$.$

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.

b$.$Find out the optimal policy and the optimal cost using probabilistic DP$.$

c$.$Calculate the percentage improvement over the fixed policy if the optimal policy would be implemented.