Linear Programming

Mathematical Principles & Theorems

Solves constrained linear optimization problems: \(\max \mathbf{c}^T \mathbf{x}\) subject to \(A\mathbf{x} \le \mathbf{b}\) and \(\mathbf{x} \ge 0\). The Fundamental Theorem of Linear Programming establishes that if an optimal solution exists, it occurs at an extreme point (vertex) of the convex polyhedron \(\mathcal{F}\). George Dantzig's Simplex Algorithm pivots between adjacent vertices along edges of steepest improvement until the gradient \(\mathbf{c}\) has no feasible improving direction.

Operating Instructions

• Enter objective function coefficients \((c_1, c_2)\) and linear constraint rows \(a_{i1}x_1 + a_{i2}x_2 \le b_i\).
• Click Solve to execute the Simplex method and compute the optimal vertex \(\mathbf{x}^*\) and value \(Z^*\).
• Inspect the 2D graphical representation showing constraint boundaries, feasible region, and objective contour lines.
• Review the Simplex tableau iterations and shadow prices (dual variables).