Chapter 3.3, Problem 26PPS

To determine

To Find: The number of day-shifts and night-shifts to minimize the cost.

Answer to Problem 26PPS

The theatre should schedule 8 day shift and 6 night shifts.

Explanation of Solution

Given information :

Employees work hour shift at theatre: 8-hour shift.

Time	12 P.M to 4 P.M.	4 P.M. to 8 P.M.	8 P.M to 12 A.M
Number of Employees Needed	At least 5	At least 14	6
Rate per hour	$5.50	$7.50	$7.50

Suppose x be the number of day shift and y be the number of night shift workers.

Then, the optimizing function is written as follows:

  $\begin{array}{l}\begin{array}{l}f\left(x,y\right)=5.50\left(4\right)x+7.50\left(4\right)\left(x+y\right)+7.50\left(4\right)y\hfill \\ =22x+30\left(x+y\right)+30y\hfill \end{array}\\ =52x+60y\end{array}$

The constraints are:

  $\begin{array}{l} x\ge 5\hfill \\ x+y\ge 14\hfill \\ y\ge 6\hfill \end{array}$

The graph is shown below:

  

The vertices of the feasible region are (5,9), and (8,6). The table is created as shown below:

$\left(x,y\right)$	$f\left(x,y\right)=52x+60y$
$\left(5,9\right)$	$\begin{array}{l}\begin{array}{l}f\left(5,9\right)=52x+60y\hfill \\ =52\left(5\right)+60\left(9\right)\hfill \\ =260+540\hfill \end{array} \\ =800\end{array}$
$\left(8,6\right)$	$\begin{array}{l}\begin{array}{l}f\left(8,6\right)=52\left(8\right)+60\left(6\right)\hfill \\ =416+360\hfill \end{array}\\ =776\end{array}$

The minimal cost is 776 dollars. Therefore, the theatre should schedule 8 day shift and 6 night shifts.