Abstract

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.

中文翻译：

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基于PROMETHEE的概率简化中智集的多准则决策方法

摘要

概率简化中智集\（\ left（{PSNS} \ right）\）是描述现实生活中模糊性的重要工具。在这项研究中，我们定义了PSNS并讨论了\（PSNSs \）的一些理论集合操作。我们还提出了\（PSNSs \）上模块的概念，以及两个\（PSNSs \）之间的内积和投影算子。关于这个新集合，我们引入了概率简化的中智数\（\ left（{PSNN} \ right）\）。一个\（PSNN \）具有三个组成部分，分别称为概率值真隶属度，概率值不确定性隶属度和概率值虚假隶属度。我们给出了\（PSNNs \）上的一些代数运算规则，得分函数和准确性函数。此外，我们在\（PSNNs \）上引入了两个集合算子，分别是概率简化中智加权算术平均算子和概率简化加权几何平均算子。此外，我们使用基于模糊测度的方法确定标准的权重，并开发基于偏好函数的方法来确定每个决策者的权重。我们提出一种基于\（PSNSs \）的扩展PROMETHEE方法对于小组决策问题。最后，作为该理论的应用，我们使用扩展的PROMETHEE方法为基于\（PSNNs \）的多准则群体决策问题提供了实践，以确保所提出方法的稳定性。