A parametric similarity measure between picture fuzzy sets and its applications in multi-attribute decision-making

Picture fuzzy set is an extension of intuitionistic fuzzy set, which can deal with inconsistent and uncertain information more accurately. Similarity measure, as an important  mathematical tool to evaluate the degree of similarity between picture fuzzy sets, has been widely used to deal with multi-attribute decision-making problems. But there are unreasonable  and counter-intuitive cases due to  a few undesirable properties. In order to handle these unreasonable cases, this paper proposes a parametric similarity measure based on three parameters $m_1, m_2$ and $m_3$, in which decision makers with different decision styles can  obtain  the appropriate similarity measure by adjusting parameters $m_1, m_2$ and $m_3$. Moreover, we  analyze some existing similarity measures from the perspective of mathematics and show that the proposed similarity measure is effective by   numerical examples. In the end, we use the proposed similarity measure to solve the problems of multi-attribute decision-making. Through the  comparison and analysis, we find that the proposed similarity measure is more effective  than some existing similarity measures between picture fuzzy sets.