How can linear optimization find the best allowed production plan?

Linear optimization seeks the largest or smallest value of a linear expression under stated restrictions. In a fictional workshop, x and y are amounts of batches A and B. Negative production is excluded. The quantities are divisible in this mathematical model, so a point between whole-number coordinates can also represent an allowed production amount.

Each A batch uses one material unit and two time units. Each B batch uses one of each. Six material units and eight time units give the constraints x + y ≤ 6 and 2x + y ≤ 8. A feasible point satisfies both inequalities and the nonnegative restrictions at once. Their intersection is the shaded region, not either half-plane alone.

The initial objective is 3x + 2y points. This expression assigns three points to each A batch and two to each B batch. The bounded feasible region has vertices (0,0), (4,0), (2,4) and (0,6). Their objective values are zero, twelve, fourteen and twelve. The largest is fourteen at (2,4).

For a bounded feasible polygon and a linear objective, at least one optimum occurs at a vertex. Sliding an equal-objective line toward larger values illustrates why the boundary matters. Sometimes an entire edge ties for the best value; changing the A coefficient to four produces that situation here. The optimum is a conclusion within the stated resource model, not a guarantee for an actual workshop.