1. Meaning of a Feasible Region
The feasible region is the set of all points that satisfy every constraint of a linear programming problem at the same time. It is formed by the overlapping shaded regions (half-planes) of all the inequalities.
Every point inside this region is a possible solution, and points outside it violate one or more constraints. The feasible region is where the optimal solution, if it exists, will be found.
2. How the Feasible Region is Formed
Each constraint creates a half-plane. When two or more constraints are plotted together, the feasible region is the intersection of their half-planes. Graphically, this region often forms a polygon or an unbounded shape.
2.1. Step-by-Step Process
- Plot each constraint as a boundary line.
- Shade the region that satisfies each inequality.
- The common shaded area where all shaded regions overlap is the feasible region.
3. Types of Feasible Regions
The feasible region can take different shapes depending on the constraints.
3.1. Bounded Feasible Region
This region is completely enclosed within the coordinate plane, forming a closed polygon. It always gives both maximum and minimum values of the objective function (if the objective is linear).
3.2. Unbounded Feasible Region
This region extends infinitely in at least one direction. It may still give an optimal value, but sometimes the objective function grows without limit.
3.3. No Feasible Region
If the constraints contradict each other, there is no common area that satisfies all of them. In this case, the problem has no solution.
4. Identifying Corner Points of the Feasible Region
The corners (vertices) of the feasible region are important because an optimal solution, if it exists, will lie at one of these points. These points are formed by the intersection of boundary lines.
4.1. How to Find Corner Points
- Read them directly from the graph where two lines meet.
- Or solve the two boundary equations algebraically.
5. Example of a Feasible Region
Consider the constraints:
\( x + y \le 6 \)
\( x \ge 0 \)
\( y \ge 0 \)
5.1. Understanding the Region
Plot the line \( x + y = 6 \) using intercepts: \( (6, 0) \) and \( (0, 6) \). Shade the region below the line because of \( x + y \le 6 \).
The constraints \( x \ge 0 \) and \( y \ge 0 \) keep the region in the first quadrant.
The final feasible region is a triangular area with vertices at:
- \( (0, 0) \)
- \( (6, 0) \)
- \( (0, 6) \)