5.11 Linear Programming

Let $x$ represent the number of scarves sold, and let $y$ represent the number of sweaters sold. Let $P$ represent profit. Since each scarf has a profit of $8 and each sweater has a profit of $10, the objective function is $P=8x+10y$.

Let $x$ equal the number of widgets made; let $y$ equal the number of wadgets made; let $T$ represent total time. The objective function is $T=24x+32y$.

Let $x$ represent the number of scarves sold, and let $y$ represent the number of sweaters sold. There are two constraints: the number of items the business can make in a day (a maximum of 8) and the number of bags of knitting wool they can use per day (a maximum of 27). The first constraint (total number of items in a day) is written as:

Since each scarf takes 3 bags of knitting wool and each sweater takes 4 bags of knitting wool, the second constraint, total bags of knitting wool per day, is written as:

In summary, here are the equations that represent the new business:

$P=8x+10y$; This is the profit equation: The business makes $8 per scarf and $10 per sweater.

Let $x$ equal the number of widgets made; let $y$ equal the number of wadgets made. Based on the long-term demand, we know the factory must produce a minimum of 12 widgets and 10 wadgets per day. We also know because of storage limitations, the factory cannot produce more than 20 widgets per day or 17 wadgets per day. Writing those as inequalities, we have:

$x\ge 12$

$y\ge 10$

$x\le 20$

$y\le 17$

The number of widgets made per day must be between 12 and 20, and the number of wadgets made per day must be between 10 and 17. Therefore, we have:

$12\le x\le 20$

$10\le y\le 17$

The system is:

$T=24x+32y$

$12\le x\le 20$

$10\le y\le 17$

$T$ is the variable for time; it takes 24 minutes to make a widget and 32 minutes to make a wadget.

Step 1: Compose an objective function to be minimized or maximized. From Example 5.98, the objective function is $P=8x+10y$.

Step 2: Compose inequalities representing the constraints of the system. From Example 5.98, the constraints are $x+y\le 8$ and $3x+4y\le 27$.

Step 3: Graph the system of inequalities representing the constraints. Using methods discussed in Graphing Linear Equations and Inequalities, the graphs of the constraints are shown below. Because the number of scarves ($x$) and the number of sweaters ($y$) both must be non-negative numbers (i.e., $x\ge 0$ and $y\ge 0$), we need to graph the system of inequalities in Quadrant I only. Figure 5.99 shows each constraint graphed on its own axes, while Figure 5.100 shows the graph of the system of inequalities (the two constraints graphed together). In Figure 5.100, the large shaded region represents the area where the two constraints intersect. If you are unsure how to graph these regions, refer back to Graphing Linear Equations and Inequalities.

Figure 5.99 Graphs of each constraint

Figure 5.100 Graph of the System of Inequalities

Step 4: Find the value of the objective function at each corner point of the graphed region. The “graphed region” is the area where both of the regions intersect; in Figure 5.101, it is the large shaded area. The “corner points” refer to each vertex of the shaded area. Why the corner points? Because the maximum and minimum of every objective function will occur at one (or more) of the corner points. Figure 5.101 shows the location and coordinates of each corner point.

Figure 5.101 Graph of Region with Corner Points

Three of the four points are readily found, as we used them to graph the regions; the fourth point, the intersection point of the two constraint lines, will have to be found using methods discussed in Systems of Linear Equations in Two Variables, either using substitution or elimination. As a reminder, set up the two equations of the constraint lines:

For this example, substitution will be used.

Substituting $8-x$ into the first equation for $y$, we have

Now, substituting the 5 in for $x$ in either equation to solve for $y$. Choosing the second equation, we have:

Therefore, $x=5$, and $y=3$.

To find the value of the objective function, $P=8x+10y$, put the coordinates for each corner point into the equation and solve. The largest solution found when doing this will be the maximum value, and thus will be the answer to the question originally posed: determining the number of scarves and sweaters the new business should make each day to maximize their profit.

The maximum value for the profit $P$ occurs when $x=5$ and $y=3$. This means that to maximize their profit, the new business should make 5 scarves and 3 sweaters every day.