Chapter 2.7, Problem 18E

To determine

To find: the number of nurses and nurse’s aides should be hired in order to minimize the salary cost; if not possible state whether the problem is infeasible, unbounded or has alternate optimal solutions.

Answer to Problem 18E

Nurse: 24

Aides: 12

Explanation of Solution

Given information:

Average annual salary for nurse = $35000

Average annual salary for nurse’s aide = $18000

Minimum number of nurses to be hired = 20

Maximum number of nurses to be hired = 50

Minimum aides required = 12

Number of nurses = 2 (number of aides)

Calculation:

Let the number of nurses and nurse’s aides be $x$ and $y$ respectively.

Let $P\left(x,y\right)$ be the maximum cost function. The maximum cost is as follows:

  $P\left(x,y\right)=35000x+18000y$ .

The inequalities derived from the given information are as follows:

  $\begin{array}{c}x\le 75000\\ 1000\le y\le 7000\\ x+y\le 11000\end{array}$

Use a graphing calculator to plot the inequalities as shown below.

  

The vertices are $\left(24,12\right)\text{,}\left(38,12\right)\text{ and }\left(34,16\right)$ . Now test each vertex as shown below.

  $\begin{array}{c}P\left(24,12\right)=35000\left(24\right)+18000\left(12\right)\\ =\$1056000\\ P\left(38,12\right)=35000\left(38\right)+18000\left(12\right)\\ =\$1546000\\ P\left(34,16\right)=35000\left(34\right)+18000\left(16\right)\\ =\$1478000\end{array}$

Thus, the number of nurses and nurse’s aides should be hired in order to minimize the salary cost is24 and 12 respectively.