Picture Fuzzy Soft Robust VIKOR Method and its Applications in Decision-Making

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Table 1. A PFSS $(\hat{F},\hat{E})$.

$\hat{Y}$	${ȷ}_1$	${ȷ}_2$	${ȷ}_3$	${ȷ}_4$	${ȷ}_5$
${\ell }_1$	$b_{11}$	$b_{12}$	$b_{13}$	$b_{14}$	$b_{15}$
${\ell }_2$	$b_{21}$	$b_{22}$	$b_{23}$	$b_{24}$	$b_{25}$
${\ell }_3$	$b_{31}$	$b_{32}$	$b_{33}$	$b_{34}$	$b_{35}$

Table 2. The PFSS.

$\hat{Y}$	${\ell }_1$	${\ell }_2$	${\ell }_3$	${\ell }_4$	y
${ȷ}_1$	(0.53,0.33,0.09)	(1.00,0.00,0.00)	(0.91,0.03,0.02)	(0.85,0.09,0.05)	(0.90,0.05,0.02)
${ȷ}_2$	(0.89,0.08,0.03)	(0.13,0.64,0.21)	(0.07,0.09,0.05)	(0.74,0.16,0.10)	(0.68,0.08,0.21)
${ȷ}_3$	(0.42,0.35,0.18)	(0.03,0.82,0.13)	(0.04,0.85,0.10)	(0.02,0.89,0.05)	(0.05,0.87,0.06)
${ȷ}_4$	(0.08,0.89,0.02)	(0.73,0.15,0.08)	(0.68,0.26,0.06)	(0.08,0.84,0.06)	(0.13,0.75,0.09)
${ȷ}_5$	(0.33,0.51,0.12)	(0.52,0.31,0.16)	(0.15,0.76,0.07)	(0.16,0.71,0.05)	(0.15,0.73,0.08)
${ȷ}_6$	(0.17,0.53,0.13)	(0.51,0.24,0.21)	(0.31,0.39,0.25)	(1.00,0.00,0.00)	(0.91,0.03,0.05)

Table 3. The Distance Between PFSSs.

	$D({\ell }_1,y)$	$D({\ell }_2,y)$	$D({\ell }_3,y)$	$D({\ell }_4,y)$	Rankings
$D_h^{\circ }$	0.057217	0.06965	0.06675	0.0146	${\ell }_4\succ {\ell }_1\succ {\ell }_3\succ {\ell }_2$
$D_e^{\circ }$	0.145501	0.17731	0.188662	0.030419	${\ell }_4\succ {\ell }_1\succ {\ell }_2\succ {\ell }_3$
$D_3^{\circ }$	0.221269	0.252763	0.279392	0.041733	${\ell }_4\succ {\ell }_1\succ {\ell }_2\succ {\ell }_3$
$D_4^{\circ }$	0.284051	0.30512	0.344331	0.050327	${\ell }_4\succ {\ell }_1\succ {\ell }_2\succ {\ell }_3$
$D_5^{\circ }$	0.335453	0.343242	0.392823	0.057047	${\ell }_4\succ {\ell }_1\succ {\ell }_2\succ {\ell }_3$
$D_{10}^{\circ }$	0.487938	0.441662	0.526076	0.076237	${\ell }_4\succ {\ell }_2\succ {\ell }_1\succ {\ell }_3$
$D_{20}^{\circ }$	0.599866	0.507824	0.626869	0.09076	${\ell }_4\succ {\ell }_2\succ {\ell }_1\succ {\ell }_3$
$D_{50}^{\circ }$	0.680383	0.55843	0.703378	0.101805	${\ell }_4\succ {\ell }_2\succ {\ell }_1\succ {\ell }_3$
$D_{100}^{\circ }$	0.709566	0.578649	0.731141	0.105823	${\ell }_4\succ {\ell }_2\succ {\ell }_1\succ {\ell }_3$

Table 4. The PFSS $\mathrm{\Gamma }=(\hat{F},\hat{A})$.

$\hat{Y}$	${ȷ}_1$	${ȷ}_2$	${ȷ}_3$	${ȷ}_4$	${ȷ}_5$	${ȷ}_6$
${\ell }_1$	(0.1,0.3,0.5)	(0.2,0.3,0.4)	(0.5,0.2,0.2)	(0.3,0.1,0.5)	(0.4,0.1,0.4)	(0.3,0.2,0.5)
${\ell }_2$	(0.5,0.1,0.3)	(0.3,0.1,0.3)	(0.4,0.1,0.3)	(0.3,0.2,0.4)	(0.3,0.2,0.5)	(0.4,0.1,0.4)
${\ell }_3$	(0.2,0.4,0.3)	(0.2,0.2,0.5)	(0.1,0.3,0.5)	(0.2,0.1,0.6)	(0.5,0.1,0.3)	(0.4,0.2,0.3)
${\ell }_4$	(0.6,0.1,0.2)	(0.1,0.2,0.4)	(0.5,0.1,0.2)	(0.7,0.1,0.1)	(0.2,0.2,0.5)	(0.2,0.4,0.3)
${\ell }_5$	(0.2,0.2,0.5)	(0.3,0.1,0.4)	(0.5,0.1,0.2)	(0.5,0.2,0.2)	(0.6,0.1,0.2)	(0.2,0.2,0.4)
${\ell }_6$	(0.5,0.1,0.3)	(0.2,0.2,0.5)	(0.4,0.2,0.1)	(0.4,0.1,0.3)	(0.3,0.1,0.5)	(0.5,0.1,0.3)

Table 5. The PFSS $\mathrm{\Gamma }=({\hat{F}}^{\mathrm{\prime }},\hat{A})$.

$\hat{Y}$	${ȷ}_1$	${ȷ}_2$	${ȷ}_3$	${ȷ}_4$	${ȷ}_5$	${ȷ}_6$
${\ell }_1$	(0.1,0.3,0.5)	(0.2,0.3,0.4)	(0.2,0.2,0.5)	(0.3,0.1,0.5)	(0.4,0.1,0.4)	(0.3,0.2,0.5)
${\ell }_2$	(0.5,0.1,0.3)	(0.3,0.1,0.3)	(0.3,0.1,0.4)	(0.3,0.2,0.4)	(0.3,0.2,0.5)	(0.4,0.1,0.4)
${\ell }_3$	(0.2,0.4,0.3)	(0.2,0.2,0.5)	(0.5,0.3,0.1)	(0.2,0.1,0.6)	(0.5,0.1,0.3)	(0.4,0.2,0.3)
${\ell }_4$	(0.6,0.1,0.2)	(0.1,0.2,0.4)	(0.2,0.1,0.5)	(0.7,0.1,0.1)	(0.2,0.2,0.5)	(0.2,0.4,0.3)
${\ell }_5$	(0.2,0.2,0.5)	(0.3,0.1,0.4)	(0.2,0.1,0.5)	(0.5,0.2,0.2)	(0.6,0.1,0.2)	(0.2,0.2,0.4)
${\ell }_6$	(0.5,0.1,0.3)	(0.2,0.2,0.5)	(0.1,0.2,0.4)	(0.4,0.1,0.3)	(0.3,0.1,0.5)	(0.5,0.1,0.3)

Table 6. Comparison with existing MCDM methods.

Reference	Method	Ranking
Wei [11]	PFWA operator	${\ell }_4\succ {\ell }_5\succ {\ell }_3\succ {\ell }_2\succ {\ell }_6\succ {\ell }_1$
Wei [11]	PFWG operator	${\ell }_5\succ {\ell }_2\succ {\ell }_3\succ {\ell }_4\succ {\ell }_6\succ {\ell }_1$
Thao [34]	Similarity measure	${\ell }_6\succ {\ell }_2\succ {\ell }_5\succ {\ell }_4\succ {\ell }_3\succ {\ell }_1$
Wang et al. [40]	Projection-based VIKOR	$\{{\ell }_6,{\ell }_5\}\succ {\ell }_2\succ {\ell }_4\succ {\ell }_1\succ {\ell }_3$
Zhang et al. [41]	EDAS method	${\ell }_5\succ {\ell }_3\succ {\ell }_1\succ {\ell }_4\succ {\ell }_6\succ {\ell }_2$
Proposed	Algorithm 1	${\ell }_2\succ {\ell }_5\succ {\ell }_6\succ {\ell }_4\succ {\ell }_3\succ {\ell }_1$
Proposed	Algorithm 2	$\{{\ell }_5,{\ell }_2\}\succ {\ell }_3\succ {\ell }_6\succ {\ell }_4\succ {\ell }_1$
Proposed	Algorithm 3	$\{{\ell }_5,{\ell }_2\}\succ {\ell }_3\succ {\ell }_6\succ {\ell }_4\succ {\ell }_1$
Proposed	Algorithm 4	$\{{\ell }_5,{\ell }_3\}\succ {\ell }_2\succ {\ell }_4\succ {\ell }_6\succ {\ell }_1$

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