please step by step,

HIT-AND-MISS manufacturing company The company has received an order to supply one item with stringent quality requirements. The company may have to produce more than one item to obtain an item of acceptable quality. Each produced item is acceptable with probability 1/2 and defective with probability 1/2. Production cost: $1 per item. Set up cost (denoted by K): $3 for every production run with positive production. A total of 3 production runs. Not producing an acceptable item at the end of 3 runs costs the company $16 in lost sales income and penalty costs. Goal: to figure out production quantity for each run to minimize total expected cost.

CONSIDER THESE INFORMATION:

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 $4 whereas production cost reduced to $0.5 per item. You are informed that the company always follows a fixed policy: (4,3,3) (i.e, producing 4, 3, 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. d. If the production cost would increase to $1.5 per item, do you think the percentage improvement over the fixed policy would be higher or lower than what you found in part c) ? Justify your