Abstract: Determining the ordering relation of interval-valued q-rung orthopair fuzzy numbers (IVq-ROFNs) is the key to solve the issue of interval-valued q-rung orthopair fuzzy (IVq-ROF) decision-making. In this context, possibility measure is one of the important tools. To address the counter intuitive problem that existing score functions, accuracy functions and possibility measures cannot solve, a multi-attribute decision-making method based on IVq-ROF possibility degree is proposed. Firstly, the possibility measure in IVq-ROF environment is defined, and its related properties are discussed. Then, through some counter intuitive examples, the feasibility and superiority of the IVq-ROF possibility measure are illustrated. Finally, a multi-attribute decision-making model based on IVq-ROF possibility degree is established. Also, two numerical examples are given to analyze the applicability of the existing possibility degree in IVq-ROF environment and verify the effectiveness and rationality of the proposed method.