Outlier identification and group satisfaction of rating experts: density-based spatial clustering of applications with noise based on multi-objective large-scale group decision-making evaluation

Figures & data

Figure 1. Framework of the proposed LSGDM model.

Source: Self-formulated.

Table 1. Final clustering results.

Cluster ID	Instance	Weight	Cluster ID	Instance	Weight
$G_1$	$e_1,e_2,e_3,e_4,e_5,e_6$	0.3	$G_4$	$e_7$	0.05
$G_2$	$e_8,e_{13},e_{14},e_{15},e_{16}$	0.25	$G_5$	$e_9$	0.05
$G_3$	$e_{10},e_{11},e_{12},e_{18},e_{19},e_{20}$	0.3	$G_6$	$e_{17}$	0.05

Source: Self-calculated.

Table 2. SSCI for each alternative based on FCM.

Alternatives	${\mathit{SCCI}}_{i1}$	${\mathit{SCCI}}_{i2}$	${\mathit{SCCI}}_{i3}$	$\mathit{SCC}{I}_i$
$x_1$	0.9542	0.9777	0.9835	0.9718
$x_2$	0.9411	0.9786	0.9937	0.9711
$x_3$	0.9548	0.9587	0.9808	0.9648
$x_4$	0.9544	0.9856	0.9795	0.9732
$x_5$	0.9346	0.9772	0.9686	0.9601

Source: Self-calculated.

Table 3. Ranking results of DBSCAN based LSGDM approach.

Alternatives	IF-RS	IF-RP	IF-FMF	IF-MULTIMOORA
$x_1$	5	5	5	5
$x_2$	2	1	1	1
$x_3$	4	4	4	4
$x_4$	3	3	3	3
$x_5$	1	2	2	2

Source: Self-calculated.

Table 4. Information on FCM clustering parameters.

Algorithms	Initializing the clustering centre	Number of clusters	Number of iterations	Clustering thresholds	Membership threshold
FCM	$\{e_2,e_8,e_{14}\}$	3	6	$\theta =0.05$	$\lambda =0.4$

Source: Self-calculated.

Table 5. Membership of each expert for each subcluster based on FCM.

Experts	$G_1$	$G_2$	$G_3$	Experts	$G_1$	$G_2$	$G_3$
$e_1$	0.2192	0.6389	0.1418	$e_{11}$	0.9269	0.0438	0.0293
$e_2$	0.1555	0.7016	0.1429	$e_{12}$	0.9519	0.0283	0.0198
$e_3$	0.1558	0.6573	0.1869	$e_{13}$	0.0716	0.0949	0.8335
$e_4$	0.1497	0.6339	0.2164	$e_{14}$	0.0300	0.0588	0.9113
$e_5$	0.1574	0.6435	0.1991	$e_{15}$	0.0242	0.0543	0.9215
$e_6$	0.1740	0.6431	0.1830	$e_{16}$	0.0485	0.0843	0.8672
$e_7$	0.2531	0.4101	0.3368	$e_{17}$	0.2540	0.4011	0.3449
$e_8$	0.0685	0.1068	0.8246	$e_{18}$	0.9188	0.0468	0.0343
$e_9$	0.2409	0.4406	0.3185	$e_{19}$	0.9465	0.0327	0.0208
$e_{10}$	0.8720	0.0741	0.0539	$e_{20}$	0.8847	0.0661	0.0492

Source: Self-calculated.

Table 6. Clustering results based on FCM.

Cluster ID	Instance
$G_1$	$e_1,e_2,e_3,e_4,e_5,e_6,e_7,e_9,e_{17}$
$G_2$	$e_8,e_{13},e_{14},e_{15},e_{16}$
$G_3$	$e_{10},e_{11},e_{12},e_{18},e_{19},e_{20}$

Source: Self-calculated.

Table 7. SSCI for each alternative based on FCM.

Alternatives	${\mathit{SCCI}}_{i1}$	${\mathit{SCCI}}_{i2}$	${\mathit{SCCI}}_{i3}$	$\mathit{SCC}{I}_i$
$x_1$	0.9047	0.9778	0.9832	0.9552
$x_2$	0.9153	0.9777	0.9937	0.9622
$x_3$	0.9290	0.9587	0.9811	0.9563
$x_4$	0.9153	0.9858	0.9794	0.9602
$x_5$	0.9033	0.9776	0.9688	0.9499

Source: Self-calculated.

Table 8. Ranking results of the FCM-based method.

Alternatives	IF-RS	IF-RP	IF-FMF	IF-MULTIMOORA
$x_1$	5	5	5	5
$x_2$	2	1	1	1
$x_3$	4	4	4	4
$x_4$	3	3	3	3
$x_5$	1	2	2	2

Source: Self-calculated.

Table 9. Information on K-means clustering parameters.

Algorithms	Initializing the clustering centre	Number of clusters	Number of iterations	Clustering thresholds
K-means	$\{e_1,e_{10},e_{18}\}$	3	2	$\theta =0.05$

Source: Self-calculated.

Table 10. Final clustering results.

Cluster ID	Instance
$G_1$	$e_1,e_2,e_3,e_4,e_5,e_6,e_9$
$G_2$	${e_7,e}_8,e_{13},e_{14},e_{15},e_{16},e_{17}$
$G_3$	$e_{10},e_{11},e_{12},e_{18},e_{19},e_{20}$

Source: Self-calculated.

Table 11. SSCI for each alternative for the K-means based approach.

Alternatives	${\mathit{SCCI}}_{i1}$	${\mathit{SCCI}}_{i2}$	${\mathit{SCCI}}_{i3}$	$\mathit{SCC}{I}_i$
$x_1$	0.9334	0.9396	0.9832	0.9521
$x_2$	0.9310	0.9177	0.9937	0.9475
$x_3$	0.9406	0.9437	0.9811	0.9551
$x_4$	0.9439	0.9259	0.9794	0.9497
$x_5$	0.9168	0.9322	0.9688	0.9393

Source: Self-calculated.

Table 12. Ranking results for the K-means based method.

Alternatives	IF-RS	IF-RP	IF-FMF	IF-MULTIMOORA
$x_1$	5	5	5	5
$x_2$	2	1	1	1
$x_3$	4	4	4	4
$x_4$	3	3	3	3
$x_5$	1	2	2	2

Source: Self-calculated.