A consumer electronics retailer can place orders for flat panel TVs at the beginning of each day. All orders are delivered immediately. Every time an order is placed for one or more TVs, the retailer pays a fixed cost of \($60\). Every TV ordered costs the retailer \($100\). The daily holding cost per unsold TV is \($10\). A daily shortage cost of \($180\) is incurred for each TV that is not available to satisfy demand. The retailer can accommodate a maximum inventory of two TVs. The daily demand for TVs is an independent, identically distributed random variable which has the following stationary probability distribution:

(a) Formulate this model as an MDP.

(b) Determine an optimal ordering policy that will minimize the expected total cost over the next 3 days.

(c) Use LP to determine an optimal ordering policy that will minimize the expected average cost per day over an infinite planning horizon.

(d) Use PI to determine an optimal ordering policy over an infinite planning horizon.

(e) Use LP with a discount factor of α = 0.9 to find an optimal ordering policy over an infinite planning horizon.

(f) Use PI with a discount factor of α = 0.9 to find an optimal ordering policy and the associated expected total discounted cost vector over an infinite planning horizon.  

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Daily demand d 0 1 2 Probability, p(d) 0.5 0.4 0.1