International Journal of Fuzzy Systems

, Volume 21, Issue 2, pp 388–406 | Cite as

An Extended Multi-criteria Group Decision-Making PROMETHEE Method Based on Probability Multi-valued Neutrosophic Sets

  • Peide LiuEmail author
  • Shufeng Cheng
  • Yuming Zhang


The PROMETHEE method, one of the most widely used and best known methods, takes advantage of the outranking principle to rank potential alternatives. The probability multi-valued neutrosophic sets (PMVNSs) have the power to describe complex uncertain information more comprehensively. Thus, in order to integrate the merits of PROMETHEE method and PMVNSs, this paper extends the PROMETHEE method to PMVNSs environment. Firstly, some basic preliminaries are reviewed, such as multi-valued neutrosophic sets (MVNSs), PMVNSs and classical PROMETHEE method. Then, we propose the operational laws of PMVNSs based on the operational rules of the MVNSs and probability distribution. Meanwhile, the score function and accuracy function of PMVNSs are given to simplify the comparison of any two probability multi-valued neutrosophic numbers (PMVNNs). Further, we develop a new distance measure for PMVNNs with unequal length, and then based on the distance measure and deviation maximization method, the attribute weights are determined; an extended PROMETHEE method for multi-criteria group decision-making with the information of PMVNSs is established to achieve the process for optimal alternative selection. In the end, a practical example concerning third party logistics providers is used to highlight the feasibility and superiority of the proposed approach.


PROMETHEE Outranking Probability multi-valued neutrosophic sets MCGDM 



Decision makers


Elimination and choice translating reality


Falsity-membership function


Fuzzy sets


Hesitant fuzzy sets


Indeterminacy-membership function


Intuitionistic fuzzy sets


Interval-valued fuzzy sets


Multi-criteria decision-making


Multi-criteria group decision-making


Multi-valued neutrosophic power weighted average


Multi-valued neutrosophic power weighted geometric


Multi-valued neutrosophic sets


Multi-valued neutrosophic weighted Bonferroni mean


Multi-valued neutrosophic weighted geometric Bonferroni mean


Neutrosophic sets


Probability multi-valued neutrosophic numbers


Probability multi-valued neutrosophic sets


Preference ranking organization method for enrichment evaluation


Qualitative flexible multiple criteria method


Simplified neutrosophic sets


Truth-membership function


An acronym in Portuguese of interactive and multi-criteria decision-making



This paper is supported by the National Natural Science Foundation of China (Nos. 71771140 and 71471172), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045) and Shandong Provincial Social Science Planning Project (Nos. 17BGLJ04, 16CGLJ31 and 16CKJJ27).

Compliance with Ethical Standards

Conflict of interest

We declare that we do have no commercial or associative interests that represent a conflict of interests in connection with this manuscript. There are no professional or other personal interests that can inappropriately influence our submitted work.

Human and Animal Rights

This paper does not include any researches with human participants or animals performed by any of the authors.


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Copyright information

© Taiwan Fuzzy Systems Association and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Management Science and EngineeringShandong University of Finance and EconomicsJinanChina
  2. 2.School of ManagementsShandong UniversityJinanChina

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