Linear Programming - BSEB Inter Study Guide
Linear Programming
Questions ask you to formulate constraints, graph the feasible region, and optimise a linear objective by the corner-point method. The exam tests bounded regions and simple unbounded cases.
Core rules
- Decision variables x,y are usually non-negative: x≥0, y≥0.
- Each constraint is a linear inequality; the feasible region is the common solution set.
- Corner-point theorem: if a maximum or minimum exists on a bounded feasible region, it occurs at a vertex.
- Evaluate Z=ax+by at every corner point; the largest value is the maximum, the smallest is the minimum.
- For an unbounded region, check whether the objective can increase without limit; if so, no maximum exists.
Worked example
Maximise Z=3x+2y subject to x+y≤4, x≥0, y≥0. The feasible region is the triangle with vertices (0,0), (4,0), (0,4). The vertices come from the intercepts of x+y=4 and the origin. Z(0,0)=0, Z(4,0)=12, Z(0,4)=8. Hence maximum Z=12 at (4,0).
Traps
- Evaluating only one or two corner points.
- Reversing an inequality sign when rearranging.
- Forgetting the non-negativity constraints, which changes the region.
- Assuming a maximum exists for every unbounded region.
- Not checking that the chosen point satisfies all constraints.