Integer Programming Problems (IPPs) play a critical role in solving real-world optimization scenarios where decision variables must be integers, such as in scheduling, logistics, and resource allocation. Traditional methods like Branch and Bound and Cutting Plane techniques have been widely used for solving such problems. In 2003, Bari and Ahmad introduced the NAZ-cut method, a straightforward and computationally efficient approach to solving IPPs by systematically reducing the feasible solution space through the addition of specific constraints. This paper presents an enhanced NAZ-cut framework for solving IPPs, particularly in the context of Fermatean fuzzy environments, where uncertainty in parameters is modeled using Fermatean fuzzy sets. A new approach is proposed to minimize computational effort by excluding specific integer candidate solutions based on a well-defined criterion. A supporting theorem is provided to justify this exclusion, ensuring that only feasible and potentially optimal points are considered. The method is demonstrated through a numerical example, highlighting the effectiveness of the proposed strategy in simplifying the search process and improving computational efficiency.