Intuitionistic Fuzzy Clustering Algorithms - Intuitionistic Fuzzy Aggregation and Clustering - page 248

n

1
n

d NH ( y i , y k ) =

1 | μ ij − μ kj | ,

i , k = 1 , 2 ,..., m (2.187)

j =

Obviously, the distances ( 2.186 ) and ( 2.187 ) show the closeness degrees of the
characteristics of each two alternatives y i and y k . The smaller the values of d NH (

y i , y k )

and d NH ( are, the more similar the two alternatives y i and y k .

In an intuitionistic fuzzy similarity matrix, each of its elements is an IFV. To get
an intuitionistic fuzzy closeness degrees of y i and y k , we may consider the value of
d NH (

y i , y k )

y i , y k ) as a non-membership degree v ik , and then it may be hopeful to define ˙

n

1
n

μ ik = 1 −

1 | μ ij − μ kj | ,

i , k = 1 , 2 ,..., m (2.188)

j =

as a membership degree. Now we need to check whether 0 ≤ μ ik +˙ v ik ≤ 1 holds or

not. However,

n n

1
n

1
n

μ ik +˙ v ik = 1 −

1 | μ ij − μ kj |+ 1 |

v ij − v kj |≥ 0 (2.189)

j = j =

n n

1
n

1
n

μ ik +˙ v ik = 1 −

1 | μ ij − μ kj |+ 1 |

v ij − v kj |

j = j =

n n

1
n

1
n

= 1 −

1 | (

1 − μ ij ) − ( 1 − μ kj ) |+

1 |

v ij − v kj | (2.190)

j = j =

n n

1
n

1
n

= 1 −

1 | (

v ij + π ij ) − ( v kj + π kj ) |+

1 |

v ij − v kj |

j = j =

n n

1
n

1
n

= 1 −

1 | (

v ij − v kj ) + (π ij − π kj ) |+

1 |

v ij − v kj |

j = j =

n n n

1
n

1
n

1
n

≥ 1 −

1 |

v ij − v kj |−

1 | π ij − π kj |+ 1 |

v ij − v kj |

j = j = j =

n

1
n

= 1 −

1 | π ij − π kj | ,

i , k = 1 , 2 ,..., m (2.191)

j =

where 1 cannot be guaranteed.

In the numerical analysis above, we can see that in an IFV, the membership degree
is closely related to both the non-membership and the uncertainty degree. Motivated
by this idea, we may modify Eq. ( 2.188 )as:

π ij = 1 − μ ij −˙ v ij . Thus, 0 ≤ μ ik +˙ v ik ≤