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Managing non-cooperative behaviors in consensus reaching processes: a comprehensive self-management weight generation mechanism

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Abstract

In group decision-making, a consensus-reaching process (CRP) is critical to minimize conflicts among decision-makers. Non-cooperative behaviors during the CRP may slow the consensus achievement or even lead to consensus failure. Previous research has not thoroughly identified various non-cooperative behaviors nor has it developed distinct management strategies for different CRP stages. This study aims to provide a systematic approach for identifying and addressing non-cooperative behaviors at different CRP stages, employing tailored management for each behavior type. We introduce and apply a concept named ‘comprehensive score’ to facilitate varied responses to non-cooperative behaviors throughout the CRP. A null-norm operator-based self-management weight generation mechanism is proposed to monitor experts’ historical performance, while a systematic analysis of experts’ characteristics enables detailed classification of non-cooperative behaviors. Through the research, we find that there are seven types of non-cooperative researches which needs to be respectively addressed according to its effects. The proposed management scheme improves the efficiency of CRP. Besides, the current research enriches the mechanisms for identifying and handling non-cooperative behaviors. It offers methodological references for non-cooperative behaviors management in more complex decision-making scenarios.

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Data availability and access

The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials.

Notes

  1. Considering that if \(k=1\), function \(\sigma _k\) has ever been applied to describe the relationship between the orness of the pessimistic exponential OWA operator[36, 37], we call \(\sigma _k\) the orness pessimistic exponential(OPE) function (see Fig. 9) in the current work. The reason why we choose OPE function is that: supposed that k is fixed, i) for a fixed y, the value of \(\sigma _k \left( y \right) \) decreases as value n increases, ii) for a fixed n, the value of \(\sigma _k \left( y \right) \) increases as value y increases. Taking use of these two properties, we can realize the distinguished treatment for non-cooperative behaviors in different periods of CRP by applying (12).

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (62006154), the Andalucía Excellence Research Program (ProyExcel-00257), the Spanish Ministry of Economy and Competitiveness through the Postdoctoral Fellowship Ramón y Cajal (RYC-2017-21978).

Author information

Authors and Affiliations

  1. Business School, University of Shanghai for Science and Technology, Shanghai, China

    Yaya Liu & Xiaowen Zhang

  2. Department of Computer Science, University of Jaen, Jaen, Spain

    Rosa M. Rodríguez & Luis Martínez

Authors
  1. Yaya Liu
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  2. Xiaowen Zhang
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  3. Rosa M. Rodríguez
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  4. Luis Martínez
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Contributions

Yaya Liu: Conceptualization, Methodology, Original draft preparation. Xiaowen Zhang: Conceptualization, Methodology, Programming. Rosa M. Rodriguez: Data curation, Original draft preparation, Writing-Reviewing, Validation. Luis Martinez: Supervision, Writing- Reviewing and Editing.

Corresponding author

Correspondence to Yaya Liu.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Appendices

Appendix A: Some figures and tables

Fig. 9
figure 9

OPE function \(\sigma _k\) with \(k=1\) and \(k=2\)

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Table 5 The original expert mutual evaluation score, comprehensive score and expert weight
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Table 6 The first round of expert mutual evaluation score, comprehensive score and expert weight
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1.1 A.1   List of mathematical notations

The list of mathematical notations in the proposed consensus model is presented by Table 4.

Table 7 The second round of expert mutual evaluation score, comprehensive score and expert weight
Full size table

1.2 A.2   Figures for OPE function

1.3 A.3    Tables in the case study

Tables in the case study is presented as below (see Table 5 to Table 7).

Appendix B: Adjusted preference relation matrix during the CRP process

B1. Experts’ adjusted preference relations in round 1.

$$\begin{aligned} P^{(1,1)}= & {} \left( \begin{matrix} 0.5&{} 0.5524&{} 0.9851&{} 0.7522&{} 0.4677&{} 0.6381\\ 0.4476&{} 0.5&{} 0.7952&{} 0.5599&{} 0.7749&{} 0.7\\ 0.0149&{} 0.2048&{} 0.5&{} 0.6&{} 0.6282&{} 0.6765\\ 0.2478&{} 0.4401&{} 0.4&{} 0.5&{} 0.5594&{} 0.3394\\ 0.5323&{} 0.2251&{} 0.3718&{} 0.4406&{} 0.5&{} 0.6895\\ 0.3619&{} 0.3&{} 0.3235&{} 0.6606&{} 0.3105&{} 0.5 \end{matrix} \right) \\ P^{(2,1)}= & {} \left( \begin{matrix} 0.5&{} 0.6286&{} 0.5619&{} 0.5988&{} 0.4&{} 0.2869\\ 0.3714&{} 0.5&{} 0.6654&{} 0.4597&{} 0.7205&{} 0.6\\ 0.4381&{} 0.3346&{} 0.5&{} 0.6912&{} 0.4829&{} 0.6\\ 0.4012&{} 0.5403&{} 0.3088&{} 0.5&{} 0.6924&{} 0.7236\\ 0.6&{} 0.2795&{} 0.5171&{} 0.3076&{} 0.5&{} 0.4930\\ 0.7131&{} 0.4&{} 0.4&{} 0.2764&{} 0.5070&{} 0.5 \end{matrix} \right) \\ P^{(3,1)}= & {} \left( \begin{matrix} 0.5&{} 0.5900&{} 0.4271&{} 0.8215&{} 0.4857&{} 0.9\\ 0.4100&{} 0.5&{} 0.5170&{} 0.4635&{} 0.4703&{} 0.8\\ 0.5729&{} 0.4830&{} 0.5&{} 0.7154&{} 0.6897&{} 0.7144\\ 0.1785&{} 0.5365&{} 0.2846&{} 0.5&{} 0.6812&{} 0.7779\\ 0.5143&{} 0.5297&{} 0.3103&{} 0.3188&{} 0.5&{} 0.7061\\ 0.1&{} 0.2&{} 0.2856&{} 0.2221&{} 0.2939&{} 0.5 \end{matrix} \right) \\ P^{(4,1)}= & {} \left( \begin{matrix} 0.5&{} 0.3268&{} 0.4787&{} 0.66767&{} 0.2700&{} 0.7997\\ 0.6732&{} 0.5&{} 0.5810&{} 0.8282&{} 0.6554&{} 0.7636\\ 0.5213&{} 0.4190&{} 0.5&{} 0.6935&{} 0.4719&{} 0.4600\\ 0.3324&{} 0.1718&{} 0.3065&{} 0.5&{} 0.3586&{} 0.1700\\ 0.7300&{} 0.3446&{} 0.5281&{} 0.6414&{} 0.5&{} 0.5920\\ 0.2003&{} 0.2364&{} 0.5400&{} 0.8300&{} 0.4080&{} 0.5 \end{matrix} \right) \\ P^{(5,1)}= & {} \left( \begin{matrix} 0.5&{} 0.5944&{} 0.4500&{} 0.4621&{} 0.5602&{} 0.5212\\ 0.4056&{} 0.5&{} 0.7846&{} 0.1825&{} 0.3&{} 0.7\\ 0.5500&{} 0.2154&{} 0.5&{} 0.2532&{} 0.6010&{} 0.5153\\ 0.5379&{} 0.8175&{} 0.7468&{} 0.5&{} 0.8399&{} 0.5971\\ 0.4398&{} 0.7&{} 0.3990&{} 0.1601&{} 0.5&{} 0.6852\\ 0.4788&{} 0.3&{} 0.4847&{} 0.4029&{} 0.3148&{} 0.5 \end{matrix} \right) \\ P^{(6,1)}= & {} \left( \begin{matrix} 0.5&{} 0.6548&{} 0.5566&{} 0.4595&{} 0.6253&{} 0.7818\\ 0.3452&{} 0.5&{} 0.6&{} 0.6854&{} 0.4361&{} 0.6148\\ 0.4434&{} 0.4&{} 0.5&{} 0.6664&{} 0.6910&{} 0.3500\\ 0.5405&{} 0.3146&{} 0.3336&{} 0.5&{} 0.6081&{} 0.3904\\ 0.3747&{} 0.5639&{} 0.3090&{} 0.3919&{} 0.5&{} 0.6842\\ 0.2182&{} 0.3852&{} 0.6500&{} 0.6096&{} 0.3158&{} 0.5 \end{matrix} \right) \\ P^{(7,1)}= & {} \left( \begin{matrix} 0.5&{} 0.8679&{} 0.4730&{} 0.7049&{} 0.6814&{} 0.7477\\ 0.1321&{} 0.5&{} 0.2&{} 0.5693&{} 0.7900&{} 0.7282\\ 0.5270&{} 0.8&{} 0.5&{} 0.9&{} 0.8094&{} 0.7511\\ 0.2951&{} 0.4307&{} 0.1&{} 0.5&{} 0.4223&{} 0.6235\\ 0.3186&{} 0.2100&{} 0.1906&{} 0.5777&{} 0.5&{} 0.7022\\ 0.2523&{} 0.2718&{} 0.2489&{} 0.3765&{} 0.2978&{} 0.5 \end{matrix} \right) \\ P^{(8,1)}= & {} \left( \begin{matrix} 0.5&{} 0.8842&{} 0.4413&{} 0.4348&{} 0.5822&{} 0.1302\\ 0.1158&{} 0.5&{} 0.5600&{} 0.7177&{} 0.7111&{} 0.3300\\ 0.5587&{} 0.4400&{} 0.5&{} 0.1610&{} 0.7300&{} 0.7392\\ 0.5652&{} 0.2823&{} 0.8390&{} 0.5&{} 0.5771&{} 0.8434\\ 0.4178&{} 0.2889&{} 0.2700&{} 0.4229&{} 0.5&{} 0.7643\\ 0.8698&{} 0.6700&{} 0.2608&{} 0.1566&{} 0.2357&{} 0.5 \end{matrix} \right) \end{aligned}$$

B2. Experts’ adjusted preference relations in round 2.

$$\begin{aligned} P^{(1,2)}= & {} \left( \begin{matrix} 0.5&{} 0.5921&{} 0.6919&{} 0.6366&{} 0.5082&{} 0.6026\\ 0.4079&{} 0.5&{} 0.6836&{} 0.5592&{} 0.9733&{} 0.6973\\ 0.3081&{} 0.3164&{} 0.5&{} 0.6&{} 0.6316&{} 0.6434\\ 0.3634&{} 0.4408&{} 0.4&{} 0.5&{} 0.5603&{} 0.4603\\ 0.4918&{} 0.0267&{} 0.3684&{} 0.4397&{} 0.5&{} 0.6820\\ 0.3974&{} 0.3027&{} 0.3566&{} 0.5397&{} 0.3180&{} 0.5 \end{matrix} \right) \\ P^{(2,2)}= & {} \left( \begin{matrix} 0.5&{} 0.6288&{} 0.5507&{} 0.5988&{} 0.4393&{} 0.2869\\ 0.3712&{} 0.5&{} 0.6654&{} 0.5031&{} 0.7186&{} 0.6126\\ 0.4493&{} 0.3346&{} 0.5&{} 0.5913&{} 0.5680&{} 0.6008\\ 0.4012&{} 0.4969&{} 0.4087&{} 0.5&{} 0.6191&{} 0.7098\\ 0.5607&{} 0.2814&{} 0.4320&{} 0.3809&{} 0.5&{} 0.2022\\ 0.7131&{} 0.3874&{} 0.3992&{} 0.2902&{} 0.7978&{} 0.5 \end{matrix} \right) \\ P^{(3,2)}= & {} \left( \begin{matrix} 0.5&{} 0.5902&{} 0.5332&{} 0.8129&{} 0.4996&{} 0.0301\\ 0.4098&{} 0.5&{} 0.5227&{} 0.5205&{} 0.5908&{} 0.7988\\ 0.4668&{} 0.4773&{} 0.5&{} 0.7154&{} 0.6618&{} 0.6104\\ 0.1871&{} 0.4795&{} 0.2846&{} 0.5&{} 0.6608&{} 0.6184\\ 0.5004&{} 0.4092&{} 0.3382&{} 0.3392&{} 0.5&{} 0.3346\\ 0.9699&{} 0.2012&{} 0.3896&{} 0.3816&{} 0.6654&{} 0.5 \end{matrix} \right) \\ P^{(4,2)}= & {} \left( \begin{matrix} 0.5&{} 0.3929&{} 0.5032&{} 0.6332&{} 0.3605&{} 0.7997\\ 0.6071&{} 0.5&{} 0.5816&{} 0.6707&{} 0.6364&{} 0.6727\\ 0.4968&{} 0.4184&{} 0.5&{} 0.8046&{} 0.5525&{} 0.4600\\ 0.3668&{} 0.3293&{} 0.1954&{} 0.5&{} 0.4557&{} 0.2567\\ 0.6395&{} 0.3636&{} 0.4475&{} 0.5443&{} 0.5&{} 0.5975\\ 0.2003&{} 0.3273&{} 0.5400&{} 0.7433&{} 0.4025&{} 0.5 \end{matrix} \right) \\ P^{(5,2)}= & {} \left( \begin{matrix} 0.5&{} 0.6210&{} 0.4500&{} 0.5768&{} 0.5310&{} 0.5541\\ 0.3790&{} 0.5&{} 0.7124&{} 0.3109&{} 0.3860&{} 0.6754\\ 0.5500&{} 0.2876&{} 0.5&{} 0.2845&{} 0.6010&{} 0.5345\\ 0.4232&{} 0.6891&{} 0.7155&{} 0.5&{} 0.3357&{} 0.4907\\ 0.4690&{} 0.6140&{} 0.3990&{} 0.6643&{} 0.5&{} 0.6852\\ 0.4459&{} 0.3246&{} 0.4655&{} 0.5093&{} 0.3148&{} 0.5 \end{matrix} \right) \\ P^{(6,2)}= & {} \left( \begin{matrix} 0.5&{} 0.6411&{} 0.4562&{} 0.4847&{} 0.6253&{} 0.7293\\ 0.3589&{} 0.5&{} 0.5975&{} 0.6854&{} 0.5979&{} 0.6322\\ 0.5438&{} 0.4025&{} 0.5&{} 0.6664&{} 0.6386&{} 0.4373\\ 0.5153&{} 0.3146&{} 0.3336&{} 0.5&{} 0.6066&{} 0.4561\\ 0.3747&{} 0.4021&{} 0.3614&{} 0.3934&{} 0.5&{} 0.6808\\ 0.2707&{} 0.3678&{} 0.5627&{} 0.5439&{} 0.3192&{} 0.5 \end{matrix} \right) \\ P^{(7,2)}= & {} \left( \begin{matrix} 0.5&{} 0.7950&{} 0.5414&{} 0.4659&{} 0.5106&{} 0.7324\\ 0.2050&{} 0.5&{} 0.3869&{} 0.5605&{} 0.6790&{} 0.7058\\ 0.4586&{} 0.6131&{} 0.5&{} 0.7403&{} 0.6828&{} 0.7136\\ 0.5341&{} 0.4395&{} 0.2597&{} 0.5&{} 0.4223&{} 0.5886\\ 0.4894&{} 0.3210&{} 0.3172&{} 0.5777&{} 0.5&{} 0.2499\\ 0.2676&{} 0.2942&{} 0.2864&{} 0.4114&{} 0.7501&{} 0.5 \end{matrix} \right) \\ P^{(8,2)}= & {} \left( \begin{matrix} 0.5&{} 0.6757&{} 0.5064&{} 0.6063&{} 0.4466&{} 0.3118\\ 0.3243&{} 0.5&{} 0.5723&{} 0.7142&{} 0.6173&{} 0.5238\\ 0.4936&{} 0.4277&{} 0.5&{} 0.5645&{} 0.6491&{} 0.7040\\ 0.3937&{} 0.2858&{} 0.4355&{} 0.5&{} 0.5790&{} 0.8434\\ 0.5534&{} 0.3827&{} 0.3509&{} 0.4210&{} 0.5&{} 0.7104\\ 0.6882&{} 0.4762&{} 0.2960&{} 0.1566&{} 0.2896&{} 0.5 \end{matrix} \right) \end{aligned}$$

Appendix C: Computation process when Dong et al’s approach is applied

C1. Experts’ adjusted preference relations and MMEMs in round 2.

$$\begin{aligned} P^{(1,2)}= & {} \left( \begin{matrix} 0.5&{} 0.6&{} 0.4958&{} 0.8105&{} 0.4&{} 0.9\\ 0.4&{} 0.5&{} 0.7250&{} 0.5683&{} 0.9&{} 0.6726\\ 0.5042&{} 0.2750&{} 0.5&{} 0.5966&{} 0.5994&{} 0.6570\\ 0.1895&{} 0.4317&{} 0.4034&{} 0.5&{} 0.5714&{} 0.3\\ 0.6&{} 0.1&{} 0.4006&{} 0.4286&{} 0.5&{} 0.6887\\ 0.1&{} 0.3274&{} 0.3430&{} 0.7&{} 0.3113&{} 0.5 \end{matrix} \right) \\ P^{(2,2)}= & {} \left( \begin{matrix} 0.5&{} 0.5969&{} 0.8094&{} 0.5896&{} 0.4&{} 0.6191\\ 0.4031&{} 0.5&{} 0.5649&{} 0.4&{} 0.6021&{} 0.6771\\ 0.1906&{} 0.4351&{} 0.5&{} 0.7&{} 0.3&{} 0.6679\\ 0.4104&{} 0.6&{} 0.3&{} 0.5&{} 0.8013&{} 0.9\\ 0.6&{} 0.3979&{} 0.7&{} 0.1987&{} 0.5&{} 0.2987\\ 0.3809&{} 0.3229&{} 0.3321&{} 0.1&{} 0.7013&{} 0.5 \end{matrix} \right) \\ P^{(3,2)}= & {} \left( \begin{matrix} 0.5&{} 0.5925&{} 0.2006&{} 0.5152&{} 0.4695&{} 0.9\\ 0.4075&{} 0.5&{} 0.1400&{} 0.2500&{} 0.1700&{} 0.7305\\ 0.7994&{} 0.8600&{} 0.5&{} 0.7323&{} 0.6786&{} 0.9729\\ 0.4848&{} 0.7500&{} 0.2677&{} 0.5&{} 0.5617&{} 0.9300\\ 0.5305&{} 0.8300&{} 0.3214&{} 0.4383&{} 0.5&{} 0.9194\\ 0.1&{} 0.2695&{} 0.0271&{} 0.0700&{} 0.0806&{} 0.5 \end{matrix} \right) \\ P^{(4,2)}= & {} \left( \begin{matrix} 0.5&{} 0.2919&{} 0.4600&{} 0.7300&{} 0.2700&{} 0.6742\\ 0.7081&{} 0.5&{} 0.7300&{} 0.7095&{} 0.7900&{} 0.8213\\ 0.5400&{} 0.2700&{} 0.5&{} 0.6842&{} 0.2118&{} 0.4556\\ 0.2700&{} 0.2905&{} 0.3158&{} 0.5&{} 0.1097&{} 0.6056\\ 0.7300&{} 0.2100&{} 0.7882&{} 0.8903&{} 0.5&{} 0.5400\\ 0.3258&{} 0.1787&{} 0.5444&{} 0.3944&{} 0.4600&{} 0.5 \end{matrix} \right) \\ P^{(5,2)}= & {} \left( \begin{matrix} 0.5&{} 0.6522&{} 0.4869&{} 0.1353&{} 0.6012&{} 0.4\\ 0.3478&{} 0.5&{} 0.8&{} 0.5533&{} 0.3&{} 0.7\\ 0.5131&{} 0.2&{} 0.5&{} 0.5500&{} 0.6&{} 0.6726\\ 0.8647&{} 0.4467&{} 0.4500&{} 0.5&{} 0.9&{} 0.6\\ 0.3988&{} 0.7&{} 0.4&{} 0.1&{} 0.5&{} 0.7018\\ 0.6&{} 0.3&{} 0.3274&{} 0.4&{} 0.2982&{} 0.5 \end{matrix} \right) \\ P^{(6,2)}= & {} \left( \begin{matrix} 0.5&{} 0.8&{} 0.6500&{} 0.9142&{} 0.7104&{} 0.6268\\ 0.2&{} 0.5&{} 0.6&{} 0.7&{} 0.2898&{} 0.5500\\ 0.3500&{} 0.4&{} 0.5&{} 0.6&{} 0.6212&{} 0.6734\\ 0.0858&{} 0.3&{} 0.4&{} 0.5&{} 0.5568&{} 0.3000\\ 0.2896&{} 0.7102&{} 0.3788&{} 0.4432&{} 0.5&{} 0.7\\ 0.3732&{} 0.4500&{} 0.3266&{} 0.7000&{} 0.3&{} 0.5 \end{matrix} \right) \\ P^{(7,2)}= & {} \left( \begin{matrix} 0.5&{} 0.4400&{} 0.5029&{} 0.8100&{} 0.7600&{} 0.6915\\ 0.5600&{} 0.5&{} 0.5801&{} 0.5937&{} 0.7900&{} 0.9136\\ 0.4971&{} 0.4199&{} 0.5&{} 0.9000&{} 0.8768&{} 0.9500\\ 0.1900&{} 0.4063&{} 0.1000&{} 0.5&{} 0.5749&{} 0.5939\\ 0.2400&{} 0.2100&{} 0.1232&{} 0.4251&{} 0.5&{} 0.7767\\ 0.3085&{} 0.0864&{} 0.0500&{} 0.4061&{} 0.2233&{} 0.5 \end{matrix} \right) \\ P^{(8,2)}= & {} \left( \begin{matrix} 0.5&{} 0.2200&{} 0.3800&{} 0.4300&{} 0.5283&{} 0.3451\\ 0.7800&{} 0.5&{} 0.5600&{} 0.7400&{} 0.8500&{} 0.3241\\ 0.6200&{} 0.4400&{} 0.5&{} 0.6846&{} 0.6018&{} 0.7407\\ 0.5700&{} 0.2600&{} 0.3154&{} 0.5&{} 0.5625&{} 0.9\\ 0.4717&{} 0.1500&{} 0.3982&{} 0.4375&{} 0.5&{} 0.7740\\ 0.6549&{} 0.6759&{} 0.2593&{} 0.1&{} 0.2260&{} 0.5 \end{matrix} \right) \end{aligned}$$
$$\begin{aligned} V^{(1,2)}=\left( \begin{matrix} null &{} null &{} null\\ 88&{} 94&{} 99\\ 96&{} 93&{} 93\\ 98&{} 93&{} 89\\ 98&{} 89&{} 99\\ 90&{} 97&{} 99\\ 86&{} 93&{} 100\\ 90&{} 100&{}87 \end{matrix} \right) V^{(2,2)}=\left( \begin{matrix} 87&{} 98&{} 95\\ null &{} null &{} null \\ 95&{} 88&{} 99\\ 96&{} 85&{} 97\\ 94&{} 92&{} 88\\ 85&{} 93&{} 96\\ 87&{} 89&{} 93\\ 99&{} 98&{} 97 \end{matrix} \right) \end{aligned}$$
$$\begin{aligned} V^{(3,2)}=\left( \begin{matrix} 85&{} 86&{} 85\\ 89&{} 86&{} 99\\ null &{} null &{} null\\ 93&{} 85&{} 96\\ 87&{} 99&{} 86\\ 86&{} 92&{} 91\\ 92&{} 97&{} 97\\ 99&{} 87&{} 85 \end{matrix} \right) V^{(4,2)}=\left( \begin{matrix} 89&{} 88&{} 87\\ 96&{} 89&{} 96\\ 89&{} 90&{} 87\\ null&{} null&{} null\\ 93&{} 94&{} 97\\ 99&{} 89&{} 91\\ 89&{} 98&{} 98\\ 94&{} 99&{} 85 \end{matrix} \right) \end{aligned}$$
$$\begin{aligned} V^{(5,2)}=\left( \begin{matrix} 92&{} 98&{} 93\\ 92&{} 95&{} 89\\ 91&{} 100&{} 85\\ 89&{} 92&{} 86\\ null&{}null&{}null\\ 95&{} 85&{} 95\\ 92&{} 99&{} 95\\ 86&{} 93&{} 85 \end{matrix} \right) V^{(6,2)}=\left( \begin{matrix} 96&{} 92&{} 85\\ 92&{} 99&{} 89\\ 99&{} 89&{} 96\\ 85&{} 96&{} 99\\ 87&{} 94&{} 96\\ null&{} null&{} null\\ 94&{} 96&{} 97\\ 95&{} 96&{} 96 \end{matrix} \right) \end{aligned}$$
$$\begin{aligned} V^{(7,2)}=\left( \begin{matrix} 95&{} 98&{} 100\\ 89&{} 97&{} 99\\ 86&{} 96&{} 92\\ 93&{} 95&{} 96\\ 94&{} 98&{} 92\\ 87&{} 94&{} 90\\ null&{} null&{} null\\ 89&{} 87&{} 99 \end{matrix} \right) V^{(8,2)}=\left( \begin{matrix} 83&{} 98&{} 94\\ 85&{} 96&{} 95\\ 93&{} 96&{} 83\\ 94&{} 96&{} 93\\ 95&{} 95&{} 98\\ 95&{} 93&{} 96\\ 95&{} 82&{} 97\\ null &{} null &{} null \end{matrix} \right) \end{aligned}$$

C2. Experts’ adjusted preference relations and MMEMs in round 3.

$$\begin{aligned} P^{(1,3)}= & {} \left( \begin{matrix} 0.5&{} 0.5220&{} 0.4958&{} 0.8105&{} 0.5282&{} 0.6467\\ 0.4780&{} 0.5&{} 0.7250&{} 0.5683&{} 0.5721&{} 0.6726\\ 0.5042&{} 0.2750&{} 0.5&{} 0.5885&{} 0.5994&{}0.6516\\ 0.1895&{} 0.4317&{} 0.4115&{} 0.5&{} 0.5714&{} 0.2925\\ 0.4718&{} 0.4279&{} 0.4006&{} 0.4286&{} 0.5&{} 0.6887\\ 0.3533&{} 0.3274&{} 0.3484&{} 0.7075&{} 0.3113&{} 0.5 \end{matrix} \right) \\ P^{(2,3)}= & {} \left( \begin{matrix} 0.5&{} 0.5972&{} 0.4956&{} 0.5864&{} 0.4000&{} 0.6191\\ 0.4028&{} 0.5&{} 0.5649&{} 0.4000&{} 0.5913&{} 0.6771\\ 0.5044&{} 0.4351&{} 0.5&{} 0.7000&{} 0.5714&{} 0.6679\\ 0.4136&{} 0.6000&{} 0.3000&{} 0.5&{} 0.5636&{} 0.9218\\ 0.6000&{} 0.4087&{} 0.4286&{} 0.4364&{} 0.5&{} 0.2925\\ 0.3809&{} 0.3229&{} 0.3321&{} 0.0782&{} 0.7075&{} 0.5 \end{matrix} \right) \\ P^{(3,3)}= & {} \left( \begin{matrix} 0.5&{} 0.5973&{} 0.2006&{} 0.5152&{} 0.4938&{} 0.6386\\ 0.4027&{} 0.5&{} 0.1400&{} 0.2334&{} 0.1700&{} 0.7305\\ 0.7994&{} 0.8600&{} 0.5&{} 0.6817&{} 0.5568&{} 0.9729\\ 0.4848&{} 0.7666&{} 0.3183&{} 0.5&{} 0.5612&{} 0.9300\\ 0.5062&{} 0.8300&{} 0.4432&{} 0.4388&{} 0.5&{} 0.9194\\ 0.3614&{} 0.2695&{} 0.0271&{} 0.0700&{} 0.0806&{} 0.5 \end{matrix} \right) \\ P^{(4,3)}= & {} \left( \begin{matrix} 0.5&{} 0.2839&{} 0.4593&{} 0.6206&{} 0.2700&{} 0.6457\\ 0.7161&{} 0.5&{} 0.7315&{} 0.5636&{} 0.6236&{} 0.8313\\ 0.5407&{} 0.2685&{} 0.5&{} 0.6821&{} 0.2118&{} 0.4385\\ 0.3794&{} 0.4364&{} 0.3179&{} 0.5&{} 0.2381&{} 0.6056\\ 0.7300&{} 0.3764&{} 0.7882&{} 0.7619&{} 0.5&{} 0.5759\\ 0.3543&{} 0.1687&{} 0.5615&{} 0.3944&{} 0.4241&{} 0.5 \end{matrix} \right) \\ P^{(5,3)}= & {} \left( \begin{matrix} 0.5&{} 0.6432&{} 0.4869&{} 0.6391&{} 0.5986&{} 0.4000\\ 0.3568&{} 0.5&{} 0.8075&{} 0.5670&{} 0.2916&{} 0.7000\\ 0.5131&{} 0.1925&{} 0.5&{} 0.6854&{} 0.5957&{} 0.6694\\ 0.3609&{} 0.4330&{} 0.3146&{} 0.5&{} 0.9000&{} 0.5996\\ 0.4014&{} 0.7084&{} 0.4043&{} 0.1000&{} 0.5&{} 0.6682\\ 0.6000&{} 0.3000&{} 0.3306&{} 0.4004&{} 0.3318&{} 0.5 \end{matrix} \right) \\ P^{(6,3)}= & {} \left( \begin{matrix} 0.5&{} 0.8261&{} 0.4920&{} 0.7864&{} 0.5108&{} 0.6490\\ 0.1739&{} 0.5&{} 0.6000&{} 0.5572&{} 0.5783&{} 0.5420\\ 0.5080&{} 0.4000&{} 0.5&{} 0.6833&{} 0.6212&{} 0.6734\\ 0.2136&{} 0.4428&{} 0.3167&{} 0.5&{} 0.5568&{} 0.2857\\ 0.4892&{} 0.4217&{} 0.3788&{} 0.4432&{} 0.5&{} 0.7003\\ 0.3510&{} 0.4580&{} 0.3266&{} 0.7143&{} 0.2997&{} 0.5 \end{matrix} \right) \\ P^{(7,3)}= & {} \left( \begin{matrix} 0.5&{} 0.4400&{} 0.5035&{} 0.8100&{} 0.7600&{} 0.6915\\ 0.5600&{} 0.5&{} 0.5801&{} 0.5948&{} 0.8004&{} 0.9136\\ 0.4965&{} 0.4199&{} 0.5&{} 0.8968&{} 0.8808&{} 0.7006\\ 0.1900&{} 0.4052&{} 0.1032&{} 0.5&{} 0.5749&{} 0.5939\\ 0.2400&{} 0.1996&{} 0.1192&{} 0.4251&{} 0.5&{} 0.7767\\ 0.3085&{} 0.0864&{} 0.2994&{} 0.4061&{} 0.2233&{} 0.5 \end{matrix} \right) \\ P^{(8,3)}= & {} \left( \begin{matrix} 0.5&{} 0.5467&{} 0.3800&{} 0.4300&{} 0.5283&{} 0.3451\\ 0.4533&{} 0.5&{} 0.5600&{} 0.7436&{} 0.8500&{} 0.6903\\ 0.6200&{} 0.4400&{} 0.5&{} 0.6829&{} 0.6040&{} 0.7207\\ 0.5700&{} 0.2564&{} 0.3171&{} 0.5&{} 0.5617&{} 0.6346\\ 0.4717&{} 0.1500&{} 0.3960&{} 0.4383&{} 0.5&{} 0.7199\\ 0.6549&{} 0.3097&{} 0.2793&{} 0.3654&{} 0.2801&{} 0.5 \end{matrix} \right) \end{aligned}$$
$$\begin{aligned} V^{(1,3)}=\left( \begin{matrix} null &{} null &{} null\\ 88&{} 74&{} 99\\ 96&{} 73&{} 73\\ 98&{} 73&{} 89\\ 98&{} 69&{} 79\\ 90&{} 77&{} 99\\ 86&{} 73&{} 100\\ 90&{} 80&{}67 \end{matrix} \right) V^{(2,3)}=\left( \begin{matrix} 87&{} 78&{} 95\\ null &{} null &{} null \\ 95&{}68&{} 79\\ 96&{} 65&{} 97\\ 94&{} 72&{} 68\\ 85&{} 73&{} 96\\ 87&{} 69&{} 93\\ 99&{} 78&{} 77 \end{matrix} \right) \end{aligned}$$
$$\begin{aligned} V^{(3,3)}=\left( \begin{matrix} 85&{} 66&{} 85\\ 89&{} 66&{} 99\\ null &{} null &{} null\\ 93&{} 65&{} 96\\ 87&{} 79&{} 66\\ 86&{} 72&{} 91\\ 92&{} 77&{} 97\\ 99&{} 67&{} 85 \end{matrix} \right) V^{(4,3)}=\left( \begin{matrix} 89&{} 68&{} 87\\ 96&{} 69&{} 96\\ 89&{} 70&{} 87\\ null&{} null&{} null\\ 93&{} 74&{} 77\\ 99&{} 69&{} 91\\ 89&{} 78&{} 98\\ 94&{} 79&{} 65 \end{matrix} \right) \end{aligned}$$
$$\begin{aligned} V^{(5,3)}=\left( \begin{matrix} 92&{} 78&{} 93\\ 92&{}75&{} 89\\ 91&{} 80&{} 65\\ 89&{} 72&{} 86\\ null&{}null&{}null\\ 95&{} 65&{} 95\\ 92&{} 79&{} 95\\ 86&{} 73&{} 65 \end{matrix} \right) V^{(6,3)}=\left( \begin{matrix} 96&{}72&{} 85\\ 92&{} 79&{} 89\\ 99&{} 69&{} 76\\ 85&{} 76&{} 99\\ 87&{} 74&{} 76\\ null&{} null&{} null\\ 94&{} 76&{} 97\\ 95&{} 76&{} 76 \end{matrix} \right) \end{aligned}$$
$$\begin{aligned} V^{(7,3)}=\left( \begin{matrix} 95&{} 78&{} 100\\ 89&{} 77&{} 99\\ 86&{} 76&{} 72\\ 93&{} 75&{} 96\\ 94&{} 78&{} 72\\ 87&{} 74&{} 90\\ null&{} null&{} null\\ 89&{} 67&{} 79 \end{matrix} \right) V^{(8,3)}=\left( \begin{matrix} 83&{} 78&{} 94\\ 85&{} 76&{} 95\\ 93&{} 76&{} 63\\ 94&{} 76&{} 93\\ 95&{} 75&{} 78\\ 95&{} 73&{} 96\\ 95&{} 62&{} 97\\ null &{} null &{} null \end{matrix} \right) \end{aligned}$$

C3. Experts’ adjusted preference relations and MMEMs in round 4.

$$\begin{aligned} P^{(1,4)}= & {} \left( \begin{matrix} 0.5&{} 0.5216&{} 0.4958&{} 0.8105&{} 0.5198&{} 0.6467\\ 0.4784&{} 0.5&{} 0.7250&{} 0.5683&{} 0.5724&{} 0.7193\\ 0.5042&{} 0.2750&{} 0.5&{} 0.5885&{} 0.5800&{} 0.6850\\ 0.1895&{} 0.4317&{} 0.4115&{} 0.5&{} 0.5719&{} 0.4713\\ 0.4802&{} 0.4276&{} 0.4200&{} 0.4281&{} 0.5&{} 0.6887\\ 0.3533&{} 0.2807&{} 0.3150&{} 0.5287&{} 0.3113&{} 0.5 \end{matrix} \right) \\ P^{(2,4)}= & {} \left( \begin{matrix} 0.5&{} 0.5972&{} 0.4376&{} 0.5864&{} 0.3985&{} 0.6227\\ 0.4028&{} 0.5&{} 0.5649&{} 0.4000&{} 0.5914&{} 0.6736\\ 0.5624&{} 0.4351&{} 0.5&{} 0.7000&{} 0.5714&{} 0.6679\\ 0.4136&{} 0.6000&{} 0.3000&{} 0.5&{} 0.5633&{} 0.9218\\ 0.6015&{} 0.4086&{} 0.4286&{} 0.4367&{} 0.5&{} 0.2925\\ 0.3773&{} 0.3264&{} 0.3321&{} 0.0782&{} 0.7075&{} 0.5 \end{matrix} \right) \\ P^{(3,4)}= & {} \left( \begin{matrix} 0.5&{} 0.5973&{} 0.3246&{} 0.5152&{} 0.4938&{} 0.6386\\ 0.4027&{} 0.5&{} 0.1018&{} 0.2962&{} 0.1700&{} 0.7305\\ 0.6754&{} 0.8982&{} 0.5&{} 0.6816&{} 0.5607&{} 0.9968\\ 0.4848&{}0.7038&{} 0.3184&{} 0.5&{} 0.5612&{} 0.9401\\ 0.5062&{} 0.8300&{} 0.4393&{} 0.4388&{} 0.5&{} 0.8228\\ 0.3614&{} 0.2695&{} 0.0032&{} 0.0599&{} 0.1772&{} 0.5 \end{matrix} \right) \\ P^{(4,4)}= & {} \left( \begin{matrix} 0.5&{} 0.2839&{} 0.4607&{} 0.6370&{} 0.2700&{} 0.6457\\ 0.7161&{} 0.5&{} 0.7336&{} 0.5636&{} 0.6236&{} 0.8078\\ 0.5393&{} 0.2664&{} 0.5 &{}0.6821&{} 0.4025&{} 0.4143\\ 0.3630&{} 0.4364&{} 0.3179&{} 0.5&{} 0.2148&{} 0.6057\\ 0.7300&{}0.3764&{} 0.5975&{} 0.7852&{} 0.5&{} 0.6736\\ 0.3543&{} 0.1922&{} 0.5857&{} 0.3943&{} 0.3264&{} 0.5 \end{matrix} \right) \\ P^{(5,4)}= & {} \left( \begin{matrix} 0.5&{} 0.6432&{} 0.4413&{} 0.6530&{} 0.5986&{} 0.3970\\ 0.3568&{} 0.5&{} 0.8075&{} 0.5670&{} 0.2818&{} 0.7192\\ 0.5587&{} 0.1925&{} 0.5&{} 0.6874&{} 0.5957&{} 0.6694\\ 0.3470&{} 0.4330&{} 0.3126&{} 0.5&{} 0.9000&{} 0.5991\\ 0.4014&{}0.7182&{} 0.4043&{} 0.1000&{} 0.5&{} 0.6682\\ 0.6030&{} 0.2808&{} 0.3306&{} 0.4009&{} 0.3318&{} 0.5 \end{matrix} \right) \\ P^{(6,4)}= & {} \left( \begin{matrix} 0.5&{} 0.8261&{} 0.4933&{} 0.7670&{} 0.5108&{} 0.6503\\ 0.1739&{} 0.5&{} 0.5949&{} 0.5344&{} 0.5665&{} 0.5420\\ 0.5067&{} 0.4051&{} 0.5&{} 0.6826&{} 0.6216&{} 0.6734\\ 0.2330&{} 0.4656&{} 0.3174&{} 0.5&{} 0.5604&{} 0.4299\\ 0.4892&{} 0.4335&{} 0.3784&{} 0.4396&{} 0.5&{} 0.6632\\ 0.3497&{} 0.4580&{} 0.3266&{} 0.5701&{} 0.3368&{} 0.5 \end{matrix} \right) \\ P^{(7,4)}= & {} \left( \begin{matrix} 0.5&{} 0.4400&{} 0.5035&{} 0.6509&{} 0.4935&{} 0.6916\\ 0.5600&{} 0.5&{} 0.5798&{} 0.5990&{} 0.5520&{} 0.9136\\ 0.4965&{} 0.4202&{} 0.5&{} 0.6907&{} 0.8924&{} 0.7006\\ 0.3491&{} 0.4010&{} 0.3093&{} 0.5&{} 0.5749&{} 0.5939\\ 0.5065&{} 0.4480&{} 0.1076&{} 0.4251&{} 0.5&{} 0.7767\\ 0.3084&{} 0.0864&{} 0.2994&{} 0.4061&{} 0.2233&{} 0.5 \end{matrix} \right) \\ P^{(8,4)}= & {} \left( \begin{matrix} 0.5&{} 0.5467&{} 0.3790&{} 0.4300&{} 0.5283&{} 0.3389\\ 0.4533&{} 0.5&{} 0.5600&{} 0.7479&{} 0.8500&{} 0.7181\\ 0.6210&{} 0.4400&{} 0.5&{} 0.6995&{} 0.6040&{} 0.6813\\ 0.5700&{} 0.2521&{} 0.3005&{} 0.5&{} 0.5667&{} 0.6374\\ 0.4717&{} 0.1500&{} 0.3960&{} 0.4333&{} 0.5&{} 0.6645\\ 0.6611&{} 0.2819&{} 0.3187&{} 0.3626&{} 0.3355&{} 0.5 \end{matrix} \right) \end{aligned}$$
$$\begin{aligned} V^{(1,4)}=\left( \begin{matrix} null &{} null &{} null\\ 88&{} 54&{} 99\\ 96&{} 53&{} 53\\ 98&{} 53&{} 89\\ 98&{} 49&{} 79\\ 90&{} 77&{} 99\\ 86&{} 53&{} 100\\ 90&{} 80&{}67 \end{matrix} \right) V^{(2,4)}=\left( \begin{matrix} 87&{} 78&{} 95\\ null &{} null &{} null \\ 95&{}48&{} 59\\ 96&{} 45&{} 97\\ 94&{} 52&{} 68\\ 85&{} 73&{} 96\\ 87&{} 49&{} 93\\ 99&{} 78&{} 77 \end{matrix} \right) \end{aligned}$$
$$\begin{aligned} V^{(3,4)}=\left( \begin{matrix} 85&{} 66&{} 85\\ 89&{} 46&{} 99\\ null &{} null &{} null\\ 93&{} 45&{} 96\\ 87&{} 59&{} 66\\ 86&{} 72&{} 91\\ 92&{} 57&{} 97\\ 99&{} 67&{} 85 \end{matrix} \right) V^{(4,4)}=\left( \begin{matrix} 89&{} 68&{} 87\\ 96&{} 49&{} 96\\ 89&{} 50&{} 67\\ null&{} null&{} null\\ 93&{} 54&{} 77\\ 99&{} 69&{} 91\\ 89&{} 58&{} 98\\ 94&{} 79&{} 65 \end{matrix} \right) \end{aligned}$$
$$\begin{aligned} V^{(5,4)}=\left( \begin{matrix} 92&{} 78&{} 93\\ 92&{}55&{} 89\\ 91&{} 60&{} 45\\ 89&{} 52&{} 86\\ null&{}null&{}null\\ 95&{} 65&{} 95\\ 92&{} 59&{} 95\\ 86&{} 73&{} 65 \end{matrix} \right) V^{(6,4)}=\left( \begin{matrix} 96&{}72&{} 85\\ 92&{} 59&{} 89\\ 99&{} 49&{} 56\\ 85&{} 56&{} 99\\ 87&{} 54&{} 76\\ null&{} null&{} null\\ 94&{} 56&{} 97\\ 95&{} 76&{} 76 \end{matrix} \right) \end{aligned}$$
$$\begin{aligned} V^{(7,4)}=\left( \begin{matrix} 95&{} 78&{} 100\\ 89&{} 57&{} 99\\ 86&{} 56&{} 52\\ 93&{} 55&{} 96\\ 94&{} 58&{} 72\\ 87&{} 74&{} 90\\ null&{} null&{} null\\ 89&{} 67&{} 79 \end{matrix} \right) V^{(8,4)}=\left( \begin{matrix} 83&{} 78&{} 94\\ 85&{} 56&{} 95\\ 93&{} 56&{} 43\\ 94&{} 56&{} 93\\ 95&{} 55&{} 78\\ 95&{} 73&{} 96\\ 95&{} 42&{} 97\\ null&{} null&{} null\\ \end{matrix} \right) \end{aligned}$$

C4. Collective preference and consensus degree.

In round 1, the collective preference is,

$$\begin{aligned} P^{(c,1)}=\left( \begin{matrix} 0.5&{} 0.5375&{} 0.5013&{} 0.5901&{} 0.5294&{} 0.6386\\ 0.4625&{} 0.5&{} 0.5661&{} 0.5710&{} 0.6128&{} 0.6741\\ 0.4987&{} 0.4339&{} 0.5&{} 0.6853&{} 0.6037&{} 0.6635\\ 0.4099&{} 0.4290&{} 0.3147&{} 0.5&{} 0.5637&{} 0.5940\\ 0.4706&{} 0.3872&{} 0.3963&{} 0.4363&{} 0.5&{} 0.6820\\ 0.3614&{} 0.3259&{} 0.3365&{} 0.4060&{} 0.3168&{} 0.5 \end{matrix} \right) \end{aligned}$$

.

The non-cooperative behavior matrix is,

$$\begin{aligned} NC^{(1)}=\left( \begin{matrix} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{matrix} \right) \end{aligned}$$

The consensus level reaches \(cl^{1}=0.7284\)

In round 2, the collective preference matrix is,

$$\begin{aligned} P^{(c,2)}=\left( \begin{matrix} 0.5&{} 0.5375&{} 0.5013&{} 0.5901&{} 0.5294&{} 0.6386\\ 0.4625&{} 0.5&{} 0.5661&{} 0.5710&{} 0.6128&{} 0.6741\\ 0.4987&{} 0.4339&{} 0.5&{} 0.6853&{} 0.6037&{} 0.6635\\ 0.4099&{} 0.4290&{} 0.3147&{} 0.5&{} 0.5637&{} 0.5940\\ 0.4706&{} 0.3872&{} 0.3963&{} 0.4363&{} 0.5&{} 0.6820\\ 0.3614&{} 0.3259&{} 0.3365&{} 0.4060&{} 0.3168&{} 0.5 \end{matrix} \right) \end{aligned}$$

The non-cooperative matrix is,

$$\begin{aligned} NC^{(2)}=\left( \begin{matrix} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{matrix} \right) \end{aligned}$$

The consensus degree reaches \(cl^{2}=0.7721\).

In round 3, the collective preference matrix is,

$$\begin{aligned} P^{(c,3)}=\left( \begin{matrix} 0.5&{} 0.5253&{} 0.5038&{} 0.6253&{} 0.5176&{} 0.6471\\ 0.4747&{} 0.5&{} 0.5912&{} 0.5663&{} 0.5918&{} 0.6773\\ 0.4962&{} 0.4088&{} 0.5&{} 0.6822&{} 0.5593&{} 0.7215\\ 0.3747&{} 0.4337&{} 0.3178&{} 0.5&{} 0.5772&{} 0.6355\\ 0.4824&{} 0.4082&{} 0.4407&{} 0.4228&{} 0.5&{} 0.6702\\ 0.3529&{} 0.3227&{} 0.2785&{} 0.3645&{} 0.3298&{} 0.5 \end{matrix} \right) \end{aligned}$$

The non-cooperative matrix is,

$$\begin{aligned} NC^{(3)}=\left( \begin{matrix} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ \end{matrix} \right) \end{aligned}$$

The consensus degree reaches \(cl^{3}=0.8287\).

In round 4, the collective preference matrix is,

$$\begin{aligned} P^{(c,4)}=\left( \begin{matrix} 0.5&{} 0.5585&{} 0.4427&{} 0.6522&{} 0.5122&{} 0.5779\\ 0.4415&{} 0.5&{} 0.5952&{} 0.5346&{} 0.5674&{} 0.7176\\ 0.5573&{} 0.4048&{} 0.5&{} 0.6995&{} 0.5818&{} 0.6833\\ 0.3478&{} 0.4654&{} 0.3005&{} 0.5&{} 0.5663&{} 0.5991\\ 0.4878&{} 0.4326&{} 0.4182&{} 0.4337&{} 0.5&{} 0.6655\\ 0.4221&{} 0.2824&{} 0.3167&{} 0.4009&{} 0.3345&{} 0.5 \end{matrix} \right) \end{aligned}$$

The non-cooperative behavior matrix is,

$$\begin{aligned} NC^{(4)}=\left( \begin{matrix} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{matrix} \right) \end{aligned}$$

The consensus level reaches \(cl^{4}=0.8517\).

Appendix D: The OWA operator applied in the current work

Definition 3

[48] Suppose that \(\{c_1,c_2,\dots ,c_N\}\) is a set of values to be aggregated, the OWA operator is defined by

$$\begin{aligned} OWA(c_1,c_2,\dots ,c_N)=\sum _{z=1}^{N}\pi _zb_z \end{aligned}$$
(D1)

\(b_k\) is the value of the z-th largest of \(\{c_1,c_2,\dots ,c_N\}\), \(\pi =(\pi _1,\pi _2,\dots ,\pi _N)^T\) is a weight vector, \(\pi _z\in (0,1)\) and \(\sum _{z=1}^{N}\pi _z=1\).

In [33, 34], \(\pi =(\pi _1,\pi _2,\dots ,\pi _N)^T\) can be obtained by linguistic quantifier as below.

$$\begin{aligned} \pi _z=Q\left( \frac{z}{N}\right) -Q\left( \frac{z-1}{N}\right) \end{aligned}$$
(D2)
$$\begin{aligned} Q(z)=\left\{ \begin{array}{ll} 0 &{} z<a \\ \frac{z-a}{b-a}, &{}a<z <b, a,b,z\in (0,1]\\ 1,&{} z>b \end{array}\right. \end{aligned}$$
(D3)

with parameters \(a=0.5\) and \(b=1\).

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Liu, Y., Zhang, X., Rodríguez, R.M. et al. Managing non-cooperative behaviors in consensus reaching processes: a comprehensive self-management weight generation mechanism. Appl Intell 54, 2673–2702 (2024). https://doi.org/10.1007/s10489-024-05281-9

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