<p>An <i>n</i>-polygonal interval-valued fuzzy set (<i>n</i>PIVFS) reduces the complexity of arithmetic operations on interval-valued fuzzy sets. In many decision-making problems, the ranking problem of fuzzy sets has always been a popular topic. This paper presents novel score and accuracy functions for an <i>n</i>PIVFS in terms of the concept of the centroid in the geometric approach. This ranking principle utilizes the order relationship of score and accuracy functions, respectively. The potential of the proposed ranking principle is tested by two specific numerical examples. As well as, to elucidate the applicability of the score function, an example of pattern recognition is provided for demonstration purposes. Furthermore, a new decision-making method is proposed by combining score function, TOPSIS method, and prospect theory. Finally, we provide a numerical example involving linguistic variable, addressing the problem of ranking tourist cities. This example demonstrates the effectiveness of the modified model in solving multi-attribute decision-making problems. Through comparative analysis, we have verified the practicality and feasibility of the newly proposed decision-making method. Meanwhile, we also discuss the sensitivity analysis of the proposed method, taking into account the sensitivity of decision-makers to gains and losses, in order to seek appropriate parameter values.</p>