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<p>Virtual enterprise (VE) can aggregate the core competencies of individual companies and attract many researchers’ attention for its commercial value. To select the ideal partner, the authors construct a multi-attribute hybrid model with the probabilistic interval-valued hesitant fuzzy set (PIVHFS). First, we give several new definitions of distance measurement and fuzzy entropy of PIVHFS based on rigorous mathematical proofs. By introducing the entropy of PIVHFS, the mental activity of experts during scoring is quantified and not constrained by optimistic or pessimistic attitudes. Second, we design a multi-criteria group decision-making (MCGDM) model with PIVHFS and Cloud-TOPSIS, which can visualize the results and allow us to simulate a 3D plot of the scoring. Third, an illustrative example is given to verify the feasibility and practicability of our developed approach. Compared with previous methods without PIVHFS and cloud model [Meng et al. in J Ambient Intell Hum Comput 10(12):5007–5036 (2019); Zhang et al. in Expert Syst. <a href="https://doi.org/10.1111/exsy.12424">https://doi.org/10.1111/exsy.12424</a> (2019); Song et al. in Int J Intell Syst 34(4):627–651 (2019)], our approach not only considers more experts’ mental activities but also provides a more intuitive view of the strengths and weaknesses of the alternatives. Therefore, with the proposed approach, decision-makers can select an appropriate virtual enterprise partner based on more comprehensive information.</p>

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An entropy- and Cloud-TOPSIS-Based Decision-Making Framework Under Probabilistic Interval-Valued Hesitant Fuzzy Environment: Application to Virtual Enterprise Partner Selection

  • 期刊论文

  • 发表日期：

    2026年3月9日

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  • Xue Deng,
  • Yuanyi Zhang

摘要

Virtual enterprise (VE) can aggregate the core competencies of individual companies and attract many researchers’ attention for its commercial value. To select the ideal partner, the authors construct a multi-attribute hybrid model with the probabilistic interval-valued hesitant fuzzy set (PIVHFS). First, we give several new definitions of distance measurement and fuzzy entropy of PIVHFS based on rigorous mathematical proofs. By introducing the entropy of PIVHFS, the mental activity of experts during scoring is quantified and not constrained by optimistic or pessimistic attitudes. Second, we design a multi-criteria group decision-making (MCGDM) model with PIVHFS and Cloud-TOPSIS, which can visualize the results and allow us to simulate a 3D plot of the scoring. Third, an illustrative example is given to verify the feasibility and practicability of our developed approach. Compared with previous methods without PIVHFS and cloud model [Meng et al. in J Ambient Intell Hum Comput 10(12):5007–5036 (2019); Zhang et al. in Expert Syst. https://doi.org/10.1111/exsy.12424 (2019); Song et al. in Int J Intell Syst 34(4):627–651 (2019)], our approach not only considers more experts’ mental activities but also provides a more intuitive view of the strengths and weaknesses of the alternatives. Therefore, with the proposed approach, decision-makers can select an appropriate virtual enterprise partner based on more comprehensive information.