# Probabilistic Hesitant Intuitionistic Fuzzy Linguistic Term Sets and Their Application in Multiple Attribute Group Decision Making

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The existing PHIFLTSs proposed in [27] are based on the direct calculation between subscripts of linguistic terms and their associated probabilities. However, a great deal of important defects for these operations appear in some situations. For example, let $S=\{{s}_{t}|t=0,1,\dots 6\}$ be a linguistic term set, $\lambda =2$ a positive real number, and ${E}_{s}^{1}(P)=<\{{s}_{4}(0.3),{s}_{5}(0.7)\},\{{s}_{1}(1)\}>$ and ${E}_{s}^{2}(P)=<\{{s}_{4}(1)\},\{{s}_{2}(1)\}>$ two PHIFLTSs; then, by using the operational laws given by [27], we can obtain ${E}_{s}^{1}(P)+{E}_{s}^{2}(P)=<\{{s}_{5.2},{s}_{7.5}\},\{{s}_{3}\}>$, $\lambda {E}_{s}^{2}(P)=<\{{s}_{8}\},\{{s}_{4}\}>$. Obviously, the result not only exceeds the bound $[{s}_{0},{s}_{6}]$, but also loses the associated probability information.
- (2)
- The existing forms of PHIFLTSs are based on a qualitative scale mapped to a sequence of adjacent integers that are equally distributed. In fact, unbalanced conditions are very common if we take the psychology of experts into account [28]. Torra [29] discussed the unbalanced semantics for an ordered set of linguistic terms for the first time. Afterward, many achievements were obtained about the unbalanced linguistic terms [30,31,32,33,34,35,36,37]. However, Malik et al. [27] did not consider the situation of unbalanced linguistic terms on the PHIFLTS environment.
- (3)
- The distance measure defined in [27] is also based on the subscripts of each linguistic term and the associated probability. However, on the one hand, it cannot handle the conditions of unbalanced linguistic terms over PHIFLTSs; on the other hand, unreasonable results may be produced in some specific situations.
- (4)
- Compared with the current well-known ranking techniques including TOPSIS, VIKOR, ELECTRE, PROMETHEE, AHP, and MOORA, the multiplicative multi-objective optimization by ratio analysis (MULTIMOORA), which is specific to peculiarities of three aggregation models from the aspects of fully compensatory, non-compensatory, and incompletely compensatory has separately verified the superiority concerning time consumption, robustness, simplicity, and effectiveness [38,39,40,41]. However, advances in the state of the art have shown that no studies applied the MULTIMOORA method to the probabilistic hesitant intuitionistic fuzzy linguistic term sets environment.

- (1)
- This paper proposes some novel operational laws for the probabilistic hesitant intuitionistic fuzzy linguistic term sets to enrich the computation between PHIFLTSs and to improve the applicability and methodology in multi-attribute group decision-making models.
- (2)
- This paper establishes the distance and correlation measures for the probabilistic hesitant intuitionistic fuzzy linguistic term sets, which make up for the shortage of the current distance measures.
- (3)
- This paper takes the unbalanced linguistic terms of the probabilistic hesitant intuitionistic fuzzy linguistic term sets environment into account to better describe the difference in cognitive information of experts under different situations.
- (4)
- This paper presents a MAGDM model based on the MULTIMOORA approach with the use of the developed novel operational laws and correlation measures.

## 2. Preliminaries

#### 2.1. Probabilistic Hesitant Intuitionistic Fuzzy Linguistic Term Sets

**Definition**

**1.**

_{s}(P) can be mathematically represented as:${E}_{S}(P)=\{({x}_{i},<{M}_{S}^{i}(p),{N}_{s}^{i}(p)>)|{x}_{i}\in X\}$. Here,${M}_{S}^{i}(p),{N}_{s}^{i}(p)$are any subsets of S, respectively, which denote the membership degree and non-membership degree of linguistic variable x

_{i}to the linguistic term set S and can be denoted as

_{s}(P) on X, it should satisfy the following two conditions:

#### 2.2. The Normalization of PHIFLEs

**Definition**

**2.**

#### 2.3. Intuitionistic Fuzzy Sets and the Basic Operational Laws

**Definition**

**3.**

**Definition**

**4.**

- (1)
- $\alpha \oplus \beta =<{\mu}_{\alpha}+{\mu}_{\beta}-{\mu}_{\alpha}{\mu}_{\beta},{v}_{\alpha}{v}_{\beta}>$
- (2)
- $\alpha \otimes \beta =<{\mu}_{\alpha}{\mu}_{\beta},{v}_{\alpha}+{v}_{\beta}-{v}_{\alpha}{v}_{\beta}>$
- (3)
- $\lambda \alpha =<1-{(1-{\mu}_{\alpha})}^{\lambda},{v}_{\alpha}^{\lambda}>,\lambda \ge 0$
- (4)
- ${\alpha}^{\lambda}=<{\mu}_{\alpha}^{\lambda},1-{(1-{v}_{\alpha})}^{\lambda}>,\lambda \ge 0$
- (2)
- $\overline{\alpha}=<{v}_{\alpha},{v}_{\beta}>$.

#### 2.4. Linguistic Scale Function

- (1)
- If the semantics of linguistic terms are divide equally, the LSF (the first type) is defined as follows:$$f({s}_{i})={\theta}_{i}=\frac{i}{g},\mathrm{where}i=0,1,\dots ,g\mathrm{and}{\theta}_{i}\in [0,1]$$
- (2)
- If the semantics of any adjacent linguistic term are unequally divided, the deviation values in every side of ${s}_{g/2}$ both present the tendency of increase; the LSF (the second type) is defined as follows:$$f({s}_{i})={\theta}_{i}=\{\begin{array}{l}\frac{{a}^{g/2}-{a}^{(g/2)-i}}{2{a}^{g/2}-2}\\ \frac{{a}^{g/2}+{a}^{i-(g/2)}-2}{2{a}^{g/2}-2}\end{array}\begin{array}{l}(i=0,1,2,\cdots ,\frac{g}{2})\\ (i=\frac{g}{2}+1,\frac{g}{2}+2,\cdots ,g)\end{array}$$If the scale level is $g+1$, then $a=\sqrt[g+1]{9}$.
- (3)
- If the semantics of any adjacent linguistic term are unequally divided, the deviation value in every side of ${s}_{g/2}$ present the tendency of decrease; the LSF (the third type) is defined as follows:$$f({s}_{i})={\theta}_{i}=\{\begin{array}{l}\frac{{(g/2)}^{\alpha}+{((g/2)-i)}^{\alpha}}{2{(g/2)}^{\alpha}}\\ \frac{{(g/2)}^{\beta}+{(i-(g/2))}^{\beta}}{2{(g/2)}^{\beta}}\end{array}\begin{array}{l}(i=0,1,2,\cdots ,\frac{g}{2})\\ (i=\frac{g}{2}+1,\frac{g}{2}+2,\cdots ,g)\end{array}$$

## 3. The Comparison between PHIFLEs

**Definition**

**5.**

**Definition**

**6.**

**Example**

**1.**

## 4. Some Novel Operational Laws of the PHIFLEs

^{−1}can be viewed as a transformations tool, which make the equivalent transformations between the PHIFLEs and the IFVs possible. Utilizing the linguistic scale function f, the PHIFLEs are transformed to the IFVs, and then we can use the operations (Definition 4) to calculate these transformed IFVs. Furthermore, the inverse functions f

^{−1}can be used to transform these calculation results to the PHIFLEs equivalently. Therefore, the novel operational laws for PHIFLEs are defined:

**Definition**

**7.**

^{−1}be the linguistic scale functions and the inverse functions, and$\lambda $be a positive real number. Then

- (1)
- $\begin{array}{cc}\hfill {E}_{s}^{1}(P)\oplus {E}_{s}^{2}(P)& =<{f}^{-1}\{\underset{{\gamma}_{1}^{(i)}\in f({M}_{s}^{1}),{\gamma}_{2}^{(j)}\in f({M}_{s}^{2})}{\cup}({\gamma}_{1}^{(i)}+{\gamma}_{2}^{(j)}-{\gamma}_{1}^{(i)}{\gamma}_{2}^{(j)})({P}_{1}^{(i)}{P}_{2}^{(j)})\},\hfill \\ & {f}^{-1}\{\underset{{\eta}_{1}^{(i\prime )}\in f({N}_{s}^{1}),{\eta}_{2}{}^{(j\prime )}\in f({N}_{s}^{2})}{\cup}({\eta}_{1}^{(i\prime )}{\eta}_{2}^{(j\prime )})({P}_{1}^{(i\prime )}{P}_{2}^{(j\prime )})\}>,i=1,2,\dots ,\#{L}_{{M}_{1}},i\prime =1,2,\dots ,\#{L}_{{N}_{1}},j=1,2,\dots ,\#{L}_{{M}_{2}},j\prime =1,2,\dots ,\#{L}_{{N}_{2}}\hfill \end{array}$
- (2)
- $\begin{array}{cc}\hfill {E}_{s}^{1}(P)\otimes {E}_{s}^{2}(P)& =<{f}^{-1}\{\underset{{\gamma}_{1}^{(i)}\in f({M}_{s}^{1}),{\gamma}_{2}^{(j)}\in f({M}_{s}^{2})}{\cup}({\gamma}_{1}^{(i)}{\gamma}_{2}^{(j)})({P}_{1}^{(i)}{P}_{2}^{(j)})\},\hfill \\ & {f}^{-1}\{\underset{{\eta}_{1}^{(i\prime )}\in f({N}_{s}^{1}),{\eta}_{2}{}^{(j\prime )}\in f({N}_{s}^{2})}{\cup}({\eta}_{1}^{(i\prime )}+{\eta}_{2}^{(j\prime )}-{\eta}_{1}^{(i\prime )}{\eta}_{2}^{(j\prime )})({P}_{1}^{(i\prime )}{P}_{2}^{(j\prime )})\}>,i=1,2,\dots ,\#{L}_{{M}_{1}},i\prime =1,2,\dots ,\#{L}_{{N}_{1}},j=1,2,\dots ,\#{L}_{{M}_{2}},j\prime =1,2,\dots ,\#{L}_{{N}_{2}}\hfill \end{array}$
- (3)
- $\lambda {E}_{s}^{1}(P)=<{f}^{-1}\{\underset{{\gamma}_{1}^{(i)}\in f({M}_{s}^{1})}{\cup}(1-{(1-{\gamma}_{1}^{(i)})}^{\lambda})({P}^{(i)})\},{f}^{-1}\{\underset{{\eta}_{1}^{(i\prime )}\in f({N}_{s}^{1})}{\cup}{({\eta}_{1}^{(i\prime )})}^{\lambda}({P}^{(i\prime )})\}>,i=1,2,\dots ,\#{L}_{{M}_{1}},i\prime =1,2,\dots ,\#{L}_{{N}_{1}}$
- (4)
- ${E}_{s}^{1}{(P)}^{\lambda}=<{f}^{-1}\{\underset{{\gamma}_{1}^{(i)}\in f({M}_{s}^{1})}{\cup}{({\gamma}_{1}^{(i)})}^{\lambda}({P}^{(i)})\},{f}^{-1}\{\underset{{\eta}_{1}^{(i\prime )}\in f({N}_{s}^{1})}{\cup}{(1-(1-{\eta}_{1}^{(i\prime )}))}^{\lambda}({P}^{(i\prime )})\}>,i=1,2,\dots ,\#{L}_{{M}_{1}},i\prime =1,2,\dots ,\#{L}_{{N}_{1}}$

**Theorem**

**1.**

- (1)
- ${E}_{s}^{1}(P)\oplus {E}_{s}^{2}(P)={E}_{s}^{2}(P)\oplus {E}_{s}^{1}(P)$
- (2)
- ${E}_{s}^{1}(P)\otimes {E}_{s}^{2}(P)={E}_{s}^{2}(P)\otimes {E}_{s}^{1}(P)$
- (3)
- $\lambda ({E}_{s}^{1}(P)\oplus {E}_{s}^{2}(P))=\lambda {E}_{s}^{1}(P)\oplus \lambda {E}_{s}^{2}(P)$
- (4)
- ${({E}_{s}^{1}(P)\otimes {E}_{s}^{2}(P))}^{\lambda}={({E}_{s}^{1}(P))}^{\lambda}\otimes {({E}_{s}^{2}(P))}^{\lambda}$

**Proof.**

**Example**

**2.**

- (1)
- $\begin{array}{cc}{E}_{s}^{1}(p)\oplus {E}_{s}^{2}(p)=<\hfill & {f}^{-1}\{\frac{2}{3}(0.1),\frac{7}{9}(0.25),\frac{8}{9}(0.15),\frac{3}{4}(0.1),\frac{5}{6}(0.25),\frac{11}{12}(0.15)\},{f}^{-1}\{\frac{1}{4}(0.06),\frac{1}{3}(0.38),\frac{4}{9}(0.56)\}>\hfill \\ & =<\{{s}_{4}\left(0.1\right),{s}_{4.5}\left(0.1\right),{s}_{4.67}\left(0.25\right),{s}_{5}\left(0.25\right),{s}_{5.33}\left(0.15\right),{s}_{5.5}\left(0.15\right)\},\{{s}_{1.5}\left(0.06\right),{s}_{2}\left(0.38\right),{s}_{2.67}\left(0.56\right)\}>\hfill \end{array}$
- (2)
- $\begin{array}{cc}\hfill {E}_{s}^{1}(p)\otimes {E}_{s}^{2}(p)=<& {f}^{-1}\{0(0.2),\frac{1}{36}(0.25),\frac{1}{18}(0.4),\frac{1}{9}(0.15)\},{f}^{-1}\{\frac{3}{4}(0.06),\frac{5}{6}(0.38),\frac{8}{9}(0.56)\}>\hfill \\ & =<\{{s}_{0}\left(0.2\right),{s}_{0.17}\left(0.25\right),{s}_{0.33}\left(0.4\right),{s}_{0.67}\left(0.15\right)\},\{{s}_{4.5}\left(0.06\right),{s}_{5}\left(0.38\right),{s}_{5.33}\left(0.56\right)\}>\hfill \end{array}$
- (3)
- $\lambda {E}_{s}^{1}(p)=<{f}^{-1}\{\frac{11}{36}(0.5),\frac{5}{9}(0.5)\},{f}^{-1}\{\frac{1}{4}(0.3),\frac{4}{9}(0.7)\}>=<\{{s}_{1.83}(0.5),{s}_{3.33}(0.5)\},\{{s}_{1.5}(0.3),{s}_{2.67}(0.7)\}>$
- (4)
- ${({E}_{s}^{1}(p))}^{\lambda}=<{f}^{-1}\{\frac{1}{36}(0.5),\frac{1}{9}(0.5)\},{f}^{-1}\{0(0.2),\frac{11}{36}(0.5),\frac{5}{9}(0.3)\}>=<\{{s}_{0.16}(0.5),{s}_{0.67}(0.5)\},\{{s}_{0}(0.2),{s}_{1.83}(0.5),{s}_{3.33}(0.3)\}>$

## 5. Distance and Correlation Measure of the PHIFLTSs Based on Adjusted PHIFLEs

#### 5.1. The Adjusted PHIFLEs with Different Probability Distributions

**Definition**

**8.**

**Example**

**3.**

#### 5.2. The Distance Measure of the PHIFLTSs Based on the Adjusted PHIFLEs

**Example**

**4.**

**Definition**

**9.**

**Theorem**

**2.**

- (1)
- $d({E}_{S}^{1}(P),{E}_{S}^{2}(P))=d({E}_{S}^{2}(P),{E}_{S}^{1}(P))$
- (2)
- $0\le d({E}_{S}^{1}(P),{E}_{S}^{2}(P))\le 1$
- (3)
- $d({E}_{S}^{1}(P),{E}_{S}^{2}(P))=0$, if and only if${E}_{S}^{1}(P)={E}_{S}^{2}(P)$.

**Proof.**

#### 5.3. The Correlation Measure and Correlation Coefficients of PHIFLEs

**Definition**

**10.**

**Definition**

**11.**

**Definition**

**12.**

**Theorem**

**3.**

- (1)
- ${K}_{PHIFLTS}({E}_{s}^{1}(P),{E}_{s}^{2}(P))={K}_{PHIFLTS}({E}_{s}^{2}(P),{E}_{s}^{1}(P))$
- (2)
- ${K}_{PHIFLTS}({E}_{s}^{1}(P),{E}_{s}^{1}(P))=1$
- (3)
- $0\le {K}_{PHIFLTS}({E}_{s}^{1}(P),{E}_{s}^{2}(P))\le 1$

**Proof.**

**Example**

**5.**

## 6. The MAGDM Method Based on the MULTIMOORA Approach under the PHIFLTSs Environment

#### 6.1. The Problem Statement

#### 6.2. Determine the Optimal Weights of Attributes Based on the TOPSIS and LP Optimization Method

^{+}and the negative ideal decision (NID) H

^{−}associated with the attribute ${u}_{j}(j=1,2,\dots ,n)$ as follows:

^{+}and the negative ideal decision H

^{−}, respectively. It can be obtained as follows:

^{−}and maximizing the relative similarity from the H

^{+}simultaneously.

#### 6.3. The MULTIMOORA Approach under the PHIFLTSs Environment

- Probabilistic Hesitant Intuitionistic Fuzzy Linguistic Ratio System (PHIFLRS) model

- 2.
- Probabilistic Hesitant Intuitionistic Fuzzy Linguistic Reference Point (PHIFLRP) model

- 3.
- Probabilistic Hesitant Intuitionistic Fuzzy Linguistic Full Multiplicative Form (PHIFLFMF) model

_{1}), PHIFLRP (C

_{2}), and PHIFLFMF (C

_{3}). Then, two matrices can be built based on the utility value ${U}_{y}({a}_{i})$ and the rank $Ran{k}_{y}$ associated with each attribute ${c}_{y}(y=1,2,3)$, which are the utility value matrix $U={({U}_{ij})}_{m\times 3}$ and the rank matrix $RK={(Ran{k}_{ij})}_{m\times 3}$.

Algorithm 1. The MAGDM method based on the MULTIMOORA approach |

Step 1. Collect the individual linguistic evaluations values described as PHIFLEs from experts and construct the individual decision matrices ${D}_{t}={({E}_{ij}^{t}(P))}_{m\times n}$. Go to the next step.Step 2. The collective decision matrix $R={({r}_{ij}(P))}_{m\times n}$ can be built based on the frequency or the aggregation of probabilities associated with each linguistic term that appear in each individual decision matrix ${D}_{t}$. Go to the next step.Step 3. The score function and the variance value function of PHIFLEs defined in Definitions 5 and 6 are utilized to derive the PID H^{+} and the NID H^{−} associated with each attribute in the collective decision matrix R. Go to the next step.Step 4. The PHIFLEs in R should be adjusted to the same probability set based on Definition 8 to calculate the correlation coefficients between ${({r}_{ij}(P))}_{m\times n}$ and the PID H^{+} and the NID H^{−}.Step 5. The LP optimization method should be utilized to determine the optimal weights of each attributes based on Equations (14)–(16). Subsequently, we can derive the vector normalization of the scores of the collective decision matrix R based on Equations (17) and (18).Step 6. The MULTIMOORA approach is utilized to calculate the final ranking for all alternatives. Firstly, the subordinated methods as PHIFLRS are used to obtain the utility values ${U}_{1}({a}_{i})$ and the ranks of alternatives as $Ran{k}_{1}$ based on Equation (19).Step 7. The second subordinated methods as PHIFLPR in the MULTIMOORA approach are used to obtain the utility values ${U}_{2}({a}_{i})$ and the ranks of alternatives as $Ran{k}_{2}$ based on Equation (20).Step 8. The third subordinated methods as PHIFLFMF in the MULTIMOORA approach are used to obtain the utility values ${U}_{3}({a}_{i})$ and the ranks of alternatives as $Ran{k}_{3}$ based on Equation (21).Step 9. The above three subordinate ranks of alternatives $Ran{k}_{v}({a}_{i}),v=1,2,3$ need to be aggregated to obtain the final ranking. Two matrices can be built based on the utility values ${U}_{y}({a}_{i}),y=1,2,3$ and the ranks $Ran{k}_{y},y=1,2,3$ associated with each attribute ${c}_{y}$.Step 10. The improving Borda rules are utilized to obtain the final ranking of alternatives. Firstly, the normalized vectors of utility value ${U}^{N}$ are obtained based on Equation (22). Secondly, the ordinal values need to be translated to the scores based on the Borda rules for the reasons of aggregating the cardinal of each alternative. Finally, the final ranking can be derived by the values of $F{S}_{i}$ $(i=1,2,\dots ,m)$ in descending order based on Equation (23). End. |

## 7. Application Example

_{1}: Risk factor, c

_{2}: Growth, c

_{3}: Quick refund, c

_{4}: Complicated documents, are used to evaluate five alternatives: x

_{1}: Real estate, x

_{2}: Stock market, x

_{3}: T-bills, x

_{4}: National saving scheme, and x

_{5}: Insurance company.

**In Step 1**, the HIFLEs evaluation values given by each decision maker ${d}_{t}(t=1,2,\dots ,7)$ are integrated to develop the collective decision matrix R based on the frequency associated with each linguistic term appearing in the individual decision matrices, the elements of which are listed in Table 4.

**In Step 2**, for reasons of comparison between PHIFLEs, the score function and the variance value function of PHIFLE defined in Definitions 5 and 6 are utilized to obtain the positive ideal decision H

^{+}and the negative ideal decision H

^{−}associated with each attribute ${c}_{j}(j=1,2,3,4)$ in the collective decision matrix R, which are listed in Table 5 and Table 6.

**In Step 3**, to calculate the correlation coefficients between the collective decision matrix R and the PID H

^{+}and the NID H

^{−}for all the alternatives ${a}_{i}(i=1,2,\dots ,5)$, the PHIFLEs should be adjusted to the same probability set based on Definition 8.

**In Step 4**, Equations (14) and (15) are utilized to calculate the similarity $Si{m}_{i}^{+}$ and $Si{m}_{i}^{-}$ between the adjusted PHIFLEs of the collective decision matrix $R$ and the PID H

^{+}and the NID H

^{−}for all alternatives ${a}_{i}(i=1,2,\dots ,5)$, which are listed in Table 7 and Table 8.

**In Step 5**, the LP optimization method is utilized to determine the optimal weights of the attributes ${c}_{j}(j=1,2,3,4)$ based on Equation (16), which is ${w}_{j}^{*}=(0.2653,0.2848,0.2310,0.2189)$.

**In Step 6**, based on Equations (17) and (18), we can obtain the vector normalization of the scores of the collective decision matrix R.

**In Step 7**, the MULTIMOORA approach is utilized to calculate the final ranking for all alternatives ${a}_{i}(j=1,2\dots ,5)$. Firstly, the subordinated methods as PHIFLRS are used to obtain the utility values ${U}_{1}({a}_{i})=(0.1872,0.0911,0.2384,-0.275,-0.41)$ and the ranks of alternatives as $Ran{k}_{1}=(2,3,1,4,5)$ based on Equation (19).

**In Step 8**, the second subordinated methods as PHIFLPR in the MULTIMOORA approach are used to obtain the utility values ${U}_{2}({a}_{i})=(0.1934,0.2431,0.1697,0.2263,0.3039)$ and the ranks of alternatives as $Ran{k}_{1}=(2,3,1,4,5)$ based on Equation (20).

**In Step 9**, the third subordinated methods as PHIFLFMF in the MULTIMOORA approach are used to obtain the utility values ${U}_{3}({a}_{i})=(1.1036,1.0606,1.2139,0.7781,0.6832)$ and the ranks of alternatives as $Ran{k}_{3}=(2,3,1,4,5)$ based on Equation (21).

**In Step 10**, the above three subordinate ranks of alternatives $Ran{k}_{v}({a}_{i}),v=1,2,3;i=1,2,\dots ,5$ need to be aggregated to obtain the final ranking. Two matrices can be built based on the utility values ${U}_{y}({a}_{i}),y=1,2,3;i=1,2,\dots ,5$ and the ranks $Ran{k}_{y},y=1,2,3;i=1,2,\dots ,5$ associated with each attribute ${c}_{y}(y=1,2,3)$, which are listed as follows.

**In Step 11**, the improving Borda rules are utilized to derive the final ranking of alternatives ${a}_{i}$$(i=1,2,\dots ,5)$. Firstly, the normalized vectors of utility value ${U}^{N}$ are obtained based on Equation (22). Secondly, the ordinal values need to be translated to the scores based on the Borda rules for reasons of aggregating the cardinal of each alternative. Finally, the final ranking can be derived by the values of $F{S}_{i}$ $(i=1,2,\dots ,m)$ in descending order based on Equation (23), which are listed in Table 9.

## 8. Comparison with Other Related Research Works

#### 8.1. Comparative Analysis from the Numerical Points with the existing method

- (1)
- The operations over PHIFLTSs proposed in [27] are directly based on multiplying the probabilities by the subscript of the corresponding linguistic terms, which may lose the associated probability information. For example, the PIS and NIS obtained in [27] are $(\langle \{3,3\},\{0,0\}\rangle ,\langle \{3,2.4\},\{0,0\}\rangle ,\dots )$ and $(\langle \{0,0.661\},\{2.25,1\}\rangle ,\langle \{1,1\},\{2.25,1.25\}\rangle ,\dots )$; however, the PIS ${H}^{+}$ and NIS ${H}^{-}$ obtained in this paper are $\langle \{{s}_{0}(0.14),{s}_{1}(0.36),{s}_{2}(0.22),{s}_{3}(0.14),{s}_{4}(0.14)\},\dots \rangle $ and $\langle \{{s}_{4}(0.29),{s}_{5}(0.5),{s}_{6}(0.21)\},\dots \rangle $. Obviously, the former one loses the probability information, which may lead to the failure of the final decision-making;
- (2)
- The method in [27] does not consider the situations of unbalanced linguistic terms on the PHIFLTSs environment; however, this paper selects the second type of linguistic scale function for all attributes to deal with the semantics of unbalanced LTSs;
- (3)
- Table 10 shows the difference of the ranking orders between the work in [27] and this paper, which is due to the effect of the innovation in the basic operational laws, the unbalance LTSs, the distance measure and the MULTIMOORA approach proposed in this paper. It can be clearly seen that the method proposed in this paper is an advantages and innovation to solve MAGDM problems over the PHIFLTSs environment.

#### 8.2. Comparative Analysis from the Numerical Points with HIFLTSs

## 9. Conclusions

- (1)
- Utilizing the new score function and variance value function of PHIFLTSs makes the comparison between PHIFLEs more accurate and rational;
- (2)
- The unbalanced linguistic term sets were considered during the whole decision processes;
- (3)
- The correlation coefficient of the PHIFLTSs was used to measure the similarity instead of the current distance function in order to alleviate the drawbacks;
- (4)
- The decision results are robustness, simplicity, and effectiveness based on the three aggregation subordinated methods with the use of the extension of the MULTIMOORA approach.

## Author Contributions

## Funding

## Conflicts of Interest

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c_{1}: Risk Factor | c_{2}: Growth | c_{3}: Quick Refund | c_{4}: Complicated Documents Requirement | |
---|---|---|---|---|

${a}_{1}$ | $\langle \{{s}_{3},{s}_{4},{s}_{5}\},\{{s}_{1},{s}_{2}\}\rangle $ | $\langle \{{s}_{4},{s}_{5}\},\{{s}_{0},{s}_{1}\}\rangle $ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{3},{s}_{4}\}\rangle $ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{3},{s}_{4}\}\rangle $ |

${a}_{2}$ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{3},{s}_{4}\}\rangle $ | $\langle \{{s}_{3},{s}_{4},{s}_{5}\},\{{s}_{1},{s}_{2}\}\rangle $ | $\langle \{{s}_{3},{s}_{4}\},\{{s}_{0},{s}_{1}\}\rangle $ | $\langle \{{s}_{4},{s}_{5}\},\{{s}_{1},{s}_{2}\}\rangle $ |

${a}_{3}$ | $\langle \{{s}_{4},{s}_{5}\},\{{s}_{0},{s}_{1},{s}_{2}\}\rangle $ | $\langle \{{s}_{3},{s}_{4}\},\{{s}_{1},{s}_{2}\}\rangle $ | $\langle \{{s}_{5},{s}_{6}\},\{{s}_{0}\}\rangle $ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{2},{s}_{3},{s}_{4}\}\rangle $ |

${a}_{4}$ | $\langle \{{s}_{5},{s}_{6}\},\{{s}_{0},{s}_{1}\}\rangle $ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{3},{s}_{4}\}\rangle $ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{3},{s}_{4}\}\rangle $ | $\langle \{{s}_{3},{s}_{4},{s}_{5}\},\{{s}_{1},{s}_{2}\}\rangle $ |

${a}_{5}$ | $\langle \{{s}_{6}\},\{{s}_{0}\}\rangle $ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{3},{s}_{4},{s}_{5}\}\rangle $ | $\langle \{{s}_{0},{s}_{1}\},\{{s}_{2},{s}_{3}\}\rangle $ | $\langle \{{s}_{4},{s}_{5}\},\{{s}_{1},{s}_{2}\}\rangle $ |

c_{1}: Risk Factor | c_{2}: Growth | c_{3}: Quick Refund | c_{4}: Complicated Documents Requirement | |
---|---|---|---|---|

${a}_{1}$ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{3},{s}_{4}\}\rangle $ | $\langle \{{s}_{5},{s}_{6}\},\{{s}_{0},{s}_{1}\}\rangle $ | $\langle \{{s}_{0},{s}_{1}\},\{{s}_{3},{s}_{4}\}\rangle $ | $\langle \{{s}_{3},{s}_{4}\},\{{s}_{1},{s}_{2}\}\rangle $ |

${a}_{2}$ | $\langle \{{s}_{0},{s}_{1}\},\{{s}_{2},{s}_{3}\}\rangle $ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{2},{s}_{3},{s}_{4}\}\rangle $ | $\langle \{{s}_{3},{s}_{4}\},\{{s}_{0},{s}_{1}\}\rangle $ | $\langle \{{s}_{5},{s}_{6}\},\{{s}_{0},{s}_{1}\}\rangle $ |

${a}_{3}$ | $\langle \{{s}_{3},{s}_{4}\},\{{s}_{0},{s}_{1}\}\rangle $ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{3},{s}_{4}\}\rangle $ | $\langle \{{s}_{4},{s}_{5}\},\{{s}_{1},{s}_{2}\}\rangle $ | $\langle \{{s}_{0},{s}_{1}\},\{{s}_{2},{s}_{3}\}\rangle $ |

${a}_{4}$ | $\langle \{{s}_{5},{s}_{6}\},\{{s}_{0}\}\rangle $ | $\langle \{{s}_{3},{s}_{4}\},\{{s}_{0},{s}_{1},{s}_{2}\}\rangle $ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{2},{s}_{3},{s}_{4}\}\rangle $ | $\langle \{{s}_{4},{s}_{5}\},\{{s}_{0}\}\rangle $ |

${a}_{5}$ | $\langle \{{s}_{4},{s}_{5}\},\{{s}_{1},{s}_{2}\}\rangle $ | $\langle \{{s}_{3},{s}_{4}\},\{{s}_{1},{s}_{2},{s}_{3}\}\rangle $ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{3},{s}_{4}\}\rangle $ | $\langle \{{s}_{5},{s}_{6}\},\{{s}_{0}\}\rangle $ |

c_{1}: Risk Factor | c_{2}: Growth | c_{3}: Quick Refund | c_{4}: Complicated Documents Requirement | |
---|---|---|---|---|

${a}_{1}$ | $\langle \{{s}_{4},{s}_{5}\},\{{s}_{0},{s}_{1}\}\rangle $ | $\langle \{{s}_{5},{s}_{6}\},\{{s}_{0}\}\rangle $ | $\langle \{{s}_{3},{s}_{4}\},\{{s}_{1},{s}_{2}\}\rangle $ | $\langle \{{s}_{0},{s}_{1}\},\{{s}_{3},{s}_{4}\}\rangle $ |

${a}_{2}$ | $\langle \{{s}_{3},{s}_{4}\},\{{s}_{1},{s}_{2},{s}_{3}\}\rangle $ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{3},{s}_{4}\}\rangle $ | $\langle \{{s}_{5},{s}_{6}\},\{{s}_{0}\}\rangle $ | $\langle \{{s}_{3},{s}_{4}\},\{{s}_{1},{s}_{2}\}\rangle $ |

${a}_{3}$ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{2},{s}_{3},{s}_{4}\}\rangle $ | $\langle \{{s}_{5},{s}_{6}\},\{{s}_{0}\}\rangle $ | $\langle \{{s}_{4},{s}_{5}\},\{{s}_{0},{s}_{1}\}\rangle $ | $\langle \{{s}_{0},{s}_{1}\},\{{s}_{3},{s}_{4}\}\rangle $ |

${a}_{4}$ | $\langle \{{s}_{4},{s}_{5}\},\{{s}_{1},{s}_{2}\}\rangle $ | $\langle \{{s}_{4},{s}_{5}\},\{{s}_{0},{s}_{1}\}\rangle $ | $\langle \{{s}_{0},{s}_{1},{s}_{2}\},\{{s}_{2},{s}_{3}\}\rangle $ | $\langle \{{s}_{3},{s}_{4},{s}_{5}\},\{{s}_{1},{s}_{2}\}\rangle $ |

${a}_{5}$ | $\langle \{{s}_{3},{s}_{4}\},\{{s}_{0},{s}_{1},{s}_{2}\}\rangle $ | $\langle \{{s}_{1},{s}_{2}\},\{{s}_{2},{s}_{3},{s}_{4}\}\rangle $ | $\langle \{{s}_{2},{s}_{3}\},\{{s}_{3},{s}_{4}\}\rangle $ | $\langle \{{s}_{6}\},\{{s}_{0}\}\rangle $ |

c_{1}: Risk Factor | c_{2}: Growth | |
---|---|---|

${a}_{1}$ | $\langle \begin{array}{c}\{{s}_{1}(0.12),{s}_{2}(0.12),{s}_{3}(0.18),{s}_{4}(0.29),{s}_{5}(0.29)\},\\ \{{s}_{0}(0.14),{s}_{1}(0.36),{s}_{2}(0.22),{s}_{3}(0.14),{s}_{4}(0.14)\}\end{array}\rangle $ | $\langle \begin{array}{c}\{{s}_{4}(0.21),{s}_{5}(0.5),{s}_{6}(0.29)\},\\ \{{s}_{0}(0.58),{s}_{1}(0.42)\}\end{array}\rangle $ |

${a}_{2}$ | $\langle \begin{array}{c}\{{s}_{0}(0.14),{s}_{1}(0.36),{s}_{2}(0.22),{s}_{3}(0.14),{s}_{4}(0.14)\},\\ \{{s}_{1}(0.13),{s}_{2}(0.25),{s}_{3}(0.44),{s}_{4}(0.18)\}\end{array}\rangle $ | $\langle \begin{array}{c}\{{s}_{1}(0.23),{s}_{2}(0.23),{s}_{3}(0.18),{s}_{4}(0.18),{s}_{5}(0.18)\},\\ \{{s}_{1}(0.19),{s}_{2}(0.31),{s}_{3}(0.25),{s}_{4}(0.25)\}\end{array}\rangle $ |

${a}_{3}$ | $\langle \begin{array}{c}\{{s}_{1}(0.14),{s}_{2}(0.14),{s}_{3}(0.14),{s}_{4}(0.36),{s}_{5}(0.22)\},\\ \{{s}_{0}(0.26),{s}_{1}(0.26),{s}_{2}(0.26),{s}_{3}(0.11),{s}_{4}(0.11)\}\end{array}\rangle $ | $\langle \begin{array}{c}\{{s}_{1}(0.14),{s}_{2}(0.14),{s}_{3}(0.22),{s}_{4}(0.22),{s}_{5}(0.14),{s}_{6}(0.14)\},\\ \{{s}_{0}(0.17),{s}_{1}(0.25),{s}_{2}(0.25),{s}_{3}(0.17),{s}_{4}(0.16),\}\end{array}\rangle $ |

${a}_{4}$ | $\langle \begin{array}{c}\{{s}_{4}(0.14),{s}_{5}(0.5),{s}_{5}(0.36)\},\\ \{{s}_{0}(0.42),{s}_{1}(0.42),{s}_{2}(0.16)\}\end{array}\rangle $ | $\langle \begin{array}{c}\{{s}_{1}(0.21),{s}_{2}(0.21),{s}_{3}(0.14),{s}_{4}(0.3),{s}_{5}(0.14)\},\\ \{{s}_{0}(0.25),{s}_{1}(0.25),{s}_{2}(0.12),{s}_{3}(0.19),{s}_{4}(0.19)\}\end{array}\rangle $ |

${a}_{5}$ | $\langle \begin{array}{c}\{{s}_{3}(0.18),{s}_{4}(0.36),{s}_{5}(0.18),{s}_{6}(0.28)\},\\ \{{s}_{0}(0.38),{s}_{1}(0.31),{s}_{2}(0.31)\}\end{array}\rangle $ | $\langle \begin{array}{c}\{{s}_{1}(0.36),{s}_{2}(0.36),{s}_{3}(0.14),{s}_{4}(0.14)\},\\ \{{s}_{1}(0.1),{s}_{2}(0.19),{s}_{3}(0.33),{s}_{4}(0.24),{s}_{5}(0.14)\}\end{array}\rangle $ |

c_{3}: Quick Refund | c_{4}: Complicated Documents Requirement | |

${a}_{1}$ | $\langle \begin{array}{c}\{{s}_{0}(0.14),{s}_{1}(0.36),{s}_{2}(0.22),{s}_{3}(0.14),{s}_{4}(0.14)\},\\ \{{s}_{1}(0.14),{s}_{2}(0.14),{s}_{3}(0.36),{s}_{4}(0.36)\}\end{array}\rangle $ | $\langle \begin{array}{c}\{{s}_{0}(0.14),{s}_{1}(0.36),{s}_{2}(0.22),{s}_{3}(0.14),{s}_{4}(0.14)\},\\ \{{s}_{1}(0.14),{s}_{2}(0.14),{s}_{3}(0.36),{s}_{4}(0.36)\}\end{array}\rangle $ |

${a}_{2}$ | $\langle \begin{array}{c}\{{s}_{3}(0.21),{s}_{4}(0.36),{s}_{5}(0.29),{s}_{6}(0.14)\},\\ \{{s}_{0}(0.58),{s}_{1}(0.42)\}\end{array}\rangle $ | $\langle \begin{array}{c}\{{s}_{3}(0.14),{s}_{4}(0.36),{s}_{5}(0.36),{s}_{6}(0.14)\},\\ \{{s}_{0}(0.16),{s}_{1}(0.42),{s}_{2}(0.42)\}\end{array}\rangle $ |

${a}_{3}$ | $\langle \begin{array}{c}\{{s}_{4}(0.29),{s}_{5}(0.5),{s}_{6}(0.21)\},\\ \{{s}_{0}(0.46),{s}_{1}(0.36),{s}_{2}(0.18)\}\end{array}\rangle $ | $\langle \begin{array}{c}\{{s}_{0}(0.29),{s}_{1}(0.5),{s}_{2}(0.21)\},\\ \{{s}_{2}(0.29),{s}_{3}(0.41),{s}_{4}(0.3)\}\end{array}\rangle $ |

${a}_{4}$ | $\langle \begin{array}{c}\{{s}_{0}(0.12),{s}_{1}(0.44),{s}_{2}(0.44\},\\ \{{s}_{2}(0.25),{s}_{3}(0.44),{s}_{4}(0.31)\}\end{array}\rangle $ | $\langle \begin{array}{c}\{{s}_{3}(0.26),{s}_{4}(0.37),{s}_{5}(0.37)\},\\ \{{s}_{0}(0.16),{s}_{1}(0.42),{s}_{2}(0.42)\}\end{array}\rangle $ |

${a}_{5}$ | $\langle \begin{array}{c}\{{s}_{0}(0.21),{s}_{1}(0.36),{s}_{2}(0.29),{s}_{3}(0.14)\},\\ \{{s}_{2}(0.21),{s}_{3}(0.5),{s}_{4}(0.29)\}\end{array}\rangle $ | $\langle \begin{array}{c}\{{s}_{4}(0.25),{s}_{5}(0.42),{s}_{6}(0.33)\},\\ \{{s}_{0}(0.4),{s}_{1}(0.3),{s}_{2}(0.3)\}\end{array}\rangle $ |

c_{1}: Risk Factor | c_{2}: Growth | c_{3}: Quick Refund | c_{4}: Complicated DocumentsRequirement | |
---|---|---|---|---|

${a}_{1}$ | 0.2478 | 0.7168 | −0.1667 | −0.1667 |

${a}_{2}$ | −0.1354 | 0.0406 | 0.6005 | 0.4595 |

${a}_{3}$ | 0.2690 | 0.2448 | 0.6303 | −0.3104 |

${a}_{4}$ | 0.6800 | 0.1776 | −0.2416 | 0.3934 |

${a}_{5}$ | 0.5466 | −0.1434 | −0.2490 | 0.6307 |

c_{1}: Risk Factor | c_{2}: Growth | |
---|---|---|

H^{+} | $\langle \begin{array}{c}\{{s}_{0}(0.14),{s}_{1}(0.36),{s}_{2}(0.22),{s}_{3}(0.14),{s}_{4}(0.14)\},\\ \{{s}_{1}(0.13),{s}_{2}(0.25),{s}_{3}(0.44),{s}_{4}(0.18)\}\end{array}\rangle $ | $\langle \begin{array}{c}\{{s}_{4}(0.21),{s}_{5}(0.5),{s}_{6}(0.29)\},\\ \{{s}_{0}(0.58),{s}_{1}(0.42)\}\end{array}\rangle $ |

H^{−} | $\langle \begin{array}{c}\{{s}_{4}(0.14),{s}_{5}(0.5),{s}_{5}(0.36)\},\\ \{{s}_{0}(0.42),{s}_{1}(0.42),{s}_{2}(0.16)\}\end{array}\rangle $ | $\langle \begin{array}{c}\{{s}_{1}(0.36),{s}_{2}(0.36),{s}_{3}(0.14),{s}_{4}(0.14)\},\\ \{{s}_{1}(0.1),{s}_{2}(0.19),{s}_{3}(0.33),{s}_{4}(0.24),{s}_{5}(0.14)\}\end{array}\rangle $ |

c_{3}: Quick Refund | c_{4}: Complicated Documents Requirement | |

H^{+} | $\langle \begin{array}{c}\{{s}_{4}(0.29),{s}_{5}(0.5),{s}_{6}(0.21)\},\\ \{{s}_{0}(0.46),{s}_{1}(0.36),{s}_{2}(0.18)\}\end{array}\rangle $ | $\langle \begin{array}{c}\{{s}_{0}(0.29),{s}_{1}(0.5),{s}_{2}(0.21)\},\\ \{{s}_{2}(0.29),{s}_{3}(0.41),{s}_{4}(0.3)\}\end{array}\rangle $ |

H^{−} | $\langle \begin{array}{c}\{{s}_{0}(0.21),{s}_{1}(0.36),{s}_{2}(0.29),{s}_{3}(0.14)\},\\ \{{s}_{2}(0.21),{s}_{3}(0.5),{s}_{4}(0.29)\}\end{array}\rangle $ | $\langle \begin{array}{c}\{{s}_{4}(0.25),{s}_{5}(0.42),{s}_{6}(0.33)\},\\ \{{s}_{0}(0.4),{s}_{1}(0.3),{s}_{2}(0.3)\}\end{array}\rangle $ |

c_{1}: Risk Factor | c_{2}: Growth | c_{3}: Quick Refund | c_{4}: Complicated Documents Requirement | |
---|---|---|---|---|

${a}_{1}$ | 0.7706 | 1 | 0.5357 | 0.8597 |

${a}_{2}$ | 1 | 0.9693 | 0.8759 | 0.4729 |

${a}_{3}$ | 0.7855 | 0.7877 | 1 | 1 |

${a}_{4}$ | 0.4935 | 0.6771 | 0.4553 | 0.0676 |

${a}_{5}$ | 0.5842 | 0.5479 | 0.4637 | 0.3739 |

c_{1}: Risk Factor | c_{2}: Growth | c_{3}: Quick Refund | c_{4}: Complicated Documents Requirement | |
---|---|---|---|---|

${a}_{1}$ | 0.7475 | 0.5479 | 0.6766 | 0.5277 |

${a}_{2}$ | 0.4935 | 0.9637 | 0.3791 | 0.8866 |

${a}_{3}$ | 0.7011 | 0.7938 | 0.4637 | 0.1636 |

${a}_{4}$ | 1 | 0.7377 | −0.2416 | 0.1636 |

${a}_{5}$ | 0.8995 | 1 | 1 | 1 |

c | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ | ${\mathit{a}}_{5}$ |
---|---|---|---|---|---|

FS | 0.5063 | 0.0082 | 0.8904 | −0.3070 | −0.6708 |

Ranking | ${a}_{3}\succ {a}_{1}\succ {a}_{2}\succ {a}_{4}\succ {a}_{5}$ |

Ranking | |
---|---|

Result in [27] | ${a}_{5}\succ {a}_{4}\succ {a}_{1}\succ {a}_{2}\succ {a}_{3}$ |

Proposed model | ${a}_{3}\succ {a}_{1}\succ {a}_{2}\succ {a}_{4}\succ {a}_{5}$ |

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## Share and Cite

**MDPI and ACS Style**

Peng, Y.; Tao, Y.; Wu, B.; Wang, X.
Probabilistic Hesitant Intuitionistic Fuzzy Linguistic Term Sets and Their Application in Multiple Attribute Group Decision Making. *Symmetry* **2020**, *12*, 1932.
https://doi.org/10.3390/sym12111932

**AMA Style**

Peng Y, Tao Y, Wu B, Wang X.
Probabilistic Hesitant Intuitionistic Fuzzy Linguistic Term Sets and Their Application in Multiple Attribute Group Decision Making. *Symmetry*. 2020; 12(11):1932.
https://doi.org/10.3390/sym12111932

**Chicago/Turabian Style**

Peng, You, Yifang Tao, Boyi Wu, and Xiaoxin Wang.
2020. "Probabilistic Hesitant Intuitionistic Fuzzy Linguistic Term Sets and Their Application in Multiple Attribute Group Decision Making" *Symmetry* 12, no. 11: 1932.
https://doi.org/10.3390/sym12111932