Further Study on Goal Programming Models.

In this study, we investigate various goal programming models and highlight their problematic outcomes, offering our revised versions. While previous publications relied on numerical examples to determine the optimal alternative, we provide theoretical explanations for these examples to enhance researchers' understanding of goal programming models. Additionally, we employ a fast integration approach to solve a challenging calculus problem concerning the centroid of trapezoidal fuzzy numbers. Our findings aim to assist researchers in comprehending intricate theoretical concepts and facilitate the application of these results in their own research endeavors. [ABSTRACT FROM AUTHOR]