.$$
**Step 4.**- Rank alternatives based on $S\left({b}_{i}\right)$, and get$${t}_{4}>{t}_{2}>{t}_{1}>{t}_{5}>{t}_{3}$$

#### 5.2. Procedure of Decision Making Based on WMHFLGGMSM Operator

**Step 1.**- Same as the above step 1.
**Step 2.**- Integrate the evaluation value of every alternative for four criteria and obtain the comprehensive assessment results ${b}_{i}$ for ${t}_{i}\left(i=1,2,\cdots ,5\right)$ by the WMHFLGGMSM operator (suppose $m=2$, ${p}_{1}={p}_{2}=1$).
**Step 3.**- Calculate $S\left({b}_{i}\right)$ for alternative ${t}_{i}\left(i=1,2,\cdots ,5\right)$.$$S\left({b}_{1}\right)=0.5919,S\left({b}_{2}\right)=0.5921,S\left({b}_{3}\right)=0.5109,S\left({b}_{4}\right)=0.6312,S\left({b}_{5}\right)=0.5895.$$
**Step 4.**- Rank alternatives based on $S\left({b}_{i}\right)$, and get$${t}_{4}>{t}_{2}>{t}_{1}>{t}_{5}>{t}_{3}$$

_{4}. The WMHFLGMSM operator has the dominant advantage of stressing on the influence of the entire and general data, which permits forceful supplementary among the attribute values, while the WMHFLGGMSM operator has the dominant advantage of stressing on the counterpoise and the coordination among the attribute values.

#### 5.3. Analysis the Effect of the Parameters m, p_{1}, p_{2}, …, p_{m}

- (1)
- If we keep the balance of parameter assignment (i.e., ${p}_{1}={p}_{2}=\cdots ={p}_{m}$), the ranking results of alternatives are identical in the condition of distinct parameter values. If the balance of parameter assignment is broken, the ranking results of alternatives will start to be changed, but as the value of parameter gap becomes lager and lager or the value of one parameter is much larger than other parameters, the ranking results will not be changed. So, the risk preference of experts plays an important role in real MCDM.
- (2)
- (3)
- For the WMHFLGMSM operator, the greater value of m becomes, in other words, the more interrelationships of criteria we take into account, the smaller value of score function will get. Nevertheless, for the WMHFLGGMSM operator, the results are the opposite, the greater value of m becomes, in other words, the more interrelationships of criteria we take into account, the greater value of score function will get.

_{1}= p

_{2}= 1 in practical problems, which are not only simple for operations, but also take into account the interrelations for two parameters.

#### 5.4. Comparison with the Other Methods

_{1}), FU hospital (t

_{2}), UMC hospital (t

_{3}), PLA hospital (t

_{4}). Three main criteria, denoted by $R=\left\{{r}_{1},{r}_{2},{r}_{3}\right\}$, are shown as follows: service environment (r

_{1}), diagnosis and treatment (r

_{2}) and social resource allocation (r

_{3}); The weight vector of the criteria is $w={\left(0.3,0.2,0.5\right)}^{T}$. The linguistic evaluation result is listed in Table 6 and the ranking results of different methods are shown in Table 7.

- (1)
- The final overall results from the GHFLWA operator [45] and the WHFLBM operator [33] are achieved under the condition that the repeated assessment values in Table 1 are eliminated. Obviously, this can lead to information loss, and make the ranking results unreasonable. From Table 8, we can also get this conclusion because they produced the different ranking results.
- (2)
- The approaches based on the MHFL-WGHM operator [10] and the WMHFLGMSM operator proposed in this paper produced the same ranking results because they can fully express the evaluation information, and all considered the interrelationship between two criteria. However, the proposed method based on the WMHFLGMSM operator in this paper can consider the interrelationship among any number of attributes by some parameters.

- (1)
- Compared with the approach based on GHFLWA operator proposed by [45], we can note the weakness of the GHFLWA operator is that the input assessment values are independent and does not think about the interrelationships among input arguments. Under these circumstances, the GHFLWA operator proposed by [45] is a particular example of WMHFLGMSM operator when m = 1, p = 1. Our novel method takes into account the real decision environment in which there are some relationships among some criteria. In above illustrative example, the level of management and improvement will have an effect on the breadth, specialization, variety of logistic company, that is to say we should consider the interrelationships between r
_{3}and r_{4}. Therefore, our presented approach is more suitable for dealing with actual decision problems than the presented approach based on GHFLWA operator proposed by [45]. Another weakness of the presented approach based on GHFLWA operator [45] is that it adopts HFLTSs which cannot sufficiently convey the hesitance of experts. At the same time, our presented approach can completely express the evaluation information. - (2)
- Compared with the approach based on the WHFLBM operator presented by [33], our new method can preserve the repetition of linguistic evaluation information and considers the interrelationship among more than two input arguments. However, the approach based on the WHFLBM operator only considers the interrelationship between two input arguments and the frequencies of repeated values are neglected. Therefore, as an extension of HFLSs and HFLTSs, the presented approach based on the WMHFLGMSM operator is more reasonable to aggregate the repeated linguistic information in practice. Moreover, we also noticed that the WMHFLGMSM operator with parameters will degrade into the WHFLBM operator proposed by [33] if m = 2, p
_{1}= 1, p_{2}= 1. Therefore, the WMHFLGMSM operators have more generality and are more robust. - (3)
- Compared with the approach based on the MHFL-WGHM operator presented by [10], our presented novel method can take into account the correlation among multi-inputs. However, the MHFL-WGHM operator can only consider correlation between two inputs. In many real decision-making problems, the interrelationships among multi-input arguments must be considered. Thus, the presented approach is more general and wider, and it is more adequate to deal with MCDM problems.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci.
**1975**, 8, 199–249. [Google Scholar] [CrossRef] - Chang, S.L.; Wang, R.C.; Wang, S.Y. Applying fuzzy linguistic quantifier to select supply chain partners at different phases of product life cycle. Int. J. Prod. Econ.
**2006**, 100, 348–359. [Google Scholar] [CrossRef] - Yan, H.B.; Ma, T.; Li, Y. A novel fuzzy linguistic model for priority engineering design requirements in quality function deployment under uncertainties. Int. J. Prod. Res.
**2013**, 51, 6336–6355. [Google Scholar] [CrossRef] - Lin, Q.L.; Liu, L.; Liu, H.C. Integrating hierarchical balanced scorecard with fuzzy linguistic for evaluating operating room performance in hospitals. Expert Syst. Appl.
**2013**, 40, 1917–1924. [Google Scholar] [CrossRef] - Montes, R.; Villar, P.; Herrer, F. A web tool to support decision making in the housing market using hesitant fuzzy linguistic term sets. Appl. Soft Comput.
**2015**, 35, 949–957. [Google Scholar] [CrossRef] - Fattahi, R.; Khalilzadeh, M. Risk evaluation using a novel hybrid method based on FMEA, extended MULTIMOORA, and AHP methods under fuzzy environment. Saf. Sci.
**2018**, 102, 290–300. [Google Scholar] [CrossRef] - Rodriguez, R.M.; Martinez, L.; Herrera, F. Hesitant Fuzzy Linguistic Term Sets for Decision Making. IEEE Trans. Fuzzy Syst.
**2012**, 20, 109–119. [Google Scholar] [CrossRef] - Rodriguez, R.M.; Martinez, L.; Herrera, F. A group decision making model dealing with comparative linguistic expressions based on hesitant fuzzy linguistic term sets. Inf. Sci.
**2013**, 241, 28–42. [Google Scholar] [CrossRef] - Liu, P.; Chen, S.M. Multiattribute group decision making based on intuitionistic 2-tuple linguistic information. Inf. Sci.
**2018**, 430–431, 599–619. [Google Scholar] [CrossRef] - Wang, J.; Wang, J.Q.; Tian, Z.; Zhao, D. A multi-hesitant fuzzy linguistic multicriteria decision-making approach for logistics outsourcing with incomplete weight information. Int. Trans. Oper. Res.
**2018**, 25, 831–856. [Google Scholar] [CrossRef] - Wang, J.; Wang, J.Q.; Zhang, H.Y. A likelihood-based TODIM approach based on multi-hesitant fuzzy linguistic information for evaluation in logistics outsourcing. Comput. Ind. Eng.
**2016**, 99, 287–299. [Google Scholar] [CrossRef] - Liu, P.; Su, Y. The Extended TOPSIS Based on Trapezoid Fuzzy Linguistic Variables. J. Converg. Inf. Technol.
**2010**, 5, 38–53. [Google Scholar] - Li, C.C.; Rodrıguez, R.M.; Martınez, L.; Dong, Y.; Herrera, F. Personalized individual semantics based on consistency in hesitant linguistic group decision making with comparative linguistic expressions. Knowl.-Based Syst.
**2018**, 145, 156–165. [Google Scholar] [CrossRef] - Liao, H.; Xu, Z.; Zeng, X.J. Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making. Inf. Sci.
**2014**, 271, 125–142. [Google Scholar] [CrossRef] - Chen, S.M.; Hong, J.A. Multicriteria linguistic decision making based on hesitant fuzzy linguistic term sets and the aggregation of fuzzy sets. Inf. Sci.
**2014**, 286, 63–74. [Google Scholar] [CrossRef] - Wang, J.Q.; Wang, J.; Chen, Q.H. An outranking approach for multi-criteria decision-making with hesitant fuzzy linguistic term sets. Inf. Sci.
**2014**, 280, 338–351. [Google Scholar] [CrossRef] - Zhu, B.; Xu, Z. Consistency Measures for Hesitant Fuzzy Linguistic Preference Relations. IEEE Trans. Fuzzy Syst.
**2014**, 22, 35–45. [Google Scholar] [CrossRef] - Zhang, Z.; Wu, C. On the use of multiplicative consistency in hesitant fuzzy linguistic preference relations. Knowl.-Based Syst.
**2014**, 72, 13–27. [Google Scholar] [CrossRef] - Tong, X.; Yu, L. MADM Based on distance and correlation coefficient measures with decision-maker preferences under a hesitant fuzzy environment. Soft Comput.
**2016**, 20, 4449–4461. [Google Scholar] [CrossRef] - Beg, I.; Rashid, T. TOPSIS for Hesitant Fuzzy Linguistic Term Sets. Int. J. Intell. Syst.
**2013**, 28, 1162–1171. [Google Scholar] [CrossRef] - Rashid, T.; Faizi, S.; Xu, Z.; Zafar, S. ELECTRE-Based Outranking Method for Multi-criteria Decision Making Using Hesitant Intuitionistic Fuzzy Linguistic Term Sets. Int. J. Fuzzy Syst.
**2018**, 20, 78–92. [Google Scholar] [CrossRef] - Liao, H.; Xu, Z.; Zeng, X.J. Hesitant Fuzzy Linguistic VIKOR Method and Its Application in Qualitative Multiple Criteria Decision Making. IEEE Trans. Fuzzy Syst.
**2015**, 23, 1343–1355. [Google Scholar] [CrossRef] - Yu, W.; Zhang, Z.; Zhong, Q. Extended TODIM for multi-criteria group decision making based on unbalanced hesitant fuzzy linguistic term sets. Comput. Ind. Eng.
**2017**, 114, 316–328. [Google Scholar] [CrossRef] - Farhadinia, B.; Herrera, E. Entropy Measures for Hesitant Fuzzy Linguistic Term Sets Using the Concept of Interval-Transformed Hesitant Fuzzy Elements. Int. J. Fuzzy Syst.
**2017**. [Google Scholar] [CrossRef] - Zang, Y.; Sun, W.; Han, S. Grey relational projection method for multiple attribute decision making with interval-valued dual hesitant fuzzy information. J. Intell. Fuzzy Syst.
**2017**, 33, 1053–1066. [Google Scholar] [CrossRef] - Xu, Z.S.; Pan, L.; Liao, H.C. Multi-criteria decision-making method of hesitant fuzzy linguistic term set based on improved MACBETH method. Control Decis.
**2017**, 32, 1266–1272. [Google Scholar] - Xu, Y.; Xu, A.; Wang, H. Hesitant fuzzy linguistic linear programming technique for multidimensional analysis of preference for multi-attribute group decision making. Int. J. Mach. Learn. Cybern.
**2016**, 7, 845–855. [Google Scholar] [CrossRef] - Zhang, W.; Ju, Y.; Liu, X. Multiple criteria decision analysis based on Shapley fuzzy measures and interval-valued hesitant fuzzy linguistic numbers. Comput. Ind. Eng.
**2017**, 105, 28–38. [Google Scholar] [CrossRef] - Porcel, C.; Ching-López, A. Fuzzy linguistic recommender systems for the selective diffusion of information in digital libraries. J. Inf. Process. Syst.
**2017**, 13, 653–667. [Google Scholar] [CrossRef] - Sha, J.; Zhang, Y. Fuzzy Petri Net Method for Hesitant Decision Making. In Proceedings of the 2016 International IEEE Conferences Ubiquitous Intelligence & Computing, Scalable Computing and Communications, Advanced and Trusted Computing, Cloud and Big Data Computing, Internet of People, and Smart World Congress, Toulouse, France, 18–21 July 2016; pp. 942–948. [Google Scholar]
- Wang, J.; Wang, J.Q.; Zhang, H.Y. Multi-criteria Group Decision-Making Approach Based on 2-Tuple Linguistic Aggregation Operators with Multi-hesitant Fuzzy Linguistic Information. Int. J. Fuzzy Syst.
**2016**, 18, 81–97. [Google Scholar] [CrossRef] - Wu, Z. Aggregation Operators for Hesitant Fuzzy Linguistic Term Sets with an Application in MADM. In Proceedings of the 27th Chinese Control and Decision Conference, Qingdao, China, 23–25 May 2015; pp. 879–884. [Google Scholar]
- Gou, X.; Xu, Z.; Liao, H. Multiple criteria decision making based on Bonferroni means with hesitant fuzzy linguistic information. Soft Comput.
**2017**, 21, 6515–6529. [Google Scholar] [CrossRef] - Zhu, C.; Zhu, L.; Zhang, X. Linguistic hesitant fuzzy power aggregation operators and their applications in multiple attribute decision-making. Inf. Sci.
**2016**, 367, 809–826. [Google Scholar] [CrossRef] - Wang, J.Q.; Peng, L.; Zhang, H.Y. Method of multi-criteria group decision-making based on cloud aggregation operators with linguistic information. Inf. Sci.
**2014**, 274, 177–191. [Google Scholar] [CrossRef] - Liu, P.; Mahmood, T.; Khan, Q. Multi-Attribute Decision-Making Based on Prioritized Aggregation Operator under Hesitant Intuitionistic Fuzzy Linguistic Environment. Symmetry
**2017**, 9, 270. [Google Scholar] [CrossRef] - Maclaurin, C. A second letter to Martin Folkes, Esq.: Concerning the roots of equations, with the demonstration of other rules of algebra. Philos. Trans. R. Soc. Lond. Ser. A
**1729**, 36, 59–96. [Google Scholar] - Detemple, D.; Robertson, J. On generalized symmetric means of two variables. Angew. Chem.
**1979**, 47, 4638–4660. [Google Scholar] - Liu, P.; Zhang, X. Some Maclaurin Symmetric Mean Operators for Single-Valued Trapezoidal Neutrosophic Numbers and Their Applications to Group Decision Making. Int. J. Fuzzy Syst.
**2018**, 20, 45–61. [Google Scholar] [CrossRef] - Liu, P.; Qin, X. Maclaurin symmetric mean operators of linguistic intuitionistic fuzzy numbers and their application to multiple-attribute decision-making. J. Exp. Theor. Artif. Intell.
**2017**, 29, 1173–1202. [Google Scholar] [CrossRef] - Qin, J.; Liu, X. Approaches to uncertain linguistic multiple attribute decision making based on dual Maclaurin symmetric mean. J. Intell. Fuzzy Syst.
**2015**, 29, 171–186. [Google Scholar] [CrossRef] - Zhang, Z.; Wei, F. Approaches to comprehensive evaluation with 2-tuple linguistic information. J. Intell. Fuzzy Syst.
**2015**, 28, 469–475. [Google Scholar] - Wang, J.Q.; Yang, Y.; Li, L. Multi-criteria decision-making method based on single-valued neutrosophic linguistic Maclaurin symmetric mean operators. Neural Comput. Appl.
**2016**. [Google Scholar] [CrossRef] - Liu, P.; Teng, F. Multiple attribute group decision making methods based on some normal neutrosophic number Heronian Mean operators. J. Intell. Fuzzy Syst.
**2017**, 32, 2375–2391. [Google Scholar] [CrossRef] - Zhang, Z.; Wu, C. Hesitant fuzzy linguistic aggregation operators and their applications to multiple attribute group decision making. J. Intell. Fuzzy Syst.
**2014**, 26, 2185–2202. [Google Scholar] - Taibi, A.; Atmani, B. Combining fuzzy AHP with GIS and decision rules for industrial site selection. Int. J. Interact. Multimed. Artif. Intell.
**2017**, 4, 60–69. [Google Scholar] [CrossRef] - Harish, B.S.; Kumar, S.V.A. Anomaly based intrusion detection using modified fuzzy clustering. Int. J. Interact. Multimed. Artif. Intell.
**2017**, 4, 54–59. [Google Scholar] [CrossRef] - Esmaeilpour, M.; Mohammadi, A.R.A. Analyzing the EEG signals in order to estimate the depth of anesthesia using wavelet and fuzzy neural networks. Int. J. Interact. Multimed. Artif. Intell.
**2017**, 4, 12–15. [Google Scholar] [CrossRef] - Cabrerizo, F.J.; Morente-Molinera, J.A.; Pedrycz, W.; Taghavi, A.; Herrera-Viedma, E. Granulating linguistic information in decision making under consensus and consistency. Expert Syst. Appl.
**2018**, 99, 83–92. [Google Scholar] [CrossRef] - Habib, M.S.; Sarkar, B. An Integrated Location-Allocation Model for Temporary Disaster Debris Management under an Uncertain Environment. Sustainability
**2017**, 9, 716. [Google Scholar] [CrossRef] - Zha, D.; Kavuri, A.S. Effects of technical and allocative inefficiencies on energy and nonenergy elasticities: an analysis of energy-intensive industries in China. Chin. J. Popul. Resour. Environ.
**2016**, 14, 292–297. [Google Scholar] [CrossRef] - Zhang, B.; He, M.; Pan, H. A study on the design of a hybrid policy for carbon abatement. Chin. J. Popul. Resour. Environ.
**2017**, 15, 50–57. [Google Scholar] [CrossRef] - Zhang, R.; Shi, G. Analysis of the relationship between environmental policies and air quality during major social events. Chin. J. Popul. Resour. Environ.
**2016**, 14, 167–173. [Google Scholar] [CrossRef] - Zhu, J. The 2030 Agenda for Sustainable Development and China’s implementation. Chin. J. Popul. Resour. Environ.
**2017**, 15, 142–146. [Google Scholar] [CrossRef] - Hou, Y.S. Reflection on the inheritance and development of “Wang PI opera” in Shandong. Dong Yue Trib.
**2016**, 37, 142–146. (In Chinese) [Google Scholar]

**Table 1.**Linguistic evaluation result in the form of multi-hesitant linguistic term elements (MHFLTEs).

Item | r_{1} | r_{2} | r_{3} | r_{4} |
---|---|---|---|---|

t_{1} | {F, SG, VG} | {SG, SG, EG} | {G, G, EG} | {VP, SP, F} |

t_{2} | {SG, G, G} | {G, EG, EG} | {VP, SP, SP } | {SP, G, G} |

t_{3} | {P, SP, SP} | {G, G, G} | {P, SP, SG} | {F, F, F} |

t_{4} | {VG, VG, VG} | {SP, SG, SG} | {G, EG, EG} | {VP, P, SG} |

t_{5} | {G, EG, EG} | {F, SG, SG} | {SP, SG, EG} | {VP, P, P} |

**Table 2.**Ranking results when m = 2 by weighted generalized MSM operator for MHFLTSs (WMHFLGMSM) operator.

p_{1} | p_{2} | Score Function S(b_{i}) | Ranking |
---|---|---|---|

1 | 0 | ${S}_{1}=0.8507,{S}_{2}=0.8497,{S}_{3}=0.5996,{S}_{4}=0.9710,{S}_{5}=0.9769$ | ${t}_{5}>{t}_{4}>{t}_{1}>{t}_{2}>{t}_{3}$ |

0 | 1 | ${S}_{1}=0.4057,{S}_{2}=0.4194,{S}_{3}=0.3865,{S}_{4}=0.3946,{S}_{5}=0.3209$ | ${t}_{2}>{t}_{1}>{t}_{4}>{t}_{3}>{t}_{5}$ |

1 | 2 | ${S}_{1}=0.5978,{S}_{2}=0.6047,{S}_{3}=0.5436,{S}_{4}=0.5866,{S}_{5}=0.4691$ | ${t}_{2}>{t}_{1}>{t}_{4}>{t}_{3}>{t}_{5}$ |

1 | 3 | ${S}_{1}=0.5961,{S}_{2}=0.6371,{S}_{3}=0.4797,{S}_{4}=0.5892,{S}_{5}=0.5371$ | ${t}_{2}>{t}_{1}>{t}_{4}>{t}_{3}>{t}_{5}$ |

1 | 4 | ${S}_{{}_{1}}=0.6102,{S}_{2}=0.6720,{S}_{3}=0.4980,{S}_{4}=0.5881,{S}_{5}=0.5394$ | ${t}_{2}>{t}_{1}>{t}_{4}>{t}_{3}>{t}_{5}$ |

1 | 5 | ${S}_{1}=0.6241,{S}_{2}=0.7026,{S}_{3}=0.5175,{S}_{4}=0.5894,{S}_{5}=0.5442$ | ${t}_{2}>{t}_{1}>{t}_{4}>{t}_{3}>{t}_{5}$ |

2 | 1 | ${S}_{1}=0.6952,{S}_{2}=0.7033,{S}_{3}=0.5277,{S}_{4}=0.7809,{S}_{5}=0.7526$ | ${t}_{4}>{t}_{5}>{t}_{2}>{t}_{1}>{t}_{3}$ |

3 | 1 | ${S}_{1}=0.7580,{S}_{2}=0.7672,{S}_{3}=0.5558,{S}_{4}=0.8852,{S}_{5}=0.8705$ | ${t}_{4}>{t}_{5}>{t}_{2}>{t}_{1}>{t}_{3}$ |

4 | 1 | ${S}_{1}=0.8024,{S}_{2}=0.8102,{S}_{3}=0.5751,{S}_{4}=0.8915,{S}_{5}=0.8758$ | ${t}_{4}>{t}_{5}>{t}_{2}>{t}_{1}>{t}_{3}$ |

5 | 1 | ${S}_{}=0.8356,{S}_{2}=0.8411,{S}_{3}=0.5894,{S}_{4}=0.9106,{S}_{5}=0.9054$ | ${t}_{4}>{t}_{5}>{t}_{2}>{t}_{1}>{t}_{3}$ |

0.5 | 0.5 | ${S}_{1}=0.5232,{S}_{}=0.5295,{S}_{3}=0.4702,{S}_{4}=0.5542,{S}_{5}=0.5010.$ | ${t}_{4}>{t}_{2}>{t}_{1}>{t}_{5}>{t}_{3}$ |

1 | 1 | ${S}_{1}=0.5970,{S}_{2}=0.5987,{S}_{3}=0.4820,{S}_{4}=0.6329,{S}_{5}=0.5838$ | ${t}_{4}>{t}_{2}>{t}_{1}>{t}_{5}>{t}_{3}$ |

2 | 2 | ${S}_{1}=0.6619,{S}_{2}=0.6827,{S}_{3}=0.5054,{S}_{4}=0.7118,{S}_{5}=0.6732$ | ${t}_{4}>{t}_{2}>{t}_{1}>{t}_{5}>{t}_{3}$ |

3 | 3 | ${S}_{1}=0.7010,{S}_{2}=0.7398,{S}_{3}=0.5273,{S}_{4}=0.7575,{S}_{5}=0.7266$ | ${t}_{4}>{t}_{2}>{t}_{1}>{t}_{5}>{t}_{3}$ |

4 | 4 | ${S}_{1}=0.7285,{S}_{2}=0.7811,{S}_{3}=0.5464,{S}_{4}=0.7879,{S}_{5}=0.7621$ | ${t}_{4}>{t}_{2}>{t}_{1}>{t}_{5}>{t}_{3}$ |

5 | 5 | ${S}_{1}=0.7872,{S}_{2}=0.8094,{S}_{3}=0.5623,{S}_{4}=0.8112,{S}_{5}=0.7489$ | ${t}_{4}>{t}_{2}>{t}_{1}>{t}_{5}>{t}_{3}$ |

**Note:**S

_{i}is abbreviation of score value S(b

_{i}).

p_{1} | p_{2} | Score Function S(b_{i}) | Ranking |
---|---|---|---|

1 | 0 | ${S}_{1}=0.5990,{S}_{2}=0.4033,{S}_{3}=0.3773,{S}_{4}=0.6969,{S}_{5}=0.6990$ | ${t}_{4}>{t}_{5}>{t}_{1}>{t}_{2}>{t}_{3}$ |

0 | 1 | ${S}_{1}=0.4537,{S}_{2}=0.4475,{S}_{3}=0.6103,{S}_{4}=0.4372,{S}_{5}=0.3719$ | ${t}_{3}>{t}_{1}>{t}_{2}>{t}_{4}>{t}_{5}$ |

1 | 2 | ${S}_{1}=0.5191,{S}_{2}=0.5386,{S}_{3}=0.4945,{S}_{4}=0.5286,{S}_{5}=0.4816$ | ${t}_{2}>{t}_{4}>{t}_{1}>{t}_{3}>{t}_{5}$ |

1 | 3 | ${S}_{1}=0.4690,{S}_{2}=0.4819,{S}_{3}=0.4527,{S}_{4}=0.4799,{S}_{5}=0.4363$ | ${t}_{2}>{t}_{4}>{t}_{1}>{t}_{3}>{t}_{5}$ |

1 | 4 | ${S}_{1}=0.4327,{S}_{2}=0.4470,{S}_{3}=0.4288,{S}_{4}=0.4389,{S}_{5}=0.3965$ | ${t}_{2}>{t}_{4}>{t}_{1}>{t}_{3}>{t}_{5}$ |

1 | 5 | ${S}_{1}=0.4084,{S}_{2}=0.4199,{S}_{3}=0.4050,{S}_{4}=0.4092,{S}_{5}=0.3671$ | ${t}_{2}>{t}_{4}>{t}_{1}>{t}_{3}>{t}_{5}$ |

2 | 1 | ${S}_{1}=0.6741,{S}_{2}=0.6650,{S}_{3}=0.5121,{S}_{4}=0.7551,{S}_{5}=0.7264$ | ${t}_{4}>{t}_{5}>{t}_{1}>{t}_{2}>{t}_{3}$ |

3 | 1 | ${S}_{1}=0.7461,{S}_{3}=0.7292,{S}_{3}=0.5335,{S}_{4}=0.8521,{S}_{5}=0.8302$ | $$ |