Solve the following Linear Programming Problem (LPP) graphically :

Maximize $z = 3x + 5y$

subject to the constraints :

$x + 2y \le 2000$

$x + y \le 1500$

$y \le 600$

$x \ge 0, y \ge 0$

By the corner point (fundamental) theorem of linear programming, the optimum of a linear objective over a bounded feasible region always occurs at a vertex, so we only test the corners.

Concept:

• Each inequality is graphed as a half-plane; their intersection (with $x, y \ge 0$) is the feasible region. The optimal solution lies at one of its corner points, so we evaluate z at every vertex and pick the largest.

Finding the vertices:

• The non-trivial corners come from solving pairs of boundary lines simultaneously, e.g., $x + 2y = 2000$ with $x + y = 1500$ gives $(1000, 500)$. Each candidate must be checked against all constraints before it is accepted.

Exam trap:

• Including an infeasible intersection such as $(900, 600)$ (where $x + 2y = 2100 > 2000$) as a corner, or forgetting to test all vertices.

Real-world link:

• This is a classic product-mix problem: x and y could be units of two products, the constraints being material, labour and demand limits, and z the total profit to be maximised.