# Multi-Attribute Decision Making Based on Probabilistic Neutrosophic Hesitant Fuzzy Choquet Aggregation Operators

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- (1)
- ${({N}_{1})}^{c}={\displaystyle \bigcup _{{\alpha}_{1}\in {T}_{1},{\beta}_{1}\in {I}_{1},{\gamma}_{1}\in {F}_{1}}}\{{\gamma}_{1}|{P}_{1}^{{F}_{1}},1-{\beta}_{1}|{P}_{1}^{{I}_{1}},{\alpha}_{1}|{P}_{1}^{{T}_{1}}\}$,
- (2)
- ${({N}_{1})}^{\lambda}={\displaystyle \bigcup _{{\alpha}_{1}\in {T}_{1},{\beta}_{1}\in {I}_{1},{\gamma}_{1}\in {F}_{1}}}\{\{{({\alpha}_{1})}^{\lambda}|{P}_{1}^{{T}_{1}}\},\{1-{(1-{\beta}_{1})}^{\lambda}|{P}_{1}^{{I}_{1}}\},\phantom{\rule{3.33333pt}{0ex}}\{1-{(1-{\gamma}_{1})}^{\lambda}|{P}_{1}^{{F}_{1}}\}\}$,
- (3)
- $\lambda ({N}_{1})={\displaystyle \bigcup _{{\alpha}_{1}\in {T}_{1},{\beta}_{1}\in {I}_{1},{\gamma}_{1}\in {F}_{1}}}\{\{1-{(1-{\lambda}_{1})}^{\lambda}|{P}_{1}^{{T}_{1}}\},\{{({\beta}_{1})}^{\lambda}|{P}_{1}^{{I}_{1}}\},\{{({\gamma}_{1})}^{\lambda}|{P}_{1}^{{F}_{1}}\}\}$,
- (4)
- ${N}_{1}\oplus {N}_{2}={\displaystyle \bigcup _{\begin{array}{c}{\alpha}_{1}\in {T}_{1},{\beta}_{1}\in {I}_{1},{\gamma}_{1}\in {F}_{1},\\ {\eta}_{2}\in {T}_{2},{\pi}_{2}\in {I}_{2},{\mu}_{2}\in {F}_{2}\end{array}}}\{\{{\alpha}_{1}+{\eta}_{2}-{\alpha}_{2}{\eta}_{2}|{P}_{1}^{{T}_{1}}{P}_{2}^{{T}_{2}}\},\phantom{\rule{3.33333pt}{0ex}}\{{\beta}_{1}{\pi}_{2}|{P}_{1}^{{I}_{1}}{P}_{2}^{{I}_{2}}\},\{{\gamma}_{1}{\mu}_{2}|{P}_{1}^{{F}_{1}}{P}_{2}^{{F}_{2}}\}\}$,
- (5)
- ${N}_{1}\otimes {N}_{2}={\displaystyle \bigcup _{\begin{array}{c}{\alpha}_{1}\in {T}_{1},{\beta}_{1}\in {I}_{1},{\gamma}_{1}\in {F}_{1},\\ {\eta}_{2}\in {T}_{2},{\pi}_{2}\in {I}_{2},{\mu}_{2}\in {F}_{2}\end{array}}}\{\{{\alpha}_{1}{\eta}_{2}|{P}_{1}^{{T}_{1}}{P}_{2}^{{T}_{2}}\},\{{\beta}_{1}+{\pi}_{2}-{\beta}_{1}{\pi}_{2}|{P}_{1}^{{I}_{1}}{P}_{2}^{{I}_{2}}\},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\{{\gamma}_{1}+{\mu}_{2}-{\gamma}_{1}{\mu}_{2}|{P}_{1}^{{F}_{1}}{P}_{2}^{{F}_{2}}\}\}$,

**Definition**

**5.**

- (1)
- $\mu (\varnothing )=0$, $\mu (Y)=1$;
- (2)
- $A\subseteq B$, then $\mu (A)\le \mu (B)$, $\forall A,B\subseteq P(Y)$;Fuzzy measure μ satisfies property, $\forall {Y}_{1},{Y}_{2}\in P(Y)$, $A\cap B=\varnothing $$$\mu ({Y}_{1}\cup {Y}_{2})=\mu ({Y}_{1})+\mu ({Y}_{2})+\lambda \mu ({Y}_{1})\mu ({Y}_{2})\lambda \in (-1,\infty ).$$Then, μ is described a λ-fuzzy measure.

**Theorem 1.**

**Definition**

**6.**

## 3. PNHFSs and Aggregation Operators

#### 3.1. The Comparison Method of PNHFEs

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

- (1)
- If $S({N}_{1})>S({N}_{2})$, it indicates that PNHFE ${N}_{1}$ is superior to ${N}_{2}$;
- (2)
- If $S({N}_{1})=S({N}_{2}),D({N}_{1})>D({N}_{2})$, it indicates that PNHFE ${N}_{1}$ is inferior to ${N}_{2}$;
- (3)
- If $S({N}_{1})=S({N}_{2}),D({N}_{1})=D({N}_{2})$, it indicates that PNHFE ${N}_{1}$ is equal to ${N}_{2}$.

#### 3.2. The PNHFCOA Operator and PNHFCOG Operator

**Definition**

**10.**

**Theorem**

**2.**

**Proof.**

- (1)
- When $n=1$, we have the following equation by Definition 10:$$\begin{array}{c}\hfill PNHFCOA({N}_{1})={\mu}_{\pi (1)}\oplus {N}_{\pi (1)}={N}_{1}.\end{array}$$Obviously, $PNHFCOA\{{N}_{1}\}$ is an PNHFE.
- (2)
- When $n=2$, we have$$\begin{array}{c}PNHFCOA({N}_{1},{N}_{2})=({\mu}_{\pi (1)}{N}_{\pi (1)})\oplus ({\mu}_{\pi (2)}{N}_{\pi (2)})\hfill \\ ={\displaystyle \bigcup _{{\alpha}_{\pi (1)}\in {T}_{\pi (1)},{\beta}_{\pi (1)}\in {I}_{\pi (1)},{\gamma}_{\pi (1)}\in {F}_{\pi (1)}}}\{1-{(1-{\alpha}_{\pi (1)})}^{{\mu}_{\pi (1)}}|{P}_{\pi (1)}^{{T}_{\pi (1)}},{\beta}_{\pi (1)}^{{\mu}_{\pi (1)}}|{P}_{\pi (1)}^{{I}_{\pi (1)}},{\gamma}_{\pi (1)}^{{\mu}_{\pi (1)}}|{P}_{\pi (1)}^{{F}_{\pi (1)}}\}\hfill \\ \oplus {\displaystyle \bigcup _{{\alpha}_{\pi (2)}\in {T}_{\pi (2)},{\beta}_{\pi (2)}\in {I}_{\pi (2)},{\gamma}_{\pi (2)}\in {F}_{\pi (2)}}}\{1-{(1-{\alpha}_{\pi (2)})}^{{\mu}_{\pi (2)}}|{P}_{\pi (2)}^{{T}_{\pi (2)}},{\beta}_{\pi (2)}^{{\mu}_{\pi (2)}}|{P}_{\pi (2)}^{{I}_{\pi (2)}},{\gamma}_{\pi (2)}^{{\mu}_{\pi (2)}}|{P}_{\pi (2)}^{{F}_{\pi (2)}}\}\hfill \\ ={\displaystyle \bigcup _{{\alpha}_{\pi (k)}\in {T}_{\pi (k)},{\beta}_{\pi (k)}\in {I}_{\pi (k)},{\gamma}_{\pi (k)}\in {F}_{\pi (k)}}}\{1-\prod _{\pi (k)=1}^{2}{(1-{\alpha}_{\pi (k)})}^{{\mu}_{\pi (k)}}|\prod _{\pi (k)=1}^{2}{P}_{\pi (1)}^{{T}_{\pi (k)}},\hfill \\ \prod _{\pi (k)=1}^{2}{\beta}_{\pi (k)}^{{\mu}_{\pi (k)}}|\prod _{\pi (k)=1}^{2}{P}_{\pi (k)}^{{I}_{\pi (k)}},\prod _{\pi (k)=1}^{2}{\gamma}_{\pi (k)}^{{\mu}_{\pi (k)}}|\prod _{\pi (k)=1}^{2}{P}_{\pi (k)}^{{F}_{\pi (k)}}\}.\hfill \end{array}$$Thus, we know $PNHFCOA\{{N}_{1},{N}_{2}\}$ is an PNHFE.
- (3)
- When $n=k$, Equation (9) is true, and we have$$\begin{array}{c}PNHFCOA({N}_{1},{N}_{2},\cdots ,{N}_{k})={\oplus}_{\pi (k)=1}^{k}{\mu}_{\pi (k)}{N}_{\pi (k)}\hfill \\ ={\displaystyle \bigcup _{{\alpha}_{\pi (k)}\in {T}_{\pi (k)},{\beta}_{\pi (k)}\in {I}_{\pi (k)},{\gamma}_{\pi (k)}\in {F}_{\pi (k)}}}\{\{1-\prod _{\pi (k)=1}^{k}{(1-{\alpha}_{\pi (k)})}^{{\mu}_{\pi (k)}}|\prod _{\pi (k)=1}^{k}{P}_{\pi (k)}^{{T}_{\pi (k)}}\},\hfill \\ \{\prod _{\pi (k)=1}^{k}{({\beta}_{\pi (k)})}^{{\mu}_{\pi (k)}}|\prod _{\pi (k)=1}^{k}{P}_{\pi (k)}^{{I}_{\pi (k)}}\},\{\prod _{\pi (k)=1}^{k}{({\gamma}_{\pi (k)})}^{{\mu}_{\pi (k)}}|\prod _{\pi (k)=1}^{k}{P}_{\pi (k)}^{{F}_{\pi}(k)}\}\}.\hfill \end{array}$$Thus, the next formula is obtained, $n=k+1$,$$\begin{array}{c}PNHFCOA({N}_{1},{N}_{2},\cdots ,{N}_{k},{N}_{k+1})=({\oplus}_{\pi (k)=1}^{k}{\mu}_{\pi (k)}{N}_{\pi (k)})\oplus ({\mu}_{\pi (k+1)}{N}_{\pi (k+1)})\hfill \\ ={\displaystyle \bigcup _{{\alpha}_{\pi (k)}\in {T}_{\pi (k)},{\beta}_{\pi (k)}\in {I}_{\pi (k)},{\gamma}_{\pi (k)}\in {F}_{\pi (k)}}}\{\{1-\prod _{\pi (k)=1}^{k}{(1-{\alpha}_{\pi (k)})}^{{\mu}_{\pi (k)}}|\prod _{\pi (k)=1}^{k}{P}_{\pi (k)}^{{T}_{\pi (k)}}\},\hfill \\ \phantom{\rule{142.26378pt}{0ex}}\{\prod _{\pi (k)=1}^{k}{({\beta}_{\pi (k)})}^{{\mu}_{\pi (k)}}|\prod _{\pi (k)=1}^{k}{P}_{\pi (k)}^{{I}_{\pi (k)}}\},\{\prod _{\pi (k)=1}^{k}{({\gamma}_{\pi (k)})}^{{\mu}_{\pi (k)}}|\prod _{\pi (k)=1}^{k}{P}_{\pi (k)}^{{F}_{\pi}(k)}\}\}\hfill \\ \oplus {\displaystyle \bigcup _{{\alpha}_{\pi (k+1)}\in {T}_{\pi (k+1)},{\beta}_{\pi (k+1)}\in {I}_{\pi (k+1)},{\gamma}_{\pi (k+1)}\in {F}_{\pi (k+1)}}}\{1-{(1-{\alpha}_{\pi (k+1)})}^{{\mu}_{\pi (k+1)}}|{P}_{\pi (k+1)}^{{T}_{\pi (k+1)}},\hfill \\ \phantom{\rule{199.16928pt}{0ex}}{\beta}_{\pi (k+1)}^{{\mu}_{\pi (k+1)}}|{P}_{\pi (k+1)}^{{I}_{\pi (k+1)}},{\gamma}_{\pi (k+1)}^{{\mu}_{\pi (k+1)}}|{P}_{\pi (k+1)}^{{F}_{\pi (k+1)}}\}\hfill \\ ={\displaystyle \bigcup _{{\alpha}_{\pi (k)}\in {T}_{\pi (k)},{\beta}_{\pi (k)}\in {I}_{\pi (k)},{\gamma}_{\pi (k)}\in {F}_{\pi (k)}}}\{\{1-\prod _{\pi (k)=1}^{k+1}{(1-{\alpha}_{\pi (k)})}^{{\mu}_{\pi (k)}}|\prod _{\pi (k)=1}^{k+1}{P}_{\pi (k)}^{{T}_{\pi (k)}}\},\hfill \\ \phantom{\rule{142.26378pt}{0ex}}\{\prod _{\pi (k)=1}^{k+1}{({\beta}_{\pi (k)})}^{{\mu}_{\pi (k)}}|\prod _{\pi (k)=1}^{k+1}{P}_{\pi (k)}^{{I}_{\pi (k)}}\},\{\prod _{\pi (k)=1}^{k+1}{({\gamma}_{\pi (k)})}^{{\mu}_{\pi (k)}}|\prod _{\pi (k)=1}^{k+1}{P}_{\pi (k)}^{{F}_{\pi}(k)}\}\}.\hfill \end{array}$$Thus, for any n, the conclusion is right. □

- (1)
- Assume $\mu (F)=1$, then$$\begin{array}{c}\hfill PNHFCOA({N}_{1},{N}_{2},\cdots ,{N}_{n})=max\{{N}_{1},{N}_{2},\cdots ,{N}_{n}\}.\end{array}$$
- (2)
- Assume $\mu (F)=0$, then$$\begin{array}{c}\hfill PNHFCOA({N}_{1},{N}_{2},\cdots ,{N}_{n})=min\{{N}_{1},{N}_{2},\cdots ,{N}_{n}\}.\end{array}$$
- (3)
- Assume the condition $\mu ({x}_{\pi (k)})=\mu ({F}_{\pi (k)}-{F}_{\pi (k-1)})$ is independent, the PNHFCOA operator is described an PNHFWA operator,$$\begin{array}{c}PNHFWA({N}_{1},{N}_{2},\cdots ,{N}_{n})={\oplus}_{k=1}^{n}\mu ({x}_{k}){N}_{k}\hfill \\ ={\displaystyle \bigcup _{{\alpha}_{k}\in {T}_{k},{\beta}_{k}\in {I}_{k},{\gamma}_{k}\in {F}_{k}}}\{\{1-\prod _{k=1}^{n}{(1-{\alpha}_{k})}^{\mu ({x}_{k})}|\prod _{k=1}^{n}{P}_{k}^{{T}_{k}}\},\{\prod _{k=1}^{n}{({\beta}_{k})}^{\mu ({x}_{k})}|\prod _{k=1}^{n}{P}_{k}^{{I}_{k}}\},\{\prod _{k=1}^{n}{({\gamma}_{k})}^{\mu ({x}_{k})}|\prod _{k=1}^{n}{P}_{k}^{{F}_{k}}\}\}.\hfill \end{array}$$
- (4)
- Assume the condition $\mu ({x}_{\pi (k)})=\frac{1}{n}$, the PNHFCOA operator and PNHFWA operator reduce to the PNHFA operator,$$\begin{array}{c}PNHFWA({N}_{1},{N}_{2},\cdots ,{N}_{n})={\oplus}_{k=1}^{n}\mu ({x}_{k}){N}_{k}\hfill \\ ={\displaystyle \bigcup _{{\alpha}_{k}\in {T}_{k},{\beta}_{k}\in {I}_{k},{\gamma}_{k}\in {F}_{k}}}\{\{1-\prod _{k=1}^{n}{(1-{\alpha}_{k})}^{\frac{1}{n}}|\prod _{k=1}^{n}{P}_{k}^{{T}_{k}}\},\{\prod _{k=1}^{n}{({\beta}_{k})}^{\frac{1}{n}}|\prod _{k=1}^{n}{P}_{k}^{{I}_{k}}\},\{\prod _{k=1}^{n}{({\gamma}_{k})}^{\frac{1}{n}}|\prod _{k=1}^{n}{P}_{k}^{{F}_{k}}\}\}.\hfill \end{array}$$

**Theorem 3.**

**Proof.**

**Theorem 4.**

**Proof.**

**Theorem 5.**

**Proof.**

**Theorem 6.**

**Proof.**

**Definition**

**11.**

**Theorem**

**7.**

- (1)
- Assume $\mu (F)=1$, then$$\begin{array}{c}\hfill PNHFCOG({N}_{1},{N}_{2},\cdots ,{N}_{n})=max\{{N}_{1},{N}_{2},\cdots ,{N}_{n}\}.\end{array}$$
- (2)
- Assume $\mu (F)=0$, then$$\begin{array}{c}\hfill PNHFCOG({N}_{1},{N}_{2},\cdots ,{N}_{n})=min\{{N}_{1},{N}_{2},\cdots ,{N}_{n}\}.\end{array}$$
- (3)
- Assume the prerequisite $\mu ({x}_{\pi (k)})=\mu ({F}_{\pi (k)}-{F}_{\pi}(i-1))$ is independent, the PNHFCOG operator indicates an PNHFWG operator:$$\begin{array}{c}PNHFWG({N}_{1},{N}_{2},\cdots ,{N}_{n})={\otimes}_{k=1}^{n}\mu ({x}_{k}){N}_{k}\hfill \\ ={\displaystyle \bigcup _{{\alpha}_{k}\in {T}_{k},{\beta}_{k}\in {I}_{k},{\gamma}_{k}\in {F}_{k}}}\{\{\prod _{k=1}^{n}{({\alpha}_{k})}^{\mu ({x}_{k})}|\prod _{k=1}^{n}{P}_{k}^{{T}_{k}}\},\{1-\prod _{k=1}^{n}{(1-{\beta}_{k})}^{\mu ({x}_{k})}|\prod _{k=1}^{n}{P}_{k}^{{I}_{k}}\},\{1-\prod _{k=1}^{n}{(1-{\gamma}_{k})}^{\mu ({x}_{k})}|\prod _{k=1}^{n}{P}_{k}^{{F}_{k}}\}\}.\hfill \end{array}$$
- (4)
- Assume the precondition $\mu ({x}_{\pi (k)})=\frac{1}{n}$, the PNHFFCG operator and PNHFWG operator reduce to the PNHFG operator:$$\begin{array}{c}PNHFWG({N}_{1},{N}_{2},\cdots ,{N}_{n})={\otimes}_{k=1}^{n}\mu ({x}_{k}){N}_{k}\hfill \\ ={\displaystyle \bigcup _{{\alpha}_{k}\in {T}_{k},{\beta}_{k}\in {I}_{k},{\gamma}_{k}\in {F}_{k}}}\{\{\prod _{k=1}^{n}{({\alpha}_{k})}^{\frac{1}{n}}|\prod _{k=1}^{n}{P}_{k}^{{T}_{k}}\},\{1-\prod _{k=1}^{n}{(1-{\beta}_{k})}^{\frac{1}{n}}|\prod _{k=1}^{n}{P}_{k}^{{I}_{k}}\},\{1-\prod _{k=1}^{n}{(1-{\gamma}_{k})}^{\frac{1}{n}}|\prod _{k=1}^{n}{P}_{k}^{{F}_{k}}\}\}.\hfill \end{array}$$

**Theorem**

**8.**

- (1)
- (Monotonicity) Assume ${N}_{k}=\{\{{\alpha}_{k}|{P}_{k}^{{T}_{k}}\},\{{\beta}_{k}|{P}_{k}^{{I}_{k}}\},\{{\gamma}_{k}|{P}_{k}^{{F}_{k}}\}\}$ and ${\tilde{N}}_{k}=\{\{{\tilde{\alpha}}_{k}|{P}_{k}^{{\tilde{T}}_{k}}\},\{{\tilde{\beta}}_{k}|{P}_{k}^{{\tilde{I}}_{k}}\}$, $\{{\tilde{\gamma}}_{k}|{P}_{k}^{{\tilde{F}}_{k}}\}\}$ indicate two PNHFEs. The factor $\pi (k)$ satisfies condition ${N}_{\pi (1)}\ge {N}_{\pi (2)}\ge \cdots \ge {N}_{\pi (n)}$ and ${\tilde{N}}_{\pi (1)}\ge {\tilde{N}}_{\pi (2)}\ge \cdots \ge {\tilde{N}}_{\pi (n)}$. With $\forall {N}_{\pi (k)}$ and $\forall {\tilde{N}}_{\pi (k)}$, there are ${\alpha}_{\pi (k)}\le {\tilde{\alpha}}_{\pi (k)}$, ${\beta}_{\pi (k)}\ge {\tilde{\beta}}_{\pi (k)}$, ${\gamma}_{\pi (k)}\ge {\tilde{\gamma}}_{\pi (k)}$ and ${P}_{\pi (k)}^{{T}_{\pi (k)}}={P}_{\pi (k)}^{{\tilde{T}}_{\pi (k)}}$, ${P}_{\pi (k)}^{{I}_{\pi (k)}}={P}_{\pi (k)}^{{\tilde{I}}_{\pi (k)}}$, ${P}_{\pi (k)}^{{F}_{\pi (k)}}={P}_{\pi (k)}^{{\tilde{F}}_{\pi (k)}}$. Then,$$\begin{array}{c}\hfill PNHFCOG\{{N}_{1},{N}_{2},\cdots ,{N}_{n}\}\le PNHFCOG\{{\tilde{N}}_{1},{\tilde{N}}_{2},\cdots ,{\tilde{N}}_{n}\}.\end{array}$$
- (2)
- (Boundedness) Assume ${N}_{k}=\{\{{\alpha}_{k}|{P}_{k}^{{T}_{k}}\},\{{\beta}_{k}|{P}^{{I}_{k}}\},\{{\gamma}_{k}|{P}_{k}^{{F}_{k}}\}\}$ indicates an PNHFE,$${N}^{-}=\{\{min\{{\alpha}_{k}\}|min\{{P}_{k}^{{T}_{k}}\}\},\{max\{{\beta}_{k}\}|max\{{P}_{k}^{{I}_{k}}\}\},\{max\{{\gamma}_{k}\}|max\{{P}_{k}^{{F}_{k}}\}\}\},$$$${N}^{+}=\{\{max\{{\alpha}_{k}\}|max\{{P}_{k}^{{T}_{k}}\}\},\{min\{{\beta}_{k}\}|min\{{P}_{k}^{{I}_{k}}\}\},\{min\{{\gamma}_{k}\}|min\{{P}_{k}^{{F}_{k}}\}\}\}.$$Then,$$\begin{array}{c}\hfill PNHFCOG({N}^{-})\le PNHFCOA({N}_{1},{N}_{2},\cdots ,{N}_{n})\le PNHFCOG({N}^{+}).\end{array}$$
- (3)
- (Idempotency) Assume ${N}_{k}=\{\{\alpha |{P}_{1}\},\{\beta |{P}_{2}\},\{\gamma ]|{P}_{3}\}\}$ is a normalized PNHFE, then$$\begin{array}{c}\hfill PNHFCOG({N}_{1},{N}_{2},\cdots ,{N}_{X})=\{\{\alpha |{P}_{1}\},\{\beta |{P}_{2}\},\{\gamma |{P}_{3}\}\}.\end{array}$$
- (4)
- (Commutativity) Assume $A=\{{N}_{1},{N}_{2},\cdots ,{N}_{n}\}$ and $B=\{{N}_{\lambda (1)},{N}_{\lambda (2)},\cdots ,{N}_{\lambda (n)}\}$ are two finite sets. If the position of the element in $\{{N}_{\pi (1)},{N}_{\pi (2)},\cdots ,{N}_{\pi (n)}\}$ is changed arbitrarily to get $\{{N}_{1},{N}_{2},\cdots ,{N}_{n}\}$, then:$$\begin{array}{c}\hfill PNHFCOG({N}_{1},{N}_{2},\cdots ,{N}_{n})=PNHFCOG\{{N}_{\lambda (1)},{N}_{\lambda (2)},\cdots ,{N}_{\lambda (n)}\}.\end{array}$$

**Lemma**

**1.**

**Theorem**

**9.**

**Proof.**

## 4. A MADM method in PNHF Environment

## 5. The Program of the Proposed Approach

- Step 2. Since the information of fuzzy measure is $\mu ({C}_{1})=0.3$, $\mu ({C}_{2})=0.3$, $\mu ({C}_{3})=0.3$, $\mu ({C}_{4})=0.2$, respectively. By Equation (3), we get $\lambda =-0.2317$. Thus, taking ${Z}_{1}$ as an example, we can get$$\begin{array}{c}\hfill {\mu}_{\pi (1)}=0.2477,{\mu}_{\pi (2)}=0.1732,{\mu}_{\pi (3)}=0.2791,{\mu}_{\pi (4)}=0.3.\end{array}$$
- Step 3. Utilizing the PNHFCOA operator, by Equation (9), we can get$$\begin{array}{c}\hfill S({Z}_{1})=0.6466,S({Z}_{2})=0.6436,S({Z}_{3})=0.5822,S({Z}_{4})=0.6950.\end{array}$$
- Step 4. Rank the PNHFEs by Definition 9,$$\begin{array}{c}\hfill {Z}_{4}>{Z}_{1}>{Z}_{2}>{Z}_{3}.\end{array}$$The 3PL Company ${Z}_{1}$ is an optimal option.

## 6. Comparison with Other Approaches

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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${\mathit{C}}_{1}$ | |
---|---|

${A}_{1}$ | $\{\{0.5|0.3,0.57|0.22,0.58|0.27,0.64|0.21\},\{0.43|0.25,0.48|0.2,0.49|0.30,0.55|0.25\}$, |

$\{0.41|0.27,0.47|0.23,0.52|0.23,0.46|0.27\}\}$ | |

${A}_{2}$ | $\{\{0.44|0.27,0.49|0.24,0.48|0.26,0.52|0.23\},\{0.46|0.47,0.53|0.53\},$ |

$\{0.29|0.18,0.33|0.14,0.36|0.20,0.41|0.16,0.41|0.18,0.47|0.14\}\}$ | |

${A}_{3}$ | $\{\{0.41|0.30,0.48|0.22,0.47|0.27,0.53|0.21\},\{0.46|0.23,0.49|0.26,0.49|0.24,0.53|0.27\},$ |

$\{0.39|0.24,0.41|0.25,0.48|0.26,0.45|0.25\}\}$ | |

${A}_{4}$ | $\{\{0.47|0.25,0.51|0.24,0.50|0.26,0.53|0.25\},\{0.34|0.33,0.43|0.35,0.5|0.32\},$ |

$\{0.42|0.28,0.45|0.21,0.53|0.29,0.56|0.22\}\}$ | |

${\mathit{C}}_{\mathbf{2}}$ | |

${A}_{1}$ | $\{\{0.40|0.26,0.51|0.25,0.49|0.25,0.58|0.24\},\{0.56|0.27,0.59|0.24,0.60|0.26,0.63|0.23\}$, |

$\{0.39|0.23,0.43|0.29,0.42|0.21,0.47|0.27\}\}$ | |

${A}_{2}$ | $\{\{0.51|0.53,0.54|0.47\},\{0.49|0.25,0.52|0.22,0.57|0:28,0.60|0.25\},$ |

$\{0.43|0.18,0.46|0.18,0.48|0.17,0.50|0.16,0.53|0.16,0.55|0.15\}\}$ | |

${A}_{3}$ | $\{\{0.54|0.26,0.60|0.25,0.63|0.25,0.68|0.24\},\{0.50|0.48,0.56|0.52\},$ |

$\{0.43|0.26,0.46|0.24,0.46|0.26T,0.50|0.24\}\}$ | |

${A}_{4}$ | $\{\{0.61|0.54,0.67|0.46\},\{0.43|0.27,0.50|0.26,0.46|0.24,0.53|0.23\},$ |

$\{0.42|0.23,0.50|0.24T,0.45|0.26,0.53|0.27\}\}$ | |

${\mathit{C}}_{\mathbf{3}}$ | |

${A}_{1}$ | $\{\{0.56|0.24,0.62|0.24,0.59|0.26,0.64|0.26\},\{0.33|0.25,0.36|0.24,0.37|0.26,0.41|0.25\}$, |

$\{0.36|0.33,0.42|0.36,0.45|0.31\}\}$ | |

${A}_{2}$ | $\{\{0.65|0.24T,0.69|0.27,0.67|0.23,0.71|0.26\},\{0.43|0.31,0.52|0.23,0.46|0.27,0.55|0.19\},$ |

$\{0.43|0.26,0.46|0.25,0.50|0.25,0.53|0.24\}\}$ | |

${A}_{3}$ | $\{\{0.51|0.26,0.54|0.26,0.57|0.24,0.60|0.24\},\{0.43|0.26,0.46|0.24,0.48|0.26,0.52|0.24\},$ |

$\{0.49|0.25,0.54|0.26,0.57|0.24,0.62|0.25\}\}$ | |

${A}_{4}$ | $\{\{0.57|0.24,0.66|0.28,0.66|0.22,0.73|0.26\},\{0.43|0.54,0.49|0.46\},$ |

$\{0.47|0.16,0.53|0.17,0.56|0.16,0.50|0.17,0.57|0.18,0.59|0.17\}\}$ | |

${\mathit{C}}_{\mathbf{4}}$ | |

${A}_{1}$ | $\{\{0.48|0.47,0.57|0.53\},\{0.40|0.51,0.47|0.49\}$, |

$\{0.47|0.16,0.50|0.15,0.53|0.15,0.49|0.19,0.54|0.18,0.56|0.17\}\}$ | |

${A}_{2}$ | $\{\{0.51|0.27,0.62|0.26,0.54|0.24,0.64|0.23\},\{0.40|0.25,0.46|0.28,0.46|0.22,0.53|0.25\},$ |

$\{0.39|0.33,0.42|0.37,0.45|0.30\}\}$ | |

${A}_{3}$ | $\{\{0.48|0.28,0.58|0.23,0.51|0.26,0.61|0:23\},\{0.42|0.25,0.45|0.24,0.47|0.26,0.50|0.25\},$ |

$\{0.42|0.27,0.50|0.26,0.45|0.24,0.53|0.23\}\}$ | |

${A}_{4}$ | $\{\{0.66|0.27,0.73|0.24,0.71|0:26,0.77|0.23\},\{0.43|0.38,0.49|0.33,0.54|0.29\},$ |

$\{0.36|0.27,0.41|0.24,0.39|0.26T,0.45|0.23\}\}$ |

${\mathit{D}}_{1}$ | ${\mathit{D}}_{2}$ | ${\mathit{D}}_{3}$ | ${\mathit{D}}_{4}$ | |
---|---|---|---|---|

${Z}_{1}$ | 0.6185 | 0.4700 | 0.8259 | 0.5782 |

${Z}_{2}$ | 0.6081 | 0.4885 | 0.7204 | 0.6941 |

${Z}_{3}$ | 0.5395 | 0.6181 | 0.5273 | 0.6072 |

${Z}_{4}$ | 0.5907 | 0.6825 | 0.6562 | 0.8329 |

$\mathbf{Method}$ | $\mathbf{Sort}\mathbf{of}\mathbf{Results}$ | $\mathbf{Optimal}\mathbf{Alternative}$ | $\mathbf{Worst}\mathbf{Alternative}$ |
---|---|---|---|

$TOPSIS\text{-}based\phantom{\rule{4pt}{0ex}}QUALIFLEX\phantom{\rule{3.33333pt}{0ex}}method$ [3] | ${Z}_{4}>{Z}_{2}>{Z}_{1}>{Z}_{3}$ | ${Z}_{4}$ | ${Z}_{3}$ |

$SNNPWA\phantom{\rule{4pt}{0ex}}operator$ [45] | ${Z}_{3}>{Z}_{1}>{Z}_{2}>{Z}_{4}$ | ${Z}_{3}$ | ${Z}_{4}$ |

$SNNPWG\phantom{\rule{4pt}{0ex}}operator$ [45] | ${Z}_{3}>{Z}_{2}>{Z}_{1}>{Z}_{4}$ | ${Z}_{2}$ | ${Z}_{1}$ |

$PNHFCOA\phantom{\rule{4pt}{0ex}}operator$ | ${Z}_{4}>{Z}_{1}>{Z}_{2}>{Z}_{3}$ | ${Z}_{4}$ | ${Z}_{3}$ |

$PNHFCOG\phantom{\rule{4pt}{0ex}}operator$ | ${Z}_{4}>{Z}_{1}>{Z}_{2}>{Z}_{3}.$ | ${Z}_{4}$ | ${Z}_{3}$ |

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## Share and Cite

**MDPI and ACS Style**

Shao, S.; Zhang, X.; Zhao, Q.
Multi-Attribute Decision Making Based on Probabilistic Neutrosophic Hesitant Fuzzy Choquet Aggregation Operators. *Symmetry* **2019**, *11*, 623.
https://doi.org/10.3390/sym11050623

**AMA Style**

Shao S, Zhang X, Zhao Q.
Multi-Attribute Decision Making Based on Probabilistic Neutrosophic Hesitant Fuzzy Choquet Aggregation Operators. *Symmetry*. 2019; 11(5):623.
https://doi.org/10.3390/sym11050623

**Chicago/Turabian Style**

Shao, Songtao, Xiaohong Zhang, and Quan Zhao.
2019. "Multi-Attribute Decision Making Based on Probabilistic Neutrosophic Hesitant Fuzzy Choquet Aggregation Operators" *Symmetry* 11, no. 5: 623.
https://doi.org/10.3390/sym11050623