Define the concept of feasible region in linear programming.

The feasible region in linear programming is the set of all possible solutions that satisfy the constraints.

In linear programming, the objective is to maximize or minimize a linear function subject to a set of linear constraints. The feasible region is the set of all possible solutions that satisfy these constraints. It is represented graphically as the intersection of the constraint lines or planes in the solution space.

For example, consider the following linear programming problem:

Maximize 3x + 4y
Subject to:
2x + y ≤ 10
x + 3y ≤ 15
x, y ≥ 0

The feasible region is the shaded area in the graph below:


The feasible region is bounded by the constraint lines and the non-negativity constraints. Any point within the feasible region satisfies all the constraints, while any point outside the feasible region violates at least one constraint.

The optimal solution to the linear programming problem is found by evaluating the objective function at each corner point of the feasible region and selecting the point that gives the maximum (or minimum) value. In this example, the optimal solution is at the corner point (5, 0) with a value of 15.