Linear Programming Class 12 Notes

Elements of an LPP

The Corner Point Method

• Draw each constraint as a line, shade the side that satisfies the inequality, and find the intersection of all shaded regions.
• Locate the corner (vertex) points of the feasible region.
• Evaluate Z at every corner point.
• Pick the largest value for a maximisation problem, the smallest for minimisation.

Formulating a Word Problem

• Decide the decision variables — the quantities the problem asks to choose (e.g. number of units of two products).
• Write the objective as a linear function of the variables from the profit or cost data.
• Translate each resource limit and requirement into a linear constraint.
• Include the implicit
• $x \geq 0,\ y \geq 0$
• constraints.

Special Cases

• Unbounded region: the optimum may not exist for one direction — compare with a test to decide.
• Infeasible: constraints with no common region, so no solution at all.
• Multiple optima: when Z takes the same optimal value on a whole edge of the feasible region, every point of that edge is optimal.

Solved Examples

Example: Maximise $Z = 3x + 2y$ subject to $x + y \leq 4$, $x - y \leq 2$, $x, y \geq 0$.

Solution: The feasible region has corners (0,0), (2,0), (3,1), (0,4). Evaluating Z: 0, 6, 11 and 8. The maximum is 11 at (3,1).

Example: A machine makes two items A and B with a profit of ₹50 and ₹80, and takes 4 and 6 hours per unit with 60 hours available. Maximise daily profit.

Solution: $\text{Max } Z = 50x + 80y$ subject to $4x + 6y \leq 60$, $x,y \geq 0$. Corners give (0,0)=0, (15,0)=750, (0,10)=800, so 10 units of B alone maximise profit at ₹800.