Summary

Linear Programming Problem (LPP): Optimizing (maximizing or minimizing) a linear objective function $Z = ax + by$ subject to linear constraints (inequalities) and non-negativity restrictions $x \geq 0, y \geq 0$.

Feasible Region: The set of all points $(x, y)$ satisfying all constraints. It is a convex polygon (or unbounded region) in the $xy$-plane.

Feasible Solution: Any point in the feasible region.

Optimal Solution: A feasible solution that gives the maximum/minimum value of $Z$.

Corner Point Theorem: If an LPP has an optimal solution, it occurs at one of the corner points (vertices) of the feasible region.

Method to Solve:

1. Graph all constraints to find the feasible region.
2. List all corner points of the feasible region.
3. Evaluate $Z$ at each corner point.
4. The largest (for maximization) or smallest (for minimization) value is the optimal value.

Unbounded Region: If the feasible region is unbounded, a maximum/minimum may not exist. Check by drawing a line $Z = c$ and seeing if it can be moved indefinitely in the direction of optimization.

Infeasible Problem: If no point satisfies all constraints simultaneously, the LPP has no feasible solution. …