Consider the proabilistic DP example described in the DP lecture slides: production planning for the HIT$-$AND$-$MISS manufactuting company.

Suppose the setup cost is now $$5$ whereas production cost reduced to $$0\mathrm{.}5$ per item. You are informed that the company always follows a fixed policy: $(4,4,3)($i$.$e$,$ producing $4,4,$ and $3$ items for runs $1,2,$ and $3$ respectively$),$ without taking into account the inherent stochasticity.

a$.$ Calculate the expected cost incurred for the fixed policy. $(10$ points$)$

b$.$ Find out the optimal policy and the optimal cost using probabilistic DP$.(40$ points$)$

c$.$ Calculate the percentage improvement over the fixed policy if the optimal policy would be implemented. $(10$ points$)$