Similar to the extension from intuitionistic fuzzy numbers (IFNs) to neutrosophic numbers (NNs), we extend the normal intuitionistic fuzzy numbers (NIFNs) to normal neutrosophic numbers (NNNs) to handle the incompleteness, indeterminacy and inconsistency of the evaluation information. In addition, because Heronian mean (HM) operators can capture the correlations of the aggregated arguments, we further extend the HM operator to deal with the NNNs, and propose some new HM operators and apply them to solve the multiple attribute group decision making (MAGDM) problems. Firstly, we briefly introduce the definition, the operational laws, the properties, the score function, and the accuracy function of the NNNs. Secondly, some new HM operators are introduced, such as generalized Heronian mean (GHM) operator, generalized weighted Heronian mean (GWHM) operator, improved generalized weighted Heronian mean (IGWHM) operator, generalized geometric Heronian mean (GGHM) operator, improved generalized geometric Heronian mean (IGGHM) operator, and improved generalized geometric weighted Heronian mean (IGGWHM) operator. Moreover, we propose the normal neutrosophic number improved generalized weighted Heronian mean (NNNIGWHM) operator and normal neutrosophic number improved generalized geometric weighted Heronian mean (NNNIGGWHM) operator, and discuss their properties and some special cases. Furthermore, we propose two MAGDM methods respectively based on the NNNIGWHM and NNNIGGWHM operators. Finally, we give an illustrative example to demonstrate the practicality and effectiveness of the two methods.
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