Question $3(25$ points$)$

Nurses in a large hospital work $3$ days a week. The possible work schedules and the weekly wage paid to a nurse for each schedule are given in the following table. For a given schedule, if there is a $"$ X $"$ sign on a given day, that means the nurse following that schedule works on that day. The other days are off$-$days. Each nurse works according to one schedule.

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline & $\backslash$multicolumn$\{5\}\{|$c$|\}\{$ Schedule Type $\}$ & $\backslash \backslash$

$\backslash$hline Day & $\backslash (\backslash$mathbf$\{1\}\backslash )$ & $\backslash (\backslash$mathbf$\{2\}\backslash )$ & $\backslash (\backslash$mathbf$\{3\}\backslash )$ & $\backslash (\backslash$mathbf$\{4\}\backslash )$ & $\backslash (\backslash$mathbf$\{5\}\backslash )$ & $\backslash$begin$\{$tabular$\}\{$c$\}$

Number of Nurses $\backslash \backslash$

Required

$\backslash$end$\{$tabular$\}\backslash \backslash$

$\backslash$hline Monday & X & & X & & & $20\backslash \backslash$

$\backslash$hline Tuesday & X & & & & X & $25\backslash \backslash$

$\backslash$hline Wednesday & & & X & X & & $26\backslash \backslash$

$\backslash$hline Thursday & & X & & & X & $26\backslash \backslash$

$\backslash$hline Friday & & X & & X & X & $30\backslash \backslash$

$\backslash$hline Saturday & & X & & X & & $30\backslash \backslash$

$\backslash$hline Sunday & X & & X & & & $35\backslash \backslash$

$\backslash$hline Weekly wage $(\backslash$$$)$ & $525$ & $470$ & $550$ & $500$ & $425$ & $\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$2$

The number of nurses needed for each day of the week are given in the last column.

$($a$)(15$ points$)$ Formulate an integer programming model to determine the total number of nurses required to meet the nurse requirements for each day at minimum cost. Define your decision variables clearly.

$($b$)(10$ points$)$ Solve the model you developed in part $($a$)$ using OpenSolver and report the optimal solution. Upload the excel file as well when uploading your solutions. $($Hint: Optimal objective function value is $\backslash (\backslash$$ $35,725\backslash ))$