Robust maximum expert consensus modeling with dynamic feedback mechanism under uncertain environments

Shaojian Qu; Yingying Zhou; Ying Ji; Zhenhua Dai; Zelin Wang

doi:10.3934/jimo.2024093

Article Contents

2025, Volume 21, Issue 1: 524-552. Doi: 10.3934/jimo.2024093

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Robust maximum expert consensus modeling with dynamic feedback mechanism under uncertain environments

1.
School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing, China
2.
Research Center of Risk Management and Emergency Decision Making, School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing, China
3.
Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters (CIC-FEMD), Nanjing University of Information Science and Technology, Nanjing, China
4.
School of Management, Shanghai University, Shanghai, China
5.
Business School, University of Shanghai for Science and Technology, Shanghai, China

^*Corresponding author: Ying Ji
^*Corresponding author: Ying Ji

Received: March 2024

Revised: May 2024

Early access: July 2, 2024

Published online: July 2, 2024

Abstract / Introduction Full Text(HTML) Figure(10) / Table(8) Related Papers Cited by

Abstract

The maximum expert consensus model (MECM) is an important decision method in group decision making (GDM). In previous research, decision makers' (DMs) soft consensus thresholds are often predetermined and fixed in MECM, leading to decreased effectiveness in achieving consensus. Therefore, we propose flexible MECMs with dynamic feedback adjustment mechanisms (DFA-MECMs) to guides DMs to modify consensus thresholds in CRP. However, dynamic adjustments of consensus thresholds in a real-world decision environment may lead to uncertainties in unit opinion adjustment costs. Moreover, DMs have different risk preferences on this uncertainty. To solve this problem, we employ robust optimization methods to construct three robust DFA-MECMs with different degrees of conservatism. Additionally, we develop a reference budget model to obtain a feasible budget range of robust DFA-MECMs. Furthermore, the effectiveness and superiority of robust DFA-MECMs are demonstrated using the consensus-achieving example of the carbon emission standard. Finally, sensitivity analysis and comparison analysis are presented and discussed.

Keywords:

Mathematics Subject Classification: Primary: 90B50, 90C90; Secondary: 90C11.

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Figure 1. Consensus reaching process.

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Figure 2. Solution of $ \varepsilon $ under different $ \gamma $.

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Figure 3. Solution of $ w $ under different $ \gamma $.

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Figure 4. DFA-MECM under different $ \gamma $.

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Figure 5. Influence of $ B $ on the models.

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Figure 6. Influence of $ B $ on the models.

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Figure 7. Influence of $ B $ on the models.

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Figure 8. TL under different $ \gamma $.

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Figure 9. CL under different $ \gamma $.

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Figure 10. Comparison of consensus results under different models.

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Table 1. Symbol definitions.

Notations	Meanings
$ N = \{1, 2, \ldots, n\} $	Number of DMs in GDM
$ E = \{e_1, e_2, \ldots, e_n\} $	The set of DMs
$ c_i $	$ d_i $'s unit adjustment cost
$ c_i^l $	$ c_i $'s floating value
$ w_i $	$ d_i $'s importance among DMs
$ \tau $	Uncertain level in the box set
$ \Omega $	Uncertain level in the ellipsoid set
$ \Gamma $	Uncertain level in the polyhedron set
$ o_i $	Original opinion of $ d_i $
$ \bar{o}_i $	Adjusted opinion of $ d_i $
$ \bar{o} $	The collective opinion
$ \varepsilon_i $	Soft consensus threshold
$ \gamma $	Tolerance level or consensus level threshold
$ M $	A sufficiently large number
$ x_i, \alpha_{i1}, \alpha_{i2}, A_{ki}, B_{ki} $	Binary variables
$ d_i, s_i, t_i, u_l, z_i, y_l $	Intermediate variables
$ TL_i $	$ d_i $'s tolerance level
$ CL_i $	$ d_i $'s consensus level

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Table 2. Consensus reaching process.

Input:	$ c_i, o_i, \gamma, B, M $
Output:	$ \bar{o}, \bar{o}_i, x_i, \varepsilon_i, w_i, TL_i, CL_i $
Step 1:	Solve the RB-DFA-MECM and determine whether the budget provided by the moderator is within the range of the baseline budget: (1) if $ B $ is within the range, go to step 2; (2) Otherwise, robust DFA-MECMs has no solution, which means there is no expert reach consensus, then, go to step 6.
Step 2:	Substitute basic information into robust DFA-MECM for calculation.
Step 3:	Solve the robust DFA-MECM and determine if the model has a solution at this point. If there is no solution, adjust the value of $ \gamma $ and return to Step 2, otherwise, go to Step 4.
Step 4:	Calculate the value of $ CL_i $ and $ TL_i $ based on the data obtained from Step 2.
Step 5:	Judge whether the DMs are satisfied with the solution. If DMs are not satisfied, they can further adjust their $ \varepsilon_i $ or $ \gamma $ and return to Step 2. If DMs are satisfied, then output the final solution.
Step 6:	End the dynamic feedback adjustment CRP.

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Table 3. The solution of DFA-MECM under different $ \gamma $.

$ \gamma $	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
$ n $	4	4	4	4	4	4	4	4	4	4
$ \bar{o} $	8	7.9	7.9	8	8	8	7.9	8	7.9	8

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Table 4. Robust DFA-MECM under different $ \gamma $.

$ \gamma $	P3		P4		P5		P6
$ \gamma $	$ n $	label	$ n $	label	$ n $	label	$ n $	label
0.1	4	{1, 2, 3, 4}	3	{1, 3, 4}	3	{1, 3, 4}	3	{1, 3, 4}
0.2	4	{1, 2, 3, 4}	3	{1, 3, 4}	3	{1, 3, 4}	3	{1, 3, 4}
0.3	4	{1, 2, 3, 4}	3	{1, 3, 4}	3	{1, 3, 4}	3	{1, 3, 4}
0.4	4	{1, 2, 3, 4}	3	{1, 3, 4}	3	{1, 3, 4}	3	{1, 3, 4}
0.5	4	{1, 2, 3, 4}	3	{1, 3, 4}	3	{1, 3, 4}	3	{1, 3, 4}
0.6	4	{1, 2, 3, 4}	3	{1, 3, 4}	3	{1, 3, 4}	3	{1, 3, 4}
0.7	4	{1, 2, 3, 4}	3	{1, 3, 4}	3	{1, 3, 4}	3	{1, 3, 4}
0.8	4	{1, 2, 3, 4}	-	-	3	{2, 3, 4}	3	{2, 3, 4}
0.9	4	{1, 2, 3, 4}	-	-	-	-	-	-
1	4	{1, 2, 3, 4}	-	-	-	-	-	-

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Table 5. Influence of $ c_i' $ in P4.

$ \gamma $	$ c_i^l $
$ \gamma $	1	1.1	1.2	1.3	1.4	1.5
0.1	3	3	3	3	3	3
0.2	3	3	3	3	3	3
0.3	3	3	3	3	3	3
0.4	3	3	3	3	3	3
0.5	3	3	3	3	3	3
0.6	3	3	3	3	3	3
0.7	2	2	2	2	2	2
0.8	-	-	-	-	-	-
0.9	-	-	-	-	-	-
1	-	-	-	-	-	-

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Table 6. Influence of $c_i'$ in P6 .

$\gamma$	$c_i^l$
$\gamma$	1	1.1	1.2	1.3	1.4	1.5
0.1	3	3	3	3	3	3
0.2	3	3	3	3	3	3
0.3	3	3	3	3	3	3
0.4	3	3	3	3	3	3
0.5	3	3	3	3	3	3
0.6	3	3	3	3	3	3
0.7	3	3	3	3	3	3
0.8	3	2	3	3	2	-
0.9	-	-	-	-	-	-
1	-	-	-	-	-	-

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Table 7. Influence of $ c_i' $ in P6.

$ \gamma $	$ c_i^l $
$ \gamma $	1	1.1	1.2	1.3	1.4	1.5
0.1	3	3	3	3	3	3
0.2	3	3	3	3	3	3
0.3	3	3	3	3	3	3
0.4	3	3	3	3	3	3
0.5	3	3	3	3	3	3
0.6	3	3	3	3	3	3
0.7	3	3	3	3	3	3
0.8	3	2	3	3	2	-
0.9	-	-	-	-	-	-
1	-	-	-	-	-	-

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Table 8. The solution of DFA-MECM under different $ \gamma $.

Reference	objective weights?	Variable weights?	Variable threshold?	Baseline budget?	$ Z^{Box} $ set?	$ Z^{Pol} $ set?	$ Z^{Box-pol} $ set?
[10]	No	No	No	No	Yes	No	No
[40]	No	No	No	No	Yes	No	No
[34]	No	No	No	No	Yes	Yes	No
This paper	Yes	Yes	Yes	Yes	Yes	Yes	Yes

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References

[1]	S. Alonso, E. Herrera-Viedma, F. Chiclana and F. Herrera, A web based consensus support system for group decision making problems and incomplete preferences, Inf. Sci., 180 (2010), 4477-4495. doi: 10.1016/j.ins.2010.08.005.
[2]	D. Ben-Arieh and T. Easton, Multi-criteria group consensus under linear cost opinion elasticity, Decis. Support Syst., 43 (2007), 713-721. doi: 10.1016/j.dss.2006.11.009.
[3]	D. Ben-Arieh, T. Easton and B. Evans, Minimum cost consensus with quadratic cost functions, IEEE Trans. Syst. Man Cybern. Part A Syst. Humans, 39 (2009), 210-217. doi: 10.1109/TSMCA.2008.2006373.
[4]	T. Berthold, S. Heinz and S. Vigerske, Extending a CIP framework to solve MIQCPs, The IMA Volumes in Mathematics and its Applications, New York, 2012, 427-444. doi: 10.1007/978-1-4614-1927-3_15.
[5]	X. Chen, Z. Ding, Y. Dong and H. Liang, Managing consensus with minimum adjustments in group decision making with opinions evolution, IEEE Trans. Syst. Man Cybern. Part A Syst. Humans, 51 (2021), 2299-2311. doi: 10.1109/TSMC.2019.2912231.
[6]	D. Cheng, Y. Yuan, Y. Wu, T. Hao and F. Cheng, Maximum satisfaction consensus with budget constraints considering individual tolerance and compromise limit behaviors, European J. Oper. Res., 297 (2022), 221-238. doi: 10.1016/j.ejor.2021.04.051.
[7]	D. Cheng, Z. Zhou, F. Cheng, Y. Zhou and Y. Xie, Modeling the minimum cost consensus problem in an asymmetric costs context, European J. Oper. Res., 270 (2018), 1122-1137. doi: 10.1016/j.ejor.2018.04.041.
[8]	J. Chu, Y. Wang, X. Liu and Y. Liu, Social network community analysis based large-scale group decision making approach with incomplete fuzzy preference relations, Inf. Fusion, 60 (2020), 98-120. doi: 10.1016/j.inffus.2020.02.005.
[9]	Y. Dong, H. Zhang and E. Herrera-Viedma, Integrating experts' weights generated dynamically into the consensus reaching process and its applications in managing non-cooperative behaviors, Decis. Support Syst., 84 (2016), 1-15. doi: 10.1016/j.dss.2016.01.002.
[10]	Z. Gong, W. Guo, E. Herrera-Viedma, Z. Gong and G. Wei, Consistency and consensus modeling of linear uncertain preference relations, European J. Oper. Res., 283 (2020), 290-307. doi: 10.1016/j.ejor.2019.10.035.
[11]	Z. Gong, X. Xu, F. Lu, L. Li and C. Xu, On consensus models with utility preferences and limited budget, Appl. Soft Comput., 35 (2015), 840-849. doi: 10.1016/j.asoc.2015.03.019.
[12]	Z. Gong, H. Zhang, J. Forrest, L. Li and X. Xu, Two consensus models based on the minimum cost and maximum return regarding either all individuals or one individual, European J. Oper. Res., 240 (2015), 183-192. doi: 10.1016/j.ejor.2014.06.035.
[13]	Z. Gong, N. Zhang and F. Chiclana, The optimization ordering model for intuitionistic fuzzy preference relations with utility functions, Knowl.-Based Syst., 162 (2018), 174-184. doi: 10.1016/j.knosys.2018.07.012.
[14]	W. Guo, Z. Gong, X. Xu, O. Krejcar and E. Herrera-Viedma, Linear uncertain extensions of the minimum cost consensus model based on uncertain distance and consensus utility, Inf. Fusion, 70 (2021), 12-26. doi: 10.1016/j.inffus.2020.12.002.
[15]	W. Guo, Z. Gong, W.-G. Zhang and Y. Xu, Minimum cost consensus modeling under dynamic feedback regulation mechanism considering consensus principle and tolerance level, European J. Oper. Res., 306 (2023), 1279-1295. doi: 10.1016/j.ejor.2022.08.033.
[16]	Y. Han, Y. Ji and S. Qu, A robust minimum-cost consensus model with uncertain aggregation weights based on data-driven method, IEEE Trans. Comput. Social Syst., 9 (2022), 1167-1184. doi: 10.1109/TCSS.2021.3127751.
[17]	F. Herrera, E. Herrera-Viedma and J. L. Verdegay, A model of consensus in group decision making under linguistic assessments, Fuzzy Sets Syst., 78 (1996), 73-87. doi: 10.1016/0165-0114(95)00107-7.
[18]	E. Herrera-Viedma, F. Herrera and F. Chiclana, A consensus model for multiperson decision making with different preference structures, IEEE Trans. Syst. Man Cybern. Part A Syst. Humans, 32 (2002), 394-402. doi: 10.1109/TSMCA.2002.802821.
[19]	E. Herrera-Viedma, L. Martinez, F. Mata and F. Chiclana, A consensus support system model for group decision-making problems with multigranular linguistic preference relations, IEEE Trans. Fuzzy Syst., 13 (2005), 644-658. doi: 10.1109/TFUZZ.2005.856561.
[20]	Y. Ji and Y. Ma, The robust maximum expert consensus model with risk aversion, Inf. Fusion, 99 (2023), 101866. doi: 10.1016/j.inffus.2023.101866.
[21]	G. Li, G. Kou and Y. Peng, A group decision making model for integrating heterogeneous information, IEEE Trans. Syst. Man Cybern. Part A Syst. Humans, 48 (2018), 982-992. doi: 10.1109/TSMC.2016.2627050.
[22]	H. Li, Y. Ji, Z. Gong and S. Qu, Two-stage stochastic minimum cost consensus models with asymmetric adjustment costs, Inf. Fusion, 71 (2021), 77-96. doi: 10.1016/j.inffus.2021.02.004.
[23]	P. Li, Z. Xu, J. Liu and C. Wei, Social network group decision-making for probabilistic linguistic information based on GRA, Comput. Ind. Eng., 175 (2023), 108861. doi: 10.1016/j.cie.2022.108861.
[24]	B. Liu, Uncertainty Theory, Springer, Berlin, Heidelberg, 2007.
[25]	Y. Lu, Y. Xu, E. Herrera-Viedma and Y. Han, Consensus of large-scale group decision making in social network: The minimum cost model based on robust optimization, Inf. Sci., 547 (2021), 910-930. doi: 10.1016/j.ins.2020.08.022.
[26]	M. S. Memon, Y. H. Lee and S. I. Mari, Group multi-criteria supplier selection using combined grey systems theory and uncertainty theory, Expert Syst. Appl., 42 (2015), 7951-7959. doi: 10.1016/j.eswa.2015.06.018.
[27]	F. Meng, Q. An and X. Chen, A consistency and consensus-based method to group decision making with interval linguistic preference relations, J. Oper. Res. Soc., 67 (2016), 1419-1437. doi: 10.1057/jors.2016.28.
[28]	K. Nag, T. Pal, R. Mudi and N. Pal, Robust multi objective optimization with robust consensus, IEEE Trans. Fuzzy Syst., 26 (2018), 3743-3754. doi: 10.1109/TFUZZ.2018.2848261.
[29]	I. Palomares, R. M. Rodríguez and L. Martínez, An attitude-driven web consensus support system for heterogeneous group decision making, Expert Syst. Appl., 40 (2013), 139-149. doi: 10.1016/j.eswa.2012.07.029.
[30]	Y. Peng, G. Kou, G. Wang and Y. Shi, FAMCDM: A fusion approach of MCDM methods to rank multiclass classification algorithms, Omega, 39 (2011), 677-689. doi: 10.1016/j.omega.2011.01.009.
[31]	I. J. Pérez, F. J. Cabrerizo and E. Herrera-Viedma, A mobile decision support system for dynamic group decision-making problems, IEEE Trans. Syst. Man Cybern. Part A Syst. Humans, 40 (2010), 1244-1256. doi: 10.1109/TSMCA.2010.2046732.
[32]	S. Qu, Y. Han, Z. Wu and H. Raza, Consensus modeling with asymmetric cost based on data-driven robust optimization, Group Decis. Negot., 30 (2021), 1395-1432. doi: 10.1007/s10726-020-09707-w.
[33]	S. Qu, Y. Li and Y. Ji, The mixed integer robust maximum expert consensus models for large-scale GDM under uncertainty circumstances, Appl. Soft Comput., 107 (2021), 107369. doi: 10.1016/j.asoc.2021.107369.
[34]	R. Rodríguez, L. Labella, B. Dutta and L. Martínez, Comprehensive minimum cost models for large scale group decision making with consistent fuzzy preference relations, Knowl. Based Syst., 215 (2021), 106780. doi: 10.1016/j.knosys.2021.106780.
[35]	Q. Sun, J. Wu, F. Chiclana, H. Fujita and E. Herrera-Viedma, A dynamic feedback mechanism with attitudinal consensus threshold for minimum adjust-ment cost in group decision making, IEEE Trans. Fuzzy Syst., 99 (2021), 1-1.
[36]	X. Tan, Z. Gong, F. Chiclana and N. Zhang, Consensus modeling with cost chance constraint under uncertainty opinions, Appl. Soft Comput., 49 (2017), 1-20.
[37]	J. M. Tapia Garcia, M. J. Del-Moral, M. A. Martinez and E. Herrera-Viedma, A consensus model for group decision-making problems with interval fuzzy preference relations, Int. J. Inf. Tech. Decis., 11 (2012), 709-725. doi: 10.1142/S0219622012500174.
[38]	J. Wei, S. Qu, Q. Wang, D. Luan and X. Zhao, The novel data-driven robust maximum expert mixed integer consensus models Under Multirole's opinions uncertainty by considering noncooperators, IEEE Trans. Comput. Social Syst., 10 (2023), 3454-3470. doi: 10.1109/TCSS.2022.3192897.
[39]	J. Wu, X. Li, F. Chiclana and R. Yager, An attitudinal trust recommendation mechanism to balance consensus and harmony in group decision making, IEEE Trans. Fuzzy Syst., 27 (2019), 2163-2175. doi: 10.1109/TFUZZ.2019.2895564.
[40]	Y. Wu, Y. Gao, B. Zhang and W. Pedrycz, Minimum information-loss transformations to support heterogeneous group decision making in a distributed linguistic context, Inf. Fusion, 89 (2023), 437-451. doi: 10.1016/j.inffus.2022.07.009.
[41]	Z. Xu and Q. Da, An overview of operators for aggregating information, Int. J. Intell. Syst., 18 (2010), 953-969. doi: 10.1002/int.10127.
[42]	R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Syst. Man Cybern., 18 (1988), 183-190. doi: 10.1109/21.87068.
[43]	G.-F. Yu, W. Fei and D.-F. Li, A compromise-typed variable weight decision method for hybrid multi-attribute decision making, IEEE Trans. Fuzzy Syst., 27 (2019), 861-872. doi: 10.1109/TFUZZ.2018.2880705.
[44]	Y. Yuan, D. Cheng, Z. Zhou and F. Cheng, A minimum adjustment cost consensus framework considering harmony degrees and trust propagation for social network group decision making, IEEE Trans. Syst. Man Cybern. Part A Syst. Humans, 53 (2023), 1453-1465. doi: 10.1109/TSMC.2022.3200433.
[45]	B. Zhang, Y. Dong and Y. Xu, Maximum expert consensus models with linear cost function and aggregation operators, Comput. Ind. Eng., 66 (2013), 147-157. doi: 10.1016/j.cie.2013.06.001.
[46]	G. Zhang, Y. Dong, Y. Xu and H. Li, Minimum-cost consensus models under aggregation operators, IEEE Trans. Syst. Man Cybern., 41 (2011), 1253-1261. doi: 10.1109/TSMCA.2011.2113336.
[47]	H. Zhang, G. Kou and Y. Peng, Soft consensus cost models for group decision making and economic interpretations, European J. Oper. Res., 277 (2019), 964-980. doi: 10.1016/j.ejor.2019.03.009.
[48]	H. Zhang, F. Wang, Y. Dong, F. Chiclana and E. Herrera-Viedma, Social trust driven consensus reaching model with a minimum adjustment feedback mechanism considering assessments-modifications willingness, IEEE Trans. Fuzzy Syst., 30 (2022), 2019-2031. doi: 10.1109/TFUZZ.2021.3073251.