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.

Time period

Required # of

Nurses

Wage

$($$$/$hr$)$

$12$ am $-4$ am

$20$

$15$

$4$ am $-8$ am

$30$

$16$

$8$ am $-12$ pm

$40$

$13$

$12$ pm $-4$ pm

$50$

$13$

$4$ pm $-8$ pm

$40$

$14$

$8$ pm $-12$ am

$30$

$15$

Formulate a linear model for this problem to create an optimized nurse scheduling plan. $($Don$$t forget to specify your decision variables and do not abbreviate the constraints with symbols$$write all of them out!$)$