The simplex idea: walk from vertex to vertex, uphill

The simplex algorithm avoids checking every vertex. Take two mixes with three constraints on flour, sugar, and chocolate, which together give a feasible region with several vertices. The main idea is simple: start at the origin and repeatedly step to a neighboring vertex that raises revenue, stopping as soon as no neighbor does better.

Which neighbor first? A common rule is to head in the direction that earns the most revenue per kilogram. Cookie mix earns $3$ per kg and donut mix earns $2$, so the algorithm moves along the cookie axis first. Each step runs until a resource limit binds.

From each vertex, compare revenue at neighboring vertices along the remaining axes. When every neighbor earns the same or less, the current vertex is optimal. A nice feature: had it started along the donut axis instead, moving uphill would still land on the same optimal mix amounts, as long as each step picks an improving neighbor.