i.
\(\Rightarrow \) If
\(\overline{A}\) and
\(\overline{B}\) are identical, then
\(a_{1}=a_{2},b_{1}=b_{2},c_{1}=c_{2},d_{1}=d_{2}\) and
\(\eta _{{\overline{A}}}^{1}=\eta _{{\overline{B}}}^{1}, \eta _{{\overline{A}}}^{2}=\eta _{{\overline{B}}}^{2},\ldots ,\eta _{{\overline{A}}}^{p}=\eta _{{\overline{B}}}^{p}, \vartheta _{{\overline{A}}}^{1}=\vartheta _{{\overline{B}}}^{1},\vartheta _{{\overline{A}}}^{2}=\vartheta _{{\overline{B}}}^{2},\ldots ,\vartheta _{{\overline{A}}}^{p}=\vartheta _{{\overline{B}}}^{p}.\) Thus,
\((\min \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\}) =(\max \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})\) and
\(\min \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}= \max \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1}, \eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}\) and
\(\max \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}=\min \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}.\) The degree of similarity between
\(\overline{A}\) and
\(\overline{B}\) is calculated as follows:
$$\begin{aligned} S(\overline{A},\overline{B})= & {} \frac{1}{p}\cdot \left[ \left( 1-\frac{|a_{2}-a_{1}|+|b_{2}-b_{1}|+|c_{2}-c_{1}|+|d_{2}-d_{1}|}{4}\right) \right. \\&\left. \times \frac{(\min \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})+\min \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}+ \max \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}}{(\max \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})+\max \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}+ \min \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}}\right] \\= & {} (1-0)\times 1\\= & {} 1 \end{aligned}$$
\(\Leftarrow S(\overline{A},\overline{B})=1,\) then
$$\begin{aligned} S(\overline{A},\overline{B})= & {} \frac{1}{p}\cdot \left[ \left( 1-\frac{|a_{2}-a_{1}|+|b_{2}-b_{1}|+|c_{2}-c_{1}|+|d_{2}-d_{1}|}{4}\right) \right. \\&\left. \times \frac{(\min \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})+\min \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}+ \max \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}}{(\max \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})+\max \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}+ \min \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}}\right] \\= & {} 1. \end{aligned}$$
It implies that
\(a_{1}=a_{2},b_{1}=b_{2},c_{1}=c_{2},d_{1}=d_{2}\) and
\(\eta _{{\overline{A}}}^{1}=\eta _{{\overline{B}}}^{1}, \eta _{{\overline{A}}}^{2}=\eta _{{\overline{B}}}^{2},\ldots ,\eta _{{\overline{A}}}^{p}=\eta _{{\overline{B}}}^{p}, \vartheta _{{\overline{A}}}^{1}=\vartheta _{{\overline{B}}}^{1},\vartheta _{{\overline{A}}}^{2}=\vartheta _{{\overline{B}}}^{2},\ldots ,\vartheta _{{\overline{A}}}^{p}=\vartheta _{{\overline{B}}}^{p}.\) Thus,
\((\min \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})=(\max \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})\) and
\(\min \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}= \max \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}\) and
\(\max \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}=\min \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}.\) Therefore, normalized ITFM-numbers
\(\overline{A}\) and
\(\overline{B}\) are identical.
ii.
$$\begin{aligned} S(\overline{A},\overline{B})= & {} \frac{1}{p}\cdot \left[ \left( 1-\frac{|a_{2}-a_{1}|+|b_{2}-b_{1}|+|c_{2}-c_{1}|+|d_{2}-d_{1}|}{4}\right) \right. \\&\left. \times \frac{(\min \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})+\min \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}+ \max \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}}{(\max \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})+\max \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}+ \min \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}}\right] \\= & {} \frac{1}{p}\cdot \left[ \left( 1-\frac{|a_{1}-a_{2}|+|b_{1}-b_{2}|+|c_{1}-c_{2}|+|d_{1}-d_{2}|}{4}\right) \right. \\&\left. \times \frac{(\min \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})+\min \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}+ \max \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}}{(\max \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})+\max \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}+ \min \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}}\right] \\= & {} S(\overline{B},\overline{A}). \end{aligned}$$
iii.
$$\begin{aligned} S(\overline{A},\overline{B})= & {} \frac{1}{p}\cdot \left[ \left( 1-\frac{|a_{2}-a_{1}|+|b_{2}-b_{1}|+|c_{2}-c_{1}|+|d_{2}-d_{1}|}{4}\right) \right. \\&\left. \times \frac{(\min \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})+\min \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}+ \max \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}}{(\max \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})+\max \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}+ \min \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}}\right] \\= & {} \frac{1}{p}\cdot \left[ \left( 1-\frac{|a_{2}-a_{1}|+|b_{2}-b_{1}|+|c_{2}-c_{1}|+|d_{2}-d_{1}|}{4}\right) \times 1 \right. \\= & {} 1-|d|. \end{aligned}$$
iv. If
\(\overline{A}, \overline{B},\overline{C}\in \Gamma \), then
\(\overline{A}\subseteq \overline{B}\subseteq \overline{C}\Leftrightarrow \eta _{{\overline{A}}}^{i} \le \eta _{\overline{{B}}}^{i}\le \eta _{\overline{{C}}}^{i}\) and
\( \vartheta _{{\overline{A}}}^{i}\ge \vartheta _{\overline{{B}}}^{i}\ge \vartheta _{\overline{{C}}}^{i}.\) Therefore,
$$\begin{aligned} S(\overline{A},\overline{C})= & {} \frac{1}{p}\cdot \left[ \left( 1-\frac{|a_{3}-a_{1}|+|b_{3}-b_{1}|+|c_{3}-c_{1}|+|d_{3}-d_{1}|}{4}\right) \right. \\&\left. \times \frac{(\min \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(C)^{1},P(C)^{2}, P(C)^{3},P(C)^{4}\})+\min \{(\eta _{{\overline{A}}}^{1},\ldots , \eta _{\overline{{A}}}^{P}),(\eta _{\overline{{C}}}^{1},\eta _{\overline{{C}}}^{2},\ldots , \eta _{\overline{{C}}}^{P})\}+ \max \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}), (\vartheta _{{\overline{C}}}^{1},\ldots ,\vartheta _{\overline{{C}}}^{P})\}}{(\max \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(C)^{1},P(C)^{2}, P(C)^{3},P(C)^{4}\})+\max \{(\eta _{{\overline{A}}}^{1},\ldots , \eta _{\overline{{A}}}^{P}),(\eta _{\overline{{C}}}^{1},\eta _{\overline{{C}}}^{2},\ldots , \eta _{\overline{{C}}}^{P})\}+ \min \{(\vartheta _{{\overline{A}}}^{1},\ldots , \vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{C}}}^{1},\ldots , \vartheta _{\overline{{C}}}^{P})\}}\right] \\\le & {} \frac{1}{p}\cdot \left[ \left( 1-\frac{|a_{2}-a_{1}|+|b_{2}-b_{1}|+|c_{2}- c_{1}|+|d_{2}-d_{1}|}{4}\right) \right. \\&\left. \times \frac{(\min \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})+\min \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}+ \max \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}}{(\max \{P(A)^{1},P(A)^{2},P(A)^{3},P(A)^{4},P(B)^{1},P(B)^{2},P(B)^{3},P(B)^{4}\})+\max \{(\eta _{{\overline{A}}}^{1},\ldots ,\eta _{\overline{{A}}}^{P}),(\eta _{\overline{{B}}}^{1},\eta _{\overline{{B}}}^{2},\ldots ,\eta _{\overline{{B}}}^{P})\}+ \min \{(\vartheta _{{\overline{A}}}^{1},\ldots ,\vartheta _{\overline{{A}}}^{P}),(\vartheta _{{\overline{B}}}^{1},\ldots ,\vartheta _{\overline{{B}}}^{P})\}}\right] \\= & {} S(\overline{A},\overline{B}). \end{aligned}$$
In a similar way, it is easy to prove
\(S(\overline{A},\overline{C})\le S(\overline{B},\overline{C})\).
\(\square \)