Multi-criteria group decision making with a partial-ranking-based ordinal consensus reaching process for automotive development management

Figures & data

Table 1. Some mathematical notations used in this study.

Notation	Description
$E=\{e_1,e_2,\dots ,e_m\}$	A set of $m$ experts
$X=\{x_1,x_2\cdots ,x_n\}$	A set of $n$ alternatives
$C=\{c_1,c_2,\dots ,c_J\}$	A set of $J$ expert criteria
$R^k={(r_{ij})}_{n\times J}$	The decision matrix of expert $e_k,$ where $r_{ij}$ is the criterion value for alternative $x_i$ with respect to criterion $c_j.$
$h_S^{ij}$	The hesitant fuzzy linguistic element about the evaluation of alternative $x_i$ with respect to criterion $c_j.$
$\eta ={({\eta }_1,{\eta }_2,\dots ,{\eta }_m)}^T$	A vector of expert weights
$\omega ={({\omega }_1,{\omega }_2,\dots ,{\omega }_J)}^T$	A vector of criterion weights
$P$	A preference relation
$I$	An indifference relation
$R$	An incomparability relation
$P^{-1}$	An anti-preference relation
$D_{ij}^k$	The global preference score of alternative $x_i$ under criterion $c_j$ according to expert $e_k$
$D_{ij}^C$	The collective preference score of alternative $x_i$ under criterion $c_j$
$OC{D}_{ij}$	The ordinal consensus degree for binary relations between the alternative pair $(x_i,x_j)$
$CC{D}_{ij}$	The cardinal consensus degree for binary relations between the alternative pair $(x_i,x_j)$
${\overline{\mathit{OCD}}}_{ij}$	The global ordinal consensus threshold for binary relations between the alternative pair $(x_i,x_j)$
$AP=\{(x_{i^{*}},x_{j^{*}})\}$	A set of identified alternative pairs with unacceptable consensus degrees
$E{P}_{i^{*}{j}^{*}}=\left\{e_{k^{*}}\right\}$	A set of identified experts with poor consensus level at the identified alternative pair $(x_{i^{*}},x_{j^{*}})$
$b_k^{ij}\in \{P,P^{-1},I,R\}$	The binary relation type between the alternative pair $(x_i,x_j)$ according to $e_k$
$B^k={(b_k^{ij})}_{1\times n(n-1)/2},j>i$	The individual decision result of $e_k$
$B^C={(b_C^{ij})}_{1\times n(n-1)/2},j>i$	The collective decision result
$\gamma$	A hesitant fuzzy linguistic indifference threshold
$\mu$	A preference threshold
$\sigma$	An indifference threshold

Source: created by the authors.

Figure 1. The conflict analyses in the ORESTE method.

Source: from Liao et al. (2018a).

Table 2. Comparison of fusing methods.

Methods	Calculation	Expert weights	Uniqueness	Transitivity
Union-based fusion	The union of HFLEs	×	√	√
Score-based fusion	Arithmetic/geometric mean	√	√	√
Cook et al. (1986)	Hungarian algorithm	With conditions	×	With conditions
Jabeur et al. (2004)	Axiomatic distance	√	√	√
Yoo et al. (2020)	Branch and bound algorithm	√	√	√

Source: created by the authors.

Figure 2. An illustrative example of a partial ranking.

Source: created by the authors.

Table 3. Implicit binary relations in the illustrative partial ranking.

Alternatives	$x_1$	$x_2$	$x_3$	$x_4$	$x_5$
$x_1$	$I$	$I$	$P$	$P$	$P^{-1}$
$x_2$	$I$	$I$	$P$	$P$	$P^{-1}$
$x_3$	$P^{-1}$	$P^{-1}$	$I$	$R$	$P^{-1}$
$x_4$	$P^{-1}$	$P^{-1}$	$R$	$I$	$P^{-1}$
$x_5$	$P$	$P$	$P$	$P$	$I$

Source: created by the authors.

Figure 3. A consensus measurement process in terms of two hierarches.

Source: created by the authors.

Table 4. The frequency distribution of the relation types between $(x_i,x_j)$ in the first level.

Option (relation type)		Frequency	Relative frequency	Cumulative relative frequency
Comparable		$f_1^1$	$f_1^1/m$	$F_1^1=f_1^1/m$
Incomparable		$f_2^1$	$f_2^1/m$	$F_2^1=1$

Source: created by the authors.

Table 5. The frequency distribution of the relation types between $(x_i,x_j)$ in the second level.

Option (relation type)	Frequency	Relative frequency	Cumulative relative frequency
Preference $P$	$f_1^2$	$f_1^2/f_1^1$	$F_1^2=f_1^2/f_1^1$
Indifference $I$	$f_2^2$	$f_2^2/f_1^1$	$F_2^2=(f_1^2+f_2^2)/f_1^1$
Anti-preference $P^{-1}$	$f_3^2$	$f_3^2/f_1^1$	$F_3^2=1$

Source: created by the authors.

Figure 4. The train of thinking to set the consensus threshold.

Source: created by the authors.

Table 6. The collective preference scores.

	$c_1$	$c_2$	$c_3$	$c_4$
$x_1$	0.3495	0.3156	0.3005	0.1716
$x_2$	0.3704	0.2625	0.0575	0.0681
$x_3$	0.3135	0.3835	0.2917	0.2089
$x_4$	0.3412	0.2614	0.1370	0.1494
$x_5$	0.3567	0.2553	0.0795	0.1226

Source: created by the authors.

Table 7. The frequency distribution of the relation types between $(x_4,x_5)$ in the first level.

Option (relation type)	Frequency	Relative frequency	Cumulative relative frequency
Comparable	3	3/5	3/5
Incomparable	2	2/5	1

Source: created by the authors.

Table 8. The frequency distribution of the relation types between $(x_4,x_5)$ in the second level.

Option (relation type)	Frequency	Relative frequency	Cumulative relative frequency
Preference $P$	0	0	0
Indifference $I$	3/2	1/2	1/2
Anti-preference $P^{-1}$	3/2	1/2	1

Source: created by the authors.

Table 9. The ordinal consensus degrees for $(x_i,x_j).$

$(x_i,x_j)$	$(x_1,x_2)$	$(x_1,x_3)$	$(x_1,x_4)$	$(x_1,x_5)$	$(x_2,x_3)$	$(x_2,x_4)$	$(x_2,x_5)$	$(x_3,x_4)$	$(x_3,x_5)$	$(x_4,x_5)$
$OC{D}_{ij}$	1	0.42	0.8	1	1	0.82	0.18	1	1	0.38

Source: created by the authors.

Table 10. The consensus thresholds for $(x_i,x_j).$

$(x_i,x_j)$	$(x_1,x_2)$	$(x_1,x_3)$	$(x_1,x_4)$	$(x_1,x_5)$	$(x_2,x_3)$	$(x_2,x_4)$	$(x_2,x_5)$	$(x_3,x_4)$	$(x_3,x_5)$	$(x_4,x_5)$
${\overline{\mathit{OCD}}}_{ij}$	0.6	0.56	0.6	0.6	0.6	0.54	0.58	0.6	0.6	0.52

Source: created by the authors.

Figure 5. The frequency variations of binary relations in $(x_4,x_5).$

(a) The first hierarchy (b) The second hierarchy

Source: created by the authors.

Figure 6. The average preference intensities between $x_1$ and $x_3.$

Source: created by the authors.

Figure 7. The schematic of conflict analysis.

Source: from Liao et al. (2018a).

Figure 8. The distances between binary relations.

Source: from Jabeur et al. (2004).

Table A1. The global preference scores derived from the evaluations of $e_1.$

	$c_1$	$c_2$	$c_3$	$c_4$
$x_1$	0.4602	0.2517	0.4861	0.3149
$x_2$	0.3436	0.2398	0	0.0589
$x_3$	0.3076	0.2517	0.4419	0.4021
$x_4$	0.3436	0.2357	0.3094	0.1451
$x_5$	0.2946	0.3231	0.2652	0.1863

Table A2. The average preference intensities derived from the evaluations of $e_1$

$(x_i,x_k)$	$(x_1,x_2)$	$(x_1,x_3)$	$(x_1,x_4)$	$(x_1,x_5)$	$(x_2,x_3)$	$(x_2,x_4)$	$(x_2,x_5)$	$(x_3,x_4)$	$(x_3,x_5)$	$(x_4,x_5)$
$I(x_i,x_k)$($\times {10}^{-2}$)	0	2.1792	0	1.7839	19.9256	9.888	11.8963	0.8998	1.7839	3.2158
$I(x_k,x_i)$ ($\times {10}^{-2}$)	21.7671	4.9204	11.9817	12.8787	0.8998	0.1027	1.2241	10.1403	10.1375	2.329

Table A3. The global preference scores derived from the evaluations of $e_2.$

	$c_1$	$c_2$	$c_3$	$c_4$
$x_1$	0.2946	0.2946	0.3094	0.1451
$x_2$	0.3964	0.2946	0.1326	0.0589
$x_3$	0.2946	0.3889	0.3536	0.1863
$x_4$	0.3436	0.2517	0.0884	0.1451
$x_5$	0.4602	0.2357	0	0.1062

Table A4. The average preference intensities derived from the evaluations of $e_2$

$(x_i,x_k)$	$(x_1,x_2)$	$(x_1,x_3)$	$(x_1,x_4)$	$(x_1,x_5)$	$(x_2,x_3)$	$(x_2,x_4)$	$(x_2,x_5)$	$(x_3,x_4)$	$(x_3,x_5)$	$(x_4,x_5)$
$I(x_i,x_k)$($\times {10}^{-2}$)	2.5438	4.4934	1.2241	4.1399	11.0669	2.1540	2.7787	1.2241	4.1399	2.9158
$I(x_k,x_i)$ ($\times {10}^{-2}$)	6.5735	0	6.5967	10.1786	2.5438	3.4970	4.7877	11.0902	14.672	3.5819

Table A5. The global preference scores derived from the evaluations of $e_3.$

	$c_1$	$c_2$	$c_3$	$c_4$
$x_1$	0.3076	0.3231	0.2652	0.1451
$x_2$	0.3964	0.2357	0	0.0737
$x_3$	0.2946	0.3889	0.2652	0.0589
$x_4$	0.3683	0.2946	0.0884	0.0737
$x_5$	0.3231	0.2704	0.1326	0.1451

Table A6. The average preference intensities derived from the evaluations of $e_3$

$(x_i,x_k)$	$(x_1,x_2)$	$(x_1,x_3)$	$(x_1,x_4)$	$(x_1,x_5)$	$(x_2,x_3)$	$(x_2,x_4)$	$(x_2,x_5)$	$(x_3,x_4)$	$(x_3,x_5)$	$(x_4,x_5)$
$I(x_i,x_k)$($\times {10}^{-2}$)	2.2195	1.6459	1.5171	0.3871	10.4596	3.6828	5.9686	2.2097	2.8655	2.8906
$I(x_k,x_i)$ ($\times {10}^{-2}$)	10.5994	2.4784	6.9166	4.6309	2.9121	0.7024	1.8324	6.7767	6.2768	1.7349

Table A7. The global preference scores derived from the evaluations of $e_4.$

	$c_1$	$c_2$	$c_3$	$c_4$
$x_1$	0.3076	0.4623	0.0884	0.0737
$x_2$	0.2946	0.3231	0	0.1062
$x_3$	0.323	0.5403	0.2652	0.0589
$x_4$	0.3436	0.2357	0.0884	0.4021
$x_5$	0.3076	0.2398	0.1326	0.0737

Table A8. The average preference intensities derived from the evaluations of $e_4$

$(x_i,x_k)$	$(x_1,x_2)$	$(x_1,x_3)$	$(x_1,x_4)$	$(x_1,x_5)$	$(x_2,x_3)$	$(x_2,x_4)$	$(x_2,x_5)$	$(x_3,x_4)$	$(x_3,x_5)$	$(x_4,x_5)$
$I(x_i,x_k)$($\times {10}^{-2}$)	0.8143	6.7546	9.1106	1.1049	12.77	10.8303	3.6389	9.0918	0.3683	1.2075
$I(x_k,x_i)$ ($\times {10}^{-2}$)	6.0154	0.3683	5.666	5.5633	1.1826	2.1846	2.8962	12.0334	11.213	9.1106

Table A9. The global preference scores derived from the evaluations of $e_5.$

	$c_1$	$c_2$	$c_3$	$c_4$
$x_1$	0.4602	0.2946	0.3536	0.2287
$x_2$	0.3436	0.2357	0.0884	0.0589
$x_3$	0.3683	0.3548	0.1768	0.4459
$x_4$	0.2946	0.2517	0.221	0.1451
$x_5$	0.3076	0.2357	0	0.1062

Table A10. The average preference intensities derived from the evaluations of $e_5$

$(x_i,x_k)$	$(x_1,x_2)$	$(x_1,x_3)$	$(x_1,x_4)$	$(x_1,x_5)$	$(x_2,x_3)$	$(x_2,x_4)$	$(x_2,x_5)$	$(x_3,x_4)$	$(x_3,x_5)$	$(x_4,x_5)$
$I(x_i,x_k)$($\times {10}^{-2}$)	0	6.9323	0	0	15.4771	5.8693	1.1826	1.1049	0	0.3243
$I(x_k,x_i)$ ($\times {10}^{-2}$)	15.2622	6.7179	10.617	17.1891	0	1.2241	3.1095	11.9368	17.404	6.8964

Liao, H. C., Wu, X. L., Liang, X. D., Xu, J. P., & Herrera, F. (2018a). A new hesitant fuzzy linguistic ORESTE method for hybrid multi-criteria decision making. IEEE Transactions on Fuzzy Systems, 26(6), 3793–3807. https://doi.org/10.1109/TFUZZ.2018.2849368

 Web of Science ®Google Scholar

Cook, W. D., Kress, M., & Seiford, L. (1986). Information and preference in partial orders: A bimatrix representation. Psychometrika, 51(2), 197–207. https://doi.org/10.1007/BF02293980

 Web of Science ®Google Scholar

Jabeur, K., Martel, J., & Khélifa, S. (2004). A distance-based collective preorder integrating the relative importance of the group's members. Group Decision and Negotiation, 13(4), 327–349. https://doi.org/10.1023/B:GRUP.0000042894.00775.75

 Web of Science ®Google Scholar

Yoo, Y., Escobedo, A. R., & Skolfield, J. K. (2020). A new correlation coefficient for comparing and aggregating non-strict and incomplete rankings. European Journal of Operational Research, 285(3), 1025–1041. https://doi.org/10.1016/j.ejor.2020.02.027

 Web of Science ®Google Scholar