New decision-making hybrid model: intuitionistic fuzzy N-soft rough sets

  • Muhammad Akram
  • Ghous Ali
  • José Carlos R. AlcantudEmail author


In this paper, we introduce three novel hybrid models, namely, intuitionistic fuzzy N-soft sets (IFNSSs), N-soft rough intuitionistic fuzzy sets and intuitionistic fuzzy N-soft rough sets (IFNSRSs). We discuss some of their respective properties. We also present the lower and upper intuitionistic fuzzy N-soft rough approximation operators and investigate their properties. Furthermore, we present respective new approaches to decision-making problems based on the IFNSS and IFNSRS models in an algorithmic format. Finally, we present illustrative examples in order to show that these decision-making methods can be fruitfully employed to solve real life problems.


N-soft sets IFNSSs IFNSRSs Soft set Rough set Decision-making 



The authors are very grateful to the editor and the anonymous referees for their valuable comments and helpful suggestions which helped us to improve the original version of this paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Human and animal rights

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

Informed consent

It is submitted with the consent of all the authors.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of the PunjabLahorePakistan
  2. 2.BORDA Research Unit and IMEUniversity of SalamancaSalamancaSpain

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