Finding the Feasible Region

An Interactive Applet powered by Sage and MathJax.

(By Prof. Gregory V. Bard, updated by Ryan G. Hornberger.)

Overview

Systems of inequalities are extremely important tools for industrial engineering and other problems of modern-day management. They are the principal part of the subject called "Linear Programming." Example problems can be found in topics as diverse as nutrition, transportation, logistics, manufacturing, scheduling, and investing.

Discussion

In our particular case, we will be looking at the following system of inequalities: $$ \begin{array}{rcl} 1.5 - 2x & \ge & y \\ 1-x & \ge & y \\ 0.5 - 0.5 x & \le & y \\ x & \ge & 0 \\ y & \ge & 0 \end{array} $$

Each inequality in a system of inequalities represents a constraint---some sort of requirement that the solution or "plan" must accommodate. In the above, the values of $x$ and $y$ are initially not known. The feasible region is the set of all points $(x, y)$ which satisfy every inequality. If some point violates even one inequality, then it is not permitted to be part of the feasible region. These points represent circumstances or plans that meet all of the requirements.

One of the critical steps in solving a linear program, or working with systems of inequalities in any context, is to graph them and find the feasible region. The graphing of the inequalities is straight forward, as the equations are merely lines. For example, to graph $3x + 6y \le 12$ we would only need to graph the line $3x + 6y = 12$, something all of us know how to do at this point. However, it is not so easy (after the lines are drawn) to find the feasible region. The applet below will address this challenging step.

Instructions

The following applet will, at first, present you with two sliders: one for $x$ and one for $y$. Then as you move them, it will tell you if each inequality is satisfied or not.

1. You can now click on the button to launch the applet.
2. Try to find the feasible region numerically. Move the sliders around, and this will move $x$ and $y$ around. Each inequality will then report if it is satisfied or not. Try to get all the inequalities satisfied for some point $(x,y)$.
3. Graph the system on a blank piece of paper.
4. To see if your sketch is accurate, click on "Show The Graph." Don't do this prematurely, because it will spoil the test of your line-graphing skills. The large red dot is the point $(x,y)$, and the red lines are to help you see what $x$ and $y$ values you have chosen.
5. Only when the above are completed, try to figure out which region of the graph is the feasible region.
6. Click on "Show The Colors" to see if you're right. The shaded areas are regions which violate one or more inequalities. The white area is the feasible region.

Notes

• Note that the red lines above are like a cross-hairs, showing you the point you are currently working with. The big dot in the cross-hairs is the point represented by the sliders. The red lines and red dot are not part of the system of inequalities nor their graph.
• Did this exercise seem difficult? Are you curious if this step can be avoided? It can!
• You can avoid this process when graphing a system of inequaliites by first drawing the first line, then shading the region which it forbids; then drawing the second line, and shading the region it forbids; next drawing the third line, and shading the region it forbids, and so forth.
• What we did here on this web page is draw all the lines first, and only then start shading. That is what made the problem difficult.

What is the moral of the story?

On an exam, remember: draw a line, shade, draw a line, shade, and so forth. Nonetheless, it is a good skill (and a great exam question) to show the student a graph with all the lines as above, and have the student choose which of the regions is the feasible region.