2.1. Existing score function
Sahin [13] proposed the following method for the ranking of two SVNS, \(A_{1}=\left\langle a_{1},b_{1},c_{1}\right\rangle\)and\(\text{\ \ A}_{2}=\left\langle a_{2},b_{2},c_{2}\right\rangle\). Find \(K\left(A_{1}\right)=\frac{1+a_{1}-2b_{1}-c_{1}}{2}\)and \(K\left(A_{2}\right)=\frac{1+a_{2}-2b_{2}-c_{2}}{2}\), and check that \(K\left(A_{1}\right)>K\left(A_{2}\right)\) or\(K\left(A_{1}\right)<K\left(A_{2}\right)\) or\(K\left(A_{1}\right)=K\left(A_{2}\right)\).
  1. If \(K\left(A_{1}\right)>K\left(A_{2}\right)\) then\(A_{1}>A_{2}\).
  2. If \(K\left(A_{1}\right)<K\left(A_{2}\right)\) then\(A_{1}<A_{2}\).
  3. If \(K\left(A_{1}\right)=K\left(A_{2}\right)\) then\(A_{1}=A_{2}\).
Sahin [13] also proposed the following method for the ranking of two IVNS,\(A_{1}=\left\langle\left[a_{1}^{L}{,a}_{1}^{U}\right],\left[b_{1}^{L}{,b}_{1}^{U}\right],\left[c_{1}^{L}{,c}_{1}^{U}\right]\right\rangle\)and\(A_{2}=\left\langle\left[a_{2}^{L}{,a}_{2}^{U}\right],\left[b_{2}^{L}{,b}_{2}^{U}\right],\left[c_{2}^{L},c_{2}^{U}\right]\right\rangle\). Find\(L\left(A_{1}\right)=\frac{2+a_{1}^{L}+a_{1}^{U}-2b_{1}^{L}-2b_{1}^{U}-c_{1}^{L}-c_{1}^{U}}{4}\)and\(L\left(A_{2}\right)=\frac{2+a_{2}^{L}+a_{2}^{U}-2b_{2}^{L}-2b_{2}^{U}-c_{2}^{L}-c_{2}^{U}}{4}\), and check that \(L\left(A_{1}\right)>L\left(A_{2}\right)\) or\(L\left(A_{1}\right)<L\left(A_{2}\right)\) or\(L\left(A_{1}\right)=L\left(A_{2}\right)\).
  1. If \(L\left(A_{1}\right)>L\left(A_{2}\right)\) then\(A_{1}>A_{2}\).
  2. If \(L\left(A_{1}\right)<L\left(A_{2}\right)\) then\(A_{1}<A_{2}\).
  3. If \(L\left(A_{1}\right)=L\left(A_{2}\right)\)then\(\text{\ \ A}_{1}=A_{2}\).
Nancy and Garg [10, Section 2, Def. 2.6, Ex. 2.1, pp. 379] considered two different SVNS,\(A_{1}=\left\langle 0.5,0.2,0.6\right\rangle\)and\(\text{\ \ A}_{2}=\left\langle 0.2,0.2,0.3\right\rangle\) and showed that on considering the existing method [13], the relation\(\text{\ \ A}_{1}=A_{2}\) is obtained. While, it is obvious that \(A_{1}\neq A_{2}\). On the basis of this numerical example, Nancy and Garg [10, Section 2, Def. 2.6, Ex. 2.1, pp. 379] claimed that the existing method [13] for the ranking of SVNS is not valid.
It is pertinent to mention that the SVNS,\(A_{1}=\left\langle 0.5,0.2,0.6\right\rangle\)and\(\text{\ \ A}_{2}=\left\langle 0.2,0.2,0.3\right\rangle\) can also be represented as IVNS,\(A_{1}=\left\langle\left[0.5,0.5\right],\left[0.2,0.2\right],\left[0.6,0.6\right]\right\rangle\)and\(\text{\ \ A}_{2}=\left\langle\left[0.2,0.2\right],\left[0.2,0.2\right],\left[0.3,0.3\right]\right\rangle\). It can be verified that on considering the existing method [13], the relation\(\text{\ \ A}_{1}=A_{2}\) is obtained. While, it is obvious that \(A_{1}\neq A_{2}\). Hence, the existing method [13] for the ranking of IVNS is also not valid.