# A Method of Multiple Attribute Group Decision Making Based on 2-Tuple Linguistic Dependent Maclaurin Symmetric Mean Operators

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. The Linguistic Set and 2-tuple

- (1)
- ${s}_{i}<{s}_{j}$ if and only if $i<j$;
- (2)
- $neg({s}_{i})={s}_{l-i-1}$ is a negation operator;
- (3)
- max$({s}_{i},{s}_{j})$ = ${s}_{i}$ is a maximum operator if $i\ge j$;
- (4)
- min$({s}_{i},{s}_{j})$ = ${s}_{j}$ is a minimum operator If $i\ge j$.

**Definition 1**

**([8,27,30]).**

**Definition 2**

**([8,27,30]**

**).**

- (a)
- if ${\alpha}_{1}={\alpha}_{2}$ then $\left({s}_{i},{\alpha}_{1}\right)=\left({s}_{j},{\alpha}_{2}\right)$;
- (b)
- if ${\alpha}_{1}>{\alpha}_{2}$ then $\left({s}_{i},{\alpha}_{1}\right)>\left({s}_{j},{\alpha}_{2}\right)$;
- (c)
- if ${\alpha}_{1}<{\alpha}_{2}$ then $\left({s}_{i},{\alpha}_{1}\right)<\left({s}_{j},{\alpha}_{2}\right)$.

#### 2.2. Maclaurin Symmetric Mean Operator

**Definition 3**

**([14,18]).**

- (1)
- Idempotency. If ${x}_{i}=x$ for each $i$, $MS{M}^{\left(m\right)}\left(x,x,\dots ,x\right)=x$;
- (2)
- Monotonicity. If ${x}_{i}\le {y}_{i}$ for all $i$, $MS{M}^{\left(m\right)}\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)\le MS{M}^{\left(m\right)}\left({y}_{1},{y}_{2},\dots ,{y}_{n}\right)$;
- (3)
- Boundedness. min$\left\{{x}_{1},{x}_{2},\dots ,{x}_{n}\right\}\le MS{M}^{\left(m\right)}\left\{{x}_{1},{x}_{2},\dots ,{x}_{n}\right\}\le $ max$\left\{{x}_{1},{x}_{2},\dots ,{x}_{n}\right\}$.

**Definition 4**

**([18]**

**).**

- (1)
- Idempotency. If ${x}_{i}=x$ for each $i$, and then $GMS{M}^{\left(m,{p}_{1},{p}_{2},\dots ,{p}_{m}\right)}\left(x,x,\dots ,x\right)=x$;
- (2)
- Monotonicity. If ${x}_{i}\le {y}_{i}$ for all $i$,$GMS{M}^{\left(m,{p}_{1},{p}_{2},\dots ,{p}_{m}\right)}\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)\le GMS{M}^{\left(m,{p}_{1},{p}_{2},\dots ,{p}_{m}\right)}\left({y}_{1},{y}_{2},\dots ,{y}_{n}\right)$;
- (3)
- Boundedness. $\mathrm{min}\{{x}_{1},{x}_{2},\dots {x}_{n}\}\le {GMSM}^{\left(m,{p}_{1},{p}_{2},\dots ,{p}_{m}\right)}\left({x}_{1},{x}_{2},\dots {x}_{n}\right)\le \mathrm{max}\{{x}_{1},{x}_{2},\dots {x}_{n}\}$

**Definition**

**5 ([18]).**

- (1)
- Idempotency. If $x>0$, and ${x}_{i}=x$ for each $i$, then ${G}_{eo}MS{M}^{\left(m,{p}_{1},{p}_{2},\dots ,{p}_{m}\right)}\left(x,x,\dots ,x\right)=x$;
- (2)
- Monotonicity. If ${x}_{i}\le {y}_{i}$ for all $i$, ${G}_{eo}MS{M}^{\left(m,{p}_{1},{p}_{2},\dots ,{p}_{m}\right)}\left({x}_{1},{x}_{2}\dots ,{x}_{n}\right)\le {G}_{eo}MS{M}^{\left(m,{p}_{1},{p}_{2},\dots ,{p}_{m}\right)}\left({y}_{1},{y}_{2}\dots ,{y}_{n}\right)$;
- (3)
- Boundedness. $\mathrm{min}\{{x}_{1},{x}_{2},\dots {x}_{n}\}\le {G}_{eo}{MSM}^{\left(m,{p}_{1},{p}_{2},\dots ,{p}_{m}\right)}\left({x}_{1},{x}_{2},\dots {x}_{n}\right)\le \mathrm{max}\{{x}_{1},{x}_{2},\dots {x}_{n}\}$.

## 3. The 2-tuple Linguistic Dependent Weighted Aggregation Maclaurin Symmetric Mean Operators

#### 3.1. The 2-tuple Linguistic MSM Operators

**Definition 6**

**([31]).**

- (1)
- Idempotency. If $\left({r}_{i},{a}_{i}\right)=\left(r,a\right)$, for each $i$, then $2TLMS{M}^{\left(m\right)}\left(\left({r}_{1},{a}_{1}\right),\left({r}_{2},{a}_{2}\right),\dots ,\left({r}_{n},{a}_{n}\right)\right)=\left(r,a\right)$;
- (2)
- Monotonicity. If $\left({r}_{i},{a}_{i}\right)\le \left({r}_{i}^{\u2019},{a}_{i}^{\u2019}\right)$, for all $i$, then $2TLMS{M}^{\left(m\right)}\left(\left({r}_{1},{a}_{1}\right),\left({r}_{2},{a}_{2}\right),\dots ,\left({r}_{n},{a}_{n}\right)\right)\le 2TLMS{M}^{\left(m\right)}\left(\left({r}_{1}^{\u2019},{a}_{1}^{\u2019}\right),\left({r}_{2}^{\u2019},{a}_{2}^{\u2019}\right),\dots ,\left({r}_{n}^{\u2019},{a}_{n}^{\u2019}\right)\right)$;
- (3)
- Boundedness. min $\left({r}_{i},{a}_{i}\right)\le 2TLMS{M}^{\left(m\right)}\left(\left({r}_{1},{a}_{1}\right),\left({r}_{2},{a}_{2}\right),\dots ,\left({r}_{n},{a}_{n}\right)\right)\le $ max $\left({r}_{i},{a}_{i}\right)$;

**Definition 7**

**([31])**.

**Example**

**1.**

**Definition**

**8.**

**Definition**

**9.**

#### 3.2. The 2-tuple Linguistic Dependent Weighted Aggregation Maclaurin Symmetric Mean Operators

**Definition 10**

**([22]**

**).**

**Definition 11**

**([22]**

**).**

**Definition**

**12 ([22,23]**

**).**

- (1)
- Let $x=\left\{\left({r}_{1},{a}_{1}\right),\left({r}_{2},{a}_{2}\right),\dots ,\left({r}_{n},{a}_{n}\right)\right\}$ be a collection of 2-tuples, and let $\left(\overline{r},\overline{a}\right)$ be the arithmetic mean of 2-tuples, $\left(\pi \left(1\right),\pi \left(2\right),\dots ,\pi \left(n\right)\right)$ is a permutation of $\left(1,2,\dots ,n\right)$, such that $\left({r}_{\pi \left(j-1\right)},{a}_{\pi \left(j-1\right)}\right)\ge \left({r}_{\pi \left(j\right)},{a}_{\pi \left(j\right)}\right)$ for all $j=2,\dots ,n$. If $sim\left(({r}_{\pi \left(i\right)},{a}_{\pi \left(i\right)}),\left(\overline{r},\overline{a}\right)\right)\ge sim\left(({r}_{\pi \left(j\right)},{a}_{\pi \left(\mathrm{j}\right)}),\left(\overline{r},\overline{a}\right)\right)$, then ${w}_{i}\ge {w}_{j}$.

**Definition**

**13.**

**Example**

**2.**

- (1)
- Commutativity. Let $\left\{{\left({r}_{1},{a}_{1}\right)}^{\u2019},{\left({r}_{2},{a}_{2}\right)}^{\u2019},\dots ,{\left({r}_{n},{a}_{n}\right)}^{\u2019}\right\}$ be any premutation of $\left\{\left({r}_{1},{a}_{1}\right),\left({r}_{2},{a}_{2}\right),\dots ,\left({r}_{n},{a}_{n}\right)\right\}$, then $2TLDWMS{M}_{}^{\left(m\right)}\left({\left({r}_{1},{a}_{1}\right)}^{\u2019},{\left({r}_{2},{a}_{2}\right)}^{\u2019},\dots ,{\left({r}_{n},{a}_{n}\right)}^{\u2019}\right)=2TLDWMS{M}_{}^{\left(m\right)}\left(\left({r}_{1},{a}_{1}\right),\left({r}_{2},{a}_{2}\right),\dots ,\left({r}_{n},{a}_{n}\right)\right)$

**Proof.**

**Definition**

**14.**

- (1)
- Commutativity. Let $\left\{{\left({r}_{1},{a}_{1}\right)}^{\u2019},{\left({r}_{2},{a}_{2}\right)}^{\u2019},\dots ,{\left({r}_{n},{a}_{n}\right)}^{\u2019}\right\}$ be any premutation of $\left\{\left({r}_{1},{a}_{1}\right),\left({r}_{2},{a}_{2}\right),\dots ,\left({r}_{n},{a}_{n}\right)\right\}$, then $2TLDWGMS{M}_{}^{\left(m\right)}\left({\left({r}_{1},{a}_{1}\right)}^{\u2019},{\left({r}_{2},{a}_{2}\right)}^{\u2019},\dots ,{\left({r}_{n},{a}_{n}\right)}^{\u2019}\right)=2TLDWGMS{M}_{}^{\left(m\right)}\left(\left({r}_{1},{a}_{1}\right),\left({r}_{2},{a}_{2}\right),\dots ,\left({r}_{n},{a}_{n}\right)\right)$.

**Definition**

**15.**

- (1)
- Commutativity. Let $\left\{{\left({r}_{1},{a}_{1}\right)}^{\u2019},{\left({r}_{2},{a}_{2}\right)}^{\u2019},\dots ,{\left({r}_{n},{a}_{n}\right)}^{\u2019}\right\}$ be any premutation of $\left\{\left({r}_{1},{a}_{1}\right),\left({r}_{2},{a}_{2}\right),\dots ,\left({r}_{n},{a}_{n}\right)\right\}$, then $2TLDW{G}_{eo}MS{M}_{}^{\left(m\right)}\left({\left({r}_{1},{a}_{1}\right)}^{\u2019},{\left({r}_{2},{a}_{2}\right)}^{\u2019},\dots ,{\left({r}_{n},{a}_{n}\right)}^{\u2019}\right)=2TLDW{G}_{eo}MS{M}_{}^{\left(m\right)}\left(\left({r}_{1},{a}_{1}\right),\left({r}_{2},{a}_{2}\right),\dots ,\left({r}_{n},{a}_{n}\right)\right)$.Because this property is analogous to the property of $2TLDWMS{M}_{}^{\left(m\right)}$, the proof is omitted here.

## 4. MAGDM Based on 2TLDWMSM Operator or 2TLDWGMSM Operator or 2TLDWGeoMSM Operator

**Step 1.**Normalization

- (1)
- If the type of ${C}_{j}$ is beneficial: ${r}_{{i}_{j}}^{k}={x}_{{i}_{j}}^{k}$
- (2)
- If the type of ${C}_{j}$ is cost: ${r}_{{i}_{j}}^{k}=Neg\left({x}_{{i}_{j}}^{k}\right)$

**Step 2.**Converting initial decision matrix ${R}^{k}={\left[{r}_{{i}_{j}}^{k}\right]}_{m\times n}$ into $R={\left[\left({r}_{{i}_{j}}^{k},0\right)\right]}_{m\times n}$.

**Step 3.**Aggregating all of the decision matrixes $R={\left[\left({r}_{{i}_{j}}^{k},0\right)\right]}_{m\times n}$ $\left(k=1,2,\dots ,p\right)$ into a decision matrix $R={\left[\left({r}_{{i}_{j}},{a}_{{}_{{i}_{j}}}\right)\right]}_{m\times n}$ using 2TLDWMSM operator or 2TLDWGMSM operator or 2TLDWGeoMSM operator.

**Step 4.**Calculating each alternative’s comprehensive evaluation value $({r}_{i},{a}_{i})$ by the 2TLDWMSM operator or 2TLDWGMSM operator or 2TLDWGeoMSM operator.

**Step 5.**Ranking the 2-tuple $({r}_{i},{a}_{i})\left(i=1,2,\dots ,m\right)$ in the light of the comparing of 2-tuple in Section 2.1.

**Step 6.**Sorting alternatives $A=\left\{{A}_{1},{A}_{2},\dots ,{A}_{m}\right\}$ and select the best choice with the highest performance value.

**Step 7**. End.

## 5. Illustrative Example

#### 5.1. Data and Backdrop

#### 5.2. The Method Based on the 2TLDWMSM Operator

**Step 1.**Normalizing the matrices

**Step 2.**Converting initial linguistic information decision matrices ${R}^{k}={\left[{r}_{{i}_{j}}^{k}\right]}_{m\times n}$ given in Table 1, Table 2 and Table 3 into matrices ${R}^{k}={\left[\left({r}_{{i}_{j}}^{k},0\right)\right]}_{m\times n}$ which are given in Table 4, Table 5 and Table 6.

**Step 3.**Aggregating all the decision matrixes ${R}^{k}={\left[\left({r}_{{i}_{j}}^{k},0\right)\right]}_{4\times 3}$ (k = 1,2,3) obtained in step 2 to a decision matrix $R={\left[\left({r}_{{i}_{j}}^{},{a}_{{i}_{j}}\right)\right]}_{4\times 3}$ by 2TLDWMSM, then following matrix can be obtained:

**Step 4.**Calculating each alternative’s comprehensive evaluation value ${r}_{i}=({r}_{i},{a}_{i})$ by 2TLDWMSM, we would get following result:

**Step 5.**Ranking the 2-tuple ${r}_{i}=\left({r}_{i},{a}_{i}\right)\left(i=1,2,3,4\right)$, the following sorted result can be obtained:

**Step 6.**Ranking all the alternatives $A=\left\{{A}_{1},{A}_{2},{A}_{3},{A}_{4}\right\}$ in conformity to ${r}_{i}$, the sorted results are shown below:

_{2}.

#### 5.3. The Method Based on the 2TLDWGMSM Operator

_{1}= 1, p

_{2}= 2 according to Section 4, the procedures of the method are as follows:

**Step 1.**Normalizing the matrices

**Step 2.**Converting initial linguistic information decision matrices ${R}^{k}={\left[{r}_{{i}_{j}}^{k}\right]}_{m\times n}$ given in Table 1, Table 2 and Table 3 into matrices ${R}^{k}={\left[\left({r}_{{i}_{j}}^{k},0\right)\right]}_{m\times n}$ which are displayed in Table 4, Table 5 and Table 6.

**Step 3.**Aggregating all decision matrixes ${R}^{k}={\left[\left({r}_{{i}_{j}}^{k},0\right)\right]}_{4\times 3}$ (k = 1,2,3) obtained in step 2 to the decision matrix $R={\left[\left({r}_{{i}_{j}}^{},{a}_{{i}_{j}}\right)\right]}_{4\times 3}$ by 2TLDWGMSM, then following matrix can be obtained:

**Step 4.**Calculating each alternative’s comprehensive evaluation value ${r}_{i}=({r}_{i},{a}_{i})$ by 2TLDWGMSM, we would get following result:

**Step 5.**Ranking the 2-tuple ${r}_{i}=\left({r}_{i},{a}_{i}\right)\left(i=1,2,3,4\right)$, the following sorted result can be obtained:

**Step 6.**Ranking all the alternatives $A=\left\{{A}_{1},{A}_{2},{A}_{3},{A}_{4}\right\}$ in conformity to ${r}_{i}$, the sorted results are shown below:

_{2}.

#### 5.4. The Method Based on the 2TLDWGeoMSM Operator

_{1}= 1, p

_{2}= 2, according to Section 4, the procedures of the method are as follows:

**Step 1.**Normalizing the matrices

**Step 2.**Converting linguistic decision information matrices ${R}^{k}={\left[{r}_{{i}_{j}}^{k}\right]}_{m\times n}$ given in Table 1, Table 2 and Table 3 into matrices ${R}^{k}={\left[\left({r}_{{i}_{j}}^{k},0\right)\right]}_{m\times n}$ which are displayed in Table 4, Table 5 and Table 6.

**Step 3.**Aggregating all decision matrixes (k = 1,2,3) obtained in step 2 to the decision matrix $R={\left[\left({r}_{{i}_{j}}^{},{a}_{{i}_{j}}\right)\right]}_{4\times 3}$ by 2TLDWGeoMSM, then following matrix can be obtained:

**Step 4.**Calculating each alternative’s comprehensive evaluation value ${r}_{i}=({r}_{i},{a}_{i})$ by 2TLDWGeoMSM, we would get following result:

**Step 5.**Ranking the 2-tuple ${r}_{i}=\left({r}_{i},{a}_{i}\right)\left(i=1,2,3,4\right)$, the following sorted result can be obtained:

**Step 6.**Ranking all the alternatives $A=\left\{{A}_{1},{A}_{2},{A}_{3},{A}_{4}\right\}$ in conformity to ${r}_{i}$, the sorted results are shown below:

_{2}.

#### 5.5. Comparative Analysis and Discussion

_{1}and p

_{2}when m = 2, which are displayed in Table 9. The data in Table 9 indicate the best option is A

_{2}and the worst option is A

_{1}or A

_{3}when the operator is selected. With respect to Table 9, we can discover that when m = 2, we can get the same ranking results. Except when p

_{1}= p

_{2}= 0.5, the result of ranking by 2TLDWGMSM is the same as the 2TLDWMSM when m = 2. Therefore, the results in Table 9 prove the stability of the proposed method.

_{1}, p

_{2}and p

_{3}when m = 3, which are displayed in Table 10. When p

_{1}= 0 or p

_{2}= 0 or p

_{3}= 0, the impact of one of the attributes is not considered. So, the ranking results is different from others when p

_{1}, p

_{2}and p

_{3}are not equal to zero. In addition, when the values of p

_{1}, p

_{2}, and p

_{3}are equal, we can get the ranking by 2TLDWGMSM is the same as the ranking by 2TLDWMSM. At the same time, the sorting results obtained by the 2TLDWGeoMSM are also the same. Furthermore, when two parameters are set the unchanged and equal value, the other parameter is incremented and not equal to the others, we can get two kinds of results based on the 2TLDWGMSM operator and the 2TLDWGeoMSM operator, the ordering in each class is the same. Therefore, the organizations can choose appropriate parameters and operators according to their interests and practical needs.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Options\Attributes | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ |
---|---|---|---|

${A}_{1}$ | ${s}_{5}$ | ${s}_{7}$ | ${s}_{7}$ |

${A}_{2}$ | ${s}_{6}$ | ${s}_{4}$ | ${s}_{5}$ |

${A}_{3}$ | ${s}_{3}$ | ${s}_{4}$ | ${s}_{6}$ |

${A}_{4}$ | ${s}_{6}$ | ${s}_{4}$ | ${s}_{6}$ |

Options\Attributes | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ |
---|---|---|---|

${A}_{1}$ | ${s}_{4}$ | ${s}_{5}$ | ${s}_{4}$ |

${A}_{2}$ | ${s}_{7}$ | ${s}_{6}$ | ${s}_{5}$ |

${A}_{3}$ | ${s}_{4}$ | ${s}_{5}$ | ${s}_{6}$ |

${A}_{4}$ | ${s}_{5}$ | ${s}_{4}$ | ${s}_{5}$ |

Options\Attributes | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ |
---|---|---|---|

${A}_{1}$ | ${s}_{3}$ | ${s}_{5}$ | ${s}_{4}$ |

${A}_{2}$ | ${s}_{7}$ | ${s}_{6}$ | ${s}_{5}$ |

${A}_{3}$ | ${s}_{5}$ | ${s}_{4}$ | ${s}_{7}$ |

${A}_{4}$ | ${s}_{7}$ | ${s}_{6}$ | ${s}_{5}$ |

Options\Attributes | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ |
---|---|---|---|

${A}_{1}$ | $({s}_{5},0)$ | $({s}_{7},0)$ | $({s}_{7},0)$ |

${A}_{2}$ | $({s}_{6},0)$ | $({s}_{4},0)$ | $({s}_{5},0)$ |

${A}_{3}$ | $({s}_{3},0)$ | $({s}_{4},0)$ | $({s}_{6},0)$ |

${A}_{4}$ | $({s}_{6},0)$ | $({s}_{4},0)$ | $({s}_{6},0)$ |

Options\Attributes | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ |
---|---|---|---|

${A}_{1}$ | $({s}_{4},0)$ | $({s}_{5},0)$ | $({s}_{4},0)$ |

${A}_{2}$ | $({s}_{7},0)$ | $({s}_{6},0)$ | $({s}_{5},0)$ |

${A}_{3}$ | $({s}_{4},0)$ | $({s}_{5},0)$ | $({s}_{6},0)$ |

${A}_{4}$ | $({s}_{5},0)$ | $({s}_{4},0)$ | $({s}_{5},0)$ |

Options\Attributes | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ |
---|---|---|---|

${A}_{1}$ | $({s}_{3},0)$ | $({s}_{5},0)$ | $({s}_{4},0)$ |

${A}_{2}$ | $({s}_{7},0)$ | $({s}_{6},0)$ | $({s}_{5},0)$ |

${A}_{3}$ | $({s}_{5},0)$ | $({s}_{4},0)$ | $({s}_{7},0)$ |

${A}_{4}$ | $({s}_{7},0)$ | $({s}_{6},0)$ | $({s}_{5},0)$ |

Proposed Operator | m | p_{1} | p_{2} | Ranking |
---|---|---|---|---|

$2TLDWMS{M}^{\left(m\right)}$ | 2 | - | - | ${A}_{2}>{A}_{4}>{A}_{3}>{A}_{1}$ |

$2TLDWGMS{M}^{(m,{p}_{1},{p}_{2})}$ | 2 | 1 | 2 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

$2TLDW{G}_{eo}MS{M}^{(m,{p}_{1},{p}_{2})}$ | 2 | 1 | 2 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

Methods | Operator | Ranking |
---|---|---|

Methods in this paper | $2TLDWMS{M}^{\left(m\right)}$m = 2 | ${A}_{2}>{A}_{4}>{A}_{3}>{A}_{1}$ |

Methods in this paper | $2TLDWGMS{M}^{(m,{p}_{1},{p}_{2})}$p_{1} = 1 p_{2} = 2 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

Methods in this paper | $2TLDW{G}_{eo}MS{M}^{(m,{p}_{1},{p}_{2})}$p_{1} = 1 p_{2} = 2 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

Method in [27] | $2TLH{M}^{(m,{p}_{1},{p}_{2})}$p_{1} = 1 p_{2} = 2 | ${A}_{2}>{A}_{4}>{A}_{3}>{A}_{1}$ |

Method in [34] | $2TLB{M}^{(m,{p}_{1},{p}_{2})}$p_{1} = 1 p_{2} = 2 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

p_{1} | p_{2} | $\mathbf{Ranking}\text{}\mathbf{by}\text{}2\mathit{T}\mathit{L}\mathit{D}\mathit{W}\mathit{G}\mathit{M}\mathit{S}{\mathit{M}}^{(\mathit{m},{\mathit{p}}_{1},{\mathit{p}}_{2})}$ | $\mathbf{Ranking}\text{}\mathbf{by}\text{}2\mathit{T}\mathit{L}\mathit{D}\mathit{W}{\mathit{G}}_{\mathit{e}\mathit{o}}\mathit{M}\mathit{S}{\mathit{M}}^{(\mathit{m},{\mathit{p}}_{1},{\mathit{p}}_{2})}$ |
---|---|---|---|

0.5 | 0.5 | ${A}_{2}>{A}_{4}>{A}_{3}>{A}_{1}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

1 | 0 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

0 | 1 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

1 | 2 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

1 | 3 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

2 | 1 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

2 | 2 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

2 | 3 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

3 | 1 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

3 | 2 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

3 | 3 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

4 | 4 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

5 | 5 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

p_{1} | p_{2} | p_{3} | $\mathbf{Ranking}\text{}\mathbf{by}\text{}2\mathit{T}\mathit{L}\mathit{D}\mathit{W}\mathit{G}\mathit{M}\mathit{S}{\mathit{M}}^{(\mathit{m},{\mathit{p}}_{1},{\mathit{p}}_{2})}$ | $\mathbf{Ranking}\text{}\mathbf{by}\text{}2\mathit{T}\mathit{L}\mathit{D}\mathit{W}{\mathit{G}}_{\mathit{e}\mathit{o}}\mathit{M}\mathit{S}{\mathit{M}}^{(\mathit{m},{\mathit{p}}_{1},{\mathit{p}}_{2})}$ |
---|---|---|---|---|

0.5 | 0.5 | 0.5 | ${A}_{2}>{A}_{4}>{A}_{3}>{A}_{1}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

1 | 1 | 0 | ${A}_{4}>{A}_{2}>{A}_{1}>{A}_{3}$ | ${A}_{4}>{A}_{2}>{A}_{1}>{A}_{3}$ |

0 | 1 | 0 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{1}>{A}_{4}>{A}_{3}$ |

0 | 0 | 1 | ${A}_{1}>{A}_{4}>{A}_{3}>{A}_{2}$ | ${A}_{1}>{A}_{3}>{A}_{4}>{A}_{2}$ |

1 | 1 | 2 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{1}>{A}_{4}>{A}_{2}>{A}_{3}$ |

1 | 1 | 3 | ${A}_{4}>{A}_{2}>{A}_{1}>{A}_{3}$ | ${A}_{1}>{A}_{4}>{A}_{2}>{A}_{3}$ |

1 | 1 | 4 | ${A}_{4}>{A}_{1}>{A}_{2}>{A}_{3}$ | ${A}_{1}>{A}_{4}>{A}_{3}>{A}_{2}$ |

1 | 1 | 5 | ${A}_{4}>{A}_{1}>{A}_{2}>{A}_{3}$ | ${A}_{1}>{A}_{3}>{A}_{4}>{A}_{2}$ |

1 | 2 | 1 | ${A}_{2}>{A}_{3}>{A}_{1}>{A}_{4}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

1 | 3 | 1 | ${A}_{2}>{A}_{1}>{A}_{3}>{A}_{4}$ | ${A}_{2}>{A}_{1}>{A}_{3}>{A}_{4}$ |

1 | 4 | 1 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{1}>{A}_{4}>{A}_{3}$ |

1 | 5 | 1 | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{1}>{A}_{4}>{A}_{3}$ |

2 | 1 | 1 | ${A}_{4}>{A}_{2}>{A}_{1}>{A}_{3}$ | ${A}_{4}>{A}_{3}>{A}_{2}>{A}_{1}$ |

3 | 1 | 1 | ${A}_{4}>{A}_{2}>{A}_{1}>{A}_{3}$ | ${A}_{4}>{A}_{3}>{A}_{2}>{A}_{1}$ |

4 | 1 | 1 | ${A}_{4}>{A}_{2}>{A}_{1}>{A}_{3}$ | ${A}_{4}>{A}_{3}>{A}_{2}>{A}_{1}$ |

5 | 1 | 1 | ${A}_{4}>{A}_{2}>{A}_{1}>{A}_{3}$ | ${A}_{2}>{A}_{4}>{A}_{3}>{A}_{1}$ |

1 | 1 | 1 | ${A}_{2}>{A}_{4}>{A}_{3}>{A}_{1}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

2 | 2 | 2 | ${A}_{2}>{A}_{4}>{A}_{3}>{A}_{1}$ | ${A}_{2}>{A}_{4}>{A}_{1}>{A}_{3}$ |

3 | 3 | 3 | ${A}_{2}>{A}_{4}>{A}_{3}>{A}_{1}$ | ${A}_{2}>{A}_{4}>{A}_{3}>{A}_{1}$ |

4 | 4 | 4 | ${A}_{2}>{A}_{4}>{A}_{3}>{A}_{1}$ | ${A}_{2}>{A}_{4}>{A}_{3}>{A}_{1}$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Feng, M.; Liu, P.; Geng, Y.
A Method of Multiple Attribute Group Decision Making Based on 2-Tuple Linguistic Dependent Maclaurin Symmetric Mean Operators. *Symmetry* **2019**, *11*, 31.
https://doi.org/10.3390/sym11010031

**AMA Style**

Feng M, Liu P, Geng Y.
A Method of Multiple Attribute Group Decision Making Based on 2-Tuple Linguistic Dependent Maclaurin Symmetric Mean Operators. *Symmetry*. 2019; 11(1):31.
https://doi.org/10.3390/sym11010031

**Chicago/Turabian Style**

Feng, Min, Peide Liu, and Yushui Geng.
2019. "A Method of Multiple Attribute Group Decision Making Based on 2-Tuple Linguistic Dependent Maclaurin Symmetric Mean Operators" *Symmetry* 11, no. 1: 31.
https://doi.org/10.3390/sym11010031