# Interval Intuitionistic Fuzzy Clustering Algorithm Based on Symmetric Information Entropy

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Some Basic Concepts of Intuitionistic Fuzzy Set

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- (a)
- If ${u}_{A}^{-}\le {u}_{B}^{-}$, ${u}_{A}^{+}\le {u}_{B}^{+}$, ${v}_{A}^{-}\ge {v}_{B}^{-}$ and ${v}_{A}^{+}\ge {v}_{B}^{+}$, then $\tilde{a}\le \tilde{b}$.
- (b)
- If ${u}_{A}^{-}={u}_{B}^{-}$, ${u}_{A}^{+}={u}_{B}^{+}$, ${v}_{A}^{-}={v}_{B}^{-}$ and ${v}_{A}^{+}={v}_{B}^{+}$, then $\tilde{a}=\tilde{b}$.

#### 2.2. Continuous Aggregation Operators

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

## 3. Continuous OOWIFQ Operator for Aggregating Interval Intuitionistic Fuzzy Numbers Based on Chi-Squared Deviation

**Definition**

**8.**

**Definition**

**9.**

- (a).
- When $f$ is strictly monotonic increasing, then $\delta \ge 0$, $\eta \ge 0$ holds. As a result, we have$$f({u}^{+})-j\cdot \delta \le f({u}^{+})-(j-1)\cdot \delta ,\text{}f({v}^{-})+(i-1)\cdot \eta \le f({v}^{-})-i\cdot \eta .$$Since ${f}^{-1}$ is also strictly monotonic increasing, we get$${u}_{j}={f}^{-1}(f({u}^{+})-j\cdot \delta )\le {f}^{-1}(f({u}^{+})-(j-1)\cdot \delta )={u}_{j-1}$$$${v}_{i-1}={f}^{-1}(f({v}^{-})+(i-1)\cdot \eta )\le {f}^{-1}(f({v}^{-1})+i\cdot \eta )={v}_{i}$$
- (b).
- On the contrary, when $f$ is strictly monotonic decreasing, then $\delta \le 0$, $\eta \le 0$ holds. It follows that$$f({u}^{+})-j\cdot \delta \le f({u}^{+})-(j-1)\cdot \delta ,\text{}f({v}^{-})+(i-1)\cdot \eta \le f({v}^{-})-i\cdot \eta .$$$${u}_{j}={f}^{-1}(f({u}^{+})-j\cdot \delta )\le {f}^{-1}(f({u}^{+})-(j-1)\cdot \delta )={u}_{j-1}$$$${v}_{i-1}={f}^{-1}(f({v}^{-})+(i-1)\cdot \eta )\le {f}^{-1}(f({v}^{-1})+i\cdot \eta )={v}_{i}$$

**Property**

**1.**

**Proof.**

- (a)
- If $f$ is strictly monotonic increasing, then $f({u}^{-})\text{}\le f({u}^{+})$. Thus, for all $u\in [0,1]$, we have $f({u}^{-})\text{}\le f({u}^{+})-(f({u}^{+})-f({u}^{-}))\cdot u\le f({u}^{+})$. It follows that$$f({u}^{-}){\displaystyle {\int}_{0}^{1}\frac{dQ(u)}{du}}du\le {\displaystyle {\int}_{0}^{1}\frac{dQ(u)}{du}}(f({u}^{+})-(f({u}^{+})-f({u}^{-}))\cdot u)du\le f({u}^{+}){\displaystyle {\int}_{0}^{1}\frac{dQ(u)}{du}}du$$$$\frac{1}{f({u}^{+})}{\displaystyle {\int}_{0}^{1}\frac{dQ(u)}{du}du}\le {\displaystyle {\int}_{0}^{1}\frac{\frac{dQ(u)}{du}}{f({u}^{+})-(f({u}^{+})-f({u}^{-}))\cdot u}}du\le \frac{1}{f({u}^{-})}{\displaystyle {\int}_{0}^{1}\frac{dQ(u)}{du}}du.$$

- (b)
- If $f$ is strictly monotonic decreasing, then $f({u}^{-})\text{}\ge f({u}^{+})$. Thus, for all $u\in [0,1]$, we have $f({u}^{+})\text{}\le f({u}^{+})-(f({u}^{+})-f({u}^{-}))\cdot u\le f({u}^{-})$. Similarly, we have$$f({u}^{+})\le {\displaystyle {\int}_{0}^{1}\frac{dQ(u)}{du}}(f({u}^{+})-(f({u}^{+})-f({u}^{-}))\cdot u)du\le f({u}^{-})$$$$\frac{1}{f({u}^{-})}\text{}\le {\displaystyle {\int}_{0}^{1}\frac{\frac{dQ(u)}{du}}{f({u}^{+})-[f({u}^{+})-f({u}^{-})]\cdot u}}du\le \frac{1}{f({u}^{+})}.$$

**Property**

**2.**

**Proof.**

- (a)
- When $f$ function is strictly monotonic increasing, for any $u\in [0,1]$, we have$$f({u}_{1}^{+})-(f({u}_{1}^{+})-f({u}_{1}^{-}))\cdot u\le f({u}_{2}^{+})-(f({u}_{2}^{+})-f({u}_{2}^{-}))\cdot u.$$

- (b)
- When $f$ function is strictly monotonic decreasing, for any $u\in [0,1]$, we have$$f({u}_{1}^{+})-(f({u}_{1}^{+})-f({u}_{1}^{-}))\cdot u\ge f({u}_{2}^{+})-(f({u}_{2}^{+})-f({u}_{2}^{-}))\cdot u.$$

**Property**

**3.**

**Proof.**

**Property**

**4.**

**Proof.**

- (a)
- when $f$ function is strictly monotonic increasing, then $f({u}^{-})\text{}\le f({u}^{+})$.

- (b)
- when $f$ function is strictly monotonic decreasing, then $f({u}^{-})\text{}\ge f({u}^{+})$. Since ${Q}_{1}(x)-{Q}_{2}(u)\le 0$, we have$${\int}_{0}^{1}\frac{d{Q}_{1}(u)}{du}\cdot (f({u}^{+})-(f({u}^{+})-f({u}^{-}))\cdot u)}du\ge {\displaystyle {\int}_{0}^{1}\frac{d{Q}_{2}(u)}{du}\cdot (f({u}^{+})-(f({u}^{+})-f({u}^{-}))\cdot u)}du,$$$${\int}_{0}^{1}(}\frac{\frac{d{Q}_{1}(u)}{du}}{f({u}^{+})-(f({u}^{+})-f({u}^{-}))\cdot u})\cdot du\le {\displaystyle {\int}_{0}^{1}(}\frac{\frac{d{Q}_{2}(u)}{du}}{f({u}^{+})-(f({u}^{+})-f({u}^{-}))\cdot u})\cdot du.$$

**Theorem**

**1.**

**Proof.**

## 4. Distance Measure of Interval Intuitionistic Fuzzy Numbers Based on Symmetric Information Entropy

**Theorem**

**2.**

**Theorem**

**3.**

- (a)
- $0\le d(\tilde{a},\tilde{b})\le 1$.
- (b)
- $d(\tilde{a},\tilde{b})=1$ if and only if interval intuitionistic fuzzy numbers $\tilde{a}$ and $\tilde{b}$ reduce to $\tilde{a}=(1,0)$ and $\tilde{b}=(0,1)$ or $\tilde{a}=(0,1)$ and $\tilde{b}=(1,0)$.
- (c)
- $d(\tilde{a},\tilde{b})=d(\tilde{b},\tilde{a})$.
- (d)
- $\tilde{a}\le \tilde{b}\le \tilde{c}$, then $d(\tilde{a},\tilde{b})\le d(\tilde{a},\tilde{c})$ and $d(\tilde{b},\tilde{c})\le d(\tilde{a},\tilde{c})$.

**Proof.**

- (a)
- From Equation (23), we have$$d(\tilde{a},\tilde{b})=\frac{1}{2\mathrm{ln}2}(\mathsf{\Delta}\hat{u}\mathrm{ln}\frac{\mathsf{\Delta}\hat{u}+\mathsf{\Delta}\hat{v}}{\mathsf{\Delta}\hat{u}}+\mathsf{\Delta}\hat{v}\mathrm{ln}\frac{\mathsf{\Delta}\hat{u}+\mathsf{\Delta}\hat{v}}{\mathsf{\Delta}\hat{v}}).$$
- (b)
- the proof of this property can be analyzed by the following two cases.
- (i)
- $d(\tilde{a},\tilde{b})$ is an increasing function of independent various $\mathsf{\Delta}\tilde{u}$ and $\mathsf{\Delta}\tilde{v}$, therefore the maximum value $d(\tilde{a},\tilde{b})=1$ is obtained when $\mathsf{\Delta}\tilde{u}=1$ and $\mathsf{\Delta}\tilde{v}=1$. From $\mathsf{\Delta}\tilde{u}=1$ and $\mathsf{\Delta}\tilde{v}=1$, we can get that $\tilde{a}=(1,0)$ and $\tilde{b}=(0,1)$ or $\tilde{a}=(0,1)$ and $\tilde{b}=(1,0)$.
- (ii)
- When $\tilde{a}=(1,0)$ and $\tilde{b}=(0,1)$ or $\tilde{a}=(0,1)$ and $\tilde{b}=(1,0)$, $\mathsf{\Delta}\hat{u}=\mathsf{\Delta}\hat{v}=1$ can be brought into Equation (23) to get $d(A,B)=1$.

- (c)
- According to Equation (23), obviously, the symmetry of distance measure is holds.
- (d)
- When $\tilde{a}\le \tilde{b}\le \tilde{c}$, from Definition 3, we have ${u}_{A}^{-}\le {u}_{B}^{-}\le {u}_{C}^{-}$, ${u}_{A}^{+}\le {u}_{B}^{+}\le {u}_{C}^{+}$, ${v}_{A}^{-}\ge {v}_{B}^{-}\ge {v}_{C}^{-}$ and ${v}_{A}^{+}\ge {v}_{B}^{+}\ge {v}_{C}^{+}$. Denoting $\mathsf{\Delta}{\hat{u}}_{\tilde{a},\tilde{b}}=|{\hat{u}}_{A}-{\hat{u}}_{B}|$, $\mathsf{\Delta}{\hat{v}}_{\tilde{a},\tilde{b}}=|{\hat{v}}_{A}-{\hat{v}}_{B}|$, $\mathsf{\Delta}{\hat{u}}_{\tilde{a},\tilde{c}}=|{\hat{u}}_{A}-{\hat{u}}_{C}|$ and $\mathsf{\Delta}{\hat{v}}_{\tilde{a},\tilde{c}}=|{\hat{v}}_{A}-{\hat{v}}_{C}|$, respectively. It is clear that $\mathsf{\Delta}{\tilde{u}}_{\tilde{a},\tilde{b}}\le \mathsf{\Delta}{\tilde{u}}_{\tilde{a},\tilde{c}}$ and $\mathsf{\Delta}{\tilde{v}}_{\tilde{a},\tilde{b}}\le \mathsf{\Delta}{\tilde{v}}_{\tilde{a},\tilde{c}}$. Therefore, we get $d(\tilde{a},\tilde{b})\le d(\tilde{a},\tilde{c})$. In the same way, $d(\tilde{b},\tilde{c})\le d(\tilde{a},\tilde{c})$ also holds. □

## 5. Interval Intuitionistic Fuzzy Clustering Algorithm

**Step****1**- Select an strictly monotonic function $f(x)$ and BUM function $Q(x)$, and enter the cluster number k and the sample attribute matrix $\tilde{A}={({\tilde{a}}_{ij})}_{m\times n}$, where ${\tilde{a}}_{ij}=([{u}_{ij}^{-},{u}_{ij}^{+}],[{v}_{ij}^{-},{v}_{ij}^{+}])\text{}(i=1,2,\cdots ,m,j=1,2,\cdots ,n)$.
**Step****2**- ${b}_{ij}=C{O}_{f,Q}([{u}_{ij}^{-},{u}_{ij}^{+}],[{v}_{ij}^{-},{v}_{ij}^{+}])$ is calculated by Equation (12), where $\tilde{B}={({b}_{ij})}_{m\times n}$.
**Step****3**- Select a random sample from $\tilde{B}$ as the initial clustering center ${c}_{1}$.
**Step****4**- Firstly, calculate the shortest distance between each object and the existing clustering center, denoted by $D(x)$; then calculate the probability $\frac{D{(x)}^{2}}{{\displaystyle {\sum}_{x\in X}D{(x)}^{2}}}$ that each object is selected as the next cluster center. Finally, a cluster center is selected according to the roulette method.
**Step****5**- Repeat Step 4 until $k$ cluster centers are selected.
**Step****6**- For each object ${x}_{i}$ (${x}_{i}=\{{b}_{i1},{b}_{i2},\cdots ,{b}_{in}\}$) in $\tilde{B}$, its symmetric entropy distance to the k cluster center is calculated by the Equation (23) and divided into the class corresponding to the cluster center with the smallest distance.
**Step****7**- For each category ${c}_{i}$, recalculate its cluster center ${c}_{i}=\frac{1}{\left|{c}_{i}\right|}{\displaystyle {\sum}_{x\in {x}_{i}}x}$.
**Step****8**- Repeat Steps 6 and 7 until the position of the cluster center does not change.
**Step****9**- End.

## 6. Numerical Example

**Step****1**- Input the sample attribute matrix $\tilde{A}$, set $f(x)={e}^{x}$, $Q(x)=x$, and the classification number $k=4$.
**Step****2**- Calculate each ${b}_{ij}$ with Equation (12), and the calculated result is $\tilde{B}={({b}_{ij})}_{m\times n}$.
**Step****3**- Intuitionistic fuzzy matrix $\tilde{B}$ is calculated, shown in Table 2.
**Step****4**- Randomly select a sample from $\tilde{B}$ as the initial cluster center No. 1.
**Step****5**- Calculate the shortest distance between each object and the existing clustering center, denoted by $D(x)$; then calculate the probability that each object is selected as the next cluster center. A cluster center is selected according to the roulette method.
**Step****6**- Repeat Step 4 until $k$ cluster centers are selected. An initial clustering center is shown in Table 3.
**Step****7**- For each object ${x}_{i}$ (${x}_{i}=\{{b}_{i1},{b}_{i2},\cdots ,{b}_{in}\}$) in $\tilde{B}$, the symmetric entropy distance from the $k$ cluster center is calculated by Equation (22) and divided into the class corresponding to the cluster center with the smallest distance.
**Step****8**- For each category ${c}_{i}$, recalculate its cluster center ${c}_{i}=\frac{1}{\left|{c}_{i}\right|}{\displaystyle {\sum}_{x\in {x}_{i}}x}$.

## 7. Conclusions

- (1)
- Compared with traditional clustering and fuzzy clustering, interval intuitionistic fuzzy clustering describes the fuzzy nature of things more delicately.
- (2)
- The symmetric information entropy based distance measure considers all the information in the continuous interval. Thus, the distortion and loss of information are avoided, and the result is more accurate and effective.
- (3)
- The C-OOWIFQ takes into account the preferences of decision makers.

## Author Contributions

## Funding

## Conflicts of Interest

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No. | TN | TP | OM | AN | AP | AK |
---|---|---|---|---|---|---|

1 | ([0.6,0.6], [0.3,0.3]) | ([0.1,0.7], [0.2,0.3]) | ([0.1,0.5], [0.0,0.1]) | ([0.3,0.3], [0.1,0.4]) | ([0.3,0.4], [0.1,0.1]) | ([0.2,0.7], [0.0,0.1]) |

2 | ([0.1,0.1], [0.5,0.5]) | ([0.6,0.9], [0.1,0.1]) | ([0.1,0.1], [0.1,0.1]) | ([0.2,0.2], [0.1,0.2]) | ([0.3,0.3], [0.0,0.3]) | ([0.0,0.3], [0.2,0.2]) |

3 | ([0.2,0.4], [0.1,0.2]) | ([0.6,0.8], [0.0,0.1]) | ([0.0,0.7], [0.0,0.1]) | ([0.0,0.0], [0.1,0.1]) | ([0.5,0.8], [0.1,0.2]) | ([0.5,0.9], [0.0,0.0]) |

4 | ([0.4,0.6], [0.2,0.2]) | ([0.3,0.4], [0.1,0.3]) | ([0.5,0.6], [0.2,0.3]) | ([0.0,0.1], [0.1,0.1]) | ([0.4,0.9], [0.1,0.1]) | ([0.0,0.1], [0.1,0.5]) |

5 | ([0.0,0.0], [0.6,0.7]) | ([0.0,0.5], [0.0,0.0]) | ([0.0,0.1], [0.0,0.3]) | ([0.3,0.4], [0.1,0.2]) | ([0.5,0.8], [0.0,0.0]) | ([0.1,0.3], [0.0,0.0]) |

6 | ([0.1,0.6], [0.3,0.4]) | ([0.2,0.3], [0.1,0.6]) | ([0.0,0.8], [0.2,0.2]) | ([0.5,0.5], [0.3,0.5]) | ([0.2,0.2], [0.3,0.4]) | ([0.0,0.3], [0.3,0.4]) |

7 | ([0.6,0.6], [0.1,0.1]) | ([0.2,0.4], [0.0,0.1]) | ([0.1,0.2], [0.0,0.1]) | ([0.1,0.2], [0.2,0.3]) | ([0.4,0.7], [0.0,0.2]) | ([0.1,0.2], [0.1,0.1]) |

8 | ([0.4,0.5], [0.0,0.1]) | ([0.0,0.1], [0.6,0.8]) | ([0.1,0.9], [0.0,0.0]) | ([0.2,0.4], [0.3,0.6]) | ([0.0,0.5], [0.0,0.1]) | ([0.3,0.9], [0.0,0.1]) |

9 | ([0.6,0.7], [0.1,0.2]) | ([0.0,0.6], [0.0,0.0]) | ([0.3,0.8], [0.0,0.1]) | ([0.1,0.1], [0.3,0.6]) | ([0.0,0.0], [0.0,0.2]) | ([0.1,0.6], [0.1,0.1]) |

10 | ([0.2,0.4], [0.0,0.2]) | ([0.0,0.7], [0.1,0.1]) | ([0.2,0.2], [0.2,0.4]) | ([0.5,0.8], [0.0,0.1]) | ([0.6,0.9], [0.0,0.1]) | ([0.6,0.8], [0.0,0.0]) |

11 | ([0.0,0.1], [0.3,0.6]) | ([0.4,0.9], [0.0,0.0]) | ([0.1,0.1], [0.0,0.0]) | ([0.3,0.5], [0.1,0.5]) | ([0.2,0.2], [0.1,0.6]) | ([0.1,0.1], [0.1,0.5]) |

12 | ([0.3,0.3], [0.4,0.6]) | ([0.1,0.1], [0.1,0.2]) | ([0.5,1.0], [0.0,0.0]) | ([0.4,0.7], [0.1,0.1]) | ([0.1,0.6], [0.4,0.4]) | ([0.1,0.2], [0.2,0.6]) |

13 | ([0.7,0.8], [0.0,0.0]) | ([0.2,0.9], [0.0,0.0]) | ([0.1,0.5], [0.0,0.1]) | ([0.0,0.1], [0.5,0.8]) | ([0.0,0.0], [0.3,0.7]) | ([0.0,0.1], [0.4,0.5]) |

14 | ([0.4,0.4], [0.4,0.5]) | ([0.5,1.0], [0.0,0.0]) | ([0.7,0.8], [0.1,0.1]) | ([0.3,0.4], [0.1,0.6]) | ([0.1,0.1], [0.5,0.7]) | ([0.0,0.3], [0.1,0.3]) |

15 | ([0.3,0.5], [0.1,0.3]) | ([0.4,0.6], [0.2,0.3]) | ([0.2,0.6], [0.0,0.0]) | ([0.3,0.5], [0.1,0.2]) | ([0.3,0.4], [0.0,0.1]) | ([0.0,0.0], [0.0,0.1]) |

No. | TN | TP | OM | AN | AP | AK |
---|---|---|---|---|---|---|

1 | (0.60,0.30) | (0.43,0.25) | (0.31,0.05) | (0.30,0.26) | (0.35,0.10) | (0.47,0.05) |

2 | (0.10,0.50) | (0.76,0.10) | (0.10,0.10) | (0.20,0.15) | (0.30,0.16) | (0.16,0.20) |

3 | (0.30,0.15) | (0.70,0.05) | (0.39,0.05) | (0.00,0.10) | (0.66,0.15) | (0.71,0.00) |

4 | (0.50,0.20) | (0.35,0.20) | (0.55,0.25) | (0.05,0.10) | (0.67,0.10) | (0.05,0.31) |

5 | (0.00,0.65) | (0.27,0.00) | (0.05,0.16) | (0.35,0.15) | (0.66,0.00) | (0.20,0.00) |

6 | (0.37,0.35) | (0.25,0.37) | (0.45,0.20) | (0.50,0.40) | (0.20,0.35) | (0.16,0.35) |

7 | (0.60,0.10) | (0.30,0.05) | (0.15,0.05) | (0.15,0.25) | (0.56,0.10) | (0.15,0.10) |

8 | (0.45,0.05) | (0.05,0.70) | (0.55,0.00) | (0.30,0.46) | (0.27,0.05) | (0.63,0.05) |

9 | (0.65,0.15) | (0.33,0.00) | (0.57,0.05) | (0.10,0.46) | (0.00,0.10) | (0.37,0.10) |

10 | (0.30,0.10) | (0.39,0.10) | (0.20,0.30) | (0.66,0.05) | (0.76,0.05) | (0.70,0.00) |

11 | (0.05,0.46) | (0.67,0.00) | (0.10,0.00) | (0.40,0.31) | (0.20,0.37) | (0.10,0.31) |

12 | (0.30,0.50) | (0.10,0.15) | (0.77,0.00) | (0.56,0.10) | (0.37,0.40) | (0.15,0.41) |

13 | (0.75,0.00) | (0.59,0.00) | (0.31,0.05) | (0.05,0.66) | (0.00,0.51) | (0.05,0.45) |

14 | (0.40,0.45) | (0.77,0.00) | (0.75,0.10) | (0.35,0.37) | (0.10,0.60) | (0.16,0.20) |

15 | (0.40,0.20) | (0.50,0.25) | (0.41,0.00) | (0.40,0.15) | (0.35,0.05) | (0.00,0.05) |

No. | TN | TP | OM | AN | AP | AK |
---|---|---|---|---|---|---|

1 | (0.60,0.30) | (0.43,0.25) | (0.31,0.05) | (0.30,0.26) | (0.35,0.10) | (0.47,0.05) |

2 | (0.50,0.20) | (0.35,0.20) | (0.55,0.25) | (0.05,0.10) | (0.67,0.10) | (0.05,0.31) |

3 | (0.05,0.46) | (0.67,0.00) | (0.10,0.00) | (0.40,0.31) | (0.20,0.37) | (0.10,0.31) |

4 | (0.30,0.15) | (0.70,0.05) | (0.39,0.05) | (0.00,0.10) | (0.66,0.15) | (0.71,0.00) |

No. | TN | TP | OM | AN | AK | AK |
---|---|---|---|---|---|---|

1 | (0.54,0.16) | (0.32,0.25) | (0.40,0.03) | (0.25,0.31) | (0.31,0.08) | (0.32,0.07) |

2 | (0.54,0.18) | (0.40,0.19) | (0.44,0.17) | (0.20,0.39) | (0.29,0.32) | (0.09,0.37) |

3 | (0.21,0.48) | (0.57,0.06) | (0.43,0.05) | (0.38,0.23) | (0.24,0.38) | (0.14,0.28) |

4 | (0.20,0.30) | (0.45,0.05) | (0.21,0.17) | (0.34,0.10) | (0.69,0.07) | (0.54,0.00) |

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## Share and Cite

**MDPI and ACS Style**

Lin, J.; Duan, G.; Tian, Z. Interval Intuitionistic Fuzzy Clustering Algorithm Based on Symmetric Information Entropy. *Symmetry* **2020**, *12*, 79.
https://doi.org/10.3390/sym12010079

**AMA Style**

Lin J, Duan G, Tian Z. Interval Intuitionistic Fuzzy Clustering Algorithm Based on Symmetric Information Entropy. *Symmetry*. 2020; 12(1):79.
https://doi.org/10.3390/sym12010079

**Chicago/Turabian Style**

Lin, Jian, Guanhua Duan, and Zhiyong Tian. 2020. "Interval Intuitionistic Fuzzy Clustering Algorithm Based on Symmetric Information Entropy" *Symmetry* 12, no. 1: 79.
https://doi.org/10.3390/sym12010079