Please solve stochastic processes problem attached, and show all work. Thank you.

President Helble, being so relieved that you helped to solve his problem, offers you the answer to the question that you realize you've been seeking this entire time: how to get a good night's sleep so you can wake up refreshed to ace your last nal exam. And that secret is: Your online sweater business. We've talked about it multiple times throughout the semester (since all the way back to the introductory overview lecture, and at least a few times since then). You just need to solve it! Specically, you need to determine, nally, the optimal inventory policy. If you can only nd it, then your mind will be at peace and you'll nally be able to sleep and wake up refreshed. As always, you let X; be the number of sweaters that you have at the end of week if and let Dt be the number of sweaters ordered in week it. Also as always, you assume that the distribution of D, is Poisson with a mean of 1. You only consider policies in which you start a week with at most 3 sweaters. The cost to produce 2 sweaters is $(10 + 102) if z > 0; else, it is $0 if z = 0.1 On the other hand, the cost per unit of unsatised demand is $50. (Following the approach that we learned, you can approximate the expected cost in state j using the fact that 1P[Dt = 2'] 7-3 0 for z' > 6.) (a) (3 points) What are the sets of states and decisions for this problem? Also indicate which (state, decision) pairs are not valid due to the restriction that you cannot start a week with more than 3 sweaters. (b) (4 points) For all possible (2', j, Is), what is the probability that your inventory transitions from state i to state j if you make decision It? (To save time, you can ignore (i, k) pairs that are not valid and simply set those probabilities to zero.) (c) (4 points) For all possible (25,16), what is the cost that you incur if your inventory is in state i and you make decision It? (To save time, you can ignore (i, 19) pairs that are not valid and simply set those costs to zero.) ((1) (4 points) What is the optimal policy