In this paper, a bi-level linear programming problem characterized by interval uncertainty in the coefficients of both objectives and constraints is thoroughly examined. The Karush-Kuhn-Tucker (KKT) optimality conditions for interval nonlinear programming problems have been developed to address this challenge. Utilizing these conditions, the interval bi-level programming problem has been transformed into a deterministic nonlinear programming problem. Subsequently, a comprehensive methodology has been developed to solve the transformed problem. The proposed approach has been validated through numerous illustrative examples that demonstrate its successful execution. Furthermore, the developed methodology has been effectively applied to a practical problem in supply chain planning, showcasing its relevance and applicability in real-world scenarios. (original abstract)