$($P$1)$ This question refers to the two$-$dose vaccine distribution model with $L=2$ weeks required between

doses. Suppose we expect supply at a large dispensing location to be limited for the next $12$ weeks,

with supply projections given in the chart below.

Write the LOP to determine the optimal vaccination schedule, and solve it by specifying number of

first doses $v_j$ and amount of vaccine stock held $N_j,$ for each $j=1,$cdots,$12.$ Also state the value of your

vaccination plan in person weeks of protection.