Scaling the idea: the three-mix example

To see the idea work on something bigger, add a third product. Now the goal is to choose amounts of donut, cookie, and brownie mix that maximize revenue, subject to five constraints. Three variables mean a three-axis diagram, and the constraints carve out a solid whose optimum must sit at one of its vertices.

The simplex algorithm still starts at the origin and asks which axis to leave along. This time the three mixes tie on revenue per unit, so the tie-break rule picks the first one listed, donut. From there it compares neighboring vertices and moves to the one with the most revenue.

It keeps repeating that choice, moving only when a neighbor pays more, until no neighbor improves. After a short chain of moves it reaches the optimal vertex for this problem. The point is efficiency: rather than evaluating every vertex of the solid, or exponentially many in a larger problem, the algorithm visited only a small fraction of them along one improving path.