The Simplex Method

How to read this

• Green region = every point (x, y) that satisfies all constraints. Its corners are the extreme points; the table lists each with its objective value.
• Blue lines are the constraint boundaries. A dashed green edge means the region keeps going forever in that direction.
• The orange dashed line is a level line c₁x + c₂y = z. Sliding it shows the geometric solution: the last corner it touches is optimal.
• The tableau is the same problem in algebra. Each row's basic variable equals that row's right-hand side (RHS); all other variables are 0. That basic feasible solution is the blue ring on the plot.
• Each pivot swaps one basic variable for one nonbasic one — on the plot, a hop along an edge to an adjacent corner (orange arrows).
• Minimizing f is done as maximizing z = −f. If the origin breaks a ≥ or = constraint, Phase I first drives artificial variables a₁, a₂, … to 0; the ring is dashed red until the point is feasible.

Try this

1. Default problem: predict the path before pressing play. Simplex goes O → C → B (z = 0 → 10 → 14) and never visits A, even though A (z = 12) is also a corner.
2. Change the objective to 1x + 1y. Now the level line is parallel to x + y ≤ 4: every point on edge A–B is optimal. Which corner does simplex report?
3. Load Unbounded. Watch the step where the entering column has no positive entry — no θ-ratio exists, so nothing stops the growth.
4. Load Infeasible. Phase I ends with W < 0: the artificial variables can't all reach 0, so no point satisfies both constraints.
5. Add x <= 3.5 to the default problem. Does the optimum change? (It shouldn't: B = (3, 1) already satisfies it — the new constraint is not binding.)