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As far as the problem of intuitionistic fuzzy cluster analysis is concerned, this paper proposes a new formula of similarity degree with attribute weight of each index. We conduct a fuzzy cluster analysis based on the new intuitionistic fuzzy similarity matrix, which is constructed via this new weighted similarity degree method and can be transformed into a fuzzy similarity matrix. Moreover, an example is given to demonstrate the feasibility and validity of this method.

The fuzzy set theory has been widely used in various fields of modern society since it proposed by Zadeh [

This paper constructs a new similarity degree that is based on the distance of the membership degree, non- membership degree and hesitation degree. Then, considering the risk factors [

Definition 1 [

which is characterized by a membership function:

and a non-membership function:

with the condition:

where

Moreover, for each IFS A in X, if

then

In particular, if

then A reduces to Zadeh’s fuzzy set. Thus, fuzzy sets are the special cases of IFSs.

Let

Atanassov proposed the inclusion relationship of two IFSs [

Let

(1)

(2)

Definition 2 [

Definition 3 [

(1) Reflexivity:

(2) Symmetry:

Definition 4 [

If

(1)

(2)

(3)

(4) If

Then,

In the problem of multi-attribute decision making,

is an attribute set. The

is the attribute value of scheme

ation results. Let

The attribute values of scheme

For convenience, the membership degree distance and non-membership degree distance between

Theorem 1. Let

Proof: According to the 4 conditions in Definition 4, the process of this proof are as follows:

(1) Since

Hence, we have

Thus,

Since

Therefore,

(2) When

Therefore, we get

(3) Since

Thus, we have

(4) If

And since

Similarly, we can obtain

Thus,

i.e.,

and

Hence, we have

Thus, we can get

To sum up, the proof is completed.

Therefore, the intuitionistic fuzzy similarity degree between two schemes can be obtained by the Theorem 1, which can get the IFSM

there into,

In this paper, we select a case from Literature [

G_{1} | G_{2} | G_{3} | G_{4} | G_{5} | G_{6} | |
---|---|---|---|---|---|---|

A_{1} | (0.3, 0.5) | (0.6, 0.1) | (0.4, 0.3) | (0.8, 0.1) | (0.1, 0.6) | (0.5, 0.4) |

A_{2} | (0.6, 0.3) | (0.5, 0.2) | (0.6, 0.1) | (0.7, 0.1) | (0.3, 0.6) | (0.4, 0.3) |

A_{3} | (0.4, 0.4) | (0.8, 0.1) | (0.5, 0.1) | (0.6, 0.2) | (0.4, 0.5) | (0.3, 0.2) |

A_{4} | (0.2, 0.4) | (0.4, 0.1) | (0.9, 0.0) | (0.8, 0.1) | (0.2, 0.5) | (0.7, 0.1) |

A_{5} | (0.5, 0.2) | (0.3, 0.6) | (0.6, 0.3) | (0.7, 0.1) | (0.6, 0.2) | (0.5, 0.3) |

In the problem of multi-attribute decision making, the evaluation results are impacted by each attribute index on different extent, so it is necessary to give a reasonable weight coefficient for each index. The weights of the six evaluation factors are obtained via the intuitionistic fuzzy entropy calculating the weights’ method proposed by Szmidt Eulalia [

By the formula of Theorem 1, we can get a five orders IFSM with five different vehicles between each pair of similarity degree:

In order to make the IFCA more convenient, the value of similarity degree is transformed from IFN to fuzzy numbervia Equation (1):

The results of the fuzzy clustering analysis through maximal tree method are shown in

The results of clustering analysis are as follows:

Cutting off

From the above results, it is clear that the clustering method proposed in this paper and the literature [

To sum up, we can see that the new similarity degree proposed in this paper is correct and effective, and it has the following advantages:

(1) With the form of IFN, the information of the data is fully extracted;

(2) Taking into account the weight of the index attribute, the calculation results are more reasonable;

(3) The new similarity degree takes into account the membership degree and the non-membership degree, and it fully extracts the information provided by the intuitionistic fuzzy numbers;

(4) Computational process is simple and easy to operate.

This paper proposes a new method to compute intuitionistic fuzzy similarity degree. The formula is not only considered with the weight of each index attribute but also expressed by an IFN. It makes the information fully extracted. Meanwhile, the computational process is simple and convenient. Considering the risk factor, we obtain IFSM through the new similarity degree. Then, the IFSM is converted to a fuzzy similarity matrix. Finally, we get the result of cluster analysis through the method of maximum tree. In this paper, we prove the correctness of the new similarity degree and illustrate the validity and rationality of the method with an example. This method extends the research space of the intuitionistic fuzzy similarity degree.

Supported by China West Normal University special funding for basic scientific research business expenses (no.14C004), Social Science Programming general Program of Nan Chong city (no.NC2013B027).

QunLiu,ChanghuanFeng, (2016) A New Definition of Intuitionistic Fuzzy Similarity Degree. Open Journal of Statistics,06,31-36. doi: 10.4236/ojs.2016.61005