A Hospital Needs to Determine How Many Nurses to Hire

A hospital needs to determine how many nurses to hire to cover a 24 hour period. The nurses must work 8 consecutive hours but can start work at the start of 6 different shifts. They are paid different wages depending on when they start their shifts. The number of nurses required per 4-hour time period and their wages are shown in the following table.
$\begin{array}{lcc}\text { Time period } & \text { Required \# of Nurses } & \text { Wage }(\$ / \mathrm{hr}) \\\hline 12 \mathrm{am}-4 \mathrm{am} & 20 & 15 \\4 \mathrm{am}-8 & 30 & 16 \\\text { am } & & \\8 \mathrm{am}-12 \mathrm{pm} & 40 & 13 \\12 \mathrm{pm}-4 \mathrm{pm} & 50 & 13 \\4 \mathrm{pm}-48 & 40 & 14 \\\mathrm{pm} \\8 \mathrm{pm}-12 \mathrm{am}&30 & 15 \end{array}$
Enter the numbers in the appropriate cells of ranges B6:G11 and B13:G13 in the Excel spreadsheet to solve this problem based on the following formulation.
Let $\quad X _ { i } =$ mumber of nurses warking in thme period $i ; i = 1,6$
$$\begin{array} { l l } \text { MiN: } & 1 \mathbf { X } _ { 1 } + 1 \mathbf { X } _ { \mathbf { 2 } } + 1 \mathbf { X } _ { \mathbf { 3 } } + 1 \mathbf { X } _ { 4 } + 1 \mathbf { X } _ { \mathbf { 5 } } + 1 \mathbf { X } _ { \mathbf { 6 } } \\\text { Subject ta: } & 1 \mathbf { X } _ { 1 } + 1 \mathbf { X } _ { \mathbf { z } } \geq 30 \\& 1 \mathbf { X } _ { \mathbf { 2 } } + 1 \mathbf { X } _ { \mathbf { 3 } } \geq 40 \\& 1 \mathbf { X } _ { \mathbf { 3 } } + 1 \mathbf { X } _ { 4 } \geq 50 \\& 1 \mathbf { X } _ { 4 } + 1 \mathbf { X } _ { \mathbf { 5 } } \geq 40 \\& 1 \mathbf { X } _ { 5 } + 1 \mathbf { X } _ { \mathbf { 1 } } \geq 30 \\& 1 \mathbf { X } _ { 1 } + 1 \mathbf { X } _ { \mathbf { 1 } } \geq 20 \\& \mathbf { X } _ { \mathrm { i } } \geq 0\end{array}$$