The Simplex Method, invented by the late mathematical scientist George Dantzig, is an algorithm used for solving constrained linear optimization problems (these kinds of problems are referred to as linear programming problems). Linear programming problems often arise in operations research related problems, such as finding ways to maximize profits given constraints on time and resources.

Linear programming problems consist of two key elements: an objective function and a system of linear inequalities. The fundamental goal in solving such linear programming problems is to maximize or minimize the objective function given the linear constraints on the solution. The systems of linear inequalities creates a feasible region, such that the optimal solution must lie within the feasible region. This feasible region formed by the linear inequalities is a convex polytope, where a visualization of such a region is given below.

A Convex Polytope

As seen in the image, the constraints are linear, so the convex polytope has flat edges, and the feasible region is the interior and boundary of the polytope.

A key insight is that the optimal solution to any constrained linear optimization problem is always on one of the corners of the convex polytope. It is with this insight that we can motivate the development of the Simplex Method.

The Simplex Method starts at some point within the feasible region, and then with each iteration of the algorithm maps from one adjacent corner (representing a possible solution to the problem) of the convex polytope to another, such that each step gives a better or equivalent solution than the one in the previous step.

This is hardly a satisfactory description of the Simplex Method, so if the reader wants a more insightful intuition into the method, I recommend visiting this article, or watching the first two short videos on the Simplex Method video series by PatrickJMT, which give a good introduction into the kinds of problems the Method solves and how it works.