SIMPLEX METHOD AND ITS GEOMETRICAL EXPLANATION

Abstract

Linear programming (LP) is one of the most profound and widely studied optimization techniques, underpinning much of modern operational research and computational mathematics. Its formulation allows decision-makers to determine the best possible outcomes in the presence of linear constraints, offering a structured method for resource allocation, scheduling, and decision-making in complex environments. The standard LP problem seeks to maximize or minimize a linear objective function subject to a system of linear inequalities or equations (Vasquez, 2024). The origins of LP date back to the 1940s when George Dantzig formulated the Simplex Method, revolutionizing mathematical optimization. Dantzig's algorithm provided an efficient means to traverse feasible regions defined by linear constraints, identifying optimal solutions by moving across vertices of a convex polytope (Bertsimas & Freedman, 2023). Since its inception, LP has found extensive applications in industries such as logistics, telecommunications, finance, energy, and healthcare (Khan & Rossi, 2023). The algebraic underpinnings of LP are closely intertwined with linear algebra, matrix theory, and geometry. Over time, a geometric interpretation of LP and the Simplex Method has emerged as a powerful conceptual framework, offering intuitive insights into the behavior of optimization paths. This perspective enhances comprehension, especially in high-dimensional decision spaces, where each constraint forms a hyperplane and the feasible region becomes a convex polyhedron (Chang & Liu, 2024). Despite the exponential worst-case complexity of the Simplex Method, its practical efficiency has ensured its continued use in real-world optimization, often outperforming newer methods such as interior-point algorithms in certain problem classes (Zhou & Pinto, 2025). This has led to a surge in studies exploring its geometric foundations, visualization tools, and applications in both continuous and discrete optimization contexts.