5.3 Feasible Solution

5.3 Feasible Solution

While tackling a certain problem from everyday life each linear inequality concerning the problem is named as problem constraint. The system of linear inequalities involved in the problem concerned is called problem constraints. The variables used in the system of linear inequalities relating to the problems of everyday life are non-negative and are called non-negative constraints. These non-negative constraints play an important role for taking decision. So, these variables are called decision variables. A region which is restricted to the first quadrant is referred to as a feasible region for the set of given constraints. Each point of the feasible region is called a feasible solution of the system of linear inequalities (or for the set of a given constraints).

Example 6

Shade the feasible region and find the corner points for the following system of inequalities (or subject to the following constraints).

Solution: The associated equations for the inequalities
$x - y \le 3$ ... (i) and $x + 2y \le 6$ ... (ii)
are $x - y = 3$ ... (iii) and $x + 2y = 6$ ... (iv)

As the points $(3, 0)$ and $(0, -3)$ are on the line (iii), so the graph of $x - y = 3$ is drawn by joining the points $(3, 0)$ and $(0, -3)$ by solid line.

Similarly, line (iv) is graphed by joining the points $(6, 0)$ and $(0, 3)$ by solid line. For $x = 0$ and $y = 0$, we have;

So, both the closed half-planes are on the origin sides of the lines (iii) and (iv). The intersection of these closed half-planes is partially displayed as shaded region in fig. 5.7(a).

The graph of $y \ge 0$, will be the closed upper half plane. The intersection of graph shown in figure 5.7(a) and closed upper half plane is partially displayed as shaded region in figure 5.7 (b).

The graph of $x \ge 0$ will be closed right half plane. The intersection of the graph shown in fig. 5.7(a) and closed right half plane is graphed in fig. 5.7 (c).

Finally, the graph of the given system of linear inequalities is displayed in figure 5.7 (d) which is the feasible region for the given system of linear inequalities. The points $(0, 0), (3, 0), (4, 1)$ and $(0, 3)$ are corner points of the feasible region.

[Image/Diagram: Fig. 5.7(a)–(d) — building the feasible region]

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Remember!
A point of a solution region where two of its boundary lines intersect, is called a corner point or vertex of the solution region.