# Multi-Attribute Decision Making Based on Intuitionistic Fuzzy Power Maclaurin Symmetric Mean Operators in the Framework of Dempster-Shafer Theory

^{1}

^{2}

^{*}

## Abstract

**:**

_{DST}) operator and an intuitionistic fuzzy weighted power MSM (IFPWMSM

_{DST}) operator in the framework of the DST and provide their favorable properties. Then, we propose a novel method based on the proposed operators to solve multi-attribute decision-making (MADM) problems without intermediate defuzzification when their attributes and weights are both IFNs. Finally, some examples are utilized to demonstrate that the proposed methods outperform the previous ones.

## 1. Introduction

- (1)
- (2)
- (3)

- (1)
- For overcoming the revealed drawbacks of OORs of IFN and getting more convincing aggregate results, we convert an IFN into a BI and replace operations on IFNs with operations on BI;
- (2)
- For utilizing ORs of BI to develop some PMSM operator for IFNs in the framework of DST, we convert an IFN into a BPA and replace Hamming distance and Euclidean distance with Jousselme distance (JD);
- (3)
- For further reducing the loss of information, we use the presented operators to solve MADM problems without intermediate defuzzification when attributes and their weights are all IFNs.

_{DST}) operator and intuitionistic fuzzy weighted power MSM (IFPWMSM

_{DST}) operator in the framework of DST; then, based on the IFPMSM

_{DST}operator and IFPWMSBM

_{DST}operator, we develop a new MADM method. By comparing with the previous methods based on a intuitionistic fuzzy evidential power aggregation (IFEPA) operator [11], a weighted intuitionistic fuzzy MSM (WIFMSM) operator [28], and an extended weighted intuitionistic fuzzy interaction Bonferroni mean (EWIFIBM) operator [31], the advantages of the proposed methods are discussed.

_{DST}operator and IFPWMSM

_{DST}operator based the PA operator and MSM operator. In Section 4, we develop a novel method with IFNs based on IFPMSM

_{DST}operator and IFPWMSM

_{DST}operator in the framework of DST. In Section 5, we utilize some numerical examples to demonstrate the reasonability and flexibility of presented operators. In Section 6, we discuss the conclusions.

## 2. Preliminaries

#### 2.1. IFSs

**Definition**

**1**

**.**Let $\mathrm{A}=\left\{{\alpha}_{i}|i=1,2,\cdots ,t\right\}$ be a fixed set, and then the IFS B on A can be defined as follows: $\mathrm{B}=\left\{\langle \alpha ,{u}_{\beta}\left(\alpha \right),{v}_{\beta}\left(\alpha \right)\rangle \right\}$, where ${u}_{\beta}\left(\alpha \right):\mathrm{A}\to \left[0,1\right]$ and ${v}_{\beta}\left(\alpha \right):\mathrm{A}\to \left[0,1\right]$ are the MD and NMD of $\alpha \text{}\in \text{}\mathrm{A}$ to B, respectively, and $0\le {u}_{\beta}\left(\alpha \right)+{v}_{\beta}\left(\alpha \right)\le 1$. Moreover, ${\pi}_{\beta}\left(\alpha \right)=1-{u}_{\beta}\left(\alpha \right)-{v}_{\beta}\left(\alpha \right)$ denotes the HD of $\alpha $ to B.

**Definition**

**2**

**.**Let ${\beta}_{1}=\langle {u}_{1},{v}_{1}\rangle $ and ${\beta}_{2}=\langle {u}_{2},{v}_{2}\rangle $ be two IFNs, then

- (1)
- If $SF\left({\beta}_{1}\right)<SF\left({\beta}_{2}\right)$, then ${\beta}_{1}<{\beta}_{2}$;
- (2)
- If $SF\left({\beta}_{1}\right)=SF\left({\beta}_{2}\right)$, then
- (i)
- $AF\left({h}_{1}\right)<AF\left({h}_{2}\right)$, then ${\beta}_{1}<{\beta}_{2}$;
- (ii)
- $AF\left({\beta}_{1}\right)=AF\left({\beta}_{2}\right)$, then ${\beta}_{1}={\beta}_{2}$.

- (1)
- The Equation (1) is not a constant operation—that is to say, ${\beta}_{1}<{\beta}_{2}$ cannot always generate $\left({\beta}_{1}\oplus {\beta}_{3}\right)<\left({\beta}_{2}\oplus {\beta}_{3}\right)$.

**Example**

**1.**

- (2)
- Equation (2) is not a constant operation, i.e., ${\beta}_{1}<{\beta}_{2}$ cannot always generate $\left({\beta}_{1}\otimes {\beta}_{3}\right)<\left({\beta}_{2}\otimes {\beta}_{3}\right)$.

**Example**

**2.**

- (3)
- Equation (3) is not persistent under multiplication. In other words, ${\beta}_{1}<{\beta}_{2}$ cannot always generate $\delta {\beta}_{1}<\delta {\beta}_{2}\text{}\left(\delta 0\right)$.

**Example**

**3.**

- (4)
- IFWAM is not always monotone with respect to the SF and AF. In other words, ${\beta}_{1}<{\beta}_{2}$ cannot invariably generate $IFWAM\left({\beta}_{1},{\beta}_{3}\right)<IFWAM\left({\beta}_{2},{\beta}_{3}\right)$.

**Example**

**4.**

- (5)
- IFWGM is not always monotone with respect to the SF and AF. In other words, ${\beta}_{1}>{\beta}_{2}$ cannot invariably generate $IFWGM\left({h}_{1},{h}_{3}\right)>IFWGM\left({h}_{2},{h}_{3}\right)$.

**Example**

**5.**

#### 2.2. IFS in the Framework of Dempster-Shafer Theory

**Definition**

**3**

**Definition**

**4**

**.**Given a BPA $\vartheta $ on $\Phi $, the plausibility function Pl can be defined as:

**Definition**

**5**

**.**Let $\Phi $ be a frame of discernment including $\kappa $ mutually exclusive and exhaustive hypothesis, and let ${\Lambda}_{\mathsf{\Gamma}(\Phi )}$ be the space produced by all the subsets of $\Phi $. A BPA is a vector $\overrightarrow{\vartheta}$ of ${\Lambda}_{\mathsf{\Gamma}(\Phi )}$ with coordinates $\vartheta \left({\mathsf{{\rm X}}}_{i}\right)$ such that:

**Definition**

**6**

**.**Let ${\vartheta}_{1}$ and ${\vartheta}_{2}$ be two BPAs on the same frame of discernment $\Phi $, including $\kappa $ mutually exclusive and exhaustive hypotheses. The JD between ${\vartheta}_{1}$ and ${\vartheta}_{2}$ can be defined as follows:

**Definition**

**7.**

**Definition**

**8.**

_{DST}and IFWGM

_{DST}operators in the framework of DST. Let ${\tilde{\beta}}_{i}=\left[{u}_{i},1-{v}_{i}\right]$ be BI and ${w}_{i}$ be the weight of ${\tilde{\beta}}_{i}$, $\sum _{i}^{t}{\omega}_{i}=1$. Then

- (1)
- If $S{F}_{DST}\left({\tilde{\beta}}_{1}\right)<S{F}_{DST}\left({\tilde{\beta}}_{2}\right)$, then ${\tilde{\beta}}_{1}<{\tilde{\beta}}_{2}$;
- (2)
- If $S{F}_{DST}\left({\tilde{\beta}}_{1}\right)=S{F}_{DST}\left({\tilde{\beta}}_{2}\right)$, then
- (i)
- $A{F}_{DST}\left({\tilde{\beta}}_{1}\right)>A{F}_{DST}\left({\tilde{\beta}}_{2}\right)$, then ${\tilde{\beta}}_{1}<{\tilde{\beta}}_{2}$
- (ii)
- $A{F}_{DST}\left({\tilde{\beta}}_{1}\right)=A{F}_{DST}\left({\tilde{\beta}}_{2}\right)$, then ${\tilde{\beta}}_{1}={\tilde{\beta}}_{2}$

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

_{DST}has monotonicity.

**Proof.**

**Theorem**

**5.**

_{DST}has monotonicity.

**Proof.**

#### 2.3. PA Operator

**Definition**

**9**

**.**Let ${\chi}_{\eta}\left(\eta =1,2,\cdots t\right)$ be a set of evaluated values. The PA operator can be defined as follows:

#### 2.4. MSM Operator

**Definition**

**10**

**.**Let ${\chi}_{\eta}\left(\eta =1,2,\cdots ,t\right)$ be a set of positive numbers; then, the MSM operator of ${\chi}_{\eta}\left(\eta =1,2,\cdots ,t\right)$ can be defined as follows:

- (1)
- $MS{M}^{\left(\kappa \right)}\left(0,0,\cdots ,0\right)=0$, $MS{M}^{\left(\kappa \right)}\left(\chi ,\chi ,\cdots ,\chi \right)=\chi $;
- (2)
- $MS{M}^{\left(\kappa \right)}\left({\chi}_{1},{\chi}_{2},\cdots ,{\chi}_{t}\right)\le MS{M}^{\left(\kappa \right)}\left({\chi}_{1}{}^{\prime},{\chi}_{2}{}^{\prime},\cdots ,{\chi}_{t}{}^{\prime}\right)$, if ${\chi}_{\eta}\le {\chi}_{\eta}{}^{\prime}$ for all $\eta $;
- (3)
- $\underset{\eta}{\mathrm{min}}\left\{{\chi}_{\eta}\right\}\le MS{M}^{\left(\kappa \right)}\left({\chi}_{1},{\chi}_{2},\cdots ,{\chi}_{t}\right)\le \underset{\eta}{\mathrm{max}}\left\{{\chi}_{\eta}\right\}$.

## 3. The IFPMSM_{DST} Operators

_{DST}) operator, the IF power weighted average (IFPWA

_{DST}) operator, and the IF MSM (IFMSM

_{DST}) operator in the framework of the DST. Subsequently, we combine the MSM operator with the PA operator and extend them to IFNs to present the IF power MSM (IFPMSM

_{DST}) operator and IF power weighted MSM (IFPWMSM

_{DST}) operator.

#### 3.1. PA Operator for IFNs in the Framework of DST

**Definition**

**11.**

_{DST}operator of ${\tilde{\beta}}_{1},{\tilde{\beta}}_{2},\cdots ,$ and ${\tilde{\beta}}_{t}$ can be defined as follows:

**Theorem**

**6.**

_{DST}operator is also a BI and

**Proof.**

**Definition**

**12.**

_{DST}operator is also a BI and

**Theorem**

**7.**

_{DST}operator is also a BI and

#### 3.2. MSM Operator for IFNs in the Framework of DST

**Definition**

**13.**

_{DST}operator of ${\tilde{\beta}}_{1},{\tilde{\beta}}_{2},\cdots ,$ and ${\tilde{\beta}}_{t}$ can be defined as follows:

**Theorem**

**8.**

_{DST}operator is also a BI and

**Proof.**

#### 3.3. Power MSM Operator for IFNs in the Framework of DST

**Definition**

**14.**

_{DST}operator of ${\tilde{\beta}}_{1},{\tilde{\beta}}_{2},\cdots ,$ and ${\tilde{\beta}}_{t}$ can be defined as follows:

**Theorem**

**9.**

_{DST}operator is also a BI and

**Proof.**

_{DST}operator are proposed.

**Theorem**

**10**

**.**Let $\tilde{B}=\left\{{\tilde{\beta}}_{\eta}|{\tilde{\beta}}_{\eta}=\left[{u}_{\eta},1-{v}_{\eta}\right],\eta =1,2,\cdots ,t\right\}$ be the corresponding BI set of IFS $B=\left\{{\beta}_{\eta}|{\beta}_{\eta}=\langle {u}_{\eta},{v}_{\eta}\rangle ,\eta =1,2,\cdots ,t\right\}$. If ${\tilde{B}}^{\prime}=\left\{{{\tilde{\beta}}^{\prime}}_{\eta}|{{\tilde{\beta}}^{\prime}}_{\eta}=\left[{{u}^{\prime}}_{\eta},1-{{v}^{\prime}}_{\eta}\right],\eta =1,2,\cdots ,t\right\}$ is any permutation of $\tilde{B}=\left\{{\tilde{\beta}}_{\eta}|{\tilde{\beta}}_{\eta}=\left[{u}_{\eta},1-{v}_{\eta}\right],\eta =1,2,\cdots ,t\right\}$, then

**Proof.**

**Theorem**

**11**

**.**Let $\tilde{B}=\left\{{\tilde{\beta}}_{\eta}|{\tilde{\beta}}_{\eta}=\left[{u}_{\eta},1-{v}_{\eta}\right],\text{}\eta =1,2,\cdots ,t\right\}$ be the corresponding BI set of IFS $B=\left\{{\beta}_{\eta}|{\beta}_{\eta}=\langle {u}_{\eta},{v}_{\eta}\rangle ,\text{}\eta =1,2,\cdots ,t\right\}$. Suppose ${\tilde{\beta}}_{\eta}^{+}=\left[\underset{\eta =1}{\overset{t}{\mathrm{max}}}{u}_{\eta},\underset{\eta =1}{\overset{t}{\mathrm{max}}}\left(1-{v}_{\eta}\right)\right]$ and ${\tilde{\beta}}_{\eta}^{-}=\left[\underset{\eta =1}{\overset{t}{\mathrm{min}}}{u}_{\eta},\underset{\eta =1}{\overset{t}{\mathrm{min}}}\left(1-{v}_{\eta}\right)\right]$. Then,

**Proof.**

_{DST}operators do not have an idempotent property. However, if we make a slight modification (multiply by ${\left({C}_{t}^{\kappa}\right)}^{\frac{1}{\kappa}}$), the modified IFPMSM

_{DST}(MIFPMSM

_{DST}) operator will be an idempotent operator, as follows:

_{DST}operator and the MIFPMSM

_{DST}operator in a real DM. In addition, the IFPMSM

_{DST}operator does not satisfy monotonicity because the ${d}_{BPA}$ will also make a difference if the attribute values are changed.

_{DST}operator are investigated by considering some diverse values of $\kappa $.

- (1)
- When $\kappa =1$, the IFPMSM
_{DST}operator become the IFPA_{DST}operator, that is$$\begin{array}{l}IFPMS{M}_{DST}{}^{\left(1\right)}\left({\tilde{\beta}}_{1},{\tilde{\beta}}_{2},\cdots ,{\tilde{\beta}}_{t}\right)=\left[\frac{{\displaystyle \sum _{1\le {\eta}_{1}<\cdots <{\eta}_{\kappa}\le t}{\displaystyle \prod _{j=1}^{\kappa}t{\theta}_{{\eta}_{j}}{u}_{{\eta}_{j}}}}}{{\left({C}_{t}^{1}\right)}^{2}},\frac{{\displaystyle \sum _{1\le {\eta}_{1}<\cdots <{\eta}_{\kappa}\le t}{\displaystyle \prod _{j=1}^{\kappa}t{\theta}_{{\eta}_{j}}\left(1-{v}_{{\eta}_{j}}\right)}}}{{\left({C}_{t}^{\kappa}\right)}^{2}}\right]\\ =\left[\frac{1}{t}{\displaystyle \sum _{\eta =1}^{t}{\theta}_{\eta}{u}_{\eta}},\frac{1}{t}{\displaystyle \sum _{\eta =1}^{t}{\theta}_{\eta}\left(1-{v}_{\eta}\right)}\right]=IFP{A}_{DST}\left({\tilde{\beta}}_{1},{\tilde{\beta}}_{2},\cdots ,{\tilde{\beta}}_{t}\right)\end{array}$$ - (2)
- When $\kappa =2$, that is$$\begin{array}{l}IFPMS{M}_{DST}{}^{\left(2\right)}\left({\tilde{\beta}}_{1},{\tilde{\beta}}_{2},\cdots ,{\tilde{\beta}}_{t}\right)=\left[{\left(\frac{{\displaystyle \sum _{1\le {\eta}_{1}<\cdots <{\eta}_{\kappa}\le t}{\displaystyle \prod _{j=1}^{2}t{\theta}_{{\eta}_{j}}{u}_{{\eta}_{j}}}}}{{\left({C}_{t}^{2}\right)}^{2}}\right)}^{1/2},{\left(\frac{{\displaystyle \sum _{1\le {\eta}_{1}<\cdots <{\eta}_{\kappa}\le t}{\displaystyle \prod _{j=1}^{2}t{\theta}_{{\eta}_{j}}\left(1-{v}_{{\eta}_{j}}\right)}}}{{\left({C}_{t}^{2}\right)}^{2}}\right)}^{1/2}\right]\\ =\left[{\left(\frac{{\displaystyle \sum _{1\le {\eta}_{1}<\cdots <{\eta}_{\kappa}\le t}{t}^{2}{\theta}_{{\eta}_{1}}{u}_{{\eta}_{1}}{\theta}_{{\eta}_{2}}{u}_{{\eta}_{2}}}}{{\left({C}_{t}^{2}\right)}^{2}}\right)}^{1/2},{\left(\frac{{\displaystyle \sum _{1\le {\eta}_{1}<\cdots <{\eta}_{\kappa}\le t}{t}^{2}{\theta}_{{\eta}_{1}}\left(1-{v}_{{\eta}_{1}}\right){\theta}_{{\eta}_{1}}\left(1-{v}_{{\eta}_{1}}\right)}}{{\left({C}_{t}^{2}\right)}^{2}}\right)}^{1/2}\right]\\ =\left[\frac{t}{{C}_{t}^{2}}{\left({\displaystyle \sum _{1\le {\eta}_{1}<\cdots <{\eta}_{\kappa}\le t}{\theta}_{{\eta}_{1}}{\theta}_{{\eta}_{2}}{u}_{{\eta}_{1}}{u}_{{\eta}_{2}}}\right)}^{1/2},\frac{t}{{C}_{t}^{2}}{\left({\displaystyle \sum _{1\le {\eta}_{1}<\cdots <{\eta}_{\kappa}\le t}{\theta}_{{\eta}_{1}}{\theta}_{{\eta}_{1}}\left(1-{v}_{{\eta}_{1}}\right)\left(1-{v}_{{\eta}_{1}}\right)}\right)}^{1/2}\right]\end{array}$$
- (3)
- When $\kappa =n$, that is$$\begin{array}{l}IFPMS{M}_{DST}{}^{\left(t\right)}\left({\tilde{\beta}}_{1},{\tilde{\beta}}_{2},\cdots ,{\tilde{\beta}}_{t}\right)=\left[t{\left(\frac{{\displaystyle \sum _{1\le {\eta}_{1}<\cdots <{\eta}_{t}\le t}{\displaystyle \prod _{j=1}^{t}{\theta}_{{\eta}_{j}}{u}_{{\eta}_{j}}}}}{{\left({C}_{t}^{t}\right)}^{2}}\right)}^{1/t},t{\left(\frac{{\displaystyle \sum _{1\le {\eta}_{1}<\cdots <{\eta}_{t}\le t}{\displaystyle \prod _{j=1}^{t}{\theta}_{{\eta}_{j}}\left(1-{v}_{{\eta}_{j}}\right)}}}{{\left({C}_{t}^{t}\right)}^{2}}\right)}^{1/t}\right]\\ =\left[t{\left({\displaystyle \prod _{j=1}^{t}{\theta}_{{\eta}_{j}}{u}_{{\eta}_{j}}}\right)}^{1/t},t{\left({\displaystyle \prod _{j=1}^{t}{\theta}_{{\eta}_{j}}\left(1-{v}_{{\eta}_{j}}\right)}\right)}^{1/t}\right]\end{array}$$

_{DST}operator has a definite fault, i.e., it does not take the importance of the attributes into account. However, in many practice DM environments, the weights of attributes play a crucial role in the aggregate process. Therefore, we next present the weighted IFPMSM

_{DST}(IFPWMSM

_{DST}) operator.

**Definition**

**15.**

_{DST}operator of ${\tilde{\beta}}_{1},{\tilde{\beta}}_{2},\cdots ,$ and ${\tilde{\beta}}_{t}$ can be defined as follows:

**Theorem**

**12.**

_{DST}operator is also a BI and

**Proof.**

_{DST}operator has also some desirable properties, shown as follows:

**Theorem**

**13**

**.**Let $\tilde{B}=\left\{{\tilde{\beta}}_{\eta}|{\tilde{\beta}}_{\eta}=\left[{u}_{\eta},1-{v}_{\eta}\right],\eta =1,2,\cdots ,t\right\}$ be the corresponding BI set of IFS $B=\left\{{\beta}_{\eta}|{\beta}_{\eta}=\langle {u}_{\eta},{v}_{\eta}\rangle ,\eta =1,2,\cdots ,t\right\}$. If ${\tilde{B}}^{\prime}=\left\{{{\tilde{\beta}}^{\prime}}_{\eta}|{{\tilde{\beta}}^{\prime}}_{\eta}=\left[{{u}^{\prime}}_{\eta},1-{{v}^{\prime}}_{\eta}\right],\eta =1,2,\cdots ,t\right\}$ is any permutation of $\tilde{B}=\left\{{\tilde{\beta}}_{\eta}|{\tilde{\beta}}_{\eta}=\left[{u}_{\eta},1-{v}_{\eta}\right],\eta =1,2,\cdots ,t\right\}$, then

**Theorem**

**14**

**.**Let $\tilde{B}=\left\{{\tilde{\beta}}_{\eta}|{\tilde{\beta}}_{\eta}=\left[{u}_{\eta},1-{v}_{\eta}\right],\text{}\eta =1,2,\cdots ,t\right\}$ be the corresponding BI set of IFS $B=\left\{{\beta}_{\eta}|{\beta}_{\eta}=\langle {u}_{\eta},{v}_{\eta}\rangle ,\eta =1,2,\cdots ,t\right\}$. Suppose ${\tilde{\beta}}_{\eta}^{+}=\left[\underset{\eta =1}{\overset{t}{\mathrm{max}}}{u}_{\eta},\underset{\eta =1}{\overset{t}{\mathrm{max}}}\left(1-{v}_{\eta}\right)\right]$ and ${\tilde{\beta}}_{\eta}^{-}=\left[\underset{\eta =1}{\overset{t}{\mathrm{min}}}{u}_{\eta},\underset{\eta =1}{\overset{t}{\mathrm{min}}}\left(1-{v}_{\eta}\right)\right]$. Then,

_{DST}operators do not have an idempotent property. However, if we make a slight modification (multiply by ${\left({C}_{t}^{\kappa}\right)}^{\frac{1}{\kappa}}$), the modified IFPWMSM

_{DST}(MIFPMSM

_{DST}) operator will be an idempotent operator, as follows:

_{DST}operator where the attribute weights are IFN as follows.

**Definition**

**16.**

_{DST}operator of ${\tilde{\beta}}_{1},{\tilde{\beta}}_{2},\cdots ,$ and ${\tilde{\beta}}_{t}$ can be defined as follows:

**Theorem**

**15.**

_{DST}operator is also a BI and

**Theorem**

**16**

**.**Let $\tilde{B}=\left\{{\tilde{\beta}}_{\eta}|{\tilde{\beta}}_{\eta}=\left[{u}_{\eta},1-{v}_{\eta}\right],\text{}\eta =1,2,\cdots ,t\right\}$ be the corresponding BI set of IFS $B=\left\{{\beta}_{\eta}|{\beta}_{\eta}=\langle {u}_{\eta},{v}_{\eta}\rangle ,\text{}\eta =1,2,\cdots ,t\right\}$. ${\widehat{\omega}}_{i}=\langle {\widehat{u}}_{i},{\widehat{v}}_{i}\rangle $ is the weight of ${\beta}_{i}$ in the form of IFN, ${\widehat{\omega}}_{i}=\left[{\widehat{u}}_{i},1-{\widehat{v}}_{i}\right]$ is the corresponding BI of ${\widehat{\omega}}_{i}$. $\stackrel{=}{\omega}=\left({\stackrel{=}{\omega}}_{1},{\stackrel{=}{\omega}}_{2},\cdots ,{\stackrel{=}{\omega}}_{m}\right)$ is a normalized interval weight vector. If ${\tilde{B}}^{\prime}=\left\{{{\tilde{\beta}}^{\prime}}_{\eta}|{{\tilde{\beta}}^{\prime}}_{\eta}=\left[{{u}^{\prime}}_{\eta},1-{{v}^{\prime}}_{\eta}\right],\eta =1,2,\cdots ,t\right\}$ is any permutation of $\tilde{B}=\left\{{\tilde{\beta}}_{\eta}|{\tilde{\beta}}_{\eta}=\left[{u}_{\eta},1-{v}_{\eta}\right],\eta =1,2,\cdots ,t\right\}$. Then,

**Theorem**

**17**

**.**Let $\tilde{B}=\left\{{\tilde{\beta}}_{\eta}|{\tilde{\beta}}_{\eta}=\left[{u}_{\eta},1-{v}_{\eta}\right],\eta =1,2,\cdots ,t\right\}$ be the corresponding BI set of IFS $B=\left\{{\beta}_{\eta}|{\beta}_{\eta}=\langle {u}_{\eta},{v}_{\eta}\rangle ,\eta =1,2,\cdots ,t\right\}$. ${\widehat{\omega}}_{i}=\langle {\widehat{u}}_{i},{\widehat{v}}_{i}\rangle $ is the weight of ${\beta}_{i}$ in the form of IFN, ${\widehat{\omega}}_{i}=\left[{\widehat{u}}_{i},1-{\widehat{v}}_{i}\right]$ is the corresponding BI of ${\widehat{\omega}}_{i}$. $\stackrel{=}{\omega}=\left({\stackrel{=}{\omega}}_{1},{\stackrel{=}{\omega}}_{2},\cdots ,{\stackrel{=}{\omega}}_{m}\right)$ is a normalized interval weight vector. If ${\tilde{\beta}}_{\eta}^{+}=\left[\underset{\eta =1}{\overset{t}{\mathrm{max}}}{u}_{\eta},\underset{\eta =1}{\overset{t}{\mathrm{max}}}\left(1-{v}_{\eta}\right)\right]$ and ${\tilde{\beta}}_{\eta}^{-}=\left[\underset{\eta =1}{\overset{t}{\mathrm{min}}}{u}_{\eta},\underset{\eta =1}{\overset{t}{\mathrm{min}}}\left(1-{v}_{\eta}\right)\right]$, then

## 4. A Novel MADM Method with IFNs in the Framework of DST

**Step 1.**Normalize the decision matrix $R={\left({r}_{ij}\right)}_{m\times n}$. Only the cost criterion ${c}_{j}$, ${r}_{ij}$ is normalized by using the converted formula (Note: The value converted using ${r}_{ij}=\langle {v}_{ij},{u}_{ij}\rangle $ is still denoted by ${r}_{ij}$).

**Step 2.**Convert IFN ${r}_{ij}$ to BI ${\tilde{\beta}}_{ij}$.

**Step 3.**Calculate $Sup\left({\tilde{\beta}}_{ij},{\tilde{\beta}}_{i\epsilon}\right)\left(i=1,2,\cdots m;j,\epsilon =1,2,\cdots ,n;j\ne \epsilon \right)$, that is,

**Step 4.**Calculate $T\left({\tilde{\beta}}_{ij}\right)$ of ${\tilde{\beta}}_{ij}$ by the other ${\tilde{\beta}}_{i\epsilon}$, that is,

**Step 5.**Calculate ${\overline{\theta}}_{i}$ or ${\stackrel{=}{\theta}}_{i}$. If $\overline{\omega}=\left({\overline{\omega}}_{1},{\overline{\omega}}_{2},\cdots ,{\overline{\omega}}_{n}\right)$ with ${\overline{\omega}}_{i}=\left[{u}_{i},1-{v}_{i}\right]$ for $i=1,2,\cdots ,n$, satisfies

**Step 6.**Apply the proposed ${\mathrm{IFPWMSM}}_{\mathrm{DST}}$ operator or ${\mathrm{IFP}\overline{\mathrm{W}}\mathrm{MSM}}_{\mathrm{DST}}$ operator to acquire the comprehensive value ${\tilde{\beta}}_{i}$ $\left(i=1,2,\cdots m\right)$ of each alternative.

**Step 7.**Calculate the $S{F}_{DST}\left({\tilde{\beta}}_{i}\right)$, $A{F}_{DST}\left({\tilde{\beta}}_{i}\right)$ by Equation (32) and Equation (33), respectively.

**Step 8**. Rank the alternatives and obtain the best alternative.

## 5. Practical Application

**Example**

**6.**

#### 5.1. Rank the Alternatives by the New Method Based on IFWPMSM_{DST} Operator

_{DST}operator.

**Step 1:**Normalize the IFN matrix $R={\left({r}_{ij}\right)}_{m\times n}$.

**Step 2:**Convert IFN ${r}_{ij}$ to BI ${\tilde{\beta}}_{ij}$ and BPAs.

**Step 3:**Calculating $Sup\left({\tilde{\beta}}_{ij},{\tilde{\beta}}_{i\epsilon}\right)\text{}\left(i=1,2,\cdots 4;\text{}j,\epsilon =1,2,\cdots ,4;\text{}j\ne \epsilon \right)$, we have

**Step 4:**Calculate the $T\left({\tilde{\beta}}_{ij}\right)$ of ${\tilde{\beta}}_{ij}$ by the other ${\tilde{\beta}}_{ik}$, that is,

**Step 5.**Calculate ${\overline{\theta}}_{ij}$, and we have

**Step 6:**Apply the proposed ${\mathrm{IFPWMSM}}_{\mathrm{DST}}$ operator shown in Equation (50) to get the comprehensive value ${\tilde{\beta}}_{i}$ $\left(i=1,2,\cdots 4\right)$ of each alternative $\left(\kappa =2\right)$.

**Step 7:**Calculate the $S{F}_{DST}\left({\tilde{\beta}}_{i}\right)$ by Equation (31), and we have

**Step 8:**Rank the alternatives and obtain the best alternative.

#### 5.2. The Influence of the Parameter $\kappa $ on Ranking Results

#### 5.3. The Verification of the Effectiveness

#### 5.4. The Advantages Compared with the Existing Methods

#### 5.4.1. Considering the Interrelationship among Attributes

**Example**

**7.**

#### 5.4.2. Reducing the Influence of Extreme Evaluation Values

**Example**

**8.**

#### 5.4.3. The Attribute Weights can be Denoted by IFNs

**Example**

**9.**

_{DST}operator, and based on Example 7, we initially change $\omega ={\left(0.4,0.1,0.2,0.3\right)}^{T}$ to $\widehat{w}={\left(\langle 0.3,0.3\rangle ,\langle 0.2,0.6\rangle ,\langle 0.1,0.4\rangle ,\langle 0.3,0.5\rangle \right)}^{T}$. The corresponding interval weights of $\widehat{w}$ are $\widehat{w}=\left(\left[0.3,0.7\right],\left[0.2,0.4\right],\left[0.1,0.6\right],\left[0.3,0.5\right]\right)$. From Step 5 in Section 4, we know that $\widehat{w}$ is NIWV. The ranking orders are listed in Table 10.

_{DST}operator can give a ranking order, i.e., ${e}_{3}\succ {e}_{2}\succ {e}_{4}\succ {e}_{1}$. Consequently, the operation laws of the IFS in the framework of DST proposed in this paper can augment the function of AOs.

- (1)
- With regard to method [11], on the one hand, this method does not take into account the interrelationships of attributes. In Example 7, we point out that in some real circumstances, it is meaningful to consider the interrelationships of attributes, but this method can only deal with MADM problems in which attributes are independent of each other. On the other hand, because there are not variable parameters, this method cannot manifest the decision-makers’ subject preference, so it does not apply to some experts with risk attitudes.
- (2)
- With regard to method [28], for one thing, this method cannot reduce the influence of extreme evaluation values. For another, it only considers the interrelationships between attributes, but it cannot capture the interrelationships among attributes.
- (3)
- With regard to method [31], although this method can take into account the interrelationships among attributes, it similarly cannot overcome the drawbacks from the influence of extreme evaluation values.

_{DST}operator, the most obvious limitations of methods [11,28,31] is that they cannot be calculated when attribute weights are IFNs and may obtain some unfavorable ranking orders because of operation laws.

_{DST}operator with existing methods [11,28,31]. This comparison is shown in Table 11. We can see that the presented method based on the IFPWMSM

_{DST}operator is free from the drawbacks of the three existing methods [11,28,31] and is more extensive and flexible in dealing with MADM problems.

## 6. Conclusions

_{DST}operator and an IFPWMSM

_{DST}operator. In addition, we discussed the properties of above two new aggregate operators. Then, we proposed a novel MADM method based on the IFPWMSM

_{DST}operator, which can overcome drawbacks of some existing methods [11,28,31], where they cannot be calculated when attribute weights are IFNs and may obtain some unfavorable ranking orders because of operation laws. Finally, some examples were utilized to demonstrate that the presented methods outperform the previous ones [11,28,31]. In the future, we will apply the IFPWMSM

_{DST}operator to solve multi-attribute group decision-making problems in the framework of the DST. We will also power the MSM operator to aggregate other fuzzy information, such as interval intuitionistic fuzzy sets, hesitant fuzzy sets, and so on. Further, we will use the presented method to deal with some practice MADM problems, such as green supplier selection, disease diagnosis, and so on.

## Author Contributions

## Funding

## Conflicts of Interest

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${c}_{1}$ | ${c}_{2}$ | ${c}_{3}$ | ${c}_{4}$ | |

${e}_{1}$ | $\langle 0.6,0.1\rangle $ | $\langle 0.7,0.3\rangle $ | $\langle 0.7,0.1\rangle $ | $\langle 0.4,0.3\rangle $ |

${e}_{2}$ | $\langle 0.7,0.2\rangle $ | $\langle 0.6,0.1\rangle $ | $\langle 0.5,0.4\rangle $ | $\langle 0.5,0.3\rangle $ |

${e}_{3}$ | $\langle 0.3,0.3\rangle $ | $\langle 0.6,0.2\rangle $ | $\langle 0.7,0.2\rangle $ | $\langle 0.6,0.1\rangle $ |

${e}_{4}$ | $\langle 0.6,0.3\rangle $ | $\langle 0.5,0.2\rangle $ | $\langle 0.4,0.4\rangle $ | $\langle 0.5,0.3\rangle $ |

${c}_{1}$ | ${c}_{2}$ | ${c}_{3}$ | ${c}_{4}$ | |

${b}_{1}$ | $\left[0.6,0.9\right]$ | $\left[0.7,0.7\right]$ | $\left[0.7,0.9\right]$ | $\left[0.4,0.7\right]$ |

${b}_{2}$ | $\left[0.7,0.8\right]$ | $\left[0.6,0.9\right]$ | $\left[0.5,0.6\right]$ | $\left[0.5,0.7\right]$ |

${b}_{3}$ | $\left[0.3,0.7\right]$ | $\left[0.6,0.8\right]$ | $\left[0.7,0.8\right]$ | $\left[0.6,0.9\right]$ |

${b}_{4}$ | $\left[0.6,0.7\right]$ | $\left[0.5,0.8\right]$ | $\left[0.4,0.6\right]$ | $\left[0.5,0.7\right]$ |

$s\left(T\right)$ | $s\left(F\right)$ | $s\left(T\text{}or\text{}F\right)$ | $s\left(T\right)$ | $s\left(F\right)$ | $s\left(T\text{}or\text{}F\right)$ | ||

${f}_{11}$ | 0.6 | 0.1 | 0.3 | ${f}_{31}$ | 0.3 | 0.3 | 0.4 |

${f}_{12}$ | 0.7 | 0.3 | 0 | ${f}_{32}$ | 0.6 | 0.2 | 0.2 |

${f}_{13}$ | 0.7 | 0.1 | 0.2 | ${f}_{33}$ | 0.7 | 0.2 | 0.1 |

${f}_{14}$ | 0.4 | 0.3 | 0.3 | ${f}_{34}$ | 0.6 | 0.1 | 0.3 |

$s\left(T\right)$ | $s\left(F\right)$ | $s\left(T\text{}or\text{}F\right)$ | $s\left(T\right)$ | $s\left(F\right)$ | $s\left(T\text{}or\text{}F\right)$ | ||

${f}_{21}$ | 0.7 | 0.2 | 0.1 | ${f}_{41}$ | 0.6 | 0.3 | 0.1 |

${f}_{22}$ | 0.6 | 0.1 | 0.3 | ${f}_{42}$ | 0.5 | 0.2 | 0.3 |

${f}_{23}$ | 0.5 | 0.4 | 0.1 | ${f}_{43}$ | 0.4 | 0.4 | 0.2 |

${f}_{24}$ | 0.5 | 0.3 | 0.2 | ${f}_{44}$ | 0.5 | 0.3 | 0.2 |

$\kappa $ | $S{F}_{DST}\left({\tilde{\beta}}_{i}\right)$ | Ranking Orders |

$\kappa =1$ | ${\tilde{S}}_{1}=0.1859,{\tilde{S}}_{2}=0.1605,{\tilde{S}}_{3}=0.1553,{\tilde{S}}_{4}=0.1468$ | ${e}_{1}\succ {e}_{2}\succ {e}_{4}\succ {e}_{3}$ |

$\kappa =2$ | ${\tilde{S}}_{1}=0.2173,{\tilde{S}}_{2}=0.1823,{\tilde{S}}_{3}=0.1800,{\tilde{S}}_{4}=0.1485$ | ${e}_{1}\succ {e}_{2}\succ {e}_{3}\succ {e}_{4}$ |

$\kappa =3$ | ${\tilde{S}}_{1}=0.2012,{\tilde{S}}_{2}=0.1760,{\tilde{S}}_{3}=0.1627,{\tilde{S}}_{4}=0.1377$ | ${e}_{1}\succ {e}_{2}\succ {e}_{3}\succ {e}_{4}$ |

$\kappa =4$ | ${\tilde{S}}_{1}=0.1986,{\tilde{S}}_{2}=0.1605,{\tilde{S}}_{3}=0.1613,{\tilde{S}}_{4}=0.1285$ | ${e}_{1}\succ {e}_{3}\succ {e}_{2}\succ {e}_{4}$ |

Methods | Score Values | Ranking Orders |
---|---|---|

Jiang and Wei’s method [11] based on IFEPA operator | ${S}_{1}=0.4629,\text{}{S}_{2}=0.4086,\text{}{S}_{3}=0.3807,\text{}{S}_{4}=0.3518$ | ${e}_{1}\succ {e}_{2}\succ {e}_{4}\succ {e}_{3}$ |

He and He’s method [28] based on EWIFIBM operator | ${S}_{1}=0.2950,\text{}{S}_{2}=0.2749,\text{}{S}_{3}=0.2664,\text{}{S}_{4}=0.2215$ | ${e}_{1}\succ {e}_{3}\succ {e}_{2}\succ {e}_{4}$ |

Qin and Liu’s method [31] based on WIFMSM | ${S}_{1}=0.1994,\text{}{S}_{2}=0.1806,\text{}{S}_{3}=0.1594,\text{}{S}_{4}=0.1447$ | ${e}_{1}\succ {e}_{2}\succ {e}_{3}\succ {e}_{4}$ |

The proposed method based on IFPWMSM_{DST} operator | ${\tilde{S}}_{1}=0.2173,\text{}{\tilde{S}}_{2}=0.1823,\text{}{\tilde{S}}_{3}=0.1800,\text{}{\tilde{S}}_{4}=0.1485$ | ${e}_{1}\succ {e}_{3}\succ {e}_{2}\succ {e}_{4}$ |

${c}_{1}$ | ${c}_{2}$ | ${c}_{3}$ | ${c}_{4}$ | |

${e}_{1}$ | $\langle 0.4,0.3\rangle $ | $\langle 0.5,0.2\rangle $ | $\langle 0.8,0.1\rangle $ | $\langle 0.6,0.1\rangle $ |

${e}_{2}$ | $\langle 0.6,0.3\rangle $ | $\langle 0.5,0.2\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.6,0.2\rangle $ |

${e}_{3}$ | $\langle 0.4,0.3\rangle $ | $\langle 0.5,0.3\rangle $ | $\langle 0.8,0.1\rangle $ | $\langle 0.5,0.2\rangle $ |

${e}_{4}$ | $\langle 0.6,0.3\rangle $ | $\langle 0.6,0.1\rangle $ | $\langle 0.4,0.2\rangle $ | $\langle 0.6,0.2\rangle $ |

Methods | Score Values | Ranking Orders |
---|---|---|

Jiang and Wei’s method [11] based on IFEPA operator | ${S}_{1}=0.2180,{S}_{2}=0.2219,{S}_{3}=0.2309,{S}_{4}=0.1994$ | ${e}_{3}\succ {e}_{2}\succ {e}_{1}\succ {e}_{4}$ |

Qin and Liu’s method [31] based on WIFMSM ($\kappa =2$) | ${S}_{1}=0.1769,{S}_{2}=0.1905,{S}_{3}=0.1976,{S}_{4}=0.1870$ | ${e}_{3}\succ {e}_{2}\succ {e}_{4}\succ {e}_{1}$ |

He and He’s method [28] based on EWIFIBM operator ($\lambda =1,p=1$ and $q=1$) | ${S}_{1}=0.3017,{S}_{2}=0.3428,{S}_{3}=0.3592,{S}_{4}=0.3307$ | ${e}_{3}\succ {e}_{2}\succ {e}_{4}\succ {e}_{1}$ |

The proposed method based on IFPWMSM_{DST} operator ($\kappa =1$) | ${\tilde{S}}_{1}=0.2180,{\tilde{S}}_{2}=0.2286,{\tilde{S}}_{3}=0.2375,{\tilde{S}}_{4}=0.2005$ | ${e}_{3}\succ {e}_{2}\succ {e}_{1}\succ {e}_{4}$ |

The proposed method based on IFPWMSM_{DST} operator ($\kappa =2$) | ${\tilde{S}}_{1}=0.1802,{\tilde{S}}_{2}=0.1973,{\tilde{S}}_{3}=0.2095,{\tilde{S}}_{4}=0.1965$ | ${e}_{3}\succ {e}_{2}\succ {e}_{4}\succ {e}_{1}$ |

The proposed method based on IFPWMSM_{DST} operator ($\kappa =3$) | ${\tilde{S}}_{1}=0.1758,{\tilde{S}}_{2}=0.1906,{\tilde{S}}_{3}=0.1984,{\tilde{S}}_{4}=0.1863$ | ${e}_{3}\succ {e}_{2}\succ {e}_{4}\succ {e}_{1}$ |

${c}_{1}$ | ${c}_{2}$ | ${c}_{3}$ | ${c}_{4}$ | |

${e}_{1}$ | $\langle 0.99,0.01\rangle $ | $\langle 0.5,0.2\rangle $ | $\langle 0.8,0.1\rangle $ | $\langle 0.6,0.1\rangle $ |

${e}_{2}$ | $\langle 0.6,0.3\rangle $ | $\langle 0.5,0.2\rangle $ | $\langle 0.6,0.3\rangle $ | $\langle 0.6,0.2\rangle $ |

${e}_{3}$ | $\langle 0.4,0.3\rangle $ | $\langle 0.5,0.3\rangle $ | $\langle 0.8,0.1\rangle $ | $\langle 0.5,0.2\rangle $ |

${e}_{4}$ | $\langle 0.6,0.3\rangle $ | $\langle 0.6,0.1\rangle $ | $\langle 0.4,0.2\rangle $ | $\langle 0.01,0.01\rangle $ |

Methods | Score Values | Ranking Orders |
---|---|---|

Jiang and Wei’s method [11] based on IFEPA operator | ${S}_{1}=0.2216,\text{}{S}_{2}=0.2219,\text{}{S}_{3}=0.2309,\text{}{S}_{4}=0.1549$ | ${e}_{3}\succ {e}_{2}\succ {e}_{1}\succ {e}_{4}$ |

Qin and Liu’s method [31] based on WIFMSM ($\kappa =2$) | ${S}_{1}=0.2185,\text{}{S}_{2}=0.1905,\text{}{S}_{3}=0.1976,\text{}{S}_{4}=0.1439$ | ${e}_{1}\succ {e}_{3}\succ {e}_{2}\succ {e}_{4}$ |

He and He’s method [28] based on EWIFIBM operator ($\lambda =1,p=1$ and $q=1$) | ${S}_{1}=0.3682,\text{}{S}_{2}=0.3428,\text{}{S}_{3}=0.3592,\text{}{S}_{4}=0.2805$ | ${e}_{1}\succ {e}_{3}\succ {e}_{2}\succ {e}_{4}$ |

The proposed method based on IFPWMSM_{DST} operator ($\kappa =1$) | ${\tilde{S}}_{1}=0.2281,\text{}{\tilde{S}}_{2}=0.2286,\text{}{\tilde{S}}_{3}=0.2375,\text{}{\tilde{S}}_{4}=0.1759$ | ${e}_{3}\succ {e}_{2}\succ {e}_{1}\succ {e}_{4}$ |

The proposed method based on IFPWMSM_{DST} operator ($\kappa =2$) | ${\tilde{S}}_{1}=0.1965,\text{}{\tilde{S}}_{2}=0.1973,\text{}{\tilde{S}}_{3}=0.2095,\text{}{\tilde{S}}_{4}=0.1630$ | ${e}_{3}\succ {e}_{2}\succ {e}_{4}\succ {e}_{1}$ |

The proposed method based on IFPWMSM_{DST} operator ($\kappa =3$) | ${\tilde{S}}_{1}=0.1896,\text{}{\tilde{S}}_{2}=0.1906,\text{}{\tilde{S}}_{3}=0.1984,\text{}{\tilde{S}}_{4}=0.1649$ | ${e}_{3}\succ {e}_{2}\succ {e}_{4}\succ {e}_{1}$ |

Methods | Score Values | Ranking Orders |
---|---|---|

Jiang and Wei’s method [11] based on IFEPA operator | Cannot be counted | Cannot be ranked |

He and He’s method [28] based on EWIFIBM operator | Cannot be counted | Cannot be ranked |

Qin and Liu’s method [31] based on WIFMSM | Cannot be counted | Cannot be ranked |

The proposed method based on IFPWMSM_{DST} operator | ${\tilde{S}}_{1}=0.1904,\text{}{\tilde{S}}_{2}=0.2296,\text{}{\tilde{S}}_{3}=0.2408,\text{}{\tilde{S}}_{4}=0.2075$ | ${e}_{3}\succ {e}_{2}\succ {e}_{4}\succ {e}_{1}$ |

Method | Whether it Eliminates the Effects of Biased Values | Whether it Considers the Interrelationships among Attributes | Whether Attribute Weights Can Be Denoted by IFNs | Whether it Overcomes the Drawbacks of the Ordinary Operation Laws of the IFS |
---|---|---|---|---|

Jiang and Wei’s method [11] based on IFEPA operator | Yes | No | No | No |

He and He’s method [28] based on EWIFIBM operator | No | No | No | No |

Qin and Liu’s method [31] based on WIFMSM | No | Yes | No | No |

The proposed method based on IFPWMSM_{DST} operator | Yes | Yes | Yes | Yes |

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**MDPI and ACS Style**

Gao, H.; Zhang, H.; Liu, P.
Multi-Attribute Decision Making Based on Intuitionistic Fuzzy Power Maclaurin Symmetric Mean Operators in the Framework of Dempster-Shafer Theory. *Symmetry* **2019**, *11*, 807.
https://doi.org/10.3390/sym11060807

**AMA Style**

Gao H, Zhang H, Liu P.
Multi-Attribute Decision Making Based on Intuitionistic Fuzzy Power Maclaurin Symmetric Mean Operators in the Framework of Dempster-Shafer Theory. *Symmetry*. 2019; 11(6):807.
https://doi.org/10.3390/sym11060807

**Chicago/Turabian Style**

Gao, Hui, Hui Zhang, and Peide Liu.
2019. "Multi-Attribute Decision Making Based on Intuitionistic Fuzzy Power Maclaurin Symmetric Mean Operators in the Framework of Dempster-Shafer Theory" *Symmetry* 11, no. 6: 807.
https://doi.org/10.3390/sym11060807