4. Limitations of proposed method for ranking of two IVNS
Let\(A_{1}=\left\langle\left[0.1,0.7\right],\left[0.05,0.15\right],\left[0.1,0.3\right]\right\rangle\)and\(\text{\ \ A}_{2}=\left\langle\left[0.2,0.8\right],\left[0.05,0.15\right],\left[0.2,0.4\right]\right\rangle\)be any two IVNS, then according to the proposed method [10, Section 3, Def. 3.2, pp. 381], discussed in Section 2.2,\(M{(A}_{1})=M{(A}_{2})=0.5\). Therefore, according to the proposed method [10, Section 3, Def. 3.2, pp. 381], \(A_{1}=A_{2}\). While, it is obvious that\(\text{\ \ A}_{1}\neq A_{2}\). Hence, the proposed method [10, Section 3, Def. 3.2, pp. 381] for the ranking of two IVNS is not valid.
Similarly, let\(A_{1}=\left\langle\left[0.1,0.7\right],\left[0.1,0.1\right],\left[0.1,0.3\right]\right\rangle\)and\(\text{\ \ A}_{2}=\left\langle\left[0.2,0.8\right],\left[0.1,0.1\right],\left[0.2,0.4\right]\right\rangle\)be any two IVNS, then according to the proposed method [10, Section 3, Def. 3.2, pp. 381], discussed in Section 2.2,\(M{(A}_{1})=M{(A}_{2})=0.5\). Therefore, according to the proposed method [10, Section 3, Def. 3.2, pp. 381], \(A_{1}=A_{2}\). While, it is obvious that\(\text{\ \ A}_{1}\neq A_{2}\). Hence, the proposed method [10, Section 3, Def. 3.2, pp. 381] for the ranking of two IVNS is not valid.
It is observed that several such examples are occurring like this, i.e., let\(A_{1}=\left\langle\left[0.1,0.7\right],\left[0.0,0.2\right],\left[0.1,0.3\right]\right\rangle\)and\(A_{2}=\left\langle\left[0.2,0.8\right],\left[0.0,0.2\right],\left[0.2,0.4\right]\right\rangle\)also, let another example be,\(A_{1}=\left\langle\left[0.1,0.2\right],\left[0.0,0.0\right],\left[0.1,0.2\right]\right\rangle\)and\(A_{2}=\left\langle\left[0.4,0.5\right],\left[0.0,0.0\right],\left[0.4,0.5\right]\right\rangle\)and we observe that each example according to the proposed method [10, Section 3, Def. 3.2, pp. 381], discussed in Section 2.2, results into\(M{(A}_{1})=M{(A}_{2})=0.5\) declaring that \(A_{1}=A_{2}\). While, it is obvious that\(\text{\ \ A}_{1}\neq A_{2}\). Hence, the proposed method [10, Section 3, Def. 3.2, pp. 381] for the ranking of two IVNS is not valid in its present form.