Feasible Region

Although our farmer has up to $60 per hectare that he could spend on fertilizer, it is possible for him to spend less than this. For example, he might decide to use two bags of high-grade (at $12/bag) and four bags of low-grade (at $4/bag) on each hectare. This comes to $40 per hectare. There is no reason why he could not choose to do this, or one of a number of other combinations, even though it may not be the best choice. (We'll see later how to determine the best choice.) This page requires a Java capable browser.

On the graph, all the points that the farmer could choose and still not spend more than $60 are bounded by the x-axis, the y-axis and the line 12x+4y=60. This is called the feasible region. In this situation, the feasible region is the part of the graph that satisfies a number of inequalities:

• 12x+4y≤60 (that is, the total cost can't exceed $60 per hectare)
• x≥0 (that is, there can't be a negative number of high-grade bags)
• y≥0 (that is, there can't be a negative number of low-grade bags)

Each of these inequalities is called a constraint. The last two are called non-negativity constraints. Most real-world problems include non-negativity constraints, and they are often assumed rather than stated explicitly.

Because each inequality is less than or equal to or greater than or equal to, the points on the boundaries are included in the feasible region. (We will see why this is particularly important later. For now, just remember that the feasible region includes its boundaries.)

Use the buttons on the applet to illustrate the three constraints, and the feasible region. The constraint buttons each shade out the area excluded by that constraint. After you have clicked them all, the only area left unshaded is the feasible region.

Which of the following is a feasible solution to the fertilizer problem? Move the red dot around to see which are within the feasible region.

i) 4 bags of high grade and 4 bags of low grade?
ii) 1 bag of high grade and 11 bags of low grade?
iii) 3 bags of high grade and 7 bags of low grade?
iv) 4 bags of high grade and no low grade?
v) 2 bags of high grade and 9 bags of low grade?

Exercise 2.

Use the applet to determine which of the given combinations of x and y are in the feasible region for the following linear inequalities. (You may assume non-negativity constraints for x and y.)

a) 3x+5y≤35

i) x=4, y=4?
ii) x=10, y=3?
iii) x=3, y=7?

b) 4x+6y≤54

i) x=4, y=4?
ii) x=10, y=3?
iii) x=3, y=7?

Check your answers