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The demand for a certain brand of cheese at a certain store has the following distribution: Demand 0 1 2 3 4 Probability 0.2 0.2 0.2 0.2 0.2 The owner of the store is a bit lazy and doesn’t want to bother with checking the inventory at the end of every week. Hence, he has set up an automatic delivery; 2 units of cheese will be delivered to the store at the beginning of every week. If during the week the demand exceeds the inventory, an expensive emergency delivery is made to meet the excess demand. (If 1 unit is needed, 1 unit will be delivered; if 2 units are needed, 2 units will be delivered.) This brand of cheese has a shelf life of 3 weeks. Any unit of cheese that has been in the store for 3 weeks will be returned to the producer. Hence, the oldest unit is sold first. For instance, if there are 5 units in the store during week t, one unit is delivered at the beginning of week t-2, two units are delivered at the beginning of week t-1, and two units are delivered at the beginning of week t . When a customer comes, the unit delivered on week t-2 will be sold first. If no customer comes till the end of week t, the unit delivered on week t-2 will have lived out its shelf life and will be returned to the producer. The cost of the cheese is $10 per unit and each unit is sold for $25 at the store. The cost of emergency delivery is $20 per unit including the cost of the cheese. For any returned unit, the producer pays $5 to the owner of the store. (Assume that there is no holding cost.) Define ???????? as the inventory on hand at the beginning of week t right after the automatic deliverya) (8 points) Write down the state space and the transition probability matrix. b) (7 points) Compute the steady state probabilities. c) (3 points) Compute the long-run average inventory on-hand at the end of the week. d) (2 points) What is the probability of an emergency delivery? e) (2 points) What is the probability of making a return to the producer? f) (8 points) Compute the long-run average profit per week.