Linear Programming

Concept:
In linear programming, we deal with optimizing a linear objective function subject to a set of linear constraints. These constraints form a geometric shape known as the feasible region, which represents all possible solutions that satisfy the inequalities.

The objective function is typically written as:
Z = ax + by
Where:
• Z is the value to be maximized or minimized
• a and b are constants (coefficients)
• x and y are decision variables

Key Principle:
According to the Fundamental Theorem of Linear Programming, if the objective function has an optimal value (either maximum or minimum), it always occurs at one of the corner points (also called vertices) of the feasible region.

Why Corner Points?
• The feasible region is a convex polygon (or polyhedron in higher dimensions) formed by the intersection of linear inequalities.
• A linear function like Z = ax + by will always attain its extreme values on the boundary of this region.
• More specifically, the maximum or minimum value of the objective function occurs at one (or more) of the vertices of this region.

Conclusion:
Therefore, to find the optimal value of a linear objective function subject to constraints, it is sufficient to evaluate the function at each corner point of the feasible region. The best value (maximum or minimum) among these is the solution to the linear programming problem.