2.2. Proposed score function
To resolve the shortcoming discussed in Section 2.1 of the existing methods [13], Nancy and Garg [10, Section 3, Def. 3.1, pp. 379] proposed the following method for the ranking of two SVNS,\(A_{1}=\left\langle a_{1},b_{1},c_{1}\right\rangle\) and\(A_{2}=\left\langle a_{2},b_{2},c_{2}\right\rangle\). Find\(N\left(A_{1}\right)=\frac{1+(a_{1}-2b_{1}-c_{1})({2-a}_{1}-c_{1})}{2}\)and\(N\left(A_{2}\right)=\frac{1+(a_{2}-2b_{2}-c_{2})({2-a}_{2}-c_{2})}{2}\), and check that \(N\left(A_{1}\right)>N\left(A_{2}\right)\) or\(N\left(A_{1}\right)<N\left(A_{2}\right)\) or\(N\left(A_{1}\right)=N\left(A_{2}\right)\).
  1. If \(N\left(A_{1}\right)>N\left(A_{2}\right)\) then\(A_{1}>A_{2}\).
  2. If \(N\left(A_{1}\right)<N\left(A_{2}\right)\) then\(A_{1}<A_{2}\).
  3. If \(N\left(A_{1}\right)=N\left(A_{2}\right)\) then\(A_{1}=A_{2}\).
Furthermore, Nancy and Garg [10, Section 3, Def. 3.2, pp. 381] 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\(M\left(A_{1}\right)=\frac{4+{(a}_{1}^{L}+a_{1}^{U}-c_{1}^{L}-c_{1}^{U}-2b_{1}^{L}-2b_{1}^{U})(4-a_{1}^{L}{-a}_{1}^{U}-c_{1}^{L}{-c}_{1}^{U})}{8}\)and
\(M\left(A_{2}\right)=\frac{4+{(a}_{2}^{L}+a_{2}^{U}-c_{2}^{L}-c_{2}^{U}-2b_{2}^{L}-2b_{2}^{U})(4-a_{2}^{L}{-a}_{2}^{U}-c_{2}^{L}{-c}_{2}^{U})}{8}\), and check that \(M\left(A_{1}\right)>M\left(A_{2}\right)\) or\(M\left(A_{1}\right)<M\left(A_{2}\right)\) or\(M\left(A_{1}\right)=M\left(A_{2}\right)\).
  1. If \(M\left(A_{1}\right)>M\left(A_{2}\right)\) then\(A_{1}>A_{2}\).
  2. If \(M\left(A_{1}\right)<M\left(A_{2}\right)\) then\(A_{1}<A_{2}\).
  3. If \(M\left(A_{1}\right)=M\left(A_{2}\right)\)then\(\text{\ \ A}_{1}=A_{2}\).
In this note, it is shown that, there exist two different SVNS\(\text{\ \ A}_{1}\) and \(A_{2}\ \)such that\(N{(A}_{1})=N{(A}_{2})\) as well as two different IVNS\(\text{\ \ A}_{1}\) and \(A_{2}\ \)such that\(M{(A}_{1})=M{(A}_{2})\) i.e., the shortcomings, pointed out by Nancy and Garg [10, Section 2, Def. 2.6, Ex. 2.1, Ex. 2.2, pp. 379] in the existing methods [13], is also occurring in the methods proposed by Nancy and Garg [10, Section 2, Def. 3.1, pp. 379; Def. 3.2, pp. 381]. Hence, to propose the valid methods for the ranking of two SVNS as well as the ranking of two IVNS is still an open challenging research problem.