Picture Fuzzy N-Soft Sets and Their Applications in Decision-Making Problems

Figures & data

Figure 1. Flowchart of FSs, SSs, and their generalisation.

Table 1. Information obtained from the associated data.

$\mathfrak{U}$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$\star \star \star$	$\star$	$\star \star$
${\mathfrak{u}}_2$	$\star$	$◻$	$\star \star \star$
${\mathfrak{u}}_3$	$\star \star$	$\star$	$\star \star \star \star$
${\mathfrak{u}}_4$	$\star \star \star$	$\star \star \star \star$	$\star \star \star$
${\mathfrak{u}}_5$	$\star \star$	$\star \star \star$	$◻$

Table 2. The tabular form of 5-SS.

$\left(\mathrm{I},\mathfrak{T},5\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	3	1	2
${\mathfrak{u}}_2$	1	0	3
${\mathfrak{u}}_3$	2	1	4
${\mathfrak{u}}_4$	3	4	3
${\mathfrak{u}}_5$	2	3	0

Table 3. The tabular representation of the picture fuzzy 5-soft set $\left(\mathit{Y},\left(\mathbf{I},\mathfrak{T},5\right)\right)$ in Example 1.

$\left(\mathit{Y},(\mathbf{I},\mathfrak{T},5)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$3,\left(0.56,0.1,0.2\right)$	$1,\left(0.2,0.3,0.4\right)$	$2,\left(0.5,0.15,0.3\right)$
${\mathfrak{u}}_2$	$1,\left(0.25,0.2,0.5\right)$	$0,\left(0.09,0.3,0.5\right)$	$3,\left(0.75,0.1,0.1\right)$
${\mathfrak{u}}_3$	$2,\left(0.45,0.15,0.3\right)$	$1,\left(0.27,0.5,0.1\right)$	$4,\left(0.8,0.05,0.1\right)$
${\mathfrak{u}}_4$	$3,\left(0.7,0.1,0.1\right)$	$4,\left(0.9,0.05,0.04\right)$	$3,\left(0.79,0.08,0.05\right)$
${\mathfrak{u}}_5$	$2,\left(0.4,0.2,0.3\right)$	$3,\left(0.65,0.10.15\right)$	$0,\left(0.05,0.4,0.5\right)$

Table 4. Information obtained from the associated data.

$\mathfrak{U}$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$◻$	$\star \star$	$\star \star \star$
${\mathfrak{u}}_2$	$\star \star$	$\star \star \star$	$\star \star$
${\mathfrak{u}}_3$	$\star \star$	$\star \star \star$	$\star$
${\mathfrak{u}}_4$	$◻$	$\star \star$	$\star \star$

Table 5. The tabular form of 5-SS.

$\left(\mathbf{I},\mathfrak{T},4\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	0	2	3
${\mathfrak{u}}_2$	2	3	2
${\mathfrak{u}}_3$	2	3	1
${\mathfrak{u}}_4$	0	2	2

Table 6. The tabular representation of the picture fuzzy 4-soft set $\left(\mathit{Y},\left(\mathbf{I},\mathfrak{T},4\right)\right)$ in Example 2.

$\left(\mathit{Y},\left(\mathbf{I},\mathfrak{T},4\right)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$0,\left(0.16,0.4,0.1\right)$	$2,\left(0.6,0.2,0.11\right)$	$3,\left(0.88,0.05,0.03\right)$
${\mathfrak{u}}_2$	$2,\left(0.65,0.1,0.15\right)$	$3,\left(0.8,0.13,0.05\right)$	$2,\left(0.58,0.3,0.1\right)$
${\mathfrak{u}}_3$	$2,\left(0.65,0.25,0.05\right)$	$3,\left(0.9,0.05,0.01\right)$	$1,\left(0.4,0.2,0.4\right)$
${\mathfrak{u}}_4$	$0,\left(0.1,0.4,0.4\right)$	$2,\left(0.7,0.1,0.0\right)$	$2,\left(0.7,0.05,0.2\right)$

Table 7. The tabular representation of the PF4-SS $\left({\mathit{Y}}_1,{\mathit{\Theta }}_1=\left({\mathbf{I}}_1,{\mathfrak{T}}_1,4\right)\right)$ assumed in Example 3.

$\left({\mathit{Y}}_1,\left({\mathbf{I}}_1,{\mathfrak{T}}_1,4\right)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$
${\mathfrak{u}}_1$	$2,\left(0.5,0.1,0.3\right)$	$1,\left(0.27,0.2,0.4\right)$
${\mathfrak{u}}_2$	$1,\left(0.25,0.1,0.5\right)$	$0,\left(0.05,0.1,0.7\right)$
${\mathfrak{u}}_3$	$1,\left(0.3,0.2,0.4\right)$	$1,\left(0.26,0.3,0.2\right)$
${\mathfrak{u}}_4$	$2,\left(0.7,0.1,0.1\right)$	$3,\left(0.85,0.05,0.09\right)$

Table 8. The tabular representation of the PF5-SS $\left({\mathit{Y}}_2,{\mathit{\Theta }}_2=\left({\mathbf{I}}_2,{\mathfrak{T}}_2,5\right)\right)$ assumed in Example 3.

$\left({\mathit{Y}}_2,\left({\mathbf{I}}_2,{\mathfrak{T}}_2,5\right)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$3,\left(0.56,0.1,0.2\right)$	$1,\left(0.29,0.3,0.3\right)$	$2,\left(0.5,0.15,0.3\right)$
${\mathfrak{u}}_2$	$1,\left(0.25,0.2,0.4\right)$	$0,\left(0.09,0.3,0.5\right)$	$3,\left(0.75,0.1,0.1\right)$
${\mathfrak{u}}_3$	$2,\left(0.45,0.25,0.2\right)$	$1,\left(0.27,0.5,0.1\right)$	$4,\left(0.8,0.05,0.1\right)$
${\mathfrak{u}}_4$	$3,\left(0.75,0.15,0.05\right)$	$4,\left(0.9,0.05,0.04\right)$	$3,\left(0.79,0.08,0.05\right)$

Table 9. The tabular representation of the PF5-SS $\left({\mathit{Y}}_1,{\mathit{\Theta }}_1=\left({\mathbf{I}}_1,{\mathfrak{T}}_1,5\right)\right)$ assumed in Example 4.

$\left({\mathit{Y}}_1,\left({\mathbf{I}}_1,{\mathfrak{T}}_1,5\right)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$
${\mathfrak{u}}_1$	$3,\left(0.56,0.1,0.2\right)$	$1,\left(0.29,0.3,0.3\right)$
${\mathfrak{u}}_2$	$1,\left(0.25,0.2,0.4\right)$	$0,\left(0.09,0.3,0.5\right)$
${\mathfrak{u}}_3$	$2,\left(0.45,0.25,0.2\right)$	$1,\left(0.27,0.5,0.1\right)$
${\mathfrak{u}}_4$	$3,\left(0.75,0.15,0.05\right)$	$4,\left(0.9,0.05,0.04\right)$

Table 10. The tabular representation of the PF5-SS $\left({\mathit{Y}}_2,{\mathit{\Theta }}_2=\left({\mathbf{I}}_2,{\mathfrak{T}}_2,5\right)\right)$ provided in Example 4.

$\left({\mathit{Y}}_2,\left({\mathbf{I}}_2,{\mathfrak{T}}_2,5\right)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$3,\left(0.56,0.1,0.2\right)$	$1,\left(0.29,0.3,0.3\right)$	$2,\left(0.5,0.15,0.3\right)$
${\mathfrak{u}}_2$	$1,\left(0.25,0.2,0.4\right)$	$0,\left(0.09,0.3,0.5\right)$	$3,\left(0.75,0.1,0.1\right)$
${\mathfrak{u}}_3$	$2,\left(0.45,0.25,0.2\right)$	$1,\left(0.27,0.5,0.1\right)$	$4,\left(0.8,0.05,0.1\right)$
${\mathfrak{u}}_4$	$3,\left(0.75,0.15,0.05\right)$	$4,\left(0.9,0.05,0.04\right)$	$3,\left(0.79,0.08,0.05\right)$

Table 11. The tabular representation of the PF5-SS $\left(\mathit{Y},\mathit{\Theta }=\left(\mathbf{I},\mathfrak{T},5\right)\right)$ provided in Example 5 which is efficient.

$\left(\mathit{Y},\left(\mathbf{I},\mathfrak{T},5\right)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$3,\left(0.56,0.1,0.2\right)$	$1,\left(0.29,0.3,0.3\right)$	$2,\left(0.5,0.15,0.3\right)$
${\mathfrak{u}}_2$	$1,\left(0.25,0.2,0.4\right)$	$0,\left(0.09,0.3,0.5\right)$	$3,\left(0.75,0.1,0.1\right)$
${\mathfrak{u}}_3$	$2,\left(0.45,0.25,0.2\right)$	$1,\left(0.27,0.5,0.1\right)$	$4,\left(0.8,0.05,0.1\right)$
${\mathfrak{u}}_4$	$3,\left(0.75,0.15,0.05\right)$	$4,\left(1.0,0.0,0.0\right)$	$3,\left(0.79,0.08,0.05\right)$

Table 12. The picture fuzzy weak complement of PF5-SS $\left(\mathit{Y},\mathit{\Theta }=\left(\mathbf{I},\mathfrak{T},5\right)\right)$ in Example 1.

$\left({\mathit{Y}}^{\mathit{c}},\left({\mathbf{I}}^{\mathit{c}},\mathfrak{T},5\right)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$4,\left(0.2,0.1,0.56\right)$	$2,\left(0.4,0.3,0.2\right)$	$1,\left(0.3,0.15,0.5\right)$
${\mathfrak{u}}_2$	$0,\left(0.5,0.2,0.25\right)$	$2,\left(0.5,0.3,0.09\right)$	$1,\left(0.1,0.1,0.75\right)$
${\mathfrak{u}}_3$	$4,\left(0.3,0.15,0.45\right)$	$2,\left(0.1,0.5,0.27\right)$	$1,\left(0.1,0.05,0.8\right)$
${\mathfrak{u}}_4$	$4,\left(0.1,0.1,0.7\right)$	$2,\left(0.04,0.05,0.9\right)$	$1,\left(0.05,0.08,0.79\right)$
${\mathfrak{u}}_5$	$4,\left(0.3,0.2,0.4\right)$	$2,\left(0.15,0.10.65\right)$	$1,\left(0.5,0.4,0.05\right)$

Table 13. The top picture fuzzy weak complement of PF5-SS $\left(\mathit{Y},\mathit{\Theta }=\left(\mathbf{I},\mathfrak{T},5\right)\right)$ in Example 1.

$\left({\mathit{Y}}^{\mathit{c}},\left({\mathbf{I}}^{\mathit{T}},\mathfrak{T},5\right)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$4,\left(0.2,0.1,0.56\right)$	$4,\left(0.4,0.3,0.2\right)$	$4,\left(0.3,0.15,0.5\right)$
${\mathfrak{u}}_2$	$4,\left(0.5,0.2,0.25\right)$	$4,\left(0.5,0.3,0.09\right)$	$4,\left(0.1,0.1,0.75\right)$
${\mathfrak{u}}_3$	$4,\left(0.3,0.15,0.45\right)$	$4,\left(0.1,0.5,0.27\right)$	$0,\left(0.1,0.05,0.8\right)$
${\mathfrak{u}}_4$	$4,\left(0.1,0.1,0.7\right)$	$0,\left(0.04,0.05,0.9\right)$	$4,\left(0.05,0.08,0.79\right)$
${\mathfrak{u}}_5$	$4,\left(0.3,0.2,0.4\right)$	$4,\left(0.15,0.10.65\right)$	$4,\left(0.5,0.4,0.05\right)$

Table 14. The bottom picture fuzzy weak complement of PF5-SS $\left(\mathit{Y},\mathit{\Theta }=\left(\mathbf{I},\mathfrak{T},5\right)\right)$ in Example 1.

$\left({\mathit{Y}}^{\mathit{c}},\left({\mathbf{I}}^{\mathit{B}},\mathfrak{T},5\right)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$0,\left(0.2,0.1,0.56\right)$	$0,\left(0.4,0.3,0.2\right)$	$0,\left(0.3,0.15,0.5\right)$
${\mathfrak{u}}_2$	$0,\left(0.5,0.2,0.25\right)$	$4,\left(0.5,0.3,0.09\right)$	$0,\left(0.1,0.1,0.75\right)$
${\mathfrak{u}}_3$	$0,\left(0.3,0.15,0.45\right)$	$0,\left(0.1,0.5,0.27\right)$	$4,\left(0.1,0.05,0.8\right)$
${\mathfrak{u}}_4$	$0,\left(0.1,0.1,0.7\right)$	$0,\left(0.04,0.05,0.9\right)$	$0,\left(0.05,0.08,0.79\right)$
${\mathfrak{u}}_5$	$0,\left(0.3,0.2,0.4\right)$	$0,\left(0.15,0.10.65\right)$	$0,\left(0.5,0.4,0.05\right)$

Table 15. The tabular form of PF5-SS of Example 9.

$\left({\mathit{Y}}_1,\left({\mathbf{I}}_1,{\mathfrak{T}}_1,5\right)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_4$
${\mathfrak{u}}_1$	$2,\left(0.5,0.1,0.3\right)$	$2,\left(0.57,0.2,0.2\right)$	$0,\left(0.1,0.2,0.6\right)$
${\mathfrak{u}}_2$	$1,\left(0.25,0.1,0.5\right)$	$0,\left(0.15,0.1,0.6\right)$	$4,\left(0.9,0.03,0.06\right)$
${\mathfrak{u}}_3$	$1,\left(0.3,0.2,0.4\right)$	$2,\left(0.46,0.3,0.2\right)$	$3,\left(0.76,0.1,0.1\right)$
${\mathfrak{u}}_4$	$3,\left(0.7,0.1,0.1\right)$	$4,\left(0.85,0.05,0.09\right)$	$4,\left(0.87,0.02,0.1\right)$

Table 16. The tabular form of PF6-SS of Example 9.

$\left({\mathit{Y}}_2,\left({\mathbf{I}}_2,{\mathfrak{T}}_2,6\right)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$3,\left(0.6,0.1,0.2\right)$	$2,\left(0.49,0.1,0.2\right)$	$5,\left(0.95,0.01,0.02\right)$
${\mathfrak{u}}_2$	$1,\left(0.2,0.2,0.4\right)$	$1,\left(0.2,0.3,0.4\right)$	$4,\left(0.75,0.1,0.1\right)$
${\mathfrak{u}}_3$	$5,\left(0.85,0.1,0.02\right)$	$0,\left(0.13,0.5,0.1\right)$	$1,\left(0.25,0.4,0.3\right)$
${\mathfrak{u}}_4$	$2,\left(0.45,0.15,0.3\right)$	$4,\left(0.7,0.15,0.1\right)$	$3,\left(0.69,0.1,0.1\right)$

Table 17. The restricted intersection of $\left({\mathit{Y}}_1,{\mathit{\Theta }}_1=\left({\mathbf{I}}_1,{\mathfrak{T}}_1,5\right)\right)$ and $\left({\mathit{Y}}_2,{\mathit{\Theta }}_2=\left({\mathbf{I}}_2,{\mathfrak{T}}_2,5\right)\right)$.

$\left(\mathbf{\Lambda },{\mathbf{\Theta }}_1{\cap }_{\mathrm{\Re }}{\mathbf{\Theta }}_2,5\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$
${\mathfrak{u}}_1$	$2,\left(0.5,0.1,0.3\right)$	$2,\left(0.49,0.1,0.2\right)$
${\mathfrak{u}}_2$	$1,\left(0.2,0.1,0.5\right)$	$0,\left(0.15,0.1,0.6\right)$
${\mathfrak{u}}_3$	$1,\left(0.3,0.1,0.4\right)$	$0,\left(0.13,0.3,0.2\right)$
${\mathfrak{u}}_4$	$2,\left(0.45,0.1,0.3\right)$	$4,\left(0.7,0.05,0.1\right)$

Table 18. The restricted union of $\left({\mathit{Y}}_1,{\mathit{\Theta }}_1=\left({\mathbf{I}}_1,{\mathfrak{T}}_1,5\right)\right)$ and $\left({\mathit{Y}}_2,{\mathit{\Theta }}_2=\left({\mathbf{I}}_2,{\mathfrak{T}}_2,5\right)\right)$.

$\left(\mathbf{\Delta },{\mathbf{\Theta }}_1{\cup }_{\mathrm{\Re }}{\mathbf{\Theta }}_2,6\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$
${\mathfrak{u}}_1$	$3,\left(0.6,0.1,0.2\right)$	$2,\left(0.57,0.1,0.2\right)$
${\mathfrak{u}}_2$	$1,\left(0.25,0.1,0.4\right)$	$1,\left(0.2,0.1,0.4\right)$
${\mathfrak{u}}_3$	$5,\left(0.85,0.1,0.02\right)$	$2,\left(0.46,0.3,0.1\right)$
${\mathfrak{u}}_4$	$3,\left(0.7,0.1,0.1\right)$	$4,\left(0.85,0.05,0.09\right)$

Table 19. The extended intersection of $\left({\mathit{Y}}_1,{\mathit{\Theta }}_1=\left({\mathbf{I}}_1,{\mathfrak{T}}_1,5\right)\right)$ and $\left({\mathit{Y}}_2,{\mathit{\Theta }}_2=\left({\mathbf{I}}_2,{\mathfrak{T}}_2,5\right)\right)$.

$\left(\mathbf{\Lambda },{\mathbf{\Theta }}_1{\cap }_{\mathcal{E}}{\mathbf{\Theta }}_2,6\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$	${\mathcal{t}}_4$
${\mathfrak{u}}_1$	$2,\left(0.5,0.1,0.3\right)$	$2,\left(0.49,0.1,0.2\right)$	$5,\left(0.95,0.01,0.02\right)$	$0,\left(0.1,0.2,0.6\right)$
${\mathfrak{u}}_2$	$1,\left(0.2,0.1,0.5\right)$	$0,\left(0.15,0.1,0.6\right)$	$4,\left(0.75,0.1,0.1\right)$	$4,\left(0.9,0.03,0.06\right)$
${\mathfrak{u}}_3$	$1,\left(0.3,0.1,0.4\right)$	$0,\left(0.13,0.3,0.2\right)$	$1,\left(0.25,0.4,0.3\right)$	$3,\left(0.76,0.1,0.1\right)$
${\mathfrak{u}}_4$	$2,\left(0.45,0.1,0.3\right)$	$4,\left(0.7,0.05,0.1\right)$	$3,\left(0.69,0.1,0.1\right)$	$4,\left(0.87,0.02,0.1\right)$

Table 20. The extended union of $\left({\mathit{Y}}_1,{\mathit{\Theta }}_1=\left({\mathbf{I}}_1,{\mathfrak{T}}_1,5\right)\right)$ and $\left({\mathit{Y}}_2,{\mathit{\Theta }}_2=\left({\mathbf{I}}_2,{\mathfrak{T}}_2,5\right)\right)$.

$\left(\mathbf{\Delta },{\mathbf{\Theta }}_1{\cup }_{\mathcal{E}}{\mathbf{\Theta }}_2,6\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$	${\mathcal{t}}_4$
${\mathfrak{u}}_1$	$3,\left(0.6,0.1,0.2\right)$	$2,\left(0.57,0.1,0.2\right)$	$5,\left(0.95,0.01,0.02\right)$	$0,\left(0.1,0.2,0.6\right)$
${\mathfrak{u}}_2$	$1,\left(0.25,0.1,0.4\right)$	$1,\left(0.2,0.1,0.4\right)$	$4,\left(0.75,0.1,0.1\right)$	$4,\left(0.9,0.03,0.06\right)$
${\mathfrak{u}}_3$	$5,\left(0.85,0.1,0.02\right)$	$2,\left(0.46,0.3,0.1\right)$	$1,\left(0.25,0.4,0.3\right)$	$3,\left(0.76,0.1,0.1\right)$
${\mathfrak{u}}_4$	$3,\left(0.7,0.1,0.1\right)$	$4,\left(0.85,0.05,0.09\right)$	$3,\left(0.69,0.1,0.1\right)$	$4,\left(0.87,0.02,0.1\right)$

Table 21. The tabular form of PFSS is associated with threshold 1.

$\left({\mathit{Y}}^1,\mathfrak{T}\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$1.0,0.0,0.0$	$1.0,0.0,0.0$	$1.0,0.0,0.0$
${\mathfrak{u}}_2$	$1.0,0.0,0.0$	$0.0,0.0,1.0$	$1.0,0.0,0.0$
${\mathfrak{u}}_3$	$1.0,0.0,0.0$	$1.0,0.0,0.0$	$1.0,0.0,0.0$
${\mathfrak{u}}_4$	$1.0,0.0,0.0$	$1.0,0.0,0.0$	$1.0,0.0,0.0$
${\mathfrak{u}}_5$	$1.0,0.0,0.0$	$1.0,0.0,0.0$	$0.0,0.0,1.0$

Table 22. The tabular form of PFSS is associated with threshold 2.

$\left({\mathit{Y}}^2,\mathfrak{T}\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$1.0,0.0,0.0$	$0.0,0.0,1.0$	$1.0,0.0,0.0$
${\mathfrak{u}}_2$	$0.0,0.0,1.0$	$0.0,0.0,1.0$	$1.0,0.0,0.0$
${\mathfrak{u}}_3$	$1.0,0.0,0.0$	$0.0,0.0,1.0$	$1.0,0.0,0.0$
${\mathfrak{u}}_4$	$1.0,0.0,0.0$	$1.0,0.0,0.0$	$1.0,0.0,0.0$
${\mathfrak{u}}_5$	$1.0,0.0,0.0$	$1.0,0.0,0.0$	$0.0,0.0,1.0$

Table 23. The tabular form of PFSS is associated with threshold 3.

$\left({\mathit{Y}}^3,\mathfrak{T}\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$1.0,0.0,0.0$	$0.0,0.0,1.0$	$0.0,0.0,1.0$
${\mathfrak{u}}_2$	$0.0,0.0,1.0$	$0.0,0.0,1.0$	$1.0,0.0,0.0$
${\mathfrak{u}}_3$	$0.0,0.0,1.0$	$0.0,0.0,1.0$	$1.0,0.0,0.0$
${\mathfrak{u}}_4$	$1.0,0.0,0.0$	$1.0,0.0,0.0$	$1.0,0.0,0.0$
${\mathfrak{u}}_5$	$0.0,0.0,1.0$	$1.0,0.0,0.0$	$0.0,0.0,1.0$

Table 24. The tabular form of PFSS is associated with threshold 4.

$\left({\mathit{Y}}^4,\mathfrak{T}\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$0.0,0.0,1.0$	$0.0,0.0,1.0$	$0.0,0.0,1.0$
${\mathfrak{u}}_2$	$0.0,0.0,1.0$	$0.0,0.0,1.0$	$0.0,0.0,1.0$
${\mathfrak{u}}_3$	$0.0,0.0,1.0$	$0.0,0.0,1.0$	$1.0,0.0,0.0$
${\mathfrak{u}}_4$	$0.0,0.0,1.0$	$1.0,0.0,0.0$	$0.0,0.0,1.0$
${\mathfrak{u}}_5$	$0.0,0.0,1.0$	$0.0,0.0,1.0$	$0.0,0.0,1.0$

Figure 2. The flowchart of algorithm defined in Section 4.1.

Table 25. The tabular form of MP in Example 14.

$\mathit{\xi }$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$0.56$	$0.2$	$0.5$
${\mathfrak{u}}_2$	$0.25$	$0.09$	$0.75$
${\mathfrak{u}}_3$	$0.45$	$0.27$	$0.8$
${\mathfrak{u}}_4$	$0.7$	$0.9$	$0.79$
${\mathfrak{u}}_5$	$0.4$	$0.65$	$0.05$

Table 26. The comparison table for MP in Example 14.

$.$	${\mathfrak{u}}_1$	${\mathfrak{u}}_2$	${\mathfrak{u}}_3$	${\mathfrak{u}}_4$	${\mathfrak{u}}_5$
${\mathfrak{u}}_1$	3	2	1	0	2
${\mathfrak{u}}_2$	1	3	1	0	2
${\mathfrak{u}}_3$	2	3	3	1	2
${\mathfrak{u}}_4$	3	3	2	3	3
${\mathfrak{u}}_5$	1	1	1	0	3

Table 27. The score table for MP in Example 14.

	Grade sum $\left(\sum _{\mathcal{i}=1}^5{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	Row sum $\left(\mathit{R}{\mathfrak{s}}_1\right)$	Column sum $\left(\mathit{C}{\mathfrak{s}}_1\right)$	${\mathbf{\Phi }}_1=\mathit{R}{\mathfrak{s}}_1-\mathit{C}{\mathfrak{s}}_1$
${\mathfrak{u}}_1$	6	8	10	−2
${\mathfrak{u}}_2$	4	7	12	−5
${\mathfrak{u}}_3$	7	11	8	3
${\mathfrak{u}}_4$	10	14	4	10
${\mathfrak{u}}_5$	5	6	13	−7

Table 28. The tabular representation of nMP in Example 14.

$\mathit{\delta }$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$0.1$	$0.3$	$0.15$
${\mathfrak{u}}_2$	$0.2$	$0.3$	$0.1$
${\mathfrak{u}}_3$	$0.15$	$0.5$	$0.05$
${\mathfrak{u}}_4$	$0.1$	$0.05$	$0.08$
${\mathfrak{u}}_5$	0.2	0.1	0.4

Table 29. The comparison table for nMP in Example 14.

$.$	${\mathfrak{u}}_1$	${\mathfrak{u}}_2$	${\mathfrak{u}}_3$	${\mathfrak{u}}_4$	${\mathfrak{u}}_5$
${\mathfrak{u}}_1$	3	2	1	3	1
${\mathfrak{u}}_2$	2	3	2	3	2
${\mathfrak{u}}_3$	2	1	3	2	1
${\mathfrak{u}}_4$	1	0	1	3	0
${\mathfrak{u}}_5$	2	2	2	3	3

Table 30. The score table for nMP in Example 14.

	Grade sum $\left(\sum _{\mathcal{i}=1}^5{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	Row sum $\left(\mathit{R}{\mathfrak{s}}_2\right)$	Column sum $\left(\mathit{C}{\mathfrak{s}}_2\right)$	${\mathbf{\Phi }}_2=\mathit{R}{\mathfrak{s}}_2-\mathit{C}{\mathfrak{s}}_2$
${\mathfrak{u}}_1$	6	10	10	0
${\mathfrak{u}}_2$	4	12	8	4
${\mathfrak{u}}_3$	7	9	9	0
${\mathfrak{u}}_4$	10	5	14	−9
${\mathfrak{u}}_5$	5	12	7	5

Table 31. The tabular representation of NMP in Example 14.

$\vartheta$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$0.2$	$0.4$	$0.3$
${\mathfrak{u}}_2$	$0.5$	$0.5$	$0.1$
${\mathfrak{u}}_3$	$0.3$	$0.1$	$0.1$
${\mathfrak{u}}_4$	$0.1$	$0.04$	$0.05$
${\mathfrak{u}}_5$	$0.3$	$0.15$	$0.5$

Table 32. The comparison table for NMP in Example 14.

$.$	${\mathfrak{u}}_1$	${\mathfrak{u}}_2$	${\mathfrak{u}}_3$	${\mathfrak{u}}_4$	${\mathfrak{u}}_5$
${\mathfrak{u}}_1$	3	1	2	3	2
${\mathfrak{u}}_2$	2	3	3	3	2
${\mathfrak{u}}_3$	1	1	3	3	1
${\mathfrak{u}}_4$	0	0	0	3	0
${\mathfrak{u}}_5$	2	1	3	3	3

Table 33. The score table for NMP in Example 14.

	Grade sum $\left(\sum _{\mathcal{i}=1}^5{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	Row sum $\left(\mathit{R}{\mathfrak{s}}_3\right)$	Column sum $\left(\mathit{C}{\mathfrak{s}}_3\right)$	${\mathbf{\Phi }}_3=\mathit{R}{\mathfrak{s}}_3-\mathit{C}{\mathfrak{s}}_3$
${\mathfrak{u}}_1$	6	11	8	3
${\mathfrak{u}}_2$	4	13	6	7
${\mathfrak{u}}_3$	7	9	11	−2
${\mathfrak{u}}_4$	10	3	15	−12
${\mathfrak{u}}_5$	5	12	8	4

Table 34. Add the score of nMP given in table (30) and NMP given in Table 33.

	Grade sum $\left(\sum _{\mathcal{i}=1}^5{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	${\mathbf{\Phi }}_2$	${\mathbf{\Phi }}_3$	${\mathbf{\Phi }}_4={\mathbf{\Phi }}_2+{\mathbf{\Phi }}_3$
${\mathfrak{u}}_1$	$6$	$0$	$3$	$3$
${\mathfrak{u}}_2$	$4$	$4$	$7$	$11$
${\mathfrak{u}}_3$	$7$	$0$	$-2$	$-2$
${\mathfrak{u}}_4$	$10$	$-9$	$-12$	$-21$
${\mathfrak{u}}_5$	$5$	$5$	$4$	$9$

Table 35. Table of final score with grades linked with PF5-SS of Example 14.

	Grade sum $\left(\sum _{\mathcal{i}=1}^5{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	${\mathbf{\Phi }}_1$	${\mathbf{\Phi }}_4$	FINAL SCORE $={\mathbf{\Phi }}_1-{\mathbf{\Phi }}_4$
${\mathfrak{u}}_1$	$6$	$-2$	$3$	$1$
${\mathfrak{u}}_2$	$4$	$-5$	$11$	$-16$
${\mathfrak{u}}_3$	$7$	$3$	$-2$	$5$
${\mathfrak{u}}_4$	$10$	$10$	$-21$	$31$
${\mathfrak{u}}_5$	$5$	$-7$	$9$	$-16$

Figure 3. The next generation firewall.

Table 36. Information obtained from associated data.

$\mathfrak{U}$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$◻$	$\star \star$	$\star \star \star$
${\mathfrak{u}}_2$	$\star \star$	$\star \star \star$	$\star \star$
${\mathfrak{u}}_3$	$\star \star$	$\star \star \star$	$\star$
${\mathfrak{u}}_4$	$◻$	$\star \star$	$\star \star$

Table 37. The tabular form of 4-SS.

$\left(\mathbf{I},\mathfrak{T},5\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$0$	$2$	$3$
${\mathfrak{u}}_2$	$2$	$3$	$2$
${\mathfrak{u}}_3$	$2$	$3$	$1$
${\mathfrak{u}}_4$	$0$	$2$	$2$

Table 38. The tabular representation of the picture fuzzy 4-soft set in Example 15.

$\left(\mathit{Y},\left(\mathbf{I},\mathfrak{T},4\right)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$0,\left(0.16,0.4,0.1\right)$	$2,\left(0.6,0.2,0.11\right)$	$3,\left(0.88,0.05,0.03\right)$
${\mathfrak{u}}_2$	$2,\left(0.65,0.1,0.15\right)$	$3,\left(0.8,0.13,0.05\right)$	$2,\left(0.58,0.3,0.1\right)$
${\mathfrak{u}}_3$	$2,\left(0.65,0.25,0.05\right)$	$3,\left(0.9,0.05,0.01\right)$	$1,\left(0.4,0.2,0.4\right)$
${\mathfrak{u}}_4$	$0,\left(0.1,0.4,0.4\right)$	$2,\left(0.7,0.1,0.0\right)$	$2,\left(0.7,0.05,0.2\right)$

Table 39. The tabular form of MP Example 15.

$\mathit{\xi }$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$0.16$	$0.6$	$0.88$
${\mathfrak{u}}_2$	$0.65$	$0.8$	$0.58$
${\mathfrak{u}}_3$	$0.65$	$0.9$	$0.4$
${\mathfrak{u}}_4$	$0.1$	$0.7$	$0.7$

Table 40. The comparison table for MP in Example 15.

$.$	${\mathfrak{u}}_1$	${\mathfrak{u}}_2$	${\mathfrak{u}}_3$	${\mathfrak{u}}_4$
${\mathfrak{u}}_1$	$3$	$1$	$1$	$3$
${\mathfrak{u}}_2$	$2$	$3$	$2$	$2$
${\mathfrak{u}}_3$	$2$	$2$	$3$	$2$
${\mathfrak{u}}_4$	$1$	$1$	$1$	$3$

Table 41. The score table for MP in Example 15.

	Grade sum $\left(\sum _{\mathcal{i}=1}^4{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	Row sum $\left(\mathit{R}{\mathfrak{s}}_1\right)$	Column sum $\left(\mathit{C}{\mathfrak{s}}_1\right)$	${\mathbf{\Phi }}_1=\mathit{R}{\mathfrak{s}}_1-\mathit{C}{\mathfrak{s}}_1$
${\mathfrak{u}}_1$	$5$	$8$	$8$	$0$
${\mathfrak{u}}_2$	$7$	$9$	$7$	$2$
${\mathfrak{u}}_3$	$6$	$9$	$7$	$2$
${\mathfrak{u}}_4$	$4$	$6$	$10$	$-4$

Table 42. The tabular representation of nMP in Example 15.

$\mathit{\delta }$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$0.4$	$0.2$	$0.05$
${\mathfrak{u}}_2$	$0.1$	$0.13$	$0.3$
${\mathfrak{u}}_3$	$0.25$	$0.05$	$0.2$
${\mathfrak{u}}_4$	$0.4$	$0.1$	$0.05$

Table 43. The comparison table for nMP in Example 15.

$.$	${\mathfrak{u}}_1$	${\mathfrak{u}}_2$	${\mathfrak{u}}_3$	${\mathfrak{u}}_4$
${\mathfrak{u}}_1$	$3$	$2$	$2$	$3$
${\mathfrak{u}}_2$	$1$	$3$	$2$	$2$
${\mathfrak{u}}_3$	$1$	$1$	$3$	$1$
${\mathfrak{u}}_4$	$2$	$1$	$2$	$3$

Table 44. The score table for nMP in Example 15.

	Grade sum $\left(\sum _{\mathcal{i}=1}^4{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	Row sum $\left(\mathit{R}{\mathfrak{s}}_2\right)$	Column sum $\left(\mathit{C}{\mathfrak{s}}_2\right)$	${\mathbf{\Phi }}_2=\mathit{R}{\mathfrak{s}}_2-\mathit{C}{\mathfrak{s}}_2$
${\mathfrak{u}}_1$	$5$	$10$	$7$	$3$
${\mathfrak{u}}_2$	$7$	$8$	$7$	$1$
${\mathfrak{u}}_3$	$6$	$6$	$9$	$-3$
${\mathfrak{u}}_4$	$4$	$8$	$9$	$-1$

Table 45. The tabular representation of NMP in Example 15.

$\vartheta$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$0.1$	$0.11$	$0.03$
${\mathfrak{u}}_2$	$0.15$	$0.05$	$0.1$
${\mathfrak{u}}_3$	$0.05$	$0.01$	$0.4$
${\mathfrak{u}}_4$	$0.4$	$0.0$	$0.2$

Table 46. The comparison table for NMP in Example 15.

$.$	${\mathfrak{u}}_1$	${\mathfrak{u}}_2$	${\mathfrak{u}}_3$	${\mathfrak{u}}_4$
${\mathfrak{u}}_1$	$3$	$1$	$2$	$1$
${\mathfrak{u}}_2$	$2$	$3$	$2$	$1$
${\mathfrak{u}}_3$	$1$	$1$	$3$	$2$
${\mathfrak{u}}_4$	$2$	$2$	$1$	$3$

Table 47. The score table for NMP in Example 15.

	Grade sum $\left(\sum _{\mathcal{i}=1}^4{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	Row sum $\left(\mathit{R}{\mathfrak{s}}_3\right)$	Column sum $\left(\mathit{C}{\mathfrak{s}}_3\right)$	${\mathbf{\Phi }}_3=\mathit{R}{\mathfrak{s}}_3-\mathit{C}{\mathfrak{s}}_3$
${\mathfrak{u}}_1$	$5$	$7$	$7$	$0$
${\mathfrak{u}}_2$	$7$	$8$	$7$	$1$
${\mathfrak{u}}_3$	$6$	$7$	$8$	$-1$
${\mathfrak{u}}_4$	$4$	$8$	$7$	$1$

Table 48. Add the score of nMP given in table (44) and NMP given in Table 47.

	Grade sum $\left(\sum _{\mathcal{i}=1}^4{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	${\mathbf{\Phi }}_2$	${\mathbf{\Phi }}_3$	${\mathbf{\Phi }}_4={\mathbf{\Phi }}_2+{\mathbf{\Phi }}_3$
${\mathfrak{u}}_1$	$5$	$3$	$0$	$3$
${\mathfrak{u}}_2$	$7$	$1$	$1$	$2$
${\mathfrak{u}}_3$	$6$	$-3$	$-1$	$-4$
${\mathfrak{u}}_4$	$4$	$-1$	$1$	$0$

Table 49. Table of final score with grades linked with PF6-SS of Example 15.

	Grade sum $\left(\sum _{\mathcal{i}=1}^4{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	${\mathbf{\Phi }}_1$	${\mathbf{\Phi }}_4$	FINAL SCORE $={\mathbf{\Phi }}_1-{\mathbf{\Phi }}_4$
${\mathfrak{u}}_1$	$5$	$0$	$3$	$-3$
${\mathfrak{u}}_2$	$7$	$2$	$2$	$0$
${\mathfrak{u}}_3$	$6$	$2$	$-4$	$6$
${\mathfrak{u}}_4$	$4$	$-4$	$0$	$-4$

Table 50. Information obtained from the associated data.

	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$\star \star \star \star$	$\star \star \star$	$\star \star \star \star$
${\mathfrak{u}}_2$	$\star \star \star$	$\star \star \star \star$	$\star \star \star \star$
${\mathfrak{u}}_3$	$\star$	$◻$	$\star \star \star \star$

Table 51. The tabular form of 5-SS.

$\left(\mathbf{I},\mathfrak{T},5\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$4$	$3$	$4$
${\mathfrak{u}}_2$	$3$	$4$	$4$
${\mathfrak{u}}_3$	$1$	$0$	$4$

Table 52. The tabular representation of IF5-SS $\left(\mathbf{K},\left(\mathbf{I},\mathfrak{T},5\right)\right)$ in Example 16.

$\left(\mathbf{K},\left(\mathbf{I},\mathfrak{T},5\right)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$4,\left(0.9,0.04\right)$	$3,\left(075,0.1\right)$	$4,\left(0.85,0.1\right)$
${\mathfrak{u}}_2$	$3,\left(0.75,0.2\right)$	$4,\left(0.87,0.08\right)$	$4,\left(0.89,0.04\right)$
${\mathfrak{u}}_3$	$1,\left(0.25,0.5\right)$	$0,\left(0.13,0.4\right)$	$4,\left(0.9,0.04\right)$

Table 53. The tabular representation of MP Example 16.

$\mathit{\xi }$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$0.9$	$0.75$	$0.85$
${\mathfrak{u}}_2$	$0.75$	$0.87$	$0.89$
${\mathfrak{u}}_3$	$0.25$	$0.13$	$0.9$

Table 54. The comparison table for MP in Example 16.

$.$	${\mathfrak{u}}_1$	${\mathfrak{u}}_2$	${\mathfrak{u}}_3$
${\mathfrak{u}}_1$	$3$	$1$	$2$
${\mathfrak{u}}_2$	$2$	$3$	$2$
${\mathfrak{u}}_3$	$1$	$1$	$3$

Table 55. The score table for MP in Example 16.

	Grade sum $\left(\sum _{\mathcal{i}=1}^3{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	Row sum $\left(\mathit{R}{\mathfrak{s}}_1\right)$	Column sum $\left(\mathit{C}{\mathfrak{s}}_1\right)$	${\mathbf{\Phi }}_1=\mathit{R}{\mathfrak{s}}_1-\mathit{C}{\mathfrak{s}}_1$
${\mathfrak{u}}_1$	$11$	$6$	$6$	$0$
${\mathfrak{u}}_2$	$11$	$7$	$5$	$2$
${\mathfrak{u}}_3$	$5$	$5$	$7$	$-2$

Table 56. The tabular representation of NMP in Example 16.

$\vartheta$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$0.04$	$0.1$	$0.1$
${\mathfrak{u}}_2$	$0.2$	$0.08$	$0.04$
${\mathfrak{u}}_3$	$0.5$	$0.4$	$0.04$

Table 57. The comparison table for NMP in Example 16.

$.$	${\mathfrak{u}}_1$	${\mathfrak{u}}_2$	${\mathfrak{u}}_3$
${\mathfrak{u}}_1$	$3$	$2$	$1$
${\mathfrak{u}}_2$	$1$	$3$	$1$
${\mathfrak{u}}_3$	$2$	$3$	$3$

Table 58. The score table for NMP in Example 16.

	Grade sum $\left(\sum _{\mathcal{i}=1}^3{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	Row sum $\left(\mathit{R}{\mathfrak{s}}_3\right)$	Column sum $\left(\mathit{C}{\mathfrak{s}}_3\right)$	${\mathbf{\Phi }}_3=\mathit{R}{\mathfrak{s}}_3-\mathit{C}{\mathfrak{s}}_3$
${\mathfrak{u}}_1$	$11$	$6$	$6$	$0$
${\mathfrak{u}}_2$	$11$	$5$	$8$	$-3$
${\mathfrak{u}}_3$	$5$	$8$	$5$	$3$

Table 59. Table of final score with grades linked with IF5-SS of Example 16.

	Grade sum $\left(\sum _{\mathcal{i}=1}^3{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	${\mathbf{\Phi }}_1$	${\mathbf{\Phi }}_3$	FINAL SCORE $={\mathbf{\Phi }}_1-{\mathbf{\Phi }}_3$
${\mathfrak{u}}_1$	$11$	$0$	$0$	$0$
${\mathfrak{u}}_2$	$11$	$2$	$-3$	$5$
${\mathfrak{u}}_3$	$5$	$-2$	$3$	$-5$

Table 60. The tabular form of PF5-SS of Example 16.

$\left(\mathit{Y},\left(\mathbf{I},\mathfrak{T},5\right)\right)$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$4,\left(0.9,0.05,0.04\right)$	$3,\left(075,0.01,0.1\right)$	$4,\left(0.85,0.04,0.1\right)$
${\mathfrak{u}}_2$	$3,\left(0.75,0.09,0.2\right)$	$4,\left(0.87,0.04,0.08\right)$	$4,\left(0.89,0.06,0.04\right)$
${\mathfrak{u}}_3$	$1,\left(0.25,0.08,0.5\right)$	$0,\left(0.13,0.03,0.4\right)$	$4,\left(0.9,0.05,0.04\right)$

Table 61. The tabular representation of MP in Example 16.

$\mathit{\xi }$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	$0.9$	$0.75$	$0.85$
${\mathfrak{u}}_2$	0.75	0.87	0.89
${\mathfrak{u}}_3$	0.25	0.13	0.9

Table 62. The comparison table for MP in Example 16.

.	${\mathfrak{u}}_1$	${\mathfrak{u}}_2$
${\mathfrak{u}}_1$	3	1	2
${\mathfrak{u}}_2$	2	3	2
${\mathfrak{u}}_3$	1	1	3

Table 63. The score table for MP in Example 16.

	Grade sum $\left(\sum _{\mathcal{i}=1}^3{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	Row sum $\left(\mathit{R}{\mathfrak{s}}_1\right)$	Column sum $\left(\mathit{C}{\mathfrak{s}}_1\right)$	${\mathbf{\Phi }}_1=\mathit{R}{\mathfrak{s}}_1-\mathit{C}{\mathfrak{s}}_1$
${\mathfrak{u}}_1$	11	6	6	0
${\mathfrak{u}}_2$	11	7	5	2
${\mathfrak{u}}_3$	5	5	7	−2

Table 64. The tabular form of nMP is given in Example 16.

$\mathit{\delta }$	${\mathfrak{t}}_1$	${\mathfrak{t}}_1$	${\mathfrak{t}}_1$
${\mathfrak{u}}_1$	0.05	0.01	0.045
${\mathfrak{u}}_2$	0.09	0.04	0.06
${\mathfrak{u}}_3$	0.08	0.03	0.05

Table 65. The comparison table for nMP in Example 16.

.	${\mathfrak{u}}_1$	${\mathfrak{u}}_2$	${\mathfrak{u}}_3$
${\mathfrak{u}}_1$	3	0	0
${\mathfrak{u}}_2$	3	3	3
${\mathfrak{u}}_3$	3	0	3

Table 66. The score table for nMP in Example 16.

	Grade sum $\left(\sum _{\mathcal{i}=1}^3{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	Row sum $\left(\mathit{R}{\mathfrak{s}}_2\right)$	Column sum $\left(\mathit{C}{\mathfrak{s}}_2\right)$	${\mathbf{\Phi }}_2=\mathit{R}{\mathfrak{s}}_2-\mathit{C}{\mathfrak{s}}_2$
${\mathfrak{u}}_1$	11	3	9	−6
${\mathfrak{u}}_2$	11	9	3	6
${\mathfrak{u}}_3$	5	6	6	0

Table 67. The tabular form of NMP given in Example 16.

$\vartheta$	${\mathcal{t}}_1$	${\mathcal{t}}_2$	${\mathcal{t}}_3$
${\mathfrak{u}}_1$	0.04	0.1	0.1
${\mathfrak{u}}_2$	0.2	0.08	0.04
${\mathfrak{u}}_3$	0.5	0.4	0.04

Table 68. The comparison table for NMP in Example 16.

.	${\mathfrak{u}}_1$	${\mathfrak{u}}_2$	${\mathfrak{u}}_3$
${\mathfrak{u}}_1$	3	2	1
${\mathfrak{u}}_2$	1	3	1
${\mathfrak{u}}_3$	2	3	3

Table 69. The score table for NMP in Example 16.

	Grade sum $\left(\sum _{\mathcal{i}=1}^3{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	Row sum $\left(\mathit{R}{\mathfrak{s}}_3\right)$	Column sum $\left(\mathit{C}{\mathfrak{s}}_3\right)$	${\mathbf{\Phi }}_3=\mathit{R}{\mathfrak{s}}_3-\mathit{C}{\mathfrak{s}}_3$
${\mathfrak{u}}_1$	11	6	6	0
${\mathfrak{u}}_2$	11	5	8	−3
${\mathfrak{u}}_3$	5	8	5	−3

Table 70. Add the score of nMP given in table (66) and NMP given in Table 69.

	Grade sum $\left(\sum _{\mathcal{i}=1}^3{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	${\mathbf{\Phi }}_2$	${\mathbf{\Phi }}_3$	${\mathbf{\Phi }}_4\mathbf{=}{\mathbf{\Phi }}_2+{\mathbf{\Phi }}_3$
${\mathfrak{u}}_1$	11	−6	0	−6
${\mathfrak{u}}_2$	11	6	−3	3
${\mathfrak{u}}_3$	5	0	3	−3

Table 71. Table of final score with grades linked with PF5-SS of Example 16.

	Grade sum $\left(\sum _{\mathcal{i}=1}^3{\mathcal{y}}_{{\mathcal{t}}_{\mathcal{i}}}\right)$	${\mathbf{\Phi }}_1$	${\mathbf{\Phi }}_4$	FINAL SCORE $\mathbf{=}{\mathbf{\Phi }}_1-{\mathbf{\Phi }}_4$
${\mathfrak{u}}_1$	11	0	−6	6
${\mathfrak{u}}_2$	11	2	3	−1
${\mathfrak{u}}_3$	5	−2	−3	1