TOPIC 01 · GRAPHICAL METHOD
Linear Programming
Turn constraints into a picture. Find the best feasible solution, then move a slider to see what changes.
1. Define your problem
Choose an objective and up to eight constraints. Coefficients may be negative or decimal.
Z =
Turn this off to allow negative values for both variables.
2. Explore the graph
Feasible regionOptimal pointC1C2C3
The graph shows a finite window. Shading at its edge may continue beyond the view. Equality constraints can produce a line or a single point. The solver uses the constraints, independent of the view.
3. Read the answer
Maximum Z = 10
At x = 2 and y = 2.
| x | y | Z = 3x + 2y |
|---|---|---|
| 2 | 2 | 10 |
| 1 | 3 | 9 |
| 2 | 0 | 6 |
| 0 | 3 | 6 |
| 0 | 0 | 0 |
How the graphical method works
- Draw each constraint boundary and keep the side satisfying its inequality.
- Intersect these regions to find feasible points.
- Evaluate the objective at feasible corners and check whether it can improve indefinitely.
- A finite optimum lies at a corner when corners exist. A parallel objective can give an entire optimal edge.
Calculations use floating-point arithmetic and a numerical tolerance. Extremely ill-conditioned problems may require a specialist solver.