Soft Computing

, Volume 23, Issue 21, pp 10853–10879 | Cite as

Interval-valued probabilistic hesitant fuzzy set for multi-criteria group decision-making

  • R. Krishankumar
  • K. S. Ravichandran
  • Samarjit KarEmail author
  • Pankaj Gupta
  • Mukesh Kumar Mehlawat
Methodologies and Application


As a powerful extension to fuzzy set, hesitant fuzzy set (HFS) attracted many scholars in the recent times. The HFS had the ability to accept multiple membership values for a specific instance, which helped in handling uncertainty to a certain extent. However, the previous studies on the hesitant fuzzy theory consider only single occurring probability value for each element which is problematic for decision-makers (DMs) to associate an accurate occurring probability with each element. To alleviate this issue, in this paper, a new concept called interval-valued probabilistic hesitant fuzzy set (IVPHFS) is proposed. Some desirable properties of IVPHFS are also investigated. Further, a new aggregation operator called simple interval-valued probabilistic hesitant fuzzy weighted geometry (SIVPHFWG) is presented and some interesting properties are discussed. Following this, a new extension of statistical variance (SV) is put forward under IVPHFS for calculating the weights of each criterion. A new extension to the popular VIKOR (VlseKriterijumskaOptimizacijaKompromisnoResenje) method is also presented under IVPHFS for ranking objects. The practicality of the proposed decision framework is analyzed by presenting two illustrative examples, viz., supplier selection problem and smartphone selection problem. Finally, the strength and weakness of the proposed decision framework are realized by comparison with other methods.


Group decision-making Interval numbers Probabilistic hesitant fuzzy sets Statistical variance and VIKOR method 



The first author would like to thank University Grants Commission for their financial aid from Rajiv Gandhi National Fellowship with Grant No. F./2015-17/RGNF-2015-17-TAM-83, and the second author would like to thank Department of Science and Technology, Ministry of Science and Technology, Government of India, for their cloud infrastructure under the FIST programme with Grant No. SR/FST/ETI-349/2013. The authors also thank the editor and the anonymous reviewer(s) for their insightful comments which improved the quality of the paper.

Compliance with ethical standards

Conflict of interest

All authors declare that there is no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

Informed consent was obtained from all participants included in the study.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • R. Krishankumar
    • 1
  • K. S. Ravichandran
    • 1
  • Samarjit Kar
    • 2
    Email author
  • Pankaj Gupta
    • 3
  • Mukesh Kumar Mehlawat
    • 3
  1. 1.School of ComputingSASTRA UniversityThanjavurIndia
  2. 2.Department of MathematicsNational Institute of TechnologyDurgapurIndia
  3. 3.Department of Operational ResearchUniversity of DelhiDelhiIndia

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