International Journal of Fuzzy Systems

, Volume 20, Issue 3, pp 986–999 | Cite as

Interval Complex Neutrosophic Set: Formulation and Applications in Decision-Making

  • Mumtaz Ali
  • Luu Quoc Dat
  • Le Hoang SonEmail author
  • Florentin Smarandache


Neutrosophic set is a powerful general formal framework which generalizes the concepts of classic set, fuzzy set, interval-valued fuzzy set, intuitionistic fuzzy set, etc. Recent studies have developed systems with complex fuzzy sets, for better designing and modeling real-life applications. The single-valued complex neutrosophic set, which is an extended form of the single-valued complex fuzzy set and of the single-valued complex intuitionistic fuzzy set, presents difficulties to defining a crisp neutrosophic membership degree as in the single-valued neutrosophic set. Therefore, in this paper we propose a new notion, called interval complex neutrosophic set (ICNS), and examine its characteristics. Firstly, we define several set theoretic operations of ICNS, such as union, intersection and complement, and afterward the operational rules. Next, a decision-making procedure in ICNS and its applications to a green supplier selection are investigated. Numerical examples based on real dataset of Thuan Yen JSC, which is a small-size trading service and transportation company, illustrate the efficiency and the applicability of our approach.


Green supplier selection Multi-criteria decision-making Neutrosophic set Interval complex neutrosophic set Interval neutrosophic set 



Neutrosophic set


Interval neutrosophic set


Complex fuzzy set


Complex intuitionistic fuzzy set


Interval-valued complex fuzzy set


Complex neutrosophic set


Interval-valued complex neutrosophic set, or interval complex neutrosophic set


Single-valued complex neutrosophic set


Multi-criteria decision-making


Multi-criteria group decision-making


Maximum operator (t-conorm)


Minimum operator (t-norm)



This research is funded by Graduate University of Science and Technology under grant number GUST.STS.ÐT2017-TT02. The authors are grateful for the support from the Institute of Information Technology, Vietnam Academy of Science and Technology. We received the necessary devices as experiment tools to implement proposed method.


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Copyright information

© Taiwan Fuzzy Systems Association and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Mumtaz Ali
    • 1
  • Luu Quoc Dat
    • 2
  • Le Hoang Son
    • 3
    Email author
  • Florentin Smarandache
    • 4
  1. 1.University of Southern QueenslandToowoombaAustralia
  2. 2.VNU University of Economics and BusinessVietnam National UniversityHanoiVietnam
  3. 3.VNU University of ScienceVietnam National UniversityHanoiVietnam
  4. 4.University of New MexicoGallupUSA

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