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

Probabilistic simplified neutrosophic set \( \left( {PSNS} \right) \) 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 \( PSNSs \). Also we propose the concepts of module on \( PSNSs \), as well as inner product and projection operator between two \( PSNSs \). In relation to this new set, we introduce a probabilistic simplified neutrosophic number \( \left( {PSNN} \right) \). A \( PSNN \) 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 \( PSNNs \). Furthermore, we introduce two aggregation operators called the probabilistic simplified neutrosophic weighted arithmetic average operator and the probabilistic simplified weighted geometric average operator on \( PSNNs \). 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 \( PSNSs \) 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 \( PSNNs \) by using extended PROMETHEE method to ensure stability of the proposed method.