Chapter 3, Problem 27SGR

To determine

To find: The manufacturer’s maximum profit and combination of shoes for the maximum profit.

Answer to Problem 27SGR

Therefore, to obtain maximum profit the company must produce 12 pairs of outdoor and 16 pairs of indoor shoes. The maximum profit is $\text{\$}480$ .

Explanation of Solution

Given information :

Total hours needed for Step 1 is $2x+y$ hour

Total hours needed for Step 3 is $x+3y$ hour

Calculation :

Consider to obtain maximum profit the company must make x set of outdoor and y pair of indoor shoes.

For manufacturing a set of outdoor shoes requires 2 hours in step one and 1 hour in step two whereas for indoor shoes requires 1 hour of step one 3 hours of step two.

Therefore:

Total hours needed for Step 1 is $2x+y$ hour

Total hours needed for Step 3 is $x+3y$ hour

Consider the company has labors for 40 hour in Step 2 and 60 hour in Step 2 therefore:

  $\begin{array}{l}2x+y\le 40\\ x+3y\le 60\end{array}$

And obviously:

  $x,y\ge 0$

As the company earns $\text{\$}20$ as profit from each pair of outdoor shoes and $\text{\$}15$ from each pair of indoor shoes therefore the function need to be maximized. So,

  $f(x,y)=20x+15y$

Subject to the conditions:

  $\begin{array}{l}2x+y\le 40\\ x+3y\le 60\\ x,y\ge 0\end{array}$

Draw the feasible region bounded by the above inequalities as below:

  

Consider $\text{A},\text{B},\text{C}$ are the vertices of the feasible region. As $\text{A}$ is the intersection of

  $2x+y=40,x+3y=60$ therefore:

  $A=(12,16)$

As $\text{B}$ is the intersection of $x+3y=60,x=0$ therefore:

  $B=(0,20)$

As $\text{C}$ is the intersection of $2x+y=40,y=0$ therefore:

  $C=(20,0)$

Compute the objective function $f(x,y)=20x+15y$ at $A,B,C$ as below:

  $\begin{array}{cc}f(12,16)& =20\times 12+15\times 16\\ & =480\\ f(0,20)& =20\times 0+15\times 20\\ & =300\\ f(20,0)& =20\times 20+15\times 0\\ & =400\end{array}$

Therefore, $f(x,y)=20x+15y$ is maximum at $A=(12,16)$ and the maximum value is 480.

Therefore, to obtain maximum profit the company must produce 12 pairs of outdoor and 16 pairs of indoor shoes. The maximum profit is $\text{\$}480$ .