Human judgment is inherently subjective, and cognitive limitations often lead to uncertainty in decision-making. To address this, multi-criteria group decision making (MCGDM) widely uses linguistic and probabilistic linguistic variables to represent experts' preferences. This study focuses on decision problems where criteria values are expressed as probabilistic linguistic variables. First, the concept of dual probabilistic linguistic variables is introduced, extending traditional probabilistic linguistic representations. Second, the normal cloud model characterizes the interconnection between fuzziness and randomness, enabling the derivation of the dual normal cloud framework. A transformation method is developed to convert dual probabilistic linguistic variables into dual probabilistic linguistic normal clouds. Additionally, a dual probabilistic linguistic normal cloud (DPLNC)-weighted averaging operator is proposed to aggregate multiple DPLNCs, while a dual probabilistic linguistic normal cloud-gray relational degree (DPLNC-GRD) is established to enhance decision-making accuracy. A new MCGDM approach based on DPLNC-GRD is proposed and applied to a sustainable supplier selection case study. Sensitivity analysis and comparative evaluations with established methods confirm the robustness and reliability of the proposed framework, demonstrating its effectiveness in managing uncertainty and improving decision outcomes.
Citation: Abdulrahman Almandeel, Congjun Rao, Xiaolong Zhang. Multi-criteria group decision making method with dual normal clouds under an information environment of dual probabilistic linguistic variables[J]. Journal of Industrial and Management Optimization, 2026, 22(10): 4696-4734. doi: 10.3934/jimo.2026163
Human judgment is inherently subjective, and cognitive limitations often lead to uncertainty in decision-making. To address this, multi-criteria group decision making (MCGDM) widely uses linguistic and probabilistic linguistic variables to represent experts' preferences. This study focuses on decision problems where criteria values are expressed as probabilistic linguistic variables. First, the concept of dual probabilistic linguistic variables is introduced, extending traditional probabilistic linguistic representations. Second, the normal cloud model characterizes the interconnection between fuzziness and randomness, enabling the derivation of the dual normal cloud framework. A transformation method is developed to convert dual probabilistic linguistic variables into dual probabilistic linguistic normal clouds. Additionally, a dual probabilistic linguistic normal cloud (DPLNC)-weighted averaging operator is proposed to aggregate multiple DPLNCs, while a dual probabilistic linguistic normal cloud-gray relational degree (DPLNC-GRD) is established to enhance decision-making accuracy. A new MCGDM approach based on DPLNC-GRD is proposed and applied to a sustainable supplier selection case study. Sensitivity analysis and comparative evaluations with established methods confirm the robustness and reliability of the proposed framework, demonstrating its effectiveness in managing uncertainty and improving decision outcomes.
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Abdulrahman Almandeel, Congjun Rao, Xiaolong Zhang. Multi-criteria group decision making method with dual normal clouds under an information environment of dual probabilistic linguistic variables[J]. Journal of Industrial and Management Optimization, 2026, 22(10): 4696-4734. doi: 10.3934/jimo.2026163
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