Consider the nonlinear optimization model stated below.

Min$2$X$218$X $+2$XY $+$ Y$214$Y $+53$s$.$t$.$ X $+4$Y $8$

$($a$)$

Find the minimum solution to this problem.

Nurses start work at the beginning of one of the four$-$hour shifts given below $($except for shift $6)$ and work for $8$ consecutive hours. Hence, possible start times are the start of shifts $1$ through $5.$ Also, assume that the projected required number of nurses factors in time for each nurse to have a meal break.

Formulate and solve the nurse scheduling problem as an integer program for one day for the data given below. $($Let

xt $=$

number of nurses who start work at the beginning of shift t$,$

t $=1,2,3,4,5.)$

ShiftTimeMinimum Number of Nurses Needed$112$:$00$ A$.$M$.-4$:$00$ A$.$M$\mathrm{.}1124$:$00$ A$.$M$.-8$:$00$ A$.$M$\mathrm{.}2538$:$00$ A$.$M$.-12$:$00$ P$.$M$\mathrm{.}19412$:$00$ P$.$M$.-4$:$00$ P$.$M$\mathrm{.}1154$:$00$ P$.$M$.-8$:$00$ P$.$M$\mathrm{.}2468$:$00$ P$.$M$.-12$:$00$ A$.$M$\mathrm{.}18$

Hint: Note that exceeding the minimum number of needed nurses in each shift is acceptable so long as the total number of nurses over all shifts is minimized.