Multi-criteria decision making approach based on PROMETHEE with probabilistic simplified neutrosophic sets

Probabilistic simplified neutrosophic set $$ \left( {PSNS} \right) $$ PSNS is an important tool to describe the vagueness existing in the real life. In this study, we define a PSNS and discuss some of theoretical set operations of $$ PSNSs $$ PSNSs . Also we propose the concepts of module on $$ PSNSs $$ PSNSs , as well as inner product and projection operator between two $$ PSNSs $$ PSNSs . In relation to this new set, we introduce a probabilistic simplified neutrosophic number $$ \left( {PSNN} \right) $$ PSNN . A $$ PSNN $$ PSNN has three components which is called probabilistic-valued truth membership degree, probabilistic-valued indeterminacy membership degree and probabilistic-valued falsity membership degree, respectively. We give some of algebraic operational rules, a score function and an accuracy function on $$ PSNNs $$ PSNNs . Furthermore, we introduce two aggregation operators called the probabilistic simplified neutrosophic weighted arithmetic average operator and the probabilistic simplified weighted geometric average operator on $$ PSNNs $$ PSNNs . Furthermore, we determine weights of criteria with a method that is based on fuzzy measure and develop a method based on preference function to determine weight of each decision maker. We present an extended PROMETHEE method based on $$ PSNSs $$ PSNSs for group decision problems. Finally, as an application of this theory, we give a practice on a multi-criteria group decision making problem based on $$ PSNNs $$ PSNNs by using extended PROMETHEE method to ensure stability of the proposed method.