Problem $2(20$ points$)$

Labor fluctuating needs in construction projects can be met through hiring and end

employment. Both activities incur cost. The goal is to minimize the total cost of acquiring

needed labor force.

A contractor estimates that the size of the workforce needed over the next $5$ weeks is $5,7,8,4,$

and $6$ workers. Excess labor kept on the force will cost $$300$ per worker per week, and new

hiring in any week will incur a fixed cost of $$400$ plus $$200$ per worker per week.

Find the optimal solution for this problem through dynamic programming. How many workers

require for each period and what is the minimum total cost?