# A New Fuzzy Multiple Attribute Decision Making Method Based on the Utility Transformation Functions

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- By introducing the economic utility theory, the definition of utility and the 2TLIM are combined to be the 2-tuple linguistic utility. Obviously, the 2-tuple linguistic utility is decided by the decision-makers’ attitude, the specific value can be indicated by the change of 2-tuple linguistic marginal utility (2TLMU). Then the 2TLUP is developed to measure the decision-makers’ attitude.
- (2)
- The utility transformation function are presented on the 2-tuple linguistic utility, the decision-makers’ attitude can be controlled by the value of 2TLUP, then the difference between the linguistic information can be changed with the decision-makers’ attitude.
- (3)
- Because the operational laws based on the TN and TC are closed, the 2-tuple linguistic operational laws are constructed with the utility transformation function and the extended Hamacher TN and TC. Ultimately, the 2-tuple linguistic utility weighted average (2TLUWA)operators are produced to realize the aggregation of the 2-tuple linguistic information.
- (4)
- The method of MADM with the 2-tuple linguistic information is presented with the specific step, and an application example is given to certify the availability of the utility transformation function.

## 2. Preliminaries

#### 2.1. LTSs

**Definition**

**1.**

#### 2.2. 2-Tuple Linguistic Information Model

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**1.**

- (1)
- if $k<l$, then $({s}_{k},{\alpha}_{k})<({s}_{l},{\alpha}_{l})$;
- (2)
- if $k=l$,
- when ${\alpha}_{k}={\alpha}_{l}$, then $({s}_{k},{\alpha}_{k})=({s}_{l},{\alpha}_{l})$;
- when ${\alpha}_{k}<{\alpha}_{l}$, then $({s}_{k},{\alpha}_{k})<({s}_{l},{\alpha}_{l})$;
- when ${\alpha}_{k}>{\alpha}_{l}$, then $({s}_{k},{\alpha}_{k})>({s}_{l},{\alpha}_{l})$.

#### 2.3. The Extended TN and TC

**Definition**

**5.**

**Theorem**

**2.**

**Proof.**

#### 2.4. The Economic Utility Theory

## 3. The Utility Transformation Function Based on 2-Tuple Linguistic Utility

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Example**

**1.**

**Case**

**1.**

**Case**

**2.**

**Case**

**3.**

## 4. The 2-Tuple Linguistic Operational Laws Based on the Utility Transformation Function

**Definition**

**10.**

**Proof.**

**Definition**

**11.**

**Example**

**2.**

**Definition**

**12.**

**Example**

**3.**

**Definition**

**13.**

**Example**

**4.**

## 5. The 2TLUWA Operator and the MADM Method

#### 5.1. The 2TLUWA Operator

**Definition**

**14.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

#### 5.2. The MADM Method Based on the Utility Transformation Function

**Step 1:**Normalize the evaluation value. Convert the cost type into the benefit type, the specific conversation method is listed:

**Step 2:**For each decision maker ${P}_{k}$, the corresponding 2TLUP ${\rho}_{k}$ is matched. The novel 2-tuple linguistic transformation functions ${u}_{k},{u}_{k}^{-1}$ are constructed for decision maker ${P}_{k}$ as follows:

**Step 3:**According to the 2TLUWA operator, it is easy to aggregate the attributes of the alternative, then $\tilde{D}={({s}_{\tilde{{a}_{i}^{k}}},\tilde{{\alpha}_{i}^{k}})}_{n\times q}$ is obtained:

**Step 4:**According to the 2TLUWA operator, the aggregation value of alternative $({s}_{{x}_{i}},{\alpha}_{{x}_{i}})$ is obtained:

**Step 5:**Rank the aggregation value of alternative ${R}_{{x}_{i}}$, then the best alternative is picked by the biggest ${R}_{{x}_{i}}$.

## 6. An Application Example

**Step 1:**Normalize the evaluation matrices, the attributes are both benefit type, so we skip the step.

**Step 2:**For the decision maker ${P}_{k}$ with the 2TLUP ${\rho}_{k}$, then the corresponding utility transformation functions could be constructed by the Equation (21).

**Step 3:**Aggregate the attributes of the alternative by the 2TLUAWA operator, then we can obtain $\tilde{D}={({s}_{\tilde{{a}_{i}^{k}}},\tilde{{\alpha}_{i}^{k}})}_{n\times q}$, which is shown in Table 5.

**Step 4:**Adopt the Equation (19) to get the aggregated $\rho $ and apply the 2TLUAWA operator to obtain the final aggregation value $R({x}_{i})$, it is easy to get:

**Step 5:**Rank the final aggregation value, we can get the best alternative.

#### The Discussion about the 2TLUP

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Name | The Extended TN and TC | Generated Function |
---|---|---|

Algebraic | ${T}_{A}(k,l)=\frac{kl}{t}$ | ${\phi}_{A}(k)=-ln\frac{k}{t}$ |

${S}_{A}(k,l)=(k+l)-\frac{kl}{t}$ | ${\varphi}_{A}(k)=-ln\frac{t-k}{t}$ | |

Einstein | ${T}_{E}(k,l)=\frac{2tkl}{kl+(2t-k)(2t-l)}$ | ${\phi}_{E}(k)=ln\frac{2t-k}{k}$ |

${S}_{E}(k,l)=\frac{{t}^{2}(k+l)}{{t}^{2}+kl}$ | ${\varphi}_{E}(k)=ln\frac{t+k}{t-k}$ | |

Hamacher | ${T}_{H}(k,l)=\frac{t\gamma kl}{(t\gamma +(1-\gamma )k)(t\gamma +(1-\gamma )l)+(\gamma -1)kl}$ | ${\phi}_{H}(k)=ln\frac{t\gamma +(1-\gamma )k}{k}$ |

${S}_{H}(k,l)=\frac{t(t\gamma (k+l)+\gamma (\gamma -2)kl)}{\gamma {t}^{2}+\gamma (\gamma -1)kl}$ | ${\varphi}_{H}(k)=ln\frac{t+(\gamma -1)k}{t-k}$ |

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

${x}_{1}$ | $({s}_{3},-0.1)$ | $({s}_{4},0.1)$ | $({s}_{4},-0.2)$ | $({s}_{5},0.2)$ |

${x}_{2}$ | $({s}_{3},-0.2)$ | $({s}_{4},-0.3)$ | $({s}_{4},0.3)$ | $({s}_{5},-0.4)$ |

${x}_{3}$ | $({s}_{4},0.2)$ | $({s}_{4},-0.4)$ | $({s}_{4},0.4)$ | $({s}_{3},0.1)$ |

${x}_{4}$ | $({s}_{5},0.3)$ | $({s}_{5},-0.2)$ | $({s}_{4},-0.3)$ | $({s}_{3},0.2)$ |

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

${x}_{1}$ | $({s}_{4},-0.4)$ | $({s}_{4},0.3)$ | $({s}_{4},-0.1)$ | $({s}_{5},-0.5)$ |

${x}_{2}$ | $({s}_{4},0)$ | $({s}_{3},-0.3)$ | $({s}_{3},0.2)$ | $({s}_{5},-0.5)$ |

${x}_{3}$ | $({s}_{5},0.1)$ | $({s}_{3},0.4)$ | $({s}_{4},-0.4)$ | $({s}_{3},0.1)$ |

${x}_{4}$ | $({s}_{4},0.3)$ | $({s}_{4},-0.1)$ | $({s}_{3},-0.2)$ | $({s}_{4},0.1)$ |

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

${x}_{1}$ | $({s}_{4},-0.3)$ | $({s}_{4},0.2)$ | $({s}_{3},-0.1)$ | $({s}_{5},-0.4)$ |

${x}_{2}$ | $({s}_{5},0)$ | $({s}_{2},0.3)$ | $({s}_{3},0.1)$ | $({s}_{5},-0.4)$ |

${x}_{3}$ | $({s}_{5},-0.4)$ | $({s}_{3},0.3)$ | $({s}_{4},0.1)$ | $({s}_{4},-0.2)$ |

${x}_{4}$ | $({s}_{4},0.2)$ | $({s}_{5},-0.5)$ | $({s}_{3},0.4)$ | $({s}_{3},0.1)$ |

${\mathit{s}}_{{\mathit{a}}_{\mathit{i}}^{1}}$ | ${\mathit{s}}_{{\mathit{a}}_{\mathit{i}}^{2}}$ | ${\mathit{s}}_{{\mathit{a}}_{\mathit{i}}^{3}}$ | |
---|---|---|---|

${x}_{1}$ | $({s}_{4},0.34)$ | $({s}_{4},0.14)$ | $({s}_{4},-0.05)$ |

${x}_{2}$ | $({s}_{4},0.05)$ | $({s}_{4},-0.35)$ | $({s}_{4},-0.16)$ |

${x}_{3}$ | $({s}_{4},-0.10)$ | $({s}_{4},-0.12)$ | $({s}_{4},-0.07)$ |

${x}_{4}$ | $({s}_{4},-0.48)$ | $({s}_{4},-0.18)$ | $({s}_{4},-0.12)$ |

${\mathit{\rho}}_{1}$ | ${\mathit{\rho}}_{2}$ | ${\mathit{\rho}}_{3}$ | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | Rank | |
---|---|---|---|---|---|---|---|---|

case 1 | 100 | 100 | 100 | $({s}_{5},0.08)$ | $({s}_{4},-0.14)$ | $({s}_{5},-0.04)$ | $({s}_{5},0.16)$ | ${x}_{4}>{x}_{1}>{x}_{3}>{x}_{2}$ |

case 2 | 70 | 70 | 70 | $({s}_{5},0.03)$ | $({s}_{5},-0.20)$ | $({s}_{5},-0.11)$ | $({s}_{5},0.11)$ | ${x}_{4}>{x}_{1}>{x}_{3}>{x}_{2}$ |

case 3 | 50 | 50 | 50 | $({s}_{5},-0.03)$ | $({s}_{5},-0.27)$ | $({s}_{5},-0.18)$ | $({s}_{5},0.04)$ | ${x}_{4}>{x}_{1}>{x}_{3}>{x}_{2}$ |

case 4 | 30 | 30 | 30 | $({s}_{5},-0.18)$ | $({s}_{5},-0.41)$ | $({s}_{5},-0.35)$ | $({s}_{5},-0.11)$ | ${x}_{4}>{x}_{1}>{x}_{3}>{x}_{2}$ |

case 5 | 10 | 10 | 10 | $({s}_{4},0.42)$ | $({s}_{4},0.27)$ | $({s}_{4},0.20)$ | $({s}_{4},0.47)$ | ${x}_{4}>{x}_{1}>{x}_{3}>{x}_{2}$ |

case 6 | 5 | 5 | 5 | $({s}_{4},0.26)$ | $({s}_{4},0.07)$ | $({s}_{4},0.02)$ | $({s}_{4},0.28)$ | ${x}_{4}>{x}_{1}>{x}_{3}>{x}_{2}$ |

case 7 | 1 | 1 | 1 | $({s}_{4},0.13)$ | $({s}_{4},-0.14)$ | $({s}_{4},-0.11)$ | $({s}_{4},0.11)$ | ${x}_{1}>{x}_{4}>{x}_{3}>{x}_{2}$ |

case 8 | 0.01 | 0.01 | 0.01 | $({s}_{4},0.11)$ | $({s}_{4},-0.19)$ | $({s}_{4},-0.14)$ | $({s}_{4},0.07)$ | ${x}_{1}>{x}_{4}>{x}_{3}>{x}_{2}$ |

case 9 | 0.1 | 0.1 | 0.1 | $({s}_{4},0.11)$ | $({s}_{4},-0.18)$ | $({s}_{4},-0.13)$ | $({s}_{4},0.08)$ | ${x}_{1}>{x}_{4}>{x}_{3}>{x}_{2}$ |

case 10 | 0.2 | 0.2 | 0.2 | $({s}_{5},0.48)$ | $({s}_{5},0.34)$ | $({s}_{5},0.36)$ | $({s}_{5},0.46)$ | ${x}_{1}>{x}_{4}>{x}_{3}>{x}_{2}$ |

case 11 | 0.5 | 0.5 | 0.5 | $({s}_{4},0.12)$ | $({s}_{4},-0.16)$ | $({s}_{4},-0.13)$ | $({s}_{4},0.09)$ | ${x}_{1}>{x}_{4}>{x}_{3}>{x}_{2}$ |

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

the existing method [24] | $({s}_{4},0.13)$ | $({s}_{4},-0.14)$ | $({s}_{4},-0.11)$ | $({s}_{4},0.11)$ | ${x}_{1}>{x}_{4}>{x}_{3}>{x}_{2}$ |

the novel method $\rho =5>1$ | $({s}_{4},0.26)$ | $({s}_{4},0.07)$ | $({s}_{4},0.02)$ | $({s}_{4},0.28)$ | ${x}_{4}>{x}_{1}>{x}_{3}>{x}_{2}$ |

the novel method $\rho =1$ | $({s}_{4},0.13)$ | $({s}_{4},-0.14)$ | $({s}_{4},-0.11)$ | $({s}_{4},0.11)$ | ${x}_{1}>{x}_{4}>{x}_{3}>{x}_{2}$ |

the novel method $\rho =0.2<1$ | $({s}_{5},0.48)$ | $({s}_{5},0.34)$ | $({s}_{5},0.36)$ | $({s}_{5},0.46)$ | ${x}_{1}>{x}_{4}>{x}_{3}>{x}_{2}$ |

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**MDPI and ACS Style**

Zuo, Y.; Chen, C.
A New Fuzzy Multiple Attribute Decision Making Method Based on the Utility Transformation Functions. *Symmetry* **2019**, *11*, 418.
https://doi.org/10.3390/sym11030418

**AMA Style**

Zuo Y, Chen C.
A New Fuzzy Multiple Attribute Decision Making Method Based on the Utility Transformation Functions. *Symmetry*. 2019; 11(3):418.
https://doi.org/10.3390/sym11030418

**Chicago/Turabian Style**

Zuo, Yuting, and Chunfang Chen.
2019. "A New Fuzzy Multiple Attribute Decision Making Method Based on the Utility Transformation Functions" *Symmetry* 11, no. 3: 418.
https://doi.org/10.3390/sym11030418