VIKOR method for multiple criteria group decision making under 2-tuple linguistic neutrosophic environment

Figures & data

Table 1. 2TLNNs evaluation matrix by the first expert.

	G1	G2	G3	G4
$A_1$	{(s4,0), (s2,0), (s1,0)}	{(s5,0), (s3,0), (s2,0)}	{(s4,0), (s1,0), (s1,0)}	{(s3,0), (s2,0), (s2,0)}
$A_2$	{(s5,0), (s4,0), (s4,0)}	{(s3,0), (s4,0), (s2,0)}	{(s2,0), (s1,0), (s3,0)}	{(s3,0), (s1,0), (s2,0)}
$A_3$	{(s5,0), (s4,0), (s3,0)}	{(s2,0), (s4,0), (s5,0)}	{(s3,0), (s3,0), (s4,0)}	{(s2,0), (s1,0), (s4,0)}
$A_4$	{(s3,0), (s2,0), (s3,0)}	{(s4,0), (s3,0), (s2,0)}	{(s3,0), (s3,0), (s4,0)}	{(s2,0), (s1,0), (s1,0)}
$A_5$	{(s1,0), (s4,0), (s5,0)}	{(s2,0), (s3,0), (s1,0)}	{(s3,0), (s4,0), (s5,0)}	{(s2,0), (s4,0), (s3,0)}

Table 2. 2TLNNs evaluation matrix by the second expert.

	G1	G2	G3	G4
${\eta }_1$	{(s5,0), (s1,0), (s2,0)}	{(s4,0), (s3,0), (s1,0)}	{(s4,0), (s2,0), (s1,0)}	{(s5,0), (s1,0), (s2,0)}
${\eta }_2$	{(s4,0), (s3,0), (s3,0)}	{(s3,0), (s2,0), (s4,0)}	{(s2,0), (s1,0), (s3,0)}	{(s5,0), (s4,0), (s2,0)}
${\eta }_3$	{(s3,0), (s4,0), (s3,0)}	{(s2,0), (s4,0), (s5,0)}	{(s5,0), (s1,0), (s2,0)}	{(s2,0), (s1,0), (s2,0)}
${\eta }_4$	{(s4,0), (s5,0), (s4,0)}	{(s2,0), (s3,0), (s2,0)}	{(s3,0), (s3,0), (s4,0)}	{(s4,0), (s4,0), (s5,0)}
${\eta }_5$	{(s2,0), (s4,0), (s5,0)}	{(s3,0), (s1,0), (s5,0)}	{(s2,0), (s3,0), (s4,0)}	{(s2,0), (s1,0), (s3,0)}

Table 3. 2TLNNs evaluation matrix by the third expert.

	G1	G2	G3	G4
$A_1$	{(s5,0), (s1,0), (s1,0)}	{(s4,0), (s1,0), (s2,0)}	{(s3,0), (s3,0), (s1,0)}	{(s4,0), (s2,0), (s2,0)}
$A_2$	{(s5,0), (s4,0), (s5,0)}	{(s3,0), (s2,0), (s1,0)}	{(s2,0), (s1,0), (s4,0)}	{(s4,0), (s5,0), (s3,0)}
$A_3$	{(s2,0), (s1,0), (s4,0)}	{(s5,0), (s1,0), (s3,0)}	{(s4,0), (s3,0), (s4,0)}	{(s5,0), (s2,0), (s3,0)}
$A_4$	{(s2,0), (s2,0), (s3,0)}	{(s4,0), (s1,0), (s2,0)}	{(s5,0), (s3,0), (s2,0)}	{(s1,0), (s4,0), (s5,0)}
$A_5$	{(s1,0), (s4,0), (s5,0)}	{(s2,0), (s4,0), (s4,0)}	{(s3,0), (s4,0), (s3,0)}	{(s2,0), (s4,0), (s4,0)}

Table 4. The aggregation values by 2TLNNWA operator.

	G1	G2
$A_1$	{(s5,−0.1892), (s1,0.1892), (s1,0.2746)}	{(s4,0.3182), (s2,−0.0668), (s2,−0.4308)}
$A_2$	{(s5,−0.2746), (s4,−0.3831), (s4,−0.0455)}	{(s3,0.0000), (s2,0.3784), (s2,−0.0681)}
$A_3$	{(s3,0.4425), (s2,0.2974), (s3,0.3659)}	{(s4,−0.2974), (s2,0.2974), (s4,0.0760)}
$A_4$	{(s3,0.0794), (s3,−0.2438), (s3,0.3178)}	{(s3,0.4509), (s2,−0.0668), (s2,0.0000)}
$A_5$	{(s1,0.3756), (s4,0.0000), (s5,0.0000)}	{(s2,0.3831), (s2,0.2914), (s3,0.0582)}
	G3	G4
$A_1$	{(s4,−0.3522), (s2,−0.0221), (s1,0.0000)}	{(s4,0.2634), (s2,−0.4308), (s2,0.0000)}
$A_2$	{(s2,0.0000), (s1,0.0000), (s3,0.3659)}	{(s4,0.2634), (s3,0.0925), (s2,0.3522)}
$A_3$	{(s4,0.2634), (s2,0.0423), (s3,0.1383)}	{(s4,−0.2974), (s1,0.3195), (s3,−0.2028)}
$A_4$	{(s4,0.0668), (s3,0.0000), (s3,0.0314)}	{(s3,−0.4313), (s3,−0.1716), (s3,0.3437)}
$A_5$	{(s3,−0.3178), (s4,−0.3831), (s4,−0.2303)}	{(s2,0.0000), (s2,0.4623), (s3,0.3659)}

Table 5. Ordering by the VIKOR method with 2TLNNs.

Parameter	$Q_1$	$Q_2$	$Q_3$	$Q_4$	$Q_5$	Ordering
$\rho =0.0$	0.0000	0.5621	0.2982	0.4878	1.0000	$Q_1>Q_3>Q_4>Q_2>Q_5$
$\rho =0.1$	0.0000	0.5576	0.3107	0.4973	1.0000	$Q_1>Q_3>Q_4>Q_2>Q_5$
$\rho =0.3$	0.0000	0.5487	0.3357	0.5162	1.0000	$Q_1>Q_3>Q_4>Q_2>Q_5$
$\rho =0.5$	0.0000	0.5398	0.3607	0.5351	1.0000	$Q_1>Q_3>Q_4>Q_2>Q_5$
$\rho =0.7$	0.0000	0.5309	0.3858	0.5540	1.0000	$Q_1>Q_3>Q_4>Q_2>Q_5$
$\rho =1.0$	0.0000	0.5175	0.4233	0.5823	1.0000	$Q_1>Q_3>Q_4>Q_2>Q_5$

Table 6. Alternative scores $s({\eta }_i)$ by 2TLNNWA and 2TLNNWG operators.

2TLNNWA operator	2TLNNWG operator
$s({\eta }_1)=0.7433,s({\eta }_2)=0.5863,s({\eta }_3)=0.5903,s({\eta }_4)=0.5386,s({\eta }_5)=0.3856.$	$s({\eta }_1)=0.7319,s({\eta }_2)=0.5204,s({\eta }_3)=0.5822,s({\eta }_4)=0.5267,s({\eta }_5)=0.3580.$

Table 7. Rank of Alternatives by 2TLNNWA and 2TLNNWG operators.

	order
2TLNNWA	$A_1>A_3>A_2>A_4>A_5$
2TLNNWG	$A_1>A_3>A_4>A_2>A_5$
2TLNNs VIKOR	$A_1>A_3>A_4>A_2>A_5$

Table 8. IV2TLNNs evaluation matrix.

	G1	G2
$A_1$	{[(s4,0),(s5,0)],[(s1,0),(s2,0)],[(s2,0), (s3,0)]}	{[(s3,0),(s5,0)],[(s2,0),(s3,0)],[(s1,0), (s2,0)]}
$A_2$	{[(s1,0),(s2,0)],[(s2,0),(s3,0)],[(s3,0), (s4,0)]}	{[(s1,0),(s3,0)],[(s2,0),(s4,0)],[(s3,0), (s5,0)]}
$A_3$	{[(s4,0),(s5,0)],[(s3,0),(s4,0)],[(s4,0), (s5,0)]}	{[(s2,0),(s3,0)],[(s2,0),(s4,0)],[(s1,0), (s2,0)]}
$A_4$	{[(s4,0),(s6,0)],[(s2,0),(s4,0)],[(s5,0), (s6,0)]}	{[(s3,0),(s4,0)],[(s2,0),(s3,0)],[(s1,0), (s2,0)]}
$A_5$	{[(s1,0),(s2,0)],[(s4,0),(s5,0)],[(s5,0), (s6,0)]}	{[(s2,0),(s3,0)],[(s5,0),(s6,0)],[(s4,0), (s5,0)]}
	G3	G4
$A_1$	{[(s5,0),(s6,0)],[(s1,0),(s3,0)],[(s2,0), (s3,0)]}	{[(s5,0),(s6,0)],[(s3,0),(s4,0)],[(s1,0), (s2,0)]}
$A_2$	{[(s5,0),(s6,0)],[(s4,0),(s5,0)],[(s1,0), (s2,0)]}	{[(s3,0),(s4,0)],[(s2,0),(s3,0)],[(s1,0), (s2,0)]}
$A_3$	{[(s4,0),(s5,0)],[(s3,0),(s4,0)],[(s2,0), (s3,0)]}	{[(s1,0),(s2,0)],[(s4,0),(s5,0)],[(s2,0), (s3,0)]}
$A_4$	{[(s3,0),(s4,0)],[(s4,0),(s5,0)],[(s2,0), (s3,0)]}	{[(s1,0),(s2,0)],[(s2,0),(s3,0)],[(s1,0), (s2,0)]}
$A_5$	{[(s1,0),(s3,0)],[(s2,0),(s5,0)],[(s3,0), (s4,0)]}	{[(s2,0),(s3,0)],[(s4,0),(s6,0)],[(s2,0), (s3,0)]}

Table 9. Ordering by the VIKOR method with IV2TLNNs.

Parameter	$Q_1$	$Q_2$	$Q_3$	$Q_4$	$Q_5$	Ordering
$\rho =0.0$	0.0000	0.5322	0.6667	0.4152	1.0000	$Q_1>Q_4>Q_2>Q_3>Q_5$
$\rho =0.1$	0.0000	0.5247	0.6568	0.4151	1.0000	$Q_1>Q_4>Q_2>Q_3>Q_5$
$\rho =0.3$	0.0000	0.5098	0.6372	0.4150	1.0000	$Q_1>Q_4>Q_2>Q_3>Q_5$
$\rho =0.5$	0.0000	0.4949	0.6175	0.4149	1.0000	$Q_1>Q_4>Q_2>Q_3>Q_5$
$\rho =0.7$	0.0000	0.4800	0.5979	0.4148	1.0000	$Q_1>Q_4>Q_2>Q_3>Q_5$
$\rho =1.0$	0.0000	0.4577	0.5684	0.4146	1.0000	$Q_1>Q_4>Q_2>Q_3>Q_5$

Table 10. Alternative scores $s({\tilde{\eta }}_i)$ by IV2TLNNWA and IV2TLNNWG operators.

IV2TLNNWA operator	IV2TLNNWG operator
$s({\tilde{\eta }}_1)=0.7254,s({\tilde{\eta }}_2)=0.6044,s({\tilde{\eta }}_3)=0.5029,s({\tilde{\eta }}_4)=0.6130,s({\tilde{\eta }}_5)=0.3005.$	$s({\tilde{\eta }}_1)=0.6813,s({\tilde{\eta }}_2)=0.4667,s({\tilde{\eta }}_3)=0.4273,s({\tilde{\eta }}_4)=0.4119,s({\tilde{\eta }}_5)=0.2160.$

Table 11. Rank of Alternatives by IV2TLNNWA and IV2TLNNWG operators.

	Order
IV2TLNNWA	$A_1>A_4>A_2>A_3>A_5$
IV2TLNNWG	$A_1>A_2>A_3>A_4>A_5$
IV2TLNNs VIKOR	$A_1>A_4>A_2>A_3>A_5$