Consider an infinite-period inventory problem involving a single product where, at the beginning of each period, a decision must be made about how many items to produce during that period. The setup cost is $10, and the unit production cost is $5. The holding cost for each item not sold during the period is $4 (a maximum of 2 items can be stored). The demand during each period has a known probability distribution, namely, a probability of 1/3 of 0, 1, and 2 items, respectively. If the demand exceeds the supply available during the period, then those sales are lost and a shortage cost (including lost revenue) is incurred, namely, $8 and $32 for a shortage of 1 and 2 items, respectively.
(a) Consider the policy where 2 items are produced if there are no items in inventory at the beginning of a period whereas no items are produced if there are any items in inventory. Determine the (long-run) expected average cost per period for this policy. In finding the transition matrix for the Markov chain for this policy, let the states represent the inventory levels at the beginning of the period.
(b) Identify all the feasible (stationary deterministic) inventory policies, i.e., the policies that never lead to exceeding the storage capacity.