2. Counter examples for Garg and Kumar‘s the correlation coefficient
As discussed in Section 1, Garg and Kumar [1] have shown that the existing correlation coefficients\(\left(1\right)-\left(3\right)\) fails to identify a suitable classifier for the unknown pattern\(B=\left\{\left\langle x_{1},0.1,0.1\right\rangle,\ \left\langle x_{2},1.0,0.0\right\rangle,\ \left\langle x_{3},0.0,1.0\right\rangle\right\}\)from the three known patterns,\(A_{1}=\left\{\left\langle x_{1},0.4,0.5\right\rangle,\ \left\langle x_{2},0.7,0.1\right\rangle,\ \left\langle x_{3},0.3,0.3\right\rangle\right\}\),\(A_{2}=\left\{\left\langle x_{1},0.5,0.4\right\rangle,\ \left\langle x_{2},0.7,0.2\right\rangle,\ \left\langle x_{3},0.4,0.3\right\rangle\right\}\)and\(A_{3}=\left\{\left\langle x_{1},0.4,0.5\right\rangle,\ \left\langle x_{2},0.7,0.1\right\rangle,\ \left\langle x_{3},0.4,0.3\right\rangle\right\}\). Therefore, it is inappropriate to use the existing correlation coefficients \(\left(1\right)-\left(3\right)\).
On the same direction, in this section two known patterns\(A_{1}=\left\{\left\langle x_{1},0.1,0.4\right\rangle,\ \left\langle x_{2},0.4,0.3\right\rangle,\ \left\langle x_{3},0.25,0.35\right\rangle\right\}\),\(A_{2}=\left\{\left\langle x_{1},0.4,0.1\right\rangle,\ \left\langle x_{2},0.3,0.4\right\rangle,\ \left\langle x_{3},0.35,0.25\right\rangle\right\}\)and an unknown pattern\(B=\left\{\left\langle x_{1},0.3,0.3\right\rangle,\ \left\langle x_{2},0.2,0.2\right\rangle,\ \left\langle x_{3},0.1,0.1\right\rangle\right\}\), represented by IFSs, are considered. Also, the weights of a relative importance are considered as \(\left(0.40,\ 0.45,0.15\right)\), and shown that the correlation coefficients\(\left(5\right)-\left(8\right)\), proposed by Garg and Kumar [1], also fails to identify that either \(A_{1}\) or \(A_{2}\) is a suitable classifier for the unknown pattern \(B\).
To apply the correlation coefficients\(\left(5\right)-\left(8\right)\) [1], proposed by Garg and Kumar [1], firstly, there is a need to transform each element of\(A_{1},A_{2}\), and \(B\) into a CN.
Using the expression \(\left(4\right)\), proposed by Garg and Kumar [1] for transforming an IFN into a CN, the elements\(\left\langle 0.1,0.4\right\rangle,\)\(\left\langle 0.4,0.3\right\rangle\),\(\left\langle 0.25,0.35\right\rangle\),\(\left\langle 0.4,0.1\right\rangle,\)\(\left\langle 0.3,0.4\right\rangle,\)\(\left\langle 0.35,0.25\right\rangle,\)\(\left\langle 0.3,0.3\right\rangle\),\(\left\langle 0.2,0.2\right\rangle\) and\(\left\langle 0.1,0.1\right\rangle\) can be transformed into its equivalent CNs \(\left\langle 0.06,0.58,0.36\right\rangle\),\(\left\langle 0.28,0.54,0.18\right\rangle\),\(\left\langle 0.1625,0.575,0.2625\right\rangle\),\(\left\langle 0.36,0.58,0.06\right\rangle\),\(\left\langle 0.18,0.54,0.28\right\rangle\),\(\left\langle 0.2625,0.575,0.1625\right\rangle\),\(\left\langle 0.21,0.58,0.21\right\rangle\),\(\left\langle 0.16,0.68,0.16\right\rangle\) and\(\left\langle 0.09,0.82,0.09\right\rangle\) respectively. Therefore,\(A_{1}\), \(A_{2}\) and \(B\) in terms of CNs can be rewritten as
\(A_{1}=\left\{\left\langle x_{1},0.06,0.58,0.36\right\rangle,\ \left\langle x_{2},0.28,0.54,0.18\right\rangle,\ \left\langle x_{3},0.1625,0.575,0.2625\right\rangle\right\}\),
\(A_{2}=\left\{\left\langle x_{1},0.36,0.58,0.06\right\rangle,\ \left\langle x_{2},0.18,0.54,0.28\right\rangle,\ \left\langle x_{3},0.2625,0.575,0.1625\right\rangle\right\}\),
and,
\(B=\left\{\left\langle x_{1},0.21,0.58,0.21\right\rangle,\ \left\langle x_{2},0.16,0.68,0.16\right\rangle,\ \left\langle x_{3},0.09,0.82,0.09\right\rangle\right\}\).
Now,
  1. Using the existing expression \(\left(5\right)\), proposed by Garg and Kumar [1] for evaluating the correlation coefficient between IFSs,\(K_{1}\left(A_{1},B\right)=0.946359402\) and\(K_{1}\left(A_{2},B\right)=0.946359402\). Since \(K_{1}\left(A_{1},B\right)=K_{1}\left(A_{2},\ B\right)\)so it is not possible to identify the suitable classifier for the unknown pattern \(B\) from the known patterns \(A_{1}\) and \(A_{2}\). Hence, the limitation pointed out by Garg and Kumar [1] in the existing correlation coefficients\(\left(1\right)-\left(3\right)\), is also occurring in Garg and Kumar’s expression \(\left(5\right)\) [1].
  2. Using the existing expression \(\left(6\right)\), proposed by Garg and Kumar [1] for evaluating the correlation coefficient between the IFS,
\(K_{2}\left(A_{1},B\right)=0.84530981\) and\(K_{2}\left(A_{2},B\right)=0.84530981.\)Since \(K_{2}\left(A_{1},B\right)=K_{2}\left(A_{2},B\right)\) so it is not possible to identify the suitable classifier for the unknown pattern \(B\) from the known patterns \(A_{1}\ \)and \(A_{2}\). Hence, the limitation pointed out by Garg and Kumar [1] in the existing correlation coefficients\(\left(1\right)-\left(3\right)\), is also occurring in Garg and Kumar’s expression \(\left(6\right)\) [1].
Using the existing expression \(\left(7\right)\), proposed by Garg and Kumar [1] for evaluating the correlation coefficient between the IFSs,
\(K_{3}\left(A_{1},B\right)=0.951828261\)and\(\text{\ \ }K_{3}\left(A_{2},B\right)=0.951828261\). Since \(K_{3}\left(A_{1},B\right)=K_{3}\left(A_{2},B\right)\) so it is not possible to identify the suitable classifier for the unknown pattern \(B\) from the known patterns \(A_{1}\) and \(A_{2}\). Hence, the limitation pointed out by Garg and Kumar [1] in the existing correlation coefficients\(\left(1\right)-\left(3\right)\), is also occurring in Garg and Kumar’s expression \(\left(7\right)\) [1].
Using the existing expression \(\left(8\right)\), proposed by Garg and Kumar [1] for evaluating the correlation coefficient between the IFSs,
\(K_{4}\left(A_{1},B\right)=0.881829449\) and\(K_{4}\left(A_{2},B\right)=0.881829449\).
Since \(K_{4}\left(A_{1},B\right)=K_{4}\left(A_{2},B\right)\) so it is not possible to identify the suitable classifier for the unknown pattern \(B\) from the known patterns \(A_{1}\) and \(A_{2}\).
Hence, the limitation pointed out by Garg and Kumar [1] in the existing correlation coefficients\(\left(1\right)-\left(3\right)\), is also occurring in Garg and Kumar’s expression \(\left(8\right)\) [1].
3. Advantages of existing correlation coefficients over Garg and Kumar’s correlation coefficients
It is obvious from Section \(1\) and Section \(2\) that although neither the existing correlation coefficients\(\left(1\right)-\left(3\right)\) nor the correlation coefficients\(\left(5\right)-\left(8\right)\), proposed by Garg and Kumar [1], can be used for identifying a suitable classifier. But, to apply the correlation coefficients\(\left(5\right)-\left(8\right)\), proposed by Garg and Kumar [1], there is a need to transform each element of the known patterns, represented by an intuitionistic fuzzy number, into a CN. While applying the existing correlation coefficients\(\left(1\right)-\left(3\right)\), no such transformation is required i.e., more computational efforts are required for applying the correlation coefficients \(\left(5\right)-\left(8\right)\), proposed by Garg and Kumar [1], as compared to the existing correlation coefficients \(\left(1\right)-\left(3\right)\). Therefore, it is better to use the existing correlation coefficients\(\left(1\right)-\left(3\right)\) as compared to Garg and Kumar’s correlation coefficients \(\left(5\right)-\left(8\right)\)[1].