# Combined Conflict Evidence Based on Two-Tuple IOWA Operators

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Dempster–Shafer Evidence Theory

**Definition**

**1.**

#### 2.2. Jousselme Distance

**Definition**

**2.**

#### 2.3. New Conflict Coefficient

**Definition**

**3.**

**Example**

**1.**

**Example**

**2.**

#### 2.4. OWA Operator and IOWA Operator

**Definition**

**4.**

#### 2.5. Maximum Entropy Method

## 3. Two-Tuple IOWA Operator and the Determine Weighting Vector of Multi-Source BOEs

#### 3.1. Two-Tuple IOWA Operator

**Definition**

**5.**

**Example**

**3.**

**Example**

**4.**

#### 3.2. The Determination of Associated Weight of BOEs

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

## 4. New Combination Approach of Conflict Evidence

**step1**: Calculate the Jousselme distance ${d}_{ij}$, the classic conflict coefficient ${k}_{ij}$, and construct the evidence distance matrix $DM$ and conflict coefficient matrix K.

**step2**: According to Definitions 6–8, calculate $\overline{{d}_{i}},\overline{{k}_{i}}$, ${k}_{g}$, ${d}_{g}$ and ${k}_{g}^{d}$ based on the obtained matrix $DM$ and K.

**step3**: According to Equation (30), calculate $\alpha $ base on the ${k}_{g}^{d}$ obtained in step 2. Then, the weighting vector $W=\{{w}_{1},{w}_{2},\cdots ,{w}_{n}\}$ can be obtained by MEM.

**step4**: First, construct two-tuple order inducing variable ${S}_{i}=1-\langle \overline{{d}_{i}},\overline{{k}_{i}}\rangle =\langle 1-\overline{{d}_{i}},1-\overline{{k}_{i}}\rangle ,i=1,\cdots ,n$ by $\overline{{d}_{i}}$ and $\overline{{k}_{i}}$. Then, construct a two-tuple OWA pair $\langle {S}_{i},{m}_{i}\rangle ,i=1,\cdots ,n$, where ${m}_{i}$ is the argument variable (BPA of evidence ${E}_{i}$).

**step5**: According to Equation (14), the weighted average evidence in the system is given as:

**step6**: When n pieces of evidence are combined using the classic Dempster rule, there are combined $n-1$ times. Then, the final combination result can be obtained.

## 5. Example and Analysis

**Example**

**5.**

**Example**

**6.**

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Case | ${\mathit{d}}_{\mathit{j}}$ | k | ${\mathit{k}}^{\mathit{d}}$ |
---|---|---|---|

$\left\{1\right\}$ | 0.8544 | 0.1000 | 0.4772 |

$\{1,2\}$ | 0.7416 | 0.1000 | 0.4208 |

$\{1,2,3\}$ | 0.6083 | 0.1000 | 0.3541 |

$\{1,2,3,4\}$ | 0.4359 | 0.1000 | 0.2680 |

$\{1,\cdots ,5\}$ | 0.1000 | 0.1000 | 0.1000 |

$\{1,\cdots ,6\}$ | 0.4000 | 0.1000 | 0.2500 |

$\{1,\cdots ,7\}$ | 0.5292 | 0.1000 | 0.3146 |

$\{1,\cdots ,8\}$ | 0.5990 | 0.1000 | 0.3495 |

$\{1,\cdots ,9\}$ | 0.6481 | 0.1000 | 0.3740 |

$\{1,\cdots ,10\}$ | 0.6848 | 0.1000 | 0.3924 |

$\{1,\cdots ,11\}$ | 0.7135 | 0.1000 | 0.4068 |

$\{1,\cdots ,12\}$ | 0.7365 | 0.1000 | 0.4183 |

$\{1,\cdots ,13\}$ | 0.7555 | 0.1000 | 0.4277 |

$\{1,\cdots ,14\}$ | 0.7714 | 0.1000 | 0.4357 |

$\{1,\cdots ,15\}$ | 0.7849 | 0.1000 | 0.4425 |

$\{1,\cdots ,16\}$ | 0.7965 | 0.1000 | 0.4482 |

$\{1,\cdots ,17\}$ | 0.8066 | 0.1000 | 0.4533 |

$\{1,\cdots ,18\}$ | 0.8155 | 0.1000 | 0.4577 |

$\{1,\cdots ,19\}$ | 0.8233 | 0.1000 | 0.4617 |

$\{1,\cdots ,20\}$ | 0.8304 | 0.1000 | 0.4652 |

BOEs | Approach | $\mathit{m}\left(\mathit{A}\right)$ | $\mathit{m}\left(\mathit{B}\right)$ | $\mathit{m}\left(\mathit{C}\right)$ | $\mathit{m}\left(\mathit{A}\mathit{C}\right)$ | $\mathit{m}\left(\mathsf{\Theta}\right)$ | Target |
---|---|---|---|---|---|---|---|

Dempster | 0 | 0.8571 | 0.1429 | 0 | 0 | B | |

Yager [38] | 0 | 0.1800 | 0.0300 | 0 | 0.7900 | Θ | |

${m}_{1},{m}_{2}$ | Murphy [35] | 0.1543 | 0.7469 | 0.0988 | 0 | 0 | B |

Deng [41] | 0.1543 | 0.7469 | 0.0988 | 0 | 0 | B | |

Proposed | 0.1543 | 0.7469 | 0.0988 | 0 | 0 | B | |

Dempster | 0 | 0.6316 | 0.3684 | 0 | 0 | B | |

Yager [38] | 0.4345 | 0.097 | 0.0105 | 0.2765 | 0.1815 | A | |

${m}_{1},{m}_{2},{m}_{3}$ | Murphy [35] | 0.5568 | 0.3562 | 0.0782 | 0.0088 | 0 | A |

Deng [41] | 0.6500 | 0.2547 | 0.0858 | 0.0095 | 0 | A | |

Proposed | 0.7429 | 0.1489 | 0.1019 | 0.0067 | 0 | A | |

Dempster | 0 | 0.3288 | 0.6712 | 0 | 0 | C | |

Yager [38] | 0.6430 | 0.0279 | 0.0037 | 0.1603 | 0.1652 | A | |

${m}_{1},{m}_{2},{m}_{3},{m}_{4}$ | Murphy [35] | 0.8653 | 0.0891 | 0.0382 | 0.0074 | 0 | A |

Deng [41] | 0.9305 | 0.0274 | 0.0339 | 0.0082 | 0 | A | |

Proposed | 0.9638 | 0.0049 | 0.0184 | 0.0139 | 0 | A | |

Dempster | 0 | 0.1404 | 0.8596 | 0 | 0 | C | |

Yager [38] | 0.7740 | 0.0193 | 0.0011 | 0.0977 | 0.1080 | A | |

${m}_{1},{m}_{2},{m}_{3},{m}_{4},{m}_{5}$ | Murphy [35] | 0.9688 | 0.0156 | 0.0127 | 0.0029 | 0 | A |

Deng [41] | 0.9846 | 0.0024 | 0.0098 | 0.0032 | 0 | A | |

Proposed | 0.9897 | 0.0002 | 0.0043 | 0.0058 | 0 | A |

BOEs | Approach | $\mathit{m}\left(\mathit{A}\right)$ | $\mathit{m}\left(\mathit{B}\right)$ | $\mathit{m}\left(\mathit{C}\right)$ | $\mathit{m}\left(\mathit{A}\mathit{C}\right)$ | $\mathit{m}\left(\mathsf{\Theta}\right)$ | Target |
---|---|---|---|---|---|---|---|

Dempster | 0 | 0.0577 | 0.9423 | 0 | 0 | C | |

Yager [38] | 0.0968 | 0.0193 | 0.2634 | 0 | 0.6205 | Θ | |

${m}_{1},{m}_{2},{m}_{3},{m}_{4},{m}_{5}$ | Murphy [35] | 0.7250 | 0.0491 | 0.2244 | 0.0015 | 0 | A |

Deng [41] | 0.8828 | 0.0062 | 0.1091 | 0.0019 | 0 | A | |

Proposed | 0.9039 | 0.0050 | 0.0900 | 0.0011 | 0 | A |

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**MDPI and ACS Style**

Zhou, Y.; Qin, X.; Zhao, X.
Combined Conflict Evidence Based on Two-Tuple IOWA Operators. *Symmetry* **2019**, *11*, 1369.
https://doi.org/10.3390/sym11111369

**AMA Style**

Zhou Y, Qin X, Zhao X.
Combined Conflict Evidence Based on Two-Tuple IOWA Operators. *Symmetry*. 2019; 11(11):1369.
https://doi.org/10.3390/sym11111369

**Chicago/Turabian Style**

Zhou, Ying, Xiyun Qin, and Xiaozhe Zhao.
2019. "Combined Conflict Evidence Based on Two-Tuple IOWA Operators" *Symmetry* 11, no. 11: 1369.
https://doi.org/10.3390/sym11111369