Background on Linear Programming

Linear programming tackles optimization problems where both the objective function and the constraints are linear. A typical formulation seeks to maximize or minimize $c{T}^{}x$ subject to $Ax\le b$ and $x\ge 0$. Here $c$ is a vector of coefficients, $A$ is a matrix describing constraint slopes, and $b$ contains the bounds. The feasible region defined by these inequalities forms a convex polytope, and the optimal solution always resides at one of its vertices.

The simplex method, introduced by George Dantzig in 1947, systematically moves along the edges of this polytope to locate the best vertex. Starting from an initial feasible point, it examines adjacent vertices connected by an edge and marches toward one with improved objective value. Because each step increases or maintains the objective, the algorithm terminates when no neighboring vertex offers improvement. In the worst case simplex can be slow, but for typical problems it converges quickly and remains a staple of operations research.

Although modern solvers handle thousands of variables, the essence of the simplex method can be grasped with small examples. This calculator focuses on two-variable problems with up to three constraints, allowing you to enter the coefficients directly. The code performs the pivot operations that move the algorithm from one vertex to another, displaying the optimal solution when finished. By working through this limited setting, you gain intuition for the mechanics underlying industrial-scale optimization software.

Pivoting Mechanics

Each iteration of simplex manipulates a tableau containing the coefficients of the objective function and constraints. Slack variables convert inequalities to equalities, forming an augmented matrix. The algorithm selects an entering variable—one whose coefficient in the objective row is positive for maximization—and a leaving variable determined by the minimum ratio test. Row operations then pivot the tableau so that the entering variable becomes basic and the leaving variable nonbasic. This process effectively steps to an adjacent vertex in the feasible region.

The tableau approach offers a concise bookkeeping method. For a problem with two decision variables and three constraints, the tableau has three slack variables as well. The algorithm continues until all coefficients in the objective row are non-positive, indicating no further improvement is possible. The final tableau encodes the optimal value along with the values of the basic variables.

Why Learn Simplex?

Understanding simplex reveals the geometry behind optimization. Because linear programs represent high-dimensional polytopes, visualizing them directly is difficult. However, interpreting the pivot steps as moves along edges makes the solution path concrete. The algorithm also introduces ideas such as duality, degeneracy, and unboundedness. These concepts extend to more advanced topics in convex optimization and game theory. Moreover, simplex has practical implications: it forms the core of many scheduling, allocation, and planning applications.

Despite more modern algorithms like interior-point methods, simplex remains relevant due to its simplicity and reliability on small problems. It also serves as a building block for understanding dual simplex, cutting-plane methods, and column generation. By experimenting with this calculator, you will see firsthand how the choice of entering and leaving variables shapes the path to optimality.

Using the Calculator

Input the objective vector $c$ as comma-separated numbers, then enter the constraint matrix $A$ line by line with commas between entries. The bounds $b$ follow on separate lines. After clicking Solve, the script parses the data and iteratively applies simplex steps. Intermediate tableaus are not shown to keep the interface concise, but the final optimal solution vector $x$ and the objective value $c{T}^{}x$ appear in the results area.

Try modifying coefficients to observe how the optimum shifts. If the problem is infeasible or unbounded, the calculator reports this condition. Exploring different setups will sharpen your understanding of how constraints shape the feasible region and how the algorithm navigates it.

Beyond mathematics, linear programming underlies important business decisions, from airline crew scheduling to nutrition planning. Even though large companies rely on sophisticated solvers, the simplex method remains at the heart of these tools. Mastering the basics equips you with a versatile problem-solving skill applicable to engineering, logistics, economics, and data science.