# Extension of the TODIM Method to Intuitionistic Linguistic Multiple Attribute Decision Making

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. The Intuitionistic Fuzzy Set (IFS)

**Definition**

**1.**

#### 2.2. The Linguistic Set and Uncertain Linguistic Set

**Definition**

**2.**

**Definition**

**3.**

#### 2.3. The Intuitionistic Linguistic Set (ILS) and Intuitionistic Linguistic Number (ILN)

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

- 1.
- $m\oplus n=\langle {h}_{\mathsf{\theta}(m)+\mathsf{\theta}(n)},\frac{\mathsf{\theta}\left(m\right)\mathsf{\mu}\left(m\right)+\mathsf{\theta}\left(n\right)\mathsf{\mu}\left(n\right)}{\mathsf{\theta}\left(m\right)+\mathsf{\theta}\left(n\right)},\frac{\mathsf{\theta}\left(m\right)v\left(m\right)+\mathsf{\theta}\left(n\right)v\left(n\right)}{\mathsf{\theta}\left(m\right)+\mathsf{\theta}\left(n\right)}\rangle \text{\hspace{0.05em}}$
- 2.
- $m\otimes n=\langle {h}_{\mathsf{\theta}(m)\mathsf{\theta}(n)},\mathsf{\mu}\left(m\right)\mathsf{\mu}\left(n\right),v\left(m\right)+v\left(n\right)\rangle \text{\hspace{0.05em}}$
- 3.
- $\mathsf{\lambda}m=\langle {h}_{\mathsf{\lambda}\mathsf{\theta}(m)},\mathsf{\mu}\left(m\right),v\left(m\right)\rangle \text{\hspace{0.05em}},\text{\hspace{0.05em}}\mathsf{\lambda}\ge 0\text{\hspace{0.05em}}$
- 4.
- ${m}^{\mathsf{\lambda}}=\langle {h}_{\mathsf{\theta}{(m)}^{\mathsf{\lambda}}},\mathsf{\mu}{\left(m\right)}^{\mathsf{\lambda}},1-{\left(1-v\left(m\right)\right)}^{\mathsf{\lambda}}\rangle \text{\hspace{0.05em}},\text{\hspace{0.05em}}\mathsf{\lambda}\ge 0\text{\hspace{0.05em}}$

**Definition**

**7.**

#### 2.4. The Intuitionistic Uncertain Linguistic Set (IULS) and Intuitionistic Uncertain Linguistic Number (IULN)

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

- 1.
- $\tilde{m}\oplus \tilde{n}=\langle \left[{s}_{\mathsf{\theta}(m)+\mathsf{\theta}(n)},{s}_{\mathsf{\tau}(m)+\mathsf{\tau}(n)}\right],\left(1-\left(1-\mathsf{\mu}(m)\right)\left(1-\mathsf{\mu}(n)\right),v(m)v(n)\right)\rangle $
- 2.
- $\tilde{m}\otimes \tilde{n}=\langle \left[{s}_{\mathsf{\theta}(m)\times \mathsf{\theta}(n)},{s}_{\mathsf{\tau}(m)\times \mathsf{\tau}(n)}\right],\left(\mathsf{\mu}(m)\mathsf{\mu}(n),v(m)+v(n)-v(m)v(n)\right)\rangle $
- 3.
- $\mathsf{\lambda}\tilde{m}=\langle \left[{s}_{\mathsf{\lambda}\times \mathsf{\theta}(m)},{s}_{\mathsf{\lambda}\times \mathsf{\tau}(m)}\right],\left(1-{\left(1-\mathsf{\mu}(m)\right)}^{\mathsf{\lambda}},{\left(v(m)\right)}^{\mathsf{\lambda}}\right)\rangle \text{\hspace{0.05em}},\mathsf{\lambda}\ge 0$
- 4.
- ${\tilde{m}}^{\mathsf{\lambda}}=\langle \left[{s}_{{\left(\mathsf{\theta}(m)\right)}^{\mathsf{\lambda}}},{s}_{{\left(\mathsf{\tau}(m)\right)}^{\mathsf{\lambda}}}\right],\left({\left(\mathsf{\mu}(m)\right)}^{\mathsf{\lambda}},1-{\left(1-v(m)\right)}^{\mathsf{\lambda}}\right)\rangle \text{\hspace{0.05em}},\mathsf{\lambda}\ge 0$

**Definition**

**11.**

#### 2.5. The Classical TODIM Method

**Step 1:**Define the decision matrix $X=\left[{x}_{ic}\right]n\times m$, which are the evaluations of alternatives ${A}_{i}$ according to criterion ${C}_{c}$. ${x}_{ic}$ is a crisp number, i = 1, …, n, c = 1, …, m. n and m represent the number of alternatives and the number of criteria, respectively

**Step 2:**Normalize the decision matrix $X=\left[{x}_{ic}\right]n\times m$ into $Y=\left[{y}_{ic}\right]n\times m$.

**Step 3:**Let $w=\left({w}_{1},{w}_{2},\cdots ,{w}_{m}\right)$ be the weight vector of the criteria ${C}_{1},{C}_{2},\cdots ,{C}_{m}$, where $0\le {w}_{i}\le 1$ and ${{\displaystyle \Sigma}}_{i=1}^{m}{w}_{i}=1$. It is necessary that the DM defines a reference criterion ${C}_{r}$, $1\le r\le m$, usually the reference criterion with the highest weight. Calculate relative weight ${w}_{rc}={w}_{\mathrm{c}}/{w}_{r}$, where ${C}_{c}$ is a generic criterion.

**Step 4:**Calculate the dominance of ${A}_{i}$ over ${A}_{j}$ using the following expression. The term ${\mathsf{\phi}}_{c}\left({A}_{i},{A}_{j}\right)$ represents the partial dominance. θ is the attenuation factor of the losses, and the choice of θ has an influence on the shape of the prospect value function.

**Step 5:**Normalize the dominance measurements

**Step 6:**Sort the alternatives according to the value ${\mathsf{\epsilon}}_{i}$.

## 3. IL-TODIM—An Intuitionistic Linguistic TODIM Method

**Step 1:**The criteria are normally classified into two types: benefit criteria and cost criteria, and the DM needs to evaluate the alternatives ${A}_{i\text{\hspace{0.05em}}}\left(i=1,\dots ,n\right)$ with respect to criteria ${C}_{c\text{\hspace{0.05em}}}\left(c=1,\dots ,m\right)$. Each evaluation value can be expressed by IL variable $x=\langle {h}_{\mathsf{\theta}(x)},\left(\mathsf{\mu}(x),v(x)\right)\rangle $. Then we can obtain the IL decision matrix $X=\left[{x}_{ic}\right]n\times m$ with i = 1, …, n, and c = 1, …, m, where $x=\langle {h}_{\mathsf{\theta}(x)},\left(\mathsf{\mu}(x),v(x)\right)\rangle $. Firstly, the IL decision matrix X should be normalized into $R=\left[{r}_{ic}\right]n\times m$, where $r=\langle {r}_{\mathsf{\theta}(x)},\left(\mathsf{\mu}(x),v(x)\right)\rangle $. The linguistic index ${r}_{\mathsf{\theta}(x)}$ of the normalized value ${r}_{ic}$ is calculated as

**Step 2:**Calculate the dominance degree of alternative ${A}_{i}$ over ${A}_{j}$ using the expression

**Step 3:**Sort the alternatives ${A}_{i}$ by the normalized values ${\mathsf{\epsilon}}_{i}$, calculated by Equation (5). The best alternative is the one which has the highest value ${\mathsf{\epsilon}}_{i}$.

## 4. IUL-TODIM—An Intuitionistic Uncertain Linguistic TODIM Method

**Step 1:**Normalize the IUL decision matrix $\tilde{X}=\left[{\tilde{x}}_{ic}\right]n\times m$ with i = 1, …, n and c = 1, …, m, where $\tilde{x}=\langle [{s}_{\mathsf{\theta}(x)},{s}_{\mathsf{\tau}(x)}],\left(\mathsf{\mu}(x),v(x)\right)\rangle $. The normalized uncertain linguistic index $\left[{r}_{\mathsf{\theta}(x)},{r}_{\mathsf{\tau}(x)}\right]$ is calculated as

**Step 2:**Calculate the dominance degree of alternative ${A}_{i}$ over ${A}_{j}$

**Step 3:**Sort the alternatives ${A}_{i}$ by the normalized values ${\mathsf{\epsilon}}_{i}$, calculated by Equation (5).

## 5. Numerical Example

#### 5.1. The IL-TODIM Method Decision Process and Results

_{2}(growth index) and C

_{3}(social-political impact) are the benefit criteria, while C

_{1}(risk index) and C

_{4}(environmental impact) are the cost criteria. The attenuation factor of losses θ is set to 1, which means the loss contributes with its real value to the global value.

_{2}(a computer company) followed by A

_{1}(a car company), and the last choice is A

_{3}(a TV company). In order to illustrate the influence of the parameter θ on decision-making of this example, we change the values of θ from 1 to 5 and the increment is 1. The ranking orders of the four alternatives obtained by applying the IL-TODIM method with different values of θ are always ${\mathrm{{\rm A}}}_{2}\succ {\mathrm{{\rm A}}}_{1}\succ {\mathrm{{\rm A}}}_{4}\succ {\mathrm{{\rm A}}}_{3}$. In spite of increasing the attenuation factor of the losses from 1 to 5, the preferences are maintained, with the ranking orders not suffering any alteration either, and it indicates the robustness of the computational results based on the DM’s preferences.

#### 5.2. The IUL-TODIM Method Decision Process and Results

_{2}is always the best choice according to the DM’s preference, and the only change is in the ranking order of A

_{3}and A

_{4}.

#### 5.3. Discussion

#### 5.3.1. A Comparison between IL-TODIM and Intuitionistic Fuzzy TODIM (IF-TODIM)

_{1}and A

_{4}, and the discrepancy in the performance of the two alternatives, which is calculated by the IF-TODIM method, is only 0.002. We cannot exactly say that alternative A

_{4}is better than A

_{1}because of the approximate number processing. In the actual decision-making, trapezoidal fuzzy numbers cannot be given directly for the evaluation of alternatives, so there may be some errors existing in the process of transforming the linguistic variables to the trapezoidal fuzzy numbers. The linguistic variables can express fuzzy information more directly so that the IL-TODIM method can decrease the computational complexity, and has the advantages of simplicity and reliability.

#### 5.3.2. A Comparison between IL-TODIM and Other Intuitionistic Linguistic MADM Methods

_{i}) (i = 1, 2, 3, 4) is calculated by the new score function proposed in [32], and we have h(x

_{1}) = 1.4368, h(x

_{2}) = 1.6387, h(x

_{3}) = 1.1126, and h(x

_{4}) = 1.3815. We obtain that $h\left({x}_{2}\right)>h({x}_{1})>h({x}_{4})>h({x}_{3})$, then ${\mathrm{{\rm A}}}_{2}\succ {\mathrm{{\rm A}}}_{1}\succ {\mathrm{{\rm A}}}_{4}\succ {\mathrm{{\rm A}}}_{3}$.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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A_{1} | A_{2} | A_{3} | A_{4} | |
---|---|---|---|---|

A_{1} | 0.000 | ‒1.618 | ‒0.839 | ‒0.215 |

A_{2} | ‒0.694 | 0.000 | 0.307 | ‒0.893 |

A_{3} | ‒0.941 | ‒1.298 | 0.000 | ‒1.169 |

A_{4} | ‒1.079 | ‒1.310 | ‒0.736 | 0.000 |

Ranking | Alternatives | Performance | |
---|---|---|---|

Gross | Normalized | ||

1 | A_{2} | ‒1.28 | 1.000 |

2 | A_{1} | ‒2.672 | 0.346 |

3 | A_{4} | ‒3.125 | 0.213 |

4 | A_{3} | ‒3.408 | 0.000 |

A_{1} | A_{2} | A_{3} | A_{4} | |
---|---|---|---|---|

A_{1} | 0.000 | 0.2225–1.5387/θ | 0.2275–0.9926/θ | 0.2841–0.4114/θ |

A_{2} | 0.405–1.236/θ | 0.000 | 0.1888–0.1614/θ | 0.3616–1.356/θ |

A_{3} | 0.2775–1.2638/θ | 0.0516–0.8528/θ | 0.000 | 0.258–1.3417/θ |

A_{4} | 0.1317–1.3055/θ | 0.2863–1.4384/θ | 0.2772–1.0067/θ | 0.000 |

θ = 1 | θ = 2 | θ = 3 | θ = 4 | θ = 5 | |
---|---|---|---|---|---|

A_{1} | ‒2.209 | ‒0.737 | ‒0.247 | ‒0.002 | 0.146 |

A_{2} | ‒1.798 | ‒0.421 | 0.038 | 0.267 | 0.415 |

A_{3} | ‒2.871 | ‒1.142 | ‒0.566 | ‒0.277 | ‒0.277 |

A_{4} | ‒3.055 | ‒1.180 | ‒0.555 | ‒0.242 | ‒0.055 |

Different Values of θ | Ranking Results |
---|---|

θ = 1 | ${\mathrm{{\rm A}}}_{2}\succ {\mathrm{{\rm A}}}_{1}\succ {\mathrm{{\rm A}}}_{3}\succ {\mathrm{{\rm A}}}_{4}$ |

θ = 2 | ${\mathrm{{\rm A}}}_{2}\succ {\mathrm{{\rm A}}}_{1}\succ {\mathrm{{\rm A}}}_{3}\succ {\mathrm{{\rm A}}}_{4}$ |

θ = 3 | ${\mathrm{{\rm A}}}_{2}\succ {\mathrm{{\rm A}}}_{1}\succ {\mathrm{{\rm A}}}_{4}\succ {\mathrm{{\rm A}}}_{3}$ |

θ = 4 | ${\mathrm{{\rm A}}}_{2}\succ {\mathrm{{\rm A}}}_{1}\succ {\mathrm{{\rm A}}}_{4}\succ {\mathrm{{\rm A}}}_{3}$ |

θ = 5 | ${\mathrm{{\rm A}}}_{2}\succ {\mathrm{{\rm A}}}_{1}\succ {\mathrm{{\rm A}}}_{4}\succ {\mathrm{{\rm A}}}_{3}$ |

Linguistic Labels | Linguistic Terms | Trapezoidal Fuzzy Numbers |
---|---|---|

S_{0} | Very Poor | (0,0,0,1) |

S_{1} | Poor | (0,1,2,3) |

S_{2} | Slightly Poor | (2,3,4,5) |

S_{3} | Fair | (4,5,6,7) |

S_{4} | Slightly Good | (6,7,8,9) |

S_{5} | Good | (8,9,10,11) |

S_{6} | Very Good | (10,11,11,11) |

Alternatives | IL-TODIM | IF-TODIM | ||
---|---|---|---|---|

Normalized Performance | Ranking Orders | Normalized Performance | Ranking Orders | |

A_{1} | 0.346 | 2 | 0.035 | 3 |

A_{2} | 1.000 | 1 | 1.000 | 1 |

A_{3} | 0.000 | 4 | 0.000 | 4 |

A_{4} | 0.213 | 3 | 0.037 | 2 |

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**MDPI and ACS Style**

Wang, S.; Liu, J.
Extension of the TODIM Method to Intuitionistic Linguistic Multiple Attribute Decision Making. *Symmetry* **2017**, *9*, 95.
https://doi.org/10.3390/sym9060095

**AMA Style**

Wang S, Liu J.
Extension of the TODIM Method to Intuitionistic Linguistic Multiple Attribute Decision Making. *Symmetry*. 2017; 9(6):95.
https://doi.org/10.3390/sym9060095

**Chicago/Turabian Style**

Wang, Shuwei, and Jia Liu.
2017. "Extension of the TODIM Method to Intuitionistic Linguistic Multiple Attribute Decision Making" *Symmetry* 9, no. 6: 95.
https://doi.org/10.3390/sym9060095