# An Entropy-Based Knowledge Measure for Atanassov’s Intuitionistic Fuzzy Sets and Its Application to Multiple Attribute Decision Making

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Atanassov’s Intuitionistic Fuzzy Sets

**Definition**

**1**

**[1].**

**Definition**

**2**

**[7].**

**Definition**

**3**

**[7].**

- (R1)
- $A\subseteq B\iff $$\forall x\in X$${\mu}_{A}(x)\le {\mu}_{B}(x),{v}_{A}(x)\ge {v}_{B}(x)$;
- (R2)
- $A=B\iff $$\forall x\in X$${\mu}_{A}(x)={\mu}_{B}(x),{v}_{A}(x)={v}_{B}(x)$;
- (R3)
- ${A}^{C}=\left\{\langle x,{v}_{A}(x),{\mu}_{A}(x)\rangle |x\in X\right\}$, where ${A}^{C}$ is the complement of $A$.

**Definition**

**4**

**[7].**

## 3. A New Knowledge Measure for AIFSs

**Definition**

**5.**

- (KP1)
- $K(A)=1$if and only if A is a crisp set.
- (KP2)
- $K(A)=0$if and only if${\pi}_{A}({x}_{i})=1$,$\forall i\in \{1,2,\cdots ,n\}$.
- (KP3)
- $K(A)$is increasing with${\Delta}_{A}({x}_{i})=\left|{\mu}_{A}({x}_{i})-{v}_{A}({x}_{i})\right|$and decreasing with${\pi}_{A}({x}_{i})$,$i=1,2,\cdots ,n$.
- (KP4)
- $K({A}^{C})=K(A)$.

_{1}and P

_{2}are two probability distributions for X. The K–L divergence between P

_{1}and P

_{2}is defined by the equation below [35].

_{ij}is the probability of occurrence of the value X = x

_{j}for each of the probability distribution P

_{i}, I = 1, 2. To construct a symmetric divergence measure, the divergence measure between P

_{1}and P

_{2}can be defined by the formula below.

_{S}(A) satisfies all properties in Definition 5.

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. Numerical Examples

_{S}will be validated based on two numerical examples. To illustrate the effectiveness and performance of the proposed Biparametric uncertainty measure for AIFSs, some existing intuitionistic fuzzy entropy measures will be adopted for comparison. Therefore, we first recall some widely used entropy measures for AIFSs.

**Example**

**1.**

_{1}= <x,0.5,0.5>, A

_{2}= <x,0.25,0.25>, A

_{3}= <x,0.25,0.5>, A

_{4}= <x, 0.2,0.3>.

_{ZL}, E

_{SK}, and E

_{VS}cannot discriminate the uncertainty of A

_{1}and A

_{2}. Since the entropy measure E

_{BB}is defined based on the hesitancy degree of AIFSs, it assigns zero uncertainty to A

_{1}, which is unreasonable. Moreover, the uncertainty grades of A

_{2}and A

_{4}cannot be distinguished by E

_{BB}because their hesitancy degrees are identical. It is also shown that uncertainty grades of A

_{2}and A

_{4}calculated by the entropy measure E

^{2}

_{HC}are equal to each other.

_{SKB}, K

_{N}, K

_{G}, together with our proposed measure K

_{S}can discriminate these four AIFSs well from the perspective of the amount of knowledge. For the ranking order of the knowledge amount, we see that the measures K

_{SKB}and K

_{G}yield the results K(A

_{3}) > K(A

_{1}) > K(A

_{4}) > K(A

_{2}). When the knowledge measure K

_{N}is applied, we can obtain that K(A

_{1}) > K(A

_{3}) > K(A

_{2}) > K(A

_{4}). Our proposed measure K

_{S}leads to the order K(A

_{1}) > K(A

_{3}) > K(A

_{4}) > K(A

_{2}). Comparing AIFSs A

_{1}and A

_{3}, we can see that the difference between the membership and non-membership degree of A

_{1}is less than that of A

_{3}. However, A

_{3}has more of a hesitancy degree than A

_{1}. According to the monotonicity of the knowledge measure proposed in Definition 5, the knowledge amount of A

_{1}and A

_{3}cannot be determined. But for A

_{2}and A

_{4}, it is shown that A

_{2}has less Δ, i.e., the difference between its membership and non-membership grades, and more hesitancy degree π than A

_{4}. Therefore, A

_{4}should convey a higher amount of knowledge than A

_{2}. In such a case, the knowledge measure K

_{N}is less reasonable than the other three knowledge measures.

**Example**

**2.**

_{1}= <x,0.7,0.2>, A

_{2}= <x,0.5,0.3>, A

_{3}= <x,0.5,0>, A

_{4}= <x,0.5,0.5>, A

_{5}= <x,0.5,0.4>, A

_{6}= <x,0.6,0.2>, A

_{7}= <x,0.4,0.4>, A

_{8}= <x,1,0>, and A

_{9}= <x,0,0>.

_{ZL}, E

_{BB}, E

_{SK}, E

_{VS}, and E

^{2}

_{HC}, three knowledge measures K

_{SKB}, K

_{N}, and K

_{G}, and our proposed measures K

_{S}, we can calculate the uncertainty of these nine AIFSs. The results are shown in Table 2.

_{9}= <x,0,0> is the most uncertain one with the least amount of knowledge. Therefore, its uncertainty measure should be 1 and the knowledge amount is 0. It can be seen that only E

^{2}

_{HC}cannot produce reasonable results for AIFS A

_{9}= <x,0,0>. For AIFS A

_{8}= <x,1,0>, it is the most certain one conveying the maximum knowledge amount. Hence, its uncertainty degree is 0 and the knowledge amount is 1 since the results were produced by all measures in Table 2. Moreover, we can see that all entropy measures may bring counter-intuitive results, which have been highlighted in a bold type in Table 2. These unreasonable results show that these entropy measures are not competent to distinguishing different AIFSs. The knowledge measure K

_{SKB}assigned the same knowledge amount to AIFSs A

_{3}= <x,0.5,0> and A

_{4}= <x,0.5,0.5>, which indicates the poorer discriminant ability of K

_{SKB}. By the proposed axiomatic definition, we cannot rank the knowledge amount of A

_{3}and A

_{4}since ${\Delta}_{{A}_{4}}$ and ${\pi}_{{A}_{4}}$ is greater that ${\Delta}_{{A}_{5}}$ and ${\pi}_{{A}_{5}}$, respectively. Comparing A

_{1}and A

_{3}, we find that ${\Delta}_{{A}_{1}}={\Delta}_{{A}_{3}}$ and ${\pi}_{{A}_{1}}<{\pi}_{{A}_{3}}$. Therefore, A

_{1}has a greater knowledge amount than A

_{3}, which can be yielded by all knowledge measures. For AIFSs A

_{1}and A

_{5}, they have the same hesitancy degree. The greater difference between the membership and non-membership grades of A

_{1}brings more knowledge amount than A

_{5}, as shown by the results of all knowledge measures. We can also rank the knowledge amount of AIFSs A

_{2}, A

_{6}, and A

_{7}in the same way. We note that AIFSs A

_{4}, A

_{7}, and A

_{9}have the same $\Delta $, so the less hesitancy degree indicates a greater knowledge amount. Thus, the knowledge conveyed by A

_{4}, A

_{7}, and A

_{9}should be ranked as K(A

_{4}) > K(A

_{7}) > K(A

_{9}). It is shown that all knowledge measures can provide this ranking order. This example tells us that our proposed knowledge measure is effective to distinguish the knowledge amount of different AIFSs.

**Example**

**3.**

^{0.5}, A

^{2}, A

^{3}and A

^{4}can be regarded as “More or less LARGE”, “Very LARGE”, “Quite very LARGE”, and “Very very LARGE”, respectively.

^{0.5}to A

^{4}, the uncertainty hidden in them becomes less and the knowledge amount conveyed by them increases. So the following relations hold:

_{ZL}, E

_{BB}, E

_{VS}, E

^{2}

_{HC}, and knowledge measures K

_{SKB}, K

_{N}, and K

_{G}are employed to facilitate analysis. In Table 3, we present the results obtained based on different measures to facilitate comparative analysis.

^{0.5}when entropy measures E

_{ZL}, E

_{BB}, and E

_{SK}are applied. The ranking orders obtained based on these measures are listed below.

^{2}

_{HC}and E

_{VS}perform well. This illustrates that these entropy measures are not robust enough to distinguish the uncertainty of AIFSs with linguistic information.

_{SKB}, K

_{N}, and K

_{G}are also not reasonable, which are shown as the equations below.

_{S}indicates that:

_{SVB}, K

_{N}, and K

_{G}are not suitable for differentiating the knowledge amount conveyed by AIFSs. The effectiveness of our proposed knowledge measure K

^{I}is indicated by this example once again.

_{ZL}, E

_{BB}, E

_{SK}, E

_{VS}, and E

^{2}

_{HC}perform poor because of their lack of robustness and discriminability. Our proposed knowledge measure performs much better than knowledge measures K

_{SVB}, K

_{N}, and K

_{G}.

## 5. Application in Solving Intuitionistic Fuzzy MADM

_{i}with respect to the attribute A

_{j}, provided by the decision-maker in the form of an intuitionistic fuzzy value.

_{i}can be aggregated to an intuitionistic fuzzy value z

_{i}, which is expressed as:

_{i}) and accuracy function H(z

_{i}) of z

_{i}($i=1,2,\cdots ,m$) can be calculated as:

**Example**

**4.**

- G
_{1}: A cell phone company. - G
_{2}: A food company. - G
_{3}: An automobile sales company. - G
_{4}: A computer company. - G
_{5}: a TV company.

- A
_{1}: The investment risk. - A
_{2}: The capital gain. - A
_{3}: The social and political impact. - A
_{4}: The environmental impact.

**Case**

**1.**

_{1}= <0.3738, 0.5218>, Z

_{2}= <0.6203, 0.2962>, Z

_{3}= <0.5568, 0.3327>,

_{4}= <0.4787, 0.3323>, Z

_{5}= <0.5776, 0.1990>.

_{1}) = −0.1480, S(Z

_{2}) = 0.3241, S(Z

_{3}) = 0.2240, S(Z

_{4}) = 0.1464, S(Z

_{5}) = 0.3787.

**Case**

**2.**

_{1}= <0.4259,0.4697>, Z

_{2}= <0.6459,0.2806>, Z

_{3}= <0.5561,0.3493>,

_{4}= <0.5616,0.2740>, Z

_{5}= <0.5660,0.2064>.

_{1}) = −0.0438, S(Z

_{2}) = 0.3562, S(Z

_{3}) = 0.2069, S(Z

_{4}) = 0.2876, S(Z

_{5}) = 0.3596.

_{S}and the method in Reference [44] can take G

_{5}as the best choice for investment. Even though the order between G

_{3}and G

_{4}obtained by our method is different from that obtained by Xia and Xu’ method, this difference has no effect on choosing the best company to invest. Actually, the solution of an MADM problem only concerns the best alternative. The order of other alternatives is beyond the ultimate goal of an MADM problem.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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E_{ZL} | E_{BB} | E_{SK} | E_{VS} | E^{2}_{HC} | K_{SKB} | K_{N} | K_{G} | K_{S} | |
---|---|---|---|---|---|---|---|---|---|

A_{1} = <x,0.5,0.5> | 1 | 0 | 1 | 1 | 0.5 | 0.5 | 0.8660 | 0.5000 | 0.5 |

A_{2} = <x,0.25,0.25> | 1 | 0.5 | 1 | 1 | 0.6250 | 0.25 | 0.4330 | 0.2500 | 0.1038 |

A_{3} = <x,0.25,0.5> | 0.7500 | 0.2500 | 0.6667 | 0.9387 | 0.6250 | 0.5417 | 0.7089 | 0.5313 | 0.2945 |

A_{4} = <x,0.2,0.3> | 0.9 | 0.5 | 0.8750 | 0.9855 | 0.6200 | 0.3125 | 0.4259 | 0.3250 | 0.1106 |

E_{ZL} | E_{BB} | E_{SK} | E_{VS} | E^{2}_{HC} | K_{SKB} | K_{N} | K_{G} | K_{S} | |
---|---|---|---|---|---|---|---|---|---|

A_{1} = <x,0.7,0.2> | 0.5 | 0.1 | 0.3750 | 0.7878 | 0.4600 | 0.7625 | 0.8185 | 0.7250 | 0.5344 |

A_{2} = <x,0.5,0.3> | 0.8 | 0.2 | 0.7143 | 0.9635 | 0.6200 | 0.5429 | 0.7000 | 0.5200 | 0.3211 |

A_{3} = <x,0.5,0> | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5000 | 0.5000 | 0.6250 | 0.2500 |

A_{4} = <x,0.5,0.5> | 1 | 0 | 1 | 1 | 0.5 | 0.5000 | 0.8660 | 0.5000 | 0.5000 |

A_{5} = <x,0.5,0.4> | 0.9 | 0.1 | 0.8333 | 0.992 | 0.58 | 0.5333 | 0.7810 | 0.5050 | 0.3950 |

A_{6} = <x,0.6,0.2> | 0.6 | 0.2 | 0.5 | 0.849 | 0.56 | 0.6500 | 0.7211 | 0.6400 | 0.3919 |

A_{7} = <x,0.4,0.4> | 1 | 0.2 | 1 | 1 | 0.64 | 0.4000 | 0.6928 | 0.4000 | 0.2948 |

A_{8} = <x,1,0> | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |

A_{9} = <x,0,0> | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

**Table 3.**Comparative results of all AIFSs with respect to B (counter-intuitive results are in bold type).

E_{ZL} | E_{BB} | E_{SK} | E_{VS} | E^{2}_{HC} | K_{SKB} | K_{N} | K_{G} | K_{S} | |
---|---|---|---|---|---|---|---|---|---|

A^{0.5} | 0.4291 | 0.0683 | 0.3518 | 0.5640 | 0.3355 | 0.7899 | 0.8680 | 0.7633 | 0.6532 |

A | 0.4400 | 0.0800 | 0.4073 | 0.5233 | 0.3280 | 0.7563 | 0.8641 | 0.7600 | 0.6564 |

A^{2} | 0.2160 | 0.0760 | 0.1677 | 0.3369 | 0.2891 | 0.8782 | 0.8950 | 0.8828 | 0.7579 |

A^{3} | 0.1364 | 0.0752 | 0.1101 | 0.2212 | 0.2602 | 0.9074 | 0.9108 | 0.9230 | 0.8157 |

A^{4} | 0.1082 | 0.0800 | 0.0950 | 0.1612 | 0.2397 | 0.9125 | 0.9133 | 0.9337 | 0.8395 |

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**MDPI and ACS Style**

Wang, G.; Zhang, J.; Song, Y.; Li, Q.
An Entropy-Based Knowledge Measure for Atanassov’s Intuitionistic Fuzzy Sets and Its Application to Multiple Attribute Decision Making. *Entropy* **2018**, *20*, 981.
https://doi.org/10.3390/e20120981

**AMA Style**

Wang G, Zhang J, Song Y, Li Q.
An Entropy-Based Knowledge Measure for Atanassov’s Intuitionistic Fuzzy Sets and Its Application to Multiple Attribute Decision Making. *Entropy*. 2018; 20(12):981.
https://doi.org/10.3390/e20120981

**Chicago/Turabian Style**

Wang, Gang, Jie Zhang, Yafei Song, and Qiang Li.
2018. "An Entropy-Based Knowledge Measure for Atanassov’s Intuitionistic Fuzzy Sets and Its Application to Multiple Attribute Decision Making" *Entropy* 20, no. 12: 981.
https://doi.org/10.3390/e20120981