Large-scale consensus with endo-confidence under probabilistic linguistic circumstance and its application

Figures & data

Figure 1. The visual procedure of consensus decision-making model with endo-confidence.

Source: calculated by the methods using the original data.

Table 1. The clustering results and the corresponding weights and endo-confidence levels.

$G_{11,L(p)}$	$e^a$	$\overline{G_{11}}$	$e^a$	$w_{11,\overline{G_{11,g}}}^{(0)}$	$e{c}_{11,\overline{G_{11,g}}}^{(0)}$
$G_{11,L{(p)}_1}$	$e^1,e^4,e^6,e^7,e^{13},e^{15},e^{17}$	$\overline{G_{11,1}}$	$e^1,e^4,e^6,e^7,e^{15},e^{17}$	0.3	0.4455
$G_{11,L{(p)}_2}$	$e^3,e^5,e^8,e^9,e^{10},e^{11},e^{16},e^{18},e^{20}$	$\overline{G_{11,2}}$	$e^3,e^5,e^8,e^{10},e^{20}$	0.25	0.5098
$G_{11,L{(p)}_3}$	$e^{12},e^{14}$	$\overline{G_{11,3}}$	$e^{11},e^{16},e^{18}$	0.15	0.6686
$G_{11,L{(p)}_4}$	$e^2,e^{19}$	$\overline{G_{11,4}}$	$e^2,e^{19}$	0.1	0.5099
$G_{11,ec}$	$e^a$	$\overline{G_{11,5}}$	$e^9$	0.05	0.8400
$G_{11,e{c}_1}$	$e^1,e^2,e^3,e^4,e^5,e^6,e^7,e^8,e^{10},e^{14},e^{15},e^{17},e^{19},e^{20}$	$\overline{G_{11,6}}$	$e^{12}$	0.05	0.6400
$G_{11,e{c}_2}$	$e^{11},e^{12},e^{13},e^{16},e^{18}$	$\overline{G_{11,7}}$	$e^{13}$	0.05	0.6968
$G_{11,e{c}_3}$	$e^9$	$\overline{G_{11,8}}$	$e^{14}$	0.05	0.4400

Source: calculated by the methods using the original data.

Table 2. The results of the updated weights.

$\overline{G_{11}}$	$\overline{w_{11,\overline{G_{11,g}}}^{(1)}}$	$\overline{w_{11,\overline{G_{11,g}}}^{(2)}}$	$\overline{w_{11,\overline{G_{11,g}}}^{(3)}}$	$\overline{w_{11,\overline{G_{11,g}}}^{(4)}}$	$\overline{w_{11,\overline{G_{11,g}}}^{(5)}}$
$\overline{G_{11,1}}$	0.1012	0.0998	0.0983	0.0975	0.0961
$\overline{G_{11,2}}$	0.1110	0.1095	0.1077	0.1069	0.1200
$\overline{G_{11,3}}$	0.1353	0.1335	0.1314	0.1379	0.1359
$\overline{G_{11,4}}$	0.1110	0.1095	0.1235	0.1226	0.1208
$\overline{G_{11,5}}$	0.1630	0.1608	0.1583	0.1571	0.1548
$\overline{G_{11,6}}$	0.1383	0.1364	0.1342	0.1332	0.1313
$\overline{G_{11,7}}$	0.1397	0.1378	0.1357	0.1346	0.1327
$\overline{G_{11,8}}$	0.1004	0.1127	0.1109	0.1101	0.1085

Source: calculated by the methods using the original data.

Table 3. The comparative results of Model I (case I and case II) and Model II.

	Model I with case I		Model I with case II		Model II
$\overline{G_{11}}$	$e^a$	$\overline{w_{11,\overline{G_{11,g}}}^{(5)}}$	$e^a$	$\overline{w_{11,\overline{G_{11,g}}}^{(6)}}$	$e^a$	$\overline{w_{11,\overline{G_{11,g}}}^{(3)}}$
$\overline{G_{11,1}}$	$e^1,e^4,e^6,e^7,e^{15},e^{17}$	0.0961	$e^1,e^6,e^7,e^{15},e^{17}$	0.0649	$e^1,e^4,e^6,e^7,e^{13},e^{15},e^{17}$	0.35
$\overline{G_{11,2}}$	$e^3,e^5,e^8,e^{10},e^{20}$	0.1200	$e^3,e^5,e^8,e^{20}$	0.0970	$e^3,e^5,e^8,e^9,e^{10},e^{11},e^{16},e^{18},e^{20}$	0.45
$\overline{G_{11,3}}$	$e^{11},e^{16},e^{18}$	0.1359	$e^{10},e^{11},e^{16}$	0.1090	$e^{12},e^{14}$	0.1
$\overline{G_{11,4}}$	$e^2,e^{19}$	0.1208	$e^2,e^{19}$	0.0905	$e^2,e^{19}$	0.1
$\overline{G_{11,5}}$	$e^9$	0.1548	$e^9$	0.1392
$\overline{G_{11,6}}$	$e^{12}$	0.1313	$e^{12}$	0.1323
$\overline{G_{11,7}}$	$e^{13}$	0.1327	$e^{13}$	0.1067
$\overline{G_{11,8}}$	$e^{14}$	0.1085	$e^{14}$	0.0977
$\overline{G_{11,9}}$			$e^{18}$	0.1152
$\overline{G_{11,10}}$			$e^4$	0.0475
	$\overline{{\upsilon }^i}$	Rank	$\overline{{\upsilon }^i}$	Rank	$\overline{{\upsilon }^i}$	Rank
$A_1$	3.0851	1	3.0988	1	3.0689	2
$A_2$	2.9427	3	2.9437	3	2.9437	3
$A_3$	2.7630	4	2.7694	4	2.7744	4
$A_4$	3.0749	2	3.0790	2	3.0791	1

Source: calculated by the methods using the original data.

Table A1. The meaning of notations used in the proposed model.

Notation	Meaning
$\overline{L}=\{s_l,\dots ,s_{\gamma },\dots ,s_u\}$	The normalised LTS, where $s_l$ and $s_u$ represent the lower and upper bounds of the normalised LTS respectively
$\overline{L(p_{\varsigma })}=\{\overline{L^{(k^{\prime })}}(p_{\varsigma }^{(k^{\prime })})|p_{\varsigma }^{(k^{\prime })}\ge 0,k^{\prime }=1,2,\dots ,\#\overline{L(p_{\varsigma })},{\sum }_{k^{\prime }=1}^{\#\overline{L(p_{\varsigma })}}{p}_{\varsigma }^{(k^{\prime })}\le 1\}$	The PLTS with the expanded original information
$\overline{h(p^{(k^{\prime })})}$	The normalised proportion of the probability distribution
$\overline{L(p)}=\{\overline{L^{(k^{\prime })}}(\overline{p^{(k^{\prime })}})|\overline{p^{(k^{\prime })}}\ge 0,k^{\prime }=1,2,\dots ,\#\overline{L(p)},{\sum }_{k^{\prime }=1}^{\#\overline{L(p)}}\overline{p^{(k^{\prime })}}=1\}$	The normalised PLTS
$E=\{e^1,e^2,\dots ,e^t\}$ ($T=\{1,2,\dots ,t\},$ $a\in T,$ $t\ge 20$)	The finite set of experts
$A=\{A_1,A_2,\dots ,A_n\}$ ($N=\{1,2,\dots ,n\},$ $i\in N,$ $n\ge 3$)	The finite set of potential alternatives
$C=\{c_1,c_2,\dots ,c_m\}$ ($M=\{1,2,\dots ,m\},$ $j\in M,$ $m\ge 2$)	The finite set of attributes
$e{c}^a$	The endo-confidence level of the expert $e^a$
$e{c}_p$	The endo-confidence level from the perspective of total probability
$e{c}_h$	The endo-confidence level based on hesitation
$e{c}_d$	The endo-confidence level from the perspective of preference among linguistic terms
$G_{ij,L(p)}=\{G_{ij,L{(p)}_1},G_{ij,L{(p)}_2},\dots ,G_{ij,L{(p)}_g},\dots ,G_{ij,L{(p)}_{\overline{\xi }}}\}$	The clusters of DMs based on the similarities of evaluation information
$G_{ij,ec}=\{G_{ij,e{c}_1},G_{ij,e{c}_2},\dots ,G_{ij,e{c}_g},\dots ,G_{ij,e{c}_{\overline{\xi }}}\}$	The clusters of DMs based on the similarities of endo-confidence levels among DMs
$\overline{G_{ij}}=\{\overline{G_{ij,1}},\overline{G_{ij,2}},\dots ,\overline{G_{ij,g}},\dots ,\overline{G_{ij,\sigma }}\}$	The final clustering results
${\eta }_{ij,L(p)}^{\ast }$/${\eta }_{ij,ec}^{\ast }$	The optimal classification threshold which is determined by the maximum value of the classification threshold ${{\eta }^{\prime }}_{ij,L(p)}$/${{\eta }^{\prime }}_{ij,ec}$ and the threshold ${\chi }_{ij,L(p)}$/${\chi }_{ij,ec}$ given by DMs
$\overline{w_{ij,\overline{G_{ij,g}}}}$	The weight of the cluster $\overline{G_{ij,g}}$
$\overline{w_{ij,\overline{G_{ij,g}}}^a}$	The weight of the expert $e^a$ in the cluster $\overline{G_{ij,g}}$
${\overline{L_{ij,\overline{G_{ij,g}}}^a(p)}}^{\ast }=\{\overline{L_{ij,\overline{G_{ij,g}}}^{a(k^{\prime })}}({\overline{p_{ij,\overline{G_{ij,g}}}^{(k^{\prime })}}}^{\ast })|k^{\prime }=1,2,\dots ,\#{\overline{L_{ij,\overline{G_{ij,g}}}^a(p)}}^{\ast }\}$	The rearranged PLTSs of the evaluation given by the expert $e^a$ in the cluster $\overline{G_{ij,g}}$
$\overline{L_{ij,\overline{G_{ij,g}}}(p)}$	The collective evaluation information of the cluster $\overline{G_{ij,g}}$
${\overline{L_{ij,\overline{G_{ij,g}}}(p)}}^{\ast }=\{\overline{L_{ij,\overline{G_{ij,g}}}^{(k^{\prime })}}({\overline{p_{ij}^{(k^{\prime })}}}^{\ast })|k^{\prime }=1,2,\dots ,\#{\overline{L_{ij,\overline{G_{ij,g}}}(p)}}^{\ast }\}$	The rearranged PLTS of the collective evaluation information $\overline{L_{ij,\overline{G_{ij,g}}}(p)}$ in the cluster $\overline{G_{ij,g}}$
$\overline{L_{ij}(p)}$	The overall evaluation
$OC{I}_{ij}$	The overall consensus index
$\overline{{G^{\prime }}_{ij,g}}$	The order of the clusters obtained by sorting the similarities ${\rho }_{ij,\overline{G_{ij,g}}}$
$e{c}_{ij,\overline{G_{ij,g}}}$	The endo-confidence level of the cluster $\overline{G_{ij,g}}$
$A{S}_{ij,\overline{G_{ij,g}}}=\left\{\overline{G_{ij,g^{\prime }}}\right|e{c}_{ij,\overline{G_{ij,g^{\prime }}}}\ge e{c}_{ij,\overline{G_{ij,g}}},g^{\prime }\ne g\}$	The suggested adjustment set
$\overline{w_{ij,\overline{G_{ij,g^{\prime }}}}}$	The weight of the cluster $\overline{G_{ij,g^{\prime }}}$ in $A{S}_{ij,\overline{G_{ij,g}}}$
${\overline{L_{ij,\overline{G_{ij,g^{\prime }}}}(p)}}^{\ast }=\{\overline{L_{ij,\overline{G_{ij,g^{\prime }}}}^{(k^{\prime })}}({\overline{p_{ij}^{(k^{\prime })}}}^{\ast })|k^{\prime }=1,2,\dots ,\#{\overline{L_{ij,\overline{G_{ij,g^{\prime }}}}(p)}}^{\ast }\}$	The rearranged PLTSs of the collective evaluation information $\overline{L_{ij,\overline{G_{ij,g^{\prime }}}}(p)}$ in the suggested adjustment set $A{S}_{ij,\overline{G_{ij,g}}}$
$\overline{L_{\mathit{ijA}{S}_{ij,\overline{G_{ij,g}}}}(p)}=\{\overline{L_{\mathit{ijA}{S}_{ij,\overline{G_{ij,g}}}}^{(k^{\prime })}}(\overline{p_{\mathit{ijA}{S}_{ij,\overline{G_{ij,g}}}}^{(k^{\prime })}})|\overline{p_{\mathit{ijA}{S}_{ij,\overline{G_{ij,g}}}}^{(k^{\prime })}}\ge 0,{\sum }_{k^{\prime }=1}^{\#\overline{L_{\mathit{ijA}{S}_{ij,\overline{G_{ij,g}}}}(p)}}\overline{p_{\mathit{ijA}{S}_{ij,\overline{G_{ij,g}}}}^{(k^{\prime })}}=1\}$	Part of the suggestions for the cluster $\overline{G_{ij,g}}$
${\overline{L_{ij,\overline{G_{ij,g}}}(p)}}^{(\varphi )}=\{{\overline{L_{ij,\overline{G_{ij,g}}}^{(k^{\prime })}}}^{(\varphi )}({\overline{p_{ij,\overline{G_{ij,g}}}^{(k^{\prime })}}}^{(\varphi )})|{\overline{p_{ij,\overline{G_{ij,g}}}^{(k^{\prime })}}}^{(\varphi )}\ge 0,{\sum }_{k^{\prime }=1}^{\#{\overline{L_{ij,\overline{G_{ij,g}}}(p)}}^{(\varphi )}}{\overline{p_{ij,\overline{G_{ij,g}}}^{(k^{\prime })}}}^{(\varphi )}=1\}$	The updated evaluation information of the cluster $\overline{G_{ij,g}}$ in the $\varphi$-th round
$e{c^{\prime }}_{ij,\overline{G_{ij,g}}}$	The average endo-confidence level of the cluster $\overline{G_{ij,g^{\prime }}}$ in the suggested adjustment set $A{S}_{ij,\overline{G_{ij,g}}}$
$e{c}_{ij,\overline{G_{ij,g}}}^{(\varphi )}$	The updated endo-confidence level of the cluster $\overline{G_{ij,g}}$ in the $\varphi$-th round
$\overline{w_{ij,\overline{G_{ij,g}}}^{(\varphi )}}$	The updated weight for the cluster $\overline{G_{ij,g}}$ in the $\varphi$-th round
${\overline{L_{ij}(p)}}^{\ast (\varphi )}=\{{\overline{L_{ij}^{(k^{\prime })}}}^{(\varphi )}({\overline{p_i^{(k^{\prime })}}}^{\ast (\varphi )})|k^{\prime }=1,2,\dots ,\#{\overline{L_{ij}(p)}}^{\ast (\varphi )}\}$	The rearranged PLTSs of the consensus decision information ${\overline{L_{ij}(p)}}^{(\varphi )}$
$\overline{L_i(p)}$	The overall evaluation of the alternative $A_i$
$E(\overline{L_i(p)})$	The score of the overall evaluation $\overline{L_i(p)}$

Source: calculated by the methods using the original data.