Graphical Solution of LP problems

• Feasible region: the area on a graph that satisfies every constraint at once — found by shading OUT the parts that don't work, leaving the answer unshaded and labelled R.
• Objective line method: draw a trial line for the objective function, then slide a ruler parallel to it across the region — the last vertex it touches is the optimal solution.
• Vertex method: find the coordinates of every corner of the feasible region (by solving simultaneous equations), plug each into the objective function, and pick the best value.
• Integer solutions: when decision variables must be whole numbers (you can't make 3.65 chairs), check the four integer points surrounding the true optimal solution.

Imagine you run a small furniture workshop. You make chairs (x per day) and tables (y per day). You want to make as much profit as possible — but you're limited by how much wood you have, how many hours your staff can work, and how much storage space exists.

Each of those limits is a constraint — an inequality like 3x + 2y ≤ 24. Your profit is the objective function — something like P = 30x + 40y, which you want to maximise (or in other problems, minimise, e.g. minimising cost).

A linear programming (LP) problem always has this same skeleton:

For problems with just two decision variables (x and y), we can literally see the answer by drawing a graph — that's what this whole chapter is about.

What is it?

The feasible region is the set of every point (x, y) that satisfies all the constraints simultaneously — including the "hidden" constraint that you usually can't have negative amounts of anything (x ≥ 0, y ≥ 0).

Think of it like a Venn diagram of allowed zones, except instead of circles you're overlapping half-planes cut off by straight lines. Wherever all the allowed zones overlap — that's your feasible region.

How to actually draw it — step by step