Chapter 7.6, Problem 41E

To determine

To find: the numbers of audits and tax returns will yield an optimal revenue and the optimal revenue.

Answer to Problem 41E

Optimal revenue is 20800 and it is obtained when there is 13 audits and 0 tax returns.

Explanation of Solution

Given:

An accounting firm has 780 hours of staff time and 272 hours of reviewing time available each week. The firm charges $1600 for an audit and $250 for a tax return. Each audit requires 60 hours of staff time and 16 hours of review time. Each tax return requires 10 hours of staff time and 4 hours of review time.

Calculation:

Let $x$ be the number of audits and $y$ be the number of tax returns carried out by the firm. So the objective function is given by:

  $z=1600x+250y$

The three constraints can be converted into linear inequalities as:

  $\begin{array}{l}60x+10y\le 780\\ 16x+4y\le 272\end{array}$

Since the number of pieces produced cannot be negative, we can get two more constraints of

  $x\ge 0$ and $y\ge 0$ .

The area determined by the constraints are shown below:

  

The point of intersection of two lines A, should be found by solving the equation of two lines:

  $\begin{array}{l}60x+10y=\mathrm{780............}(1)\\ 16x+4y=\mathrm{272.............}(2)\end{array}$

From equation (1),

  $x=13-\frac{1}{6}y\mathrm{............}(3)$

From equation (2),

  $x=17-\frac{1}{4}y\mathrm{................}(4)$  ..........(4)

At the intersection point $x$ values are same. So we can equate (3) and (4) and we get , $y$ as 48 substituting this value in equation (4), we get the value of $x$ as 5.

So the co-ordinates of the intersection point A is $\left(5,48\right)$

At the four vertices of the region formed by the constraints the objective function has the following values:

At $\left(0,0\right):z=1600\left(0\right)+250\left(0\right)=0$

At $\left(0,68\right):z=1600\left(0\right)+250\left(68\right)=17000$

At $\left(5,48\right):z=1600\left(5\right)+250\left(48\right)=20000$

At $\left(13,0\right):z=1600\left(13\right)+250\left(0\right)=20800$ Maximum value of $z$

Optimal revenue is 20800 and it is obtained when there is 13 audits and 0 tax returns.